Finite Group Databasebeta

← #33146081

Subgroups — #33146081

Order 18688.

904 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988792C211yesmaximal, characteristic
2C2×C4×#11924016C211yesmaximal, characteristic
3#22613707C211yesmaximal, characteristic, complemented
4#22611475C211yesmaximal, characteristic, complemented
5#22611432C211yesmaximal, characteristic, complemented
6#22611340C211yesmaximal, characteristic, complemented
7#22611177C211yesmaximal, characteristic, complemented
8#22610606C211yesmaximal, characteristic, complemented
9#22610490C211yesmaximal, characteristic, complemented
10#22610265C211yesmaximal, characteristic
11#22610265C211yesmaximal, characteristic
12#22610265C211yesmaximal, characteristic
13#22610198C211yesmaximal, characteristic, complemented
14#22610192C211yesmaximal, characteristic
15D4×C4⋊C4×C73C211yesmaximal, characteristic
16C22×#11924016C2211yescharacteristic
17C22×#11924016C2211yescharacteristic
18C2×#14556591C2211yescharacteristic
19C2×C4×Dic146C2211yescharacteristic
20#17988792—24no—
21C4×#11924016—24no—
22C2×#14556522C2211yescharacteristic
23C2×#14556522C2211yescharacteristic
24C2×C4⋊#10596831C2211yescharacteristic
25C2×#14556514C2211yescharacteristic
26C2×#14556514C2211yescharacteristic
27C2×#14556514C2211yescharacteristic
28C2×#14556514C2211yescharacteristic
29C2×#14556514C2211yescharacteristic
30C2×#14556514C2211yescharacteristic
31C2×#14556512C2211yescharacteristic
32C2×#14556512C2211yescharacteristic
33C2×C4×#10596831C2211yescharacteristic
34#17988529—21no—
35#17988520—21no—
36#17988493C2211yescharacteristic
37#17988493C2211yescharacteristic
38#17988493C2211yescharacteristic
39#17988485C2211yescharacteristic
40#17988474—21no—
41#17988473—21no—
42#17988349C2211yescharacteristic
43#17988349C2211yescharacteristic
44#17988349C2211yescharacteristic
45#17988349C2211yescharacteristic
46#17988342C2211yescharacteristic
47#17988342C2211yescharacteristic
48#17988342C2211yescharacteristic
49#17988342C2211yescharacteristic
50C2×C4×D4×C73C2211yescharacteristic
51C2×C4×D4×C73C2211yescharacteristic
52C2×C4×D4×C73C2211yescharacteristic
53C22×C4⋊C4×C73—21no—
54#389×C73C2211yescharacteristic
55#389×C73C2211yescharacteristic
56#385×C73—21no—
57#385×C73—21no—
58#384×C73C2211yescharacteristic
59#380×C73—21no—
60C4×C4⋊C4×C73C2211yescharacteristic
61C22×Dic146C2311yescharacteristic
62C22×Dic146C2311yescharacteristic
63C2×#11924016—22no—
64C2×#11924016—22no—
65C2×#11924016—22no—
66C2×#11924016—22no—
67C2×#11924016—22no—
68C2×#11924016D414yes—
69C22×#10596831C2311yescharacteristic
70C22×#10596831C2311yescharacteristic
71C22×#10596831C2311yescharacteristic
72C22×#10596831C2311yescharacteristic
73#14556591—22no—
74#14556591—41no—
75C4×Dic146—22no—
76C4×Dic146—41no—
77C2×C4⋊Dic73—21no—
78C2×C4⋊Dic73—21no—
79C2×C4⋊Dic73C2311yescharacteristic
80C2×C4⋊Dic73C2311yescharacteristic
81C2×#11924007—21no—
82C2×#11924007—21no—
83C2×C4×Dic73—21no—
84#14556522—22no—
85#14556522—41no—
86C4⋊#10596831—22no—
87#14556514—22no—
88#14556514—22no—
89#14556514—22no—
90#14556514—22no—
91#14556514—22no—
92#14556514—41no—
93#14556512—22no—
94#14556512—22no—
95C4×#10596831—22no—
96#14556498—21no—
97#14556498—21no—
98#14556498—21no—
99#14556498—21no—
100#14556498—21no—
101#14556498—21no—
102#14556498—21no—
103C22×D4×C73C2311yescharacteristic
104C23×C4×C73—21no—
105C23×C4×C73—21no—
106C23×C4×C73—21no—
107C4×D4×C73—22no—
108C4×D4×C73—22no—
109C4×D4×C73—22no—
110C4×D4×C73—22no—
111C4×D4×C73—41no—
112C4×D4×C73—41no—
113C2×C4⋊C4×C73—21no—
114C2×C4⋊C4×C73—21no—
115C2×C4⋊C4×C73—21no—
116C2×C4⋊C4×C73—21no—
117C2×C4⋊C4×C73—44no—
118C2×C4⋊C4×C73D411yescharacteristic
119C2×C4⋊C4×C73D411yescharacteristic
120C2×C4⋊C4×C73C2311yescharacteristic
121C2×C4⋊C4×C73C2311yescharacteristic
122C2×C4⋊C4×C73C2311yescharacteristic
123C2×C22⋊C4×C73—21no—
124C2×C22⋊C4×C73—21no—
125C2×C22⋊C4×C73—21no—
126C2×C42×C73C2311yescharacteristic
127C2×C42×C73C2311yescharacteristic
128C2×C42×C73C2311yescharacteristic
129#95×C73—21no—
130#95×C73—21no—
131#95×C73D411yescharacteristic
132#95×C73D411yescharacteristic
133C2×Dic146—21no—
134C2×Dic146—21no—
135C2×Dic146—21no—
136C2×Dic146—21no—
137C2×Dic146—21no—
138C2×Dic146—21no—
139C2×Dic146—21no—
140C2×Dic146—21no—
141C2×Dic146—21no—
142C2×Dic146—21no—
143C2×Dic146—21no—
144C2×Dic146C2×D412yes—
145C22×Dic73—21no—
146C22×Dic73—21no—
147C22×Dic73—21no—
148C22×Dic73—21no—
149#11924016—28no—
150#11924016—44no—
151C2×#10596831—21no—
152C2×#10596831—21no—
153C2×#10596831—21no—
154C2×#10596831—21no—
155C2×#10596831—21no—
156C2×#10596831—21no—
157C2×#10596831—21no—
158C2×#10596831—21no—
159C2×#10596831—21no—
160C2×#10596831—21no—
161C2×#10596831—21no—
162C2×#10596831C2×D412yes—
163C4⋊Dic73—21no—
164C4⋊Dic73—21no—
165C4⋊Dic73—21no—
166C4⋊Dic73—21no—
167C4⋊Dic73—42no—
168C4⋊Dic73—42no—
169#11924007—42no—
170#11924007—42no—
171C4×Dic73—42no—
172C24×C73—21no—
173C2×D4×C73—21no—
174C2×D4×C73—21no—
175C2×D4×C73—21no—
176C2×D4×C73C2×D412yes—
177C2×D4×C73#4112yes—
178C2×D4×C73#4112yes—
179C22×C4×C73—24no—
180C22×C4×C73—21no—
181C22×C4×C73—21no—
182C22×C4×C73—21no—
183C22×C4×C73—42no—
184C22×C4×C73—42no—
185C22×C4×C73—42no—
186C22×C4×C73—42no—
187C22×C4×C73—42no—
188C22×C4×C73—42no—
189C22×C4×C73—81no—
190C22×C4×C73—81no—
191C22×C4×C73C2×D411yescharacteristic
192C22×C4×C73C2×D411yescharacteristic
193C22×C4×C73C2×D411yescharacteristic
194C22×C4×C73C2×D411yescharacteristic
195C22×C4×C73C2×D411yescharacteristic
196C22×C4×C73C2×D411yescharacteristic
197C22×C4×C73C2411yescharacteristic
198C4⋊C4×C73—21no—
199C4⋊C4×C73—21no—
200C4⋊C4×C73—21no—
201C4⋊C4×C73—21no—
202C4⋊C4×C73—21no—
203C4⋊C4×C73—41no—
204C4⋊C4×C73—42no—
205C4⋊C4×C73—42no—
206C4⋊C4×C73—42no—
207C4⋊C4×C73—81no—
208C4⋊C4×C73SD1614yescomplemented
209C4⋊C4×C73#4112yes—
210C22⋊C4×C73—24no—
211C22⋊C4×C73—42no—
212C22⋊C4×C73—42no—
213C42×C73—21no—
214C42×C73—21no—
215C42×C73—21no—
216C42×C73—21no—
217C42×C73—21no—
218C42×C73—21no—
219C23×C73—22no—
220C23×C73—22no—
221C23×C73—22no—
222C23×C73—41no—
223C23×C73—41no—
224C23×C73—41no—
225C23×C73—41no—
226C23×C73C22×D411yescharacteristic
227D4×C73—22no—
228D4×C73—22no—
229D4×C73—22no—
230D4×C73#12314yes—
231C2×C4×C73—21no—
232C2×C4×C73—21no—
233C2×C4×C73—21no—
234C2×C4×C73—21no—
235C2×C4×C73—21no—
236C2×C4×C73—21no—
237C2×C4×C73—21no—
238C2×C4×C73—22no—
239C2×C4×C73—21no—
240C2×C4×C73—21no—
241C2×C4×C73—21no—
242C2×C4×C73—21no—
243C2×C4×C73—21no—
244C2×C4×C73—21no—
245C2×C4×C73—21no—
246C2×C4×C73—21no—
247C2×C4×C73—48no—
248C2×C4×C73—41no—
249C2×C4×C73—41no—
250C2×C4×C73—41no—
251C2×C4×C73—41no—
252C2×C4×C73—41no—
253C2×C4×C73—41no—
254C2×C4×C73—41no—
255C2×C4×C73—41no—
256C2×C4×C73—84no—
257C2×C4×C73—84no—
258C2×C4×C73C2×SD1612yes—
259C2×C4×C73C2×SD1612yes—
260C2×C4×C73C2×SD1612yes—
261C2×C4×C73C22×D411yescharacteristic
262C2×C4×C73C2×#4111yescharacteristic
263C2×C4×C73C2×#4111yescharacteristic
264C2×C4×C73C2×#4111yescharacteristic
265C2×C4×C73H(5,𝔽2)11yescharacteristic
266C2×C4×C732−1+411yescharacteristic
267C2×Dic73—22no—
268C2×Dic73—41no—
269C2×Dic73—41no—
270C2×Dic73—41no—
271C2×Dic73—41no—
272C2×Dic73—41no—
273C2×Dic73—41no—
274C2×Dic73—41no—
275C2×Dic73—41no—
276C2×Dic73—41no—
277C2×Dic73—41no—
278C2×Dic73—41no—
279Dic146—22no—
280Dic146—22no—
281Dic146—41no—
282Dic146—41no—
283Dic146—41no—
284Dic146—41no—
285Dic146—41no—
286Dic146—41no—
287#10596831—22no—
288#10596831—41no—
289#10596831—41no—
290#10596831—41no—
291C22×C73—24no—
292C22×C73—42no—
293C22×C73—42no—
294C22×C73—42no—
295C22×C73—42no—
296C22×C73—42no—
297C22×C73—42no—
298C22×C73#52511yescharacteristic
299C22×C73D4211yescharacteristic
300C22×C73#54811yescharacteristic
301C22×C73C22×SD1611yescharacteristic
302C22×C73#57511yescharacteristic
303C22×C73#57711yescharacteristic
304C22×C73#57811yescharacteristic
305C4×C73—22no—
306C4×C73—22no—
307C4×C73—22no—
308C4×C73—42no—
309C4×C73—41no—
310C4×C73—41no—
311C4×C73—41no—
312C4×C73—81no—
313C4×C73—81no—
314C4×C73C2×#12311yescharacteristic
315C4×C73#53911yescharacteristic
316C4×C73#54711yescharacteristic
317C4×C73#55311yescharacteristic
318Dic73—42no—
319Dic73—81no—
320Dic73—81no—
321Dic73—81no—
322#26166—731nomaximal, Sylow
323C2×C73—44no—
324C2×C73#244311yescharacteristic
325C2×C73#310111yescharacteristic
326C2×C73#312011yescharacteristic
327C2×C73#326911yescharacteristic
328C2×C73D4×SD1611yescharacteristic
329C2×C73#335911yescharacteristic
330C2×C73#337611yescharacteristic
331C2×#461—731no—
332C2×C4×SD16—731no—
333#2446—731no—
334D4×C4⋊C4D7311yescharacteristic
335#2096—731no—
336#2088—731no—
337#2072—731no—
338#2043—731no—
339#1933—731no—
340#1916—731no—
341#1877—731no—
342#1877—731no—
343#1877—731no—
344#1865—731no—
345#1864—731no—
346C73#2616611yescharacteristic, complemented, Sylow
347C22×SD16—731no—
348C22×SD16—731no—
349C2×C4⋊Q8—731no—
350C2×C4×Q8—731no—
351C2×C4×D4C2×D7311yescharacteristic
352C2×C4×D4C2×D7311yescharacteristic
353C2×C4×D4C2×D7311yescharacteristic
354C22×C4⋊C4—21no—
355#461—1464no—
356C4×SD16—1464no—
357C2×#106—731no—
358C2×#106—731no—
359C2×C4⋊C8—731no—
360C2×#103—731no—
361C2×#103—731no—
362C2×#103—731no—
363C2×#103—731no—
364C2×#103—731no—
365C2×#103—731no—
366C2×#102—731no—
367C2×#102—731no—
368C2×C4×C8—731no—
369#400—1461no—
370#398—1461no—
371#391—731no—
372#391—731no—
373#391—731no—
374#389—731no—
375#389C2×D7311yescharacteristic
376#389C2×D7311yescharacteristic
377#387—1461no—
378#387—1461no—
379#385—21no—
380#385—21no—
381#384C2×D7311yescharacteristic
382#380—21no—
383C4×C4⋊C4C2×D7311yescharacteristic
384#340—731no—
385#340—731no—
386#340—731no—
387#340—731no—
388#336—731no—
389#336—731no—
390#336—731no—
391#336—731no—
392C22×Q8—731no—
393C22×Q8—731no—
394C22×D4C22×D7311yescharacteristic
395C23×C4—21no—
396C23×C4—21no—
397C23×C4—21no—
398C2×SD16—734no—
399C2×SD16—1462no—
400C2×SD16—1462no—
401C2×SD16—1462no—
402C2×SD16—1462no—
403C2×SD16—1462no—
404C22×C8—731no—
405C22×C8—731no—
406C22×C8—731no—
407C22×C8—731no—
408C4⋊Q8—2921no—
409C4⋊Q8—1462no—
410C4×Q8—2921no—
411C4×Q8—1462no—
412C4×D4—22no—
413C4×D4—22no—
414C4×D4—22no—
415C4×D4—22no—
416C4×D4—41no—
417C4×D4—41no—
418C2×C4⋊C4—21no—
419C2×C4⋊C4—21no—
420C2×C4⋊C4—21no—
421C2×C4⋊C4—21no—
422C2×C4⋊C4—44no—
423C2×C4⋊C4—731no—
424C2×C4⋊C4—731no—
425C2×C4⋊C4—1461no—
426C2×C4⋊C4—1461no—
427C2×C4⋊C4—1461no—
428C2×C4⋊C4—1461no—
429C2×C4⋊C4C22⋊D7311yescharacteristic
430C2×C4⋊C4C22⋊D7311yescharacteristic
431C2×C4⋊C4C22×D7311yescharacteristic
432C2×C4⋊C4C22×D7311yescharacteristic
433C2×C4⋊C4C22×D7311yescharacteristic
434C2×C22⋊C4—21no—
435C2×C22⋊C4—21no—
436C2×C22⋊C4—21no—
437C2×C42—1461no—
438C2×C42C22×D7311yescharacteristic
439C2×C42C22×D7311yescharacteristic
440C2×C42C22×D7311yescharacteristic
441#106—2921no—
442#106—1462no—
443C4⋊C8—1462no—
444#103—2921no—
445#103—1462no—
446#103—1462no—
447#103—1462no—
448#103—1462no—
449#103—1462no—
450#102—1462no—
451#102—1462no—
452C4×C8—1462no—
453#95—21no—
454#95—21no—
455#95—1461no—
456#95—1461no—
457#95—1461no—
458#95—1461no—
459#95—1461no—
460#95—1461no—
461#95—1461no—
462#95C22⋊D7311yescharacteristic
463#95C22⋊D7311yescharacteristic
464C24—21no—
465C2×Q8—732no—
466C2×Q8—1461no—
467C2×Q8—1461no—
468C2×Q8—1461no—
469C2×Q8—1461no—
470C2×Q8—1461no—
471C2×Q8—1461no—
472C2×Q8—1461no—
473C2×Q8—1461no—
474C2×Q8—1461no—
475C2×Q8—1461no—
476C2×Q8—1461no—
477C2×D4—21no—
478C2×D4—21no—
479C2×D4—21no—
480C2×D4D4×D7312yes—
481C2×D4#1192402812yes—
482C2×D4#1192402912yes—
483C22×C4—24no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—42no—
491C22×C4—42no—
492C22×C4—42no—
493C22×C4—81no—
494C22×C4—81no—
495C22×C4—1461no—
496C22×C4—1461no—
497C22×C4—1461no—
498C22×C4—1461no—
499C22×C4C2×C22⋊D7311yescharacteristic
500C22×C4C2×C22⋊D7311yescharacteristic
501C22×C4C2×C22⋊D7311yescharacteristic
502C22×C4C2×C22⋊D7311yescharacteristic
503C22×C4C2×C22⋊D7311yescharacteristic
504C22×C4C2×C22⋊D7311yescharacteristic
505C22×C4C23×D7311yescharacteristic
506SD16—2924no—
507SD16—1468no—
508C2×C8—732no—
509C2×C8—1461no—
510C2×C8—1461no—
511C2×C8—1461no—
512C2×C8—1461no—
513C2×C8—1461no—
514C2×C8—1461no—
515C2×C8—1461no—
516C2×C8—1461no—
517C2×C8—1461no—
518C2×C8—1461no—
519C2×C8—1461no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—41no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—81no—
530C4⋊C4—2922no—
531C4⋊C4—2922no—
532C4⋊C4—2922no—
533C4⋊C4—2922no—
534C4⋊C4—1461no—
535C4⋊C4—1461no—
536C4⋊C4—1461no—
537C4⋊C4—1461no—
538C4⋊C4#1192401614yescomplemented
539C4⋊C4#1192402812yes—
540C22⋊C4—24no—
541C22⋊C4—42no—
542C22⋊C4—42no—
543C42—21no—
544C42—21no—
545C42—21no—
546C42—21no—
547C42—21no—
548C42—21no—
549C42—2922no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23C22×C22⋊D7311yescharacteristic
558Q8—2921no—
559Q8—2921no—
560Q8—2921no—
561Q8—2921no—
562Q8—2921no—
563Q8—2921no—
564Q8—1462no—
565Q8—1462no—
566D4—22no—
567D4—22no—
568D4—22no—
569D4#1455657614yes—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—48no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—84no—
596C2×C4—84no—
597C2×C4—2921no—
598C2×C4—2921no—
599C2×C4—2921no—
600C2×C4—2921no—
601C2×C4—2921no—
602C2×C4—2921no—
603C2×C4—2921no—
604C2×C4—2921no—
605C2×C4—2921no—
606C2×C4—2921no—
607C2×C4—2921no—
608C2×C4—1462no—
609C2×C4C2×#1192401612yes—
610C2×C4C2×#1192401612yes—
611C2×C4C2×#1192401612yes—
612C2×C4C2×D4×D7311yescharacteristic
613C2×C4C2×#1192402811yescharacteristic
614C2×C4C2×#1192402811yescharacteristic
615C2×C4C2×#1192402911yescharacteristic
616C2×C4#1455663611yescharacteristic
617C2×C4#1455663911yescharacteristic
618C8—2921no—
619C8—2921no—
620C8—2921no—
621C8—1462no—
622C22—24no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22#1798909811yescharacteristic
630C22D4×C22⋊D7311yescharacteristic
631C22#1798924811yescharacteristic
632C22C22×#1192401611yescharacteristic
633C22#1798938511yescharacteristic
634C22#1798939811yescharacteristic
635C22#1798940211yescharacteristic
636C4—22no—
637C4—22no—
638C4—22no—
639C4—42no—
640C4—41no—
641C4—41no—
642C4—41no—
643C4—81no—
644C4—81no—
645C4—2922no—
646C4—5841no—
647C4—5841no—
648C4—5841no—
649C4C2×#1455657611yescharacteristic
650C4#1798918411yescharacteristic
651C4#1798923811yescharacteristic
652C4#1798928511yescharacteristic
653C2—44no—
654C2#2261366611yescharacteristic
655C2#2261993911yescharacteristic
656C2#2262014511yescharacteristic
657C2#2262148211yescharacteristic
658C2D4×#1192401611yescharacteristic
659C2#2262257311yescharacteristic
660C2#2262275811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingD4×C4⋊C4×C73C2
socleC23×C73C22×D4
prime radicalC22×D4×C73C23
hypercenterD4×C4⋊C4D73
hypocenterC73#26166
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ D4×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #22613707C2semidirect
4. #22611475C2semidirect
5. #22611432C2semidirect
6. #22611340C2semidirect
7. #22611177C2semidirect
8. #22610606C2semidirect
9. #22610490C2semidirect
13. #22610198C2semidirect
208. C4⋊C4×C73SD16semidirect
346. C73#26166semidirect
538. C4⋊C4#11924016semidirect

Maximal subgroups

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

subgroupindex
C2×#179887922
C2×C4×#119240162
#226137072
#226114752
#226114322
#226113402
#226111772
#226106062
#226104902
#226102652
#226101982
#226101922
D4×C4⋊C4×C732
#2616673