Finite Group Databasebeta

← #29738877

Subgroups — #29738877

Order 15616.

818 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840170C211yesmaximal, characteristic, complemented
2C2×#16839805C211yesmaximal, characteristic
3#21449526C211yesmaximal, characteristic
4#21449522C211yesmaximal, characteristic
5#21447299C211yesmaximal, characteristic, complemented
6#21447293C211yesmaximal, characteristic, complemented
7#21447156C211yesmaximal, characteristic, complemented
8#21447150C211yesmaximal, characteristic, complemented
9#21446556C211yesmaximal, characteristic, complemented
10#21446551C211yesmaximal, characteristic, complemented
11#21446014C211yesmaximal, characteristic
12#21445974C211yesmaximal, characteristic, complemented
13#21445850C211yesmaximal, characteristic
14#21445605C211yesmaximal, characteristic
15C61×#1854C211yesmaximal, characteristic
16C22×#11758194C2211yescharacteristic
17C2×#14519776C2211yescharacteristic
18C2×#14519768C2211yescharacteristic
19C2×#14519754—21no—
20C22×C4⋊Dic61C2211yescharacteristic
21#16840170—24no—
22#16839805—24no—
23C2×#14519710C2211yescharacteristic
24C2×#14519710C2211yescharacteristic
25C2×#14519710C2211yescharacteristic
26C2×#14519709C2211yescharacteristic
27C2×#14519709C2211yescharacteristic
28C2×#14519708C2211yescharacteristic
29C2×#14519708C2211yescharacteristic
30C2×#14519708C2211yescharacteristic
31C2×#14519687C2211yescharacteristic
32C2×C8×Dic61C2211yescharacteristic
33#16839698—21no—
34#16839696—21no—
35#16839685—21no—
36#16839683—21no—
37#16839661C2211yescharacteristic
38#16839661C2211yescharacteristic
39#16839660—21no—
40#16839658C2211yescharacteristic
41#16839658C2211yescharacteristic
42#16839646—21no—
43#16839634—21no—
44#16839631C2211yescharacteristic
45#16839631C2211yescharacteristic
46#16839622C2211yescharacteristic
47C4×C22⋊Dic61C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839516C2211yescharacteristic
53#16839516C2211yescharacteristic
54C2×#117×C61C2211yescharacteristic
55C2×#107×C61C2211yescharacteristic
56C2×#106×C61C2211yescharacteristic
57C2×C22⋊C8×C61C2211yescharacteristic
58C61×#380C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C22×Dic122—21no—
62C23×Dic61—21no—
63C2×#11758194—22no—
64C2×#11758194—22no—
65C2×#11758194Q814yescomplemented
66C22×#92418C2311yescharacteristic
67C22×#92418C2311yescharacteristic
68#14519776—22no—
69#14519776—41no—
70#14519768—22no—
71#14519768—41no—
72#14519754—44no—
73C2×C4⋊Dic61—22no—
74C2×C4⋊Dic61—21no—
75C2×C4⋊Dic61—21no—
76C2×C4⋊Dic61—21no—
77C2×C4⋊Dic61—21no—
78C2×C4⋊Dic61—41no—
79C2×C4⋊Dic61C2311yescharacteristic
80C2×C4⋊Dic61C2311yescharacteristic
81C2×C4⋊Dic61C2311yescharacteristic
82C2×C4⋊Dic61C2311yescharacteristic
83C2×#11758196—21no—
84C2×#11758196—21no—
85C2×#11758196—21no—
86C2×#11758196—21no—
87C2×#11758196—21no—
88C2×#11758196—21no—
89C2×#11758196—21no—
90C2×#11758196—21no—
91C2×C22⋊Dic61—21no—
92C2×C22⋊Dic61D411yescharacteristic
93C2×C22⋊Dic61D411yescharacteristic
94C2×C4×Dic61C2311yescharacteristic
95C2×C4×Dic61C2311yescharacteristic
96#14519710—22no—
97#14519710—22no—
98#14519710D414yes—
99#14519709—22no—
100#14519709—41no—
101#14519708—22no—
102#14519708—22no—
103#14519708—22no—
104#14519687—22no—
105C8×Dic61—22no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684D411yescharacteristic
112#14519684D411yescharacteristic
113C23×C4×C61C2311yescharacteristic
114C22×C8×C61C2311yescharacteristic
115C22×C8×C61C2311yescharacteristic
116#117×C61—22no—
117#117×C61—41no—
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C4⋊C4×C61C2311yescharacteristic
120C2×C4⋊C4×C61C2311yescharacteristic
121C2×C4⋊C4×C61C2311yescharacteristic
122C2×C22⋊C4×C61—21no—
123C2×C22⋊C4×C61—21no—
124C2×C42×C61—21no—
125#107×C61—22no—
126#106×C61—41no—
127C22⋊C8×C61Q814yescomplemented
128#95×C61—21no—
129C24×C61C2×D411yescharacteristic
130C22×C4×C61—21no—
131C22×C4×C61—21no—
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61—21no—
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2×Q812yes—
140C22×C4×C61C2×Q812yes—
141C22×C4×C61C2411yescharacteristic
142C22×Dic61—21no—
143C22×Dic61—21no—
144C22×Dic61—21no—
145C22×Dic61—21no—
146C22×Dic61—42no—
147C22×Dic61—42no—
148C22×Dic61—81no—
149C22×Dic61C2×D411yescharacteristic
150C22×Dic61C2×D411yescharacteristic
151C22×Dic61C2×D411yescharacteristic
152C22×Dic61C2×D411yescharacteristic
153C2×Dic122—42no—
154C2×Dic122—42no—
155C2×Dic122—81no—
156C2×C8×C61—21no—
157C2×C8×C61—21no—
158C2×C8×C61—21no—
159C2×C8×C61—21no—
160C2×C8×C61C2×Q812yes—
161C2×C8×C61C2×Q812yes—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61C2×D412yes—
170C22⋊C4×C61—42no—
171C22⋊C4×C61—42no—
172C42×C61—42no—
173C22⋊Dic61—22no—
174C22⋊Dic61—22no—
175C22⋊Dic61—81no—
176C4⋊Dic61—21no—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—42no—
184C4⋊Dic61—42no—
185C4⋊Dic61—42no—
186C4⋊Dic61—42no—
187C4⋊Dic61C2×D412yes—
188#11758196—42no—
189#11758196—42no—
190#11758196—42no—
191#11758196—42no—
192#11758196—42no—
193#11758196—81no—
194#11758196—81no—
195#11758196—81no—
196C4×Dic61—21no—
197C4×Dic61—21no—
198C4×Dic61—21no—
199C4×Dic61—21no—
200#11758194—28no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418C2×D412yes—
204C2×#92418C2×Q812yes—
205C2×#92418C2×Q812yes—
206C2×#92418#4112yes—
207C23×C61—21no—
208C23×C61—21no—
209C23×C61—21no—
210C23×C61—21no—
211C23×C61—21no—
212C23×C61—21no—
213C23×C61C4⋊Q812yes—
214C23×C61C22×D411yescharacteristic
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—22no—
218C2×C4×C61—22no—
219C2×C4×C61—22no—
220C2×C4×C61—22no—
221C2×C4×C61—22no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61C4⋊Q812yes—
234C2×C4×C61C22×D411yescharacteristic
235C2×C4×C61C22×Q811yescharacteristic
236C2×C4×C61C22×Q811yescharacteristic
237C2×C4×C61C2×#4111yescharacteristic
238C2×C4×C61H(5,𝔽2)11yescharacteristic
239C2×C4×C612−1+411yescharacteristic
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—21no—
245C2×Dic61—21no—
246C2×Dic61—22no—
247C2×Dic61—21no—
248C2×Dic61—21no—
249C2×Dic61—21no—
250C2×Dic61—21no—
251C2×Dic61—41no—
252C2×Dic61—41no—
253C2×Dic61—41no—
254C2×Dic61—41no—
255C2×Dic61—41no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61—84no—
263C2×Dic61AGL(1,ℤ/8)12yes—
264C2×Dic61#13712yes—
265Dic122—84no—
266C8×C61—22no—
267C8×C61—22no—
268#92418—22no—
269#92418C22⋊Q814yescomplemented
270#26078—611nomaximal, Sylow
271C22×C61—22no—
272C22×C61—22no—
273C22×C61—22no—
274C22×C61—41no—
275C22×C61—41no—
276C22×C61—41no—
277C22×C61—41no—
278C22×C61C2×C4⋊Q811yescharacteristic
279C22×C61D4211yescharacteristic
280C22×C61#54711yescharacteristic
281C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
282C22×C61C2×#13711yescharacteristic
283C22×C61#57711yescharacteristic
284C22×C61#57711yescharacteristic
285C4×C61—22no—
286C4×C61—42no—
287C4×C61—81no—
288C4×C61—81no—
289C4×C61—81no—
290C4×C61C2×C22⋊Q811yescharacteristic
291C4×C61#54411yescharacteristic
292C4×C61D4×Q811yescharacteristic
293C4×C61#55411yescharacteristic
294Dic61—22no—
295Dic61—22no—
296Dic61—41no—
297Dic61—42no—
298Dic61—41no—
299Dic61—81no—
300Dic61—81no—
301Dic61—81no—
302C2×#501—611no—
303C2×#428—611no—
304#2441—611no—
305#2440—611no—
306#2084—611no—
307#2083—611no—
308#2059—611no—
309#2058—611no—
310#1943—611no—
311#1942—611no—
312#1854D6111yescharacteristic
313#1853—611no—
314#1844—611no—
315#1809—611no—
316#1753—611no—
317C2×C61—42no—
318C2×C61#244011yescharacteristic
319C2×C61#316311yescharacteristic
320C2×C61#316511yescharacteristic
321C2×C61#327211yescharacteristic
322C2×C61#327811yescharacteristic
323C2×C61#334411yescharacteristic
324C2×C61#334511yescharacteristic
325C22×M16—611no—
326C2×C4⋊Q8—611no—
327C2×#125—611no—
328C2×C22⋊Q8—1221no—
329C2×#117C2×D6111yescharacteristic
330C22×C4⋊C4—611no—
331#501—1224no—
332#428—1224no—
333C2×#107—611no—
334C2×#107—611no—
335C2×#107—611no—
336C2×#107C2×D6111yescharacteristic
337C2×#106—611no—
338C2×#106—611no—
339C2×#106—611no—
340C2×#106—611no—
341C2×#106—611no—
342C2×#106C2×D6111yescharacteristic
343C2×C22⋊C8C2×D6111yescharacteristic
344C2×#97—611no—
345C2×C4×C8—611no—
346#399—1221no—
347#398—1221no—
348#396—1221no—
349#395—1221no—
350#389—611no—
351#389—611no—
352#389—611no—
353#389—611no—
354#389—1221no—
355#387—1221no—
356#385—1221no—
357#384—611no—
358#384—611no—
359#380—611no—
360#380C2×D6111yescharacteristic
361C4×C22⋊C4—611no—
362#340—611no—
363#340—611no—
364#340—611no—
365#340—611no—
366#340C2×D6111yescharacteristic
367#340C2×D6111yescharacteristic
368#336—611no—
369#336—611no—
370C61#2607811yescharacteristic, complemented, Sylow
371C22×Q8—1221no—
372C23×C4—1221no—
373C23×C4C22×D6111yescharacteristic
374C2×M16—614no—
375C2×M16—1222no—
376C2×M16—1222no—
377C22×C8—611no—
378C22×C8—611no—
379C22×C8C22×D6111yescharacteristic
380C22×C8C22×D6111yescharacteristic
381C4⋊Q8—1222no—
382C4⋊Q8—2441no—
383#125—1222no—
384#125—2441no—
385C22⋊Q8—2444no—
386#117—22no—
387#117—41no—
388C2×C4⋊C4—611no—
389C2×C4⋊C4—611no—
390C2×C4⋊C4—611no—
391C2×C4⋊C4—611no—
392C2×C4⋊C4—1222no—
393C2×C4⋊C4—1221no—
394C2×C4⋊C4—1221no—
395C2×C4⋊C4—1221no—
396C2×C4⋊C4—1221no—
397C2×C4⋊C4—1221no—
398C2×C4⋊C4—1221no—
399C2×C4⋊C4—1221no—
400C2×C4⋊C4—1221no—
401C2×C4⋊C4—1221no—
402C2×C4⋊C4—1221no—
403C2×C4⋊C4—1221no—
404C2×C4⋊C4—1221no—
405C2×C4⋊C4—2441no—
406C2×C4⋊C4C22×D6111yescharacteristic
407C2×C4⋊C4C22×D6111yescharacteristic
408C2×C4⋊C4C22×D6111yescharacteristic
409C2×C4⋊C4C22×D6111yescharacteristic
410C2×C22⋊C4—21no—
411C2×C22⋊C4—21no—
412C2×C22⋊C4—611no—
413C2×C22⋊C4—611no—
414C2×C22⋊C4—1221no—
415C2×C42—21no—
416C2×C42—611no—
417C2×C42—611no—
418#107—22no—
419#107—614no—
420#107—1222no—
421#107—1222no—
422#106—41no—
423#106—1222no—
424#106—1222no—
425#106—1222no—
426#106—1222no—
427#106—2441no—
428C22⋊C8Dic12214yescomplemented
429#97—1222no—
430C4×C8—1222no—
431#95—21no—
432#95—611no—
433#95—611no—
434#95—1221no—
435#95—1221no—
436#95—1221no—
437#95—1221no—
438#95—1221no—
439C24D4×D6111yescharacteristic
440C2×Q8—4881no—
441C2×Q8—2442no—
442C2×Q8—2442no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—611no—
452C22×C4—611no—
453C22×C4—611no—
454C22×C4—611no—
455C22×C4—1221no—
456C22×C4—1221no—
457C22×C4—1221no—
458C22×C4—1221no—
459C22×C4—4881no—
460C22×C4—2442no—
461C22×C4—2442no—
462C22×C4C2×Dic12212yes—
463C22×C4D4×D6111yescharacteristic
464C22×C4Q8×D6112yes—
465C22×C4C23×D6111yescharacteristic
466M16—1228no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—612no—
472C2×C8—612no—
473C2×C8—612no—
474C2×C8—612no—
475C2×C8—1221no—
476C2×C8—1221no—
477C2×C8C2×Dic12212yes—
478C2×C8C2×Dic12212yes—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—612no—
487C4⋊C4—1221no—
488C4⋊C4—1221no—
489C4⋊C4—1221no—
490C4⋊C4—1221no—
491C4⋊C4—1221no—
492C4⋊C4—1221no—
493C4⋊C4—1221no—
494C4⋊C4—4881no—
495C4⋊C4—4881no—
496C4⋊C4—4881no—
497C4⋊C4—2442no—
498C4⋊C4—2442no—
499C4⋊C4—2442no—
500C4⋊C4—2442no—
501C4⋊C4—2442no—
502C4⋊C4—2442no—
503C4⋊C4—2442no—
504C4⋊C4—2442no—
505C4⋊C4—2442no—
506C4⋊C4D4×D6112yes—
507C22⋊C4—42no—
508C22⋊C4—42no—
509C22⋊C4—1222no—
510C22⋊C4—1222no—
511C22⋊C4—4881no—
512C42—42no—
513C42—1221no—
514C42—1221no—
515C42—1221no—
516C42—1221no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23#1451977612yes—
524C23C2×D4×D6111yescharacteristic
525Q8—4884no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—612no—
545C2×C4—612no—
546C2×C4—1221no—
547C2×C4—1221no—
548C2×C4—1221no—
549C2×C4—1221no—
550C2×C4—1221no—
551C2×C4—1221no—
552C2×C4—1221no—
553C2×C4—1221no—
554C2×C4—1221no—
555C2×C4—1221no—
556C2×C4—1222no—
557C2×C4—4884no—
558C2×C4—2441no—
559C2×C4—2441no—
560C2×C4—2441no—
561C2×C4—2441no—
562C2×C4—2441no—
563C2×C4—2441no—
564C2×C4—2441no—
565C2×C4—2441no—
566C2×C4—2441no—
567C2×C4—2441no—
568C2×C4—2441no—
569C2×C4#1451977612yes—
570C2×C4C2×D4×D6111yescharacteristic
571C2×C4C2×Q8×D6111yescharacteristic
572C2×C4C22×Dic12211yescharacteristic
573C2×C4#41×D6111yescharacteristic
574C2×C4#1451982211yescharacteristic
575C2×C4#1451982411yescharacteristic
576C8—22no—
577C8—22no—
578C8—614no—
579C8—1222no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—41no—
584C22—41no—
585C22—41no—
586C22—41no—
587C22C2×#1451977611yescharacteristic
588C22#1684039411yescharacteristic
589C22#1684041011yescharacteristic
590C22AGL(1,ℤ/8)×D6111yescharacteristic
591C22#137×D6111yescharacteristic
592C22#1684056911yescharacteristic
593C22#1684056911yescharacteristic
594C4—22no—
595C4—42no—
596C4—81no—
597C4—81no—
598C4—81no—
599C4—1222no—
600C4—1222no—
601C4—4881no—
602C4—4881no—
603C4—4881no—
604C4—2441no—
605C4—2442no—
606C4—2441no—
607C4C22⋊Q8×D6111yescharacteristic
608C4#1684039111yescharacteristic
609C4D4×Dic12211yescharacteristic
610C4#1684047111yescharacteristic
611C2—42no—
612C2#2144952011yescharacteristic
613C2#2145635111yescharacteristic
614C2#2145636311yescharacteristic
615C2#2145741211yescharacteristic
616C2#2145747811yescharacteristic
617C2#2145826711yescharacteristic
618C2#2145829211yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1854C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
hypercenter#1854D61
hypocenterC61#26078
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1854

Complements

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

NH ≅ G/Ntype
1. C2×#16840170C2semidirect
5. #21447299C2semidirect
6. #21447293C2semidirect
7. #21447156C2semidirect
8. #21447150C2semidirect
9. #21446556C2semidirect
10. #21446551C2semidirect
12. #21445974C2semidirect
65. C2×#11758194Q8semidirect
127. C22⋊C8×C61Q8semidirect
269. #92418C22⋊Q8semidirect
370. C61#26078semidirect
428. C22⋊C8Dic122semidirect

Maximal subgroups

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

subgroupindex
C2×#168401702
C2×#168398052
#214495262
#214495222
#214472992
#214472932
#214471562
#214471502
#214465562
#214465512
#214460142
#214459742
#214458502
#214456052
C61×#18542
#2607861