Finite Group Databasebeta

← #29748102

Subgroups — #29748102

Order 15616.

826 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21450182C211yesmaximal, characteristic
2#21448822C211yesmaximal, characteristic, complemented
3#21447137C211yesmaximal, characteristic, complemented
4#21447016C211yesmaximal, characteristic, complemented
5#21446961C211yesmaximal, characteristic, complemented
6#21446824C211yesmaximal, characteristic
7#21446770C211yesmaximal, characteristic, complemented
8#21446502C211yesmaximal, characteristic, complemented
9#21446447C211yesmaximal, characteristic, complemented
10#21446405C211yesmaximal, characteristic, complemented
11#21446173C211yesmaximal, characteristic, complemented
12#21446134C211yesmaximal, characteristic
13#21445910C211yesmaximal, characteristic, complemented
14C4×#14519702C211yesmaximal, characteristic, complemented
15C61×#1923C211yesmaximal, characteristic, complemented
16C22×#11758190C2211yescharacteristic
17C2×#14519777C2211yescharacteristic
18C2×#14519763—21no—
19C2×#14519756C2211yescharacteristic
20C2×#14519751C2211yescharacteristic
21C2×C4×Dic122C2211yescharacteristic
22C22×C4⋊Dic61C2211yescharacteristic
23C2×#14519709C2211yescharacteristic
24C2×#14519702C2211yescharacteristic
25C2×#14519702C2211yescharacteristic
26C2×#14519702C2211yescharacteristic
27C2×#14519699C2211yescharacteristic
28C2×#14519699C2211yescharacteristic
29C2×#14519699C2211yescharacteristic
30#16839678—21no—
31#16839675—21no—
32#16839668C2211yescharacteristic
33#16839664C2211yescharacteristic
34#16839664C2211yescharacteristic
35#16839660C2211yescharacteristic
36#16839659—21no—
37#16839654—21no—
38#16839649—21no—
39#16839634—21no—
40#16839622C2211yescharacteristic
41C4×C4⋊Dic61C2211yescharacteristic
42C4×C4⋊Dic61C2211yescharacteristic
43C22⋊C4×Dic61—21no—
44#16839525C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839525C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839525C2211yescharacteristic
51#16839525C2211yescharacteristic
52C2×C4⋊C8×C61C2211yescharacteristic
53C2×C22⋊C8×C61C2211yescharacteristic
54C2×C22⋊C8×C61C2211yescharacteristic
55C2×C4×C8×C61C2211yescharacteristic
56C4×C22⋊C4×C61C2211yescharacteristic
57C61×#336C2211yescharacteristic
58C61×#336C2211yescharacteristic
59C22×Dic122—21no—
60C22×Dic122C2311yescharacteristic
61C22×D244C2311yescharacteristic
62C22×C22⋊D61—21no—
63C23×Dic61—21no—
64C2×#11758190—22no—
65C2×#11758190—41no—
66C2×#11758190—41no—
67#14519777—22no—
68#14519777—41no—
69#14519763—44no—
70#14519756—22no—
71#14519756—22no—
72#14519751—22no—
73#14519751—41no—
74C4×Dic122—22no—
75C4×Dic122—41no—
76C2×C4⋊Dic61—22no—
77C2×C4⋊Dic61—21no—
78C2×C4⋊Dic61—21no—
79C2×C4⋊Dic61—41no—
80C2×C4⋊Dic61C2311yescharacteristic
81C2×C4⋊Dic61C2311yescharacteristic
82C2×C4⋊Dic61C2311yescharacteristic
83C2×C4⋊Dic61C2311yescharacteristic
84C2×C4⋊Dic61C2311yescharacteristic
85C2×C4⋊Dic61C2311yescharacteristic
86C2×#11758196—21no—
87C2×#11758196—21no—
88C2×#11758196—21no—
89C2×C22⋊Dic61—21no—
90C2×C22⋊Dic61—21no—
91C2×#11758198—21no—
92C2×#11758198—21no—
93C2×#11758198—21no—
94C2×#11758198—21no—
95C2×#11758198—21no—
96C2×C4×Dic61—21no—
97C2×C4×Dic61—21no—
98C2×C4×Dic61—21no—
99#14519709—22no—
100#14519702—22no—
101#14519702—22no—
102#14519702—41no—
103#14519699—22no—
104#14519699—22no—
105#14519699—41no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112C23×C4×C61C2311yescharacteristic
113C22×C8×C61C2311yescharacteristic
114C22×C8×C61C2311yescharacteristic
115C22×C8×C61C2311yescharacteristic
116C22×C8×C61C2311yescharacteristic
117C2×C22⋊C4×C61D411yescharacteristic
118C2×C22⋊C4×C61D411yescharacteristic
119C2×C42×C61C2311yescharacteristic
120C2×C42×C61C2311yescharacteristic
121C4⋊C8×C61—41no—
122C22⋊C8×C61—41no—
123C22⋊C8×C61—41no—
124C4×C8×C61—41no—
125#95×C61D411yescharacteristic
126#95×C61D411yescharacteristic
127C24×C61C2×D411yescharacteristic
128C23×D61—21no—
129C22×C4×C61—21no—
130C22×C4×C61—21no—
131C22×C4×C61—21no—
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2411yescharacteristic
141C2×C22⋊D61—42no—
142C2×C22⋊D61—42no—
143C2×C22⋊D61—81no—
144C22×Dic61—21no—
145C22×Dic61—21no—
146C22×Dic61—21no—
147C22×Dic61—21no—
148C22×Dic61—21no—
149C22×Dic61—21no—
150C22×Dic61—21no—
151C22×Dic61—42no—
152C22×Dic61—42no—
153C22×Dic61—81no—
154C2×D244—21no—
155C2×D244—21no—
156C2×D244—21no—
157C2×D244—21no—
158C2×D244—21no—
159C2×D244—21no—
160C2×Dic122—21no—
161C2×Dic122—21no—
162C2×Dic122—21no—
163C2×Dic122—21no—
164C2×Dic122—21no—
165C2×Dic122—42no—
166C2×Dic122—42no—
167C2×Dic122—81no—
168C2×Dic122#4112yes—
169C2×C8×C61—21no—
170C2×C8×C61—21no—
171C2×C8×C61—21no—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C2×C8×C61—21no—
178C2×C8×C61—21no—
179C2×C8×C61—21no—
180C2×C8×C61—21no—
181C22⋊C4×C61—41no—
182C22⋊C4×C61SD1614yescomplemented
183C42×C61—21no—
184C42×C61—21no—
185C42×C61—21no—
186C42×C61—21no—
187C22⋊Dic61—24no—
188C22⋊Dic61—42no—
189#11758198—42no—
190#11758198—42no—
191#11758198—42no—
192#11758198—42no—
193#11758198—81no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—42no—
204C4⋊Dic61—42no—
205C4⋊Dic61#4112yes—
206C4⋊Dic61#4112yes—
207C4⋊Dic61#4112yes—
208#11758196—24no—
209#11758196—42no—
210#11758196—81no—
211C4×Dic61—42no—
212C4×Dic61—42no—
213C4×Dic61—81no—
214#11758190—44no—
215C23×C61—21no—
216C23×C61—21no—
217C23×C61—21no—
218C23×C61—21no—
219C23×C61—21no—
220C23×C61—21no—
221C23×C61C2×SD1612yes—
222C23×C61C22×D411yescharacteristic
223C22×D61—41no—
224C22×D61—41no—
225C22×D61—41no—
226C22×D61—41no—
227C22×D61—41no—
228C22×D61—41no—
229C22×D61—41no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61C2×SD1612yes—
248C2×C4×C61C2×SD1612yes—
249C2×C4×C61C2×#4111yescharacteristic
250C2×C4×C61C2×#4111yescharacteristic
251C2×C4×C61C2×#4111yescharacteristic
252C2×C4×C61C2×#4111yescharacteristic
253C2×C4×C61H(5,𝔽2)11yescharacteristic
254C2×C4×C61H(5,𝔽2)11yescharacteristic
255C22⋊D61—84no—
256C2×Dic61—22no—
257C2×Dic61—22no—
258C2×Dic61—22no—
259C2×Dic61—22no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—84no—
278D244—41no—
279D244—41no—
280D244—41no—
281D244—41no—
282Dic122—22no—
283Dic122—22no—
284Dic122—41no—
285Dic122—41no—
286Dic122—84no—
287C8×C61—41no—
288C8×C61—41no—
289C8×C61—41no—
290C8×C61—41no—
291#26724—611nomaximal, Sylow
292C22×C61—22no—
293C22×C61—22no—
294C22×C61—22no—
295C22×C61—41no—
296C22×C61—41no—
297C22×C61—41no—
298C22×C61—41no—
299C22×C61#52511yescharacteristic
300C22×C61#54611yescharacteristic
301C22×C61#54611yescharacteristic
302C22×C61C22×SD1611yescharacteristic
303C22×C61#57511yescharacteristic
304C22×C61#57611yescharacteristic
305C22×C61#57811yescharacteristic
306C2×D61—81no—
307C2×D61—81no—
308C2×D61—81no—
309C2×D61—81no—
310C2×D61—81no—
311C2×D61—81no—
312C2×D61—81no—
313C4×C61—22no—
314C4×C61—22no—
315C4×C61—41no—
316C4×C61—41no—
317C4×C61—41no—
318C4×C61#52911yescharacteristic
319C4×C61#53811yescharacteristic
320C4×C61#55511yescharacteristic
321C4×C61#55911yescharacteristic
322Dic61—42no—
323Dic61—42no—
324Dic61—42no—
325Dic61—42no—
326Dic61—81no—
327Dic61—81no—
328Dic61—81no—
329#2491—611no—
330#2360—611no—
331#2055—611no—
332#2035—611no—
333#2024—611no—
334#2000—611no—
335#1988—611no—
336#1934—611no—
337#1926—611no—
338#1923D6111yescharacteristic, complemented
339#1920—611no—
340#1880—611no—
341#1874—611no—
342#1824—611no—
343C4×#103—611no—
344C2×C61—42no—
345C2×C61#245911yescharacteristic
346C2×C61#310311yescharacteristic
347C2×C61#312411yescharacteristic
348C2×C61#329511yescharacteristic
349C2×C61#329811yescharacteristic
350C2×C61#331911yescharacteristic
351C2×C61#332711yescharacteristic
352D61—161no—
353C22×SD16—611no—
354C2×C4⋊Q8—611no—
355C2×#124—1221no—
356C2×C22⋊Q8—611no—
357C2×#121—611no—
358C2×C4×Q8—611no—
359C22×C4⋊C4—611no—
360C2×#106—611no—
361C2×C4⋊C8C2×D6111yescharacteristic
362C2×#103—611no—
363C2×#103—611no—
364C2×#103—611no—
365C2×#102—611no—
366C2×#102—611no—
367C2×#102—611no—
368C2×C22⋊C8C2×D6111yescharacteristic
369C2×C22⋊C8C2×D6111yescharacteristic
370C2×C4×C8C2×D6111yescharacteristic
371#394—1221no—
372#393—1221no—
373#391—611no—
374#390—611no—
375#390—611no—
376#389—611no—
377#389—1221no—
378#388—1221no—
379#387—1221no—
380#385—1221no—
381#380—611no—
382C4×C4⋊C4—611no—
383C4×C4⋊C4—611no—
384C4×C22⋊C4—1221no—
385C4×C22⋊C4C2×D6111yescharacteristic
386#340—611no—
387#340—611no—
388#340—611no—
389#340—611no—
390#340—611no—
391#340—611no—
392#340—611no—
393#340—611no—
394#336C2×D6111yescharacteristic
395#336C2×D6111yescharacteristic
396C61#2672411yescharacteristic, complemented, Sylow
397C22×Q8—611no—
398C22×Q8—1221no—
399C22×D4—611no—
400C22×D4—1221no—
401C23×C4—1221no—
402C23×C4C22×D6111yescharacteristic
403C2×SD16—1222no—
404C2×SD16—2441no—
405C2×SD16—2441no—
406C22×C8C22×D6111yescharacteristic
407C22×C8C22×D6111yescharacteristic
408C22×C8C22×D6111yescharacteristic
409C22×C8C22×D6111yescharacteristic
410C4⋊Q8—1222no—
411C4⋊Q8—2441no—
412#124—2444no—
413C22⋊Q8—1222no—
414C22⋊Q8—1222no—
415#121—1222no—
416#121—2441no—
417C4×Q8—1222no—
418C4×Q8—2441no—
419C2×C4⋊C4—611no—
420C2×C4⋊C4—611no—
421C2×C4⋊C4—611no—
422C2×C4⋊C4—611no—
423C2×C4⋊C4—611no—
424C2×C4⋊C4—611no—
425C2×C4⋊C4—1222no—
426C2×C4⋊C4—1221no—
427C2×C4⋊C4—1221no—
428C2×C4⋊C4—1221no—
429C2×C4⋊C4—1221no—
430C2×C4⋊C4—1221no—
431C2×C4⋊C4—2441no—
432C2×C22⋊C4—1221no—
433C2×C22⋊C4—1221no—
434C2×C22⋊C4—1221no—
435C2×C22⋊C4—1221no—
436C2×C22⋊C4—1221no—
437C2×C22⋊C4—1221no—
438C2×C22⋊C4—1221no—
439C2×C22⋊C4D24411yescharacteristic
440C2×C22⋊C4D24411yescharacteristic
441C2×C42—1221no—
442C2×C42—1221no—
443C2×C42—1221no—
444C2×C42C22×D6111yescharacteristic
445C2×C42C22×D6111yescharacteristic
446#106—1222no—
447C4⋊C8—41no—
448#103—1222no—
449#103—1222no—
450#103—2441no—
451#102—1222no—
452#102—1222no—
453#102—2441no—
454C22⋊C8—41no—
455C22⋊C8—41no—
456C4×C8—41no—
457#95—1221no—
458#95—1221no—
459#95—1221no—
460#95—1221no—
461#95—1221no—
462#95—1221no—
463#95D24411yescharacteristic
464#95D24411yescharacteristic
465C24—1221no—
466C24C2×D24411yescharacteristic
467C2×Q8—612no—
468C2×Q8—1221no—
469C2×Q8—1221no—
470C2×Q8—1221no—
471C2×Q8—1221no—
472C2×Q8—1221no—
473C2×Q8—4881no—
474C2×Q8—2442no—
475C2×Q8—2442no—
476C2×D4—1221no—
477C2×D4—1221no—
478C2×D4—1221no—
479C2×D4—1221no—
480C2×D4—1221no—
481C2×D4—1221no—
482C2×D4—4881no—
483C2×D4—2442no—
484C2×D4—2442no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—1221no—
492C22×C4—1221no—
493C22×C4—1221no—
494C22×C4—1221no—
495C22×C4—1221no—
496C22×C4—1221no—
497C22×C4—1221no—
498C22×C4—4881no—
499C22×C4—2442no—
500C22×C4—2442no—
501C22×C4C2×D24411yescharacteristic
502C22×C4C2×D24411yescharacteristic
503C22×C4C2×D24411yescharacteristic
504C22×C4C2×D24411yescharacteristic
505C22×C4C2×D24411yescharacteristic
506C22×C4C23×D6111yescharacteristic
507SD16—2444no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—21no—
517C2×C8—21no—
518C2×C8—21no—
519C2×C8—21no—
520C4⋊C4—612no—
521C4⋊C4—612no—
522C4⋊C4—612no—
523C4⋊C4—1224no—
524C4⋊C4—1221no—
525C4⋊C4—1221no—
526C4⋊C4—1221no—
527C4⋊C4—1221no—
528C4⋊C4—1221no—
529C4⋊C4—1221no—
530C4⋊C4—1221no—
531C4⋊C4—1221no—
532C4⋊C4—1221no—
533C4⋊C4—4881no—
534C4⋊C4—2442no—
535C4⋊C4—2442no—
536C4⋊C4—2442no—
537C22⋊C4—41no—
538C22⋊C4—1224no—
539C22⋊C4—4881no—
540C22⋊C4—2442no—
541C22⋊C4—2442no—
542C22⋊C4—2442no—
543C22⋊C4—2442no—
544C22⋊C4—2442no—
545C22⋊C4#1175819014yescomplemented
546C42—21no—
547C42—21no—
548C42—21no—
549C42—21no—
550C42—4881no—
551C42—2442no—
552C42—2442no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—2441no—
560C23—2441no—
561C23—2441no—
562C23—2441no—
563C23—2441no—
564C23—2441no—
565C23—2441no—
566C23C2×#1175819012yes—
567C23C22×D24411yescharacteristic
568Q8—1222no—
569Q8—1222no—
570Q8—4884no—
571Q8—2441no—
572Q8—2441no—
573D4—4884no—
574D4—2441no—
575D4—2441no—
576D4—2441no—
577D4—2441no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—1222no—
596C2×C4—1222no—
597C2×C4—1222no—
598C2×C4—1222no—
599C2×C4—4884no—
600C2×C4—2441no—
601C2×C4—2441no—
602C2×C4—2441no—
603C2×C4—2441no—
604C2×C4—2441no—
605C2×C4—2441no—
606C2×C4—2441no—
607C2×C4—2441no—
608C2×C4—2441no—
609C2×C4—2441no—
610C2×C4—2441no—
611C2×C4—2441no—
612C2×C4—2441no—
613C2×C4—2441no—
614C2×C4—2441no—
615C2×C4—2441no—
616C2×C4—2441no—
617C2×C4C2×#1175819012yes—
618C2×C4C2×#1175819012yes—
619C2×C4#41×D6111yescharacteristic
620C2×C4#41×D6111yescharacteristic
621C2×C4#41×D6111yescharacteristic
622C2×C4#41×D6111yescharacteristic
623C2×C4#1451982211yescharacteristic
624C2×C4#1451982211yescharacteristic
625C8—41no—
626C8—41no—
627C8—41no—
628C8—41no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—41no—
633C22—41no—
634C22—41no—
635C22—41no—
636C22—4881no—
637C22—4881no—
638C22—4881no—
639C22—4881no—
640C22—4881no—
641C22—4881no—
642C22—4881no—
643C22#1684027411yescharacteristic
644C22#1684040711yescharacteristic
645C22#1684040711yescharacteristic
646C22C22×#1175819011yescharacteristic
647C22#1684056211yescharacteristic
648C22#1684056711yescharacteristic
649C22#1684057911yescharacteristic
650C4—22no—
651C4—22no—
652C4—41no—
653C4—41no—
654C4—41no—
655C4—4881no—
656C4—4881no—
657C4—4881no—
658C4—2442no—
659C4—2442no—
660C4—2442no—
661C4—2442no—
662C4#1684029511yescharacteristic
663C4#1684034411yescharacteristic
664C4#1684047411yescharacteristic
665C4#1684049611yescharacteristic
666C2—42no—
667C2—9761no—
668C2#2144975411yescharacteristic
669C2#2145585111yescharacteristic
670C2#2145608311yescharacteristic
671C2#2145764411yescharacteristic
672C2#2145768511yescharacteristic
673C2#2145797711yescharacteristic
674C2#2145808911yescharacteristic
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1923C2
socleC23×C61C22×D4
prime radicalC2×#14519751C22
hypercenter#1923D61
hypocenterC61#26724
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1923

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #21448822C2semidirect
3. #21447137C2semidirect
4. #21447016C2semidirect
5. #21446961C2semidirect
7. #21446770C2semidirect
8. #21446502C2semidirect
9. #21446447C2semidirect
10. #21446405C2semidirect
11. #21446173C2semidirect
13. #21445910C2semidirect
14. C4×#14519702C2semidirect
15. C61×#1923C2semidirect
182. C22⋊C4×C61SD16semidirect
338. #1923D61semidirect
396. C61#26724semidirect
545. C22⋊C4#11758190semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#214501822
#214488222
#214471372
#214470162
#214469612
#214468242
#214467702
#214465022
#214464472
#214464052
#214461732
#214461342
#214459102
C4×#145197022
C61×#19232
#2672461