Finite Group Databasebeta

← #11675931

Subgroups — #11675931

Order 768.

1674 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3502C211yesmaximal, characteristic
2#86258C211yesmaximal, characteristic
3#83709C214yesmaximal, complemented
4#83684C214yesmaximal, complemented
5#83684C214yesmaximal, complemented
6#83683C214yesmaximal, complemented
7#83480C212yesmaximal
8#83472C212yesmaximal
9#83367C211yesmaximal, characteristic
10#83340C211yesmaximal, characteristic
11#83329C211yesmaximal, characteristic
12#82550C212yesmaximal
13#82549C212yesmaximal
14#82431C211yesmaximal, characteristic
15C2×#5030C211yesmaximal, characteristic
16#61379—31nomaximal, Sylow
17C3×Q82—21no—
18C3×Q82C2212yes—
19C3×#557—21no—
20C3×#557—22no—
21C3×#557C2214yes—
22C3×#557C2212yes—
23C3×#544—22no—
24C3×#544C2212yes—
25C3×#533—22no—
26C3×#533C2211yescharacteristic
27C3×#533C2211yescharacteristic
28C3×#519C2212yes—
29C2×C3×C4×Q8C2211yescharacteristic
30C22×#216C2211yescharacteristic
31#5722—21no—
32#5721—21no—
33#5685—22no—
34#5623—22no—
35#5622—22no—
36#5567C2212yes—
37#5567C2214yes—
38Q8×Dic6C2212yes—
39#5519C2212yes—
40#5516C2212yes—
41#5470C2211yescharacteristic
42C2×C4×Dic6C2211yescharacteristic
43#5225C2212yes—
44C2×#835C2211yescharacteristic
45#5094C2214yes—
46#5094C2212yes—
47#5091C2214yes—
48#5091C2212yes—
49#5081C2214yes—
50#5081C2212yes—
51#5081C2212yes—
52#5080C2214yes—
53#5065C2214yes—
54#5064C2214yes—
55#5064C2212yes—
56#5064C2212yes—
57#5063C2214yes—
58#5063C2214yes—
59#5063C2214yes—
60#5030C2214yes—
61#5030C2214yes—
62#5030C2214yes—
63#5030C2214yes—
64C4×#216C2214yes—
65C4×#216C2214yes—
66#5024C2214yes—
67#5024C2214yes—
68#5022C2214yes—
69#5022C2214yes—
70#5022C2214yes—
71#5021C2214yes—
72#5021C2212yes—
73#5020C2214yes—
74#5020C2212yes—
75#5004C2212yes—
76#4999C2212yes—
77#4998C2212yes—
78#4998C2212yes—
79#4997C2212yes—
80#4995C2212yes—
81C2×#810C2211yescharacteristic
82#4964C2212yes—
83C2×#808C2211yescharacteristic
84#4924C2211yescharacteristic
85C2×#803C2211yescharacteristic
86#3502—31no—
87#3502D311yescharacteristic
88#3390—34no—
89#3390—34no—
90#3387—34no—
91#3387—34no—
92#3163—32no—
93#3162—31no—
94#3082—32no—
95#3082—32no—
96#3078—31no—
97#3078—31no—
98#2961—31no—
99#2950—32no—
100C2×#441—31no—
101C22×C3×Q8C2311yescharacteristic
102C22×Dic6C2311yescharacteristic
103C3×C4⋊Q8—24no—
104C3×C4⋊Q8—22no—
105C3×C4⋊Q8—22no—
106C3×C4⋊Q8—22no—
107C3×C4⋊Q8—24no—
108C3×C4⋊Q8D411yescharacteristic
109C3×C4⋊Q8D411yescharacteristic
110C3×C4⋊Q8C2311yescharacteristic
111C3×C4⋊Q8C2314yes—
112C3×C4⋊Q8C2311yescharacteristic
113C3×#125—24no—
114C3×#125—24no—
115C3×#125—24no—
116C3×#125—22no—
117C3×#125D412yes—
118C3×#125C2312yes—
119C3×#125C2314yes—
120C3×C22⋊Q8—21no—
121C3×C22⋊Q8—21no—
122C3×C22⋊Q8—24no—
123C3×C22⋊Q8D412yes—
124C3×C22⋊Q8D412yes—
125C3×C4×Q8—22no—
126C3×C4×Q8—22no—
127C3×C4×Q8—22no—
128C3×C4×Q8—22no—
129C3×C4×Q8—24no—
130C3×C4×Q8—21no—
131C3×C4×Q8—21no—
132C3×C4×Q8—21no—
133C3×C4×Q8—21no—
134C3×C4×Q8C2314yes—
135C3×C4×Q8C2312yes—
136C3×C4×Q8C2312yes—
137C3×C4×Q8C2314yes—
138C3×C4×Q8C2312yes—
139C3×C4×Q8C2312yes—
140C3×#117—24no—
141C3×#117C2312yes—
142C3×#117C2312yes—
143C2×C3×C4⋊C4—21no—
144C2×C3×C4⋊C4C2311yescharacteristic
145C2×C3×C42—21no—
146C2×C3×C42C2311yescharacteristic
147Q8×Dic3—22no—
148#944—22no—
149#944—24no—
150C2×#216Q818yescomplemented
151C2×#216C2314yes—
152#926C2312yes—
153C2×C4⋊Dic3C2311yescharacteristic
154#924—21no—
155#924—21no—
156C2×Dic3⋊C4—21no—
157C2×C4×Dic3—21no—
158C2×#208C2312yes—
159C22×C3⋊C8C2311yescharacteristic
160C22×C3⋊C8C2311yescharacteristic
161#890—22no—
162#889—24no—
163#889—24no—
164#889—24no—
165#888—22no—
166#888—24no—
167#888—22no—
168#887—22no—
169#887—22no—
170#887—24no—
171#887—22no—
172#878—24no—
173#877—24no—
174#870C2312yes—
175#870C2314yes—
176#869C2311yescharacteristic
177#869C2311yescharacteristic
178#869C2314yes—
179C4×Dic6—21no—
180C4×Dic6—21no—
181C4×Dic6—21no—
182C4×Dic6—21no—
183C4×Dic6C2312yes—
184C4×Dic6C2312yes—
185C4×Dic6C2314yes—
186C4×Dic6C2314yes—
187#835C2314yes—
188#835C2314yes—
189#835C2314yes—
190#810C2314yes—
191#810C2314yes—
192#810C2314yes—
193#808C2314yes—
194#808C2314yes—
195#808C2314yes—
196#808C2312yes—
197#808C2312yes—
198#808C2312yes—
199#808C2312yes—
200#807C2314yes—
201#807C2314yes—
202#807C2312yes—
203#807C2312yes—
204#807C2312yes—
205#807C2312yes—
206C4⋊(C3⋊C8)C2312yes—
207C4⋊(C3⋊C8)C2312yes—
208C4⋊(C3⋊C8)C2314yes—
209C4⋊(C3⋊C8)C2314yes—
210#803C2312yes—
211#803C2312yes—
212#803C2314yes—
213C4×C3⋊C8C2314yes—
214C22×Q16—31no—
215Q82—21no—
216Q82—32no—
217Q82—61no—
218Q82C2×D312yes—
219#557—21no—
220#557—22no—
221#557—34no—
222#557—32no—
223#557—61no—
224#557—62no—
225#557C2×D314yes—
226#557C2×D312yes—
227#544—22no—
228#544—32no—
229#544—62no—
230#544C2×D312yes—
231#533—22no—
232#533—31no—
233#533—62no—
234#533C2×D311yescharacteristic
235#533C2×D311yescharacteristic
236#519—32no—
237#519C2×D312yes—
238C2×C4×Q8—31no—
239C2×C4×Q8C2×D311yescharacteristic
240#501—32no—
241#501—34no—
242#501—32no—
243#501—32no—
244#501—34no—
245#501—32no—
246#499—34no—
247#498—34no—
248#479—34no—
249#479—34no—
250#479—34no—
251#479—34no—
252#479—34no—
253#479—34no—
254#477—32no—
255#477—34no—
256#477—32no—
257#477—34no—
258#475—32no—
259#475—34no—
260#475—32no—
261#475—34no—
262#446—34no—
263#446—34no—
264#441—34no—
265#441—34no—
266#441—34no—
267#441—34no—
268C4×Q16—34no—
269C4×Q16—34no—
270#428—32no—
271#428—32no—
272#427—32no—
273#427—32no—
274C2×#106—31no—
275#424—32no—
276#423—31no—
277#419—32no—
278#419—32no—
279C2×#103—31no—
280C2×#103—31no—
281#405—32no—
282C2×#97—31no—
283C2×C3×Q8—21no—
284C2×C3×Q8—21no—
285C2×C3×Q8—21no—
286C2×C3×Q8—24no—
287C2×C3×Q8—22no—
288C2×C3×Q8C2×D412yes—
289C2×C3×Q8C2×D412yes—
290C2×C3×Q8C2×Q814yes—
291C2×C3×Q8C2412yes—
292C22×C3×C4—21no—
293C22×C3×C4—21no—
294C22×C3×C4C2×D411yescharacteristic
295C22×C3×C4C2×D411yescharacteristic
296C22×C3×C4C2411yescharacteristic
297C22×Dic3—21no—
298C22×Dic3—21no—
299C2×Dic6—21no—
300C2×Dic6—22no—
301C2×Dic6—21no—
302C2×Dic6—21no—
303C2×Dic6—24no—
304C2×Dic6C2×Q814yes—
305C2×Dic6C2412yes—
306C3×C4⋊C4—22no—
307C3×C4⋊C4—22no—
308C3×C4⋊C4—21no—
309C3×C4⋊C4—24no—
310C3×C4⋊C4—24no—
311C3×C4⋊C4—22no—
312C3×C4⋊C4—21no—
313C3×C4⋊C4—21no—
314C3×C4⋊C4—24no—
315C3×C4⋊C4—24no—
316C3×C4⋊C4—22no—
317C3×C4⋊C4—24no—
318C3×C4⋊C4C2×D412yes—
319C3×C4⋊C4C2×D412yes—
320C3×C4⋊C4C2×D414yes—
321C3×C4⋊C4C2×D412yes—
322C3×C4⋊C4C2×D414yes—
323C3×C4⋊C4C2×D412yes—
324C3×C4⋊C4C2412yes—
325C3×C4⋊C4C2412yes—
326C3×C4⋊C4C2412yes—
327C3×C22⋊C4—22no—
328C3×C22⋊C4C2×D414yes—
329C3×C42—22no—
330C3×C42—22no—
331C3×C42—24no—
332C3×C42—24no—
333C3×C42—22no—
334C3×C42C2×D411yescharacteristic
335C3×C42C2×D411yescharacteristic
336C3×C42C2414yes—
337C3×C42C2411yescharacteristic
338C3×C42C2411yescharacteristic
339#217—22no—
340#216C2×Q8116yes—
341C4⋊Dic3—21no—
342C4⋊Dic3—21no—
343C4⋊Dic3—21no—
344C4⋊Dic3—24no—
345C4⋊Dic3C2412yes—
346C4⋊Dic3C2412yes—
347C4⋊Dic3C2412yes—
348Dic3⋊C4—22no—
349Dic3⋊C4—22no—
350Dic3⋊C4—22no—
351Dic3⋊C4—22no—
352Dic3⋊C4—24no—
353Dic3⋊C4—24no—
354Dic3⋊C4—24no—
355Dic3⋊C4—24no—
356C4×Dic3—22no—
357C4×Dic3—22no—
358C4×Dic3—22no—
359C4×Dic3—24no—
360C4×Dic3—24no—
361#208—24no—
362C2×C3⋊C8—22no—
363C2×C3⋊C8C2×Q814yes—
364C2×C3⋊C8C2414yes—
365C2×C3⋊C8C2412yes—
366C2×C3⋊C8C2412yes—
367C22×Q8—31no—
368C22×Q8C22×D311yescharacteristic
369C2×Q16—38no—
370C2×Q16—34no—
371C2×M16—32no—
372C22×C8—31no—
373C22×C8—31no—
374C4⋊Q8—22no—
375C4⋊Q8—24no—
376C4⋊Q8—22no—
377C4⋊Q8—24no—
378C4⋊Q8—22no—
379C4⋊Q8—31no—
380C4⋊Q8—31no—
381C4⋊Q8—34no—
382C4⋊Q8—62no—
383C4⋊Q8—64no—
384C4⋊Q8—62no—
385C4⋊Q8—62no—
386C4⋊Q8—64no—
387C4⋊Q8#6711yescharacteristic
388C4⋊Q8#6711yescharacteristic
389C4⋊Q8C22×D311yescharacteristic
390C4⋊Q8C22×D311yescharacteristic
391C4⋊Q8C22×D314yes—
392#125—24no—
393#125—24no—
394#125—24no—
395#125—22no—
396#125—32no—
397#125—34no—
398#125—64no—
399#125—62no—
400#125—64no—
401#125—64no—
402#125#6712yes—
403#125C22×D312yes—
404#125C22×D314yes—
405C22⋊Q8—21no—
406C22⋊Q8—21no—
407C22⋊Q8—24no—
408C22⋊Q8—61no—
409C22⋊Q8—61no—
410C22⋊Q8—64no—
411C22⋊Q8#6712yes—
412C22⋊Q8#6712yes—
413C4×Q8—22no—
414C4×Q8—22no—
415C4×Q8—22no—
416C4×Q8—24no—
417C4×Q8—22no—
418C4×Q8—21no—
419C4×Q8—21no—
420C4×Q8—21no—
421C4×Q8—21no—
422C4×Q8—32no—
423C4×Q8—32no—
424C4×Q8—34no—
425C4×Q8—34no—
426C4×Q8—62no—
427C4×Q8—62no—
428C4×Q8—62no—
429C4×Q8—64no—
430C4×Q8—62no—
431C4×Q8—61no—
432C4×Q8—61no—
433C4×Q8—61no—
434C4×Q8—61no—
435C4×Q8C22×D312yes—
436C4×Q8C22×D312yes—
437C4×Q8C22×D314yes—
438C4×Q8C22×D314yes—
439C4×Q8C22×D312yes—
440C4×Q8C22×D312yes—
441#117—24no—
442#117—32no—
443#117—64no—
444#117C22×D312yes—
445#117C22×D312yes—
446C2×C4⋊C4—21no—
447C2×C4⋊C4—31no—
448C2×C4⋊C4—61no—
449C2×C4⋊C4C22×D311yescharacteristic
450C2×C42—21no—
451C2×C42—61no—
452C2×C42C22×D311yescharacteristic
453#107—32no—
454#107—32no—
455#107—32no—
456#107—32no—
457#107—34no—
458#107—34no—
459#106—34no—
460#106—32no—
461#106—32no—
462#106—32no—
463#106—32no—
464#106—34no—
465#106—34no—
466C4⋊C8—32no—
467C4⋊C8—32no—
468C4⋊C8—34no—
469C4⋊C8—34no—
470#103—34no—
471#103—34no—
472#103—34no—
473#103—34no—
474#103—34no—
475#103—34no—
476#97—32no—
477#97—32no—
478#97—34no—
479C4×C8—34no—
480C23×C3C22×D411yescharacteristic
481C3×Q8—24no—
482C3×Q8—22no—
483C3×Q8—42no—
484C3×Q8—42no—
485C3×Q8—42no—
486C3×Q8—81no—
487C3×Q8—84no—
488C3×Q8C22×Q814yes—
489C2×C3×C4—24no—
490C2×C3×C4—22no—
491C2×C3×C4—22no—
492C2×C3×C4—22no—
493C2×C3×C4—22no—
494C2×C3×C4—41no—
495C2×C3×C4—41no—
496C2×C3×C4—81no—
497C2×C3×C4C22×D411yescharacteristic
498C2×C3×C4C22×D411yescharacteristic
499C2×C3×C4C22×D414yes—
500C2×C3×C4C22×D414yes—
501C2×C3×C4C22×D411yescharacteristic
502C2×C3×C4C22×D411yescharacteristic
503C2×C3×C4C22×D411yescharacteristic
504C2×C3×C4C22×Q812yes—
505C2×C3×C42−1+412yes—
506C2×C3×C4C2511yescharacteristic
507C2×Dic3—24no—
508C2×Dic3—22no—
509C2×Dic3—22no—
510C2×Dic3—22no—
511C2×Dic3—22no—
512C2×Dic3—81no—
513Dic6—24no—
514Dic6—22no—
515Dic6—42no—
516Dic6—81no—
517Dic6—84no—
518Dic6C22×Q814yes—
519C3⋊C8—22no—
520C3⋊C8—24no—
521C3⋊C8C22×Q814yes—
522C2×Q8—22no—
523C2×Q8—21no—
524C2×Q8—21no—
525C2×Q8—21no—
526C2×Q8—24no—
527C2×Q8—34no—
528C2×Q8—32no—
529C2×Q8—62no—
530C2×Q8—61no—
531C2×Q8—61no—
532C2×Q8—61no—
533C2×Q8—64no—
534C2×Q8D3×Q814yes—
535C2×Q8C2×#6712yes—
536C2×Q8C2×#6712yes—
537C2×Q8C23×D312yes—
538C22×C4—21no—
539C22×C4—21no—
540C22×C4—61no—
541C22×C4—61no—
542C22×C4C2×#6711yescharacteristic
543C22×C4C2×#6711yescharacteristic
544C22×C4C23×D311yescharacteristic
545Q16—316no—
546M16—64no—
547C2×C8—34no—
548C2×C8—32no—
549C2×C8—34no—
550C2×C8—32no—
551C2×C8—62no—
552C4⋊C4—22no—
553C4⋊C4—22no—
554C4⋊C4—21no—
555C4⋊C4—24no—
556C4⋊C4—24no—
557C4⋊C4—22no—
558C4⋊C4—21no—
559C4⋊C4—21no—
560C4⋊C4—24no—
561C4⋊C4—24no—
562C4⋊C4—22no—
563C4⋊C4—24no—
564C4⋊C4—32no—
565C4⋊C4—32no—
566C4⋊C4—32no—
567C4⋊C4—61no—
568C4⋊C4—62no—
569C4⋊C4—62no—
570C4⋊C4—64no—
571C4⋊C4—64no—
572C4⋊C4—62no—
573C4⋊C4—61no—
574C4⋊C4—64no—
575C4⋊C4—62no—
576C4⋊C4—64no—
577C4⋊C4—61no—
578C4⋊C4—64no—
579C4⋊C4C2×#6712yes—
580C4⋊C4C2×#6712yes—
581C4⋊C4C2×#6714yes—
582C4⋊C4C2×#6712yes—
583C4⋊C4C2×#6712yes—
584C4⋊C4C2×#6714yes—
585C4⋊C4C23×D312yes—
586C4⋊C4C23×D312yes—
587C4⋊C4C23×D312yes—
588C22⋊C4—22no—
589C22⋊C4—62no—
590C22⋊C4C2×#6714yes—
591C42—22no—
592C42—22no—
593C42—24no—
594C42—24no—
595C42—22no—
596C42—62no—
597C42—62no—
598C42—62no—
599C42—64no—
600C42—64no—
601C42C2×#6711yescharacteristic
602C42C2×#6711yescharacteristic
603C42C23×D314yes—
604C42C23×D311yescharacteristic
605C42C23×D311yescharacteristic
606C22×C3—21no—
607C22×C3—21no—
608C22×C3D4×Q812yes—
609C22×C3C23×D411yescharacteristic
610C3×C4—21no—
611C3×C4—21no—
612C3×C4—24no—
613C3×C4—21no—
614C3×C4—21no—
615C3×C4—24no—
616C3×C4—24no—
617C3×C4—42no—
618C3×C4—42no—
619C3×C4—42no—
620C3×C4D4×Q812yes—
621C3×C4C23×Q811yescharacteristic
622C3×C4C2×2−1+411yescharacteristic
623Dic3—24no—
624Dic3—42no—
625Dic3—42no—
626Dic3—42no—
627C23C22×#6711yescharacteristic
628Q8—22no—
629Q8—24no—
630Q8—34no—
631Q8—42no—
632Q8—42no—
633Q8—42no—
634Q8—62no—
635Q8—64no—
636Q8—81no—
637Q8—84no—
638Q8—122no—
639Q8—241no—
640Q8—244no—
641Q8C2×D3×Q814yes—
642C2×C4—22no—
643C2×C4—24no—
644C2×C4—22no—
645C2×C4—22no—
646C2×C4—22no—
647C2×C4—41no—
648C2×C4—41no—
649C2×C4—62no—
650C2×C4—64no—
651C2×C4—62no—
652C2×C4—62no—
653C2×C4—62no—
654C2×C4—81no—
655C2×C4—241no—
656C2×C4C2×D3×Q812yes—
657C2×C4#100712yes—
658C2×C4C22×#6711yescharacteristic
659C2×C4C22×#6711yescharacteristic
660C2×C4C22×#6714yes—
661C2×C4C22×#6714yes—
662C2×C4C22×#6711yescharacteristic
663C2×C4C22×#6711yescharacteristic
664C2×C4C22×#6711yescharacteristic
665C2×C4C24×D311yescharacteristic
666C8—34no—
667C8—62no—
668C8—64no—
669C2×C3—22no—
670C2×C3C2×D4×Q811yescharacteristic
671C2×C3#358611yescharacteristic
672C2×C3#358611yescharacteristic
673C22—21no—
674C22—21no—
675C22Q8×#6712yes—
676C22C23×#6711yescharacteristic
677C4—21no—
678C4—21no—
679C4—24no—
680C4—24no—
681C4—21no—
682C4—21no—
683C4—24no—
684C4—42no—
685C4—42no—
686C4—42no—
687C4—64no—
688C4—122no—
689C4—122no—
690C4—122no—
691C4Q8×#6712yes—
692C4C22×D3×Q811yescharacteristic
693C4C2×#100711yescharacteristic
694C3#6137911yescharacteristic, complemented, Sylow
695C2—22no—
696C2C2×Q8×#6711yescharacteristic
697C2#8728711yescharacteristic
698C2#8729911yescharacteristic
subgroupG/H
centerC22C23×#67
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC3×#3502C2
socleC22×C3C23×D4
prime radicalC23×C3C22×D4
hypercenter#3502D3
hypocenterC3#61379
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ #533 ◁ #3502

Complements

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

NH ≅ G/Ntype
3. #83709C2semidirect
4. #83684C2semidirect
5. #83684C2semidirect
6. #83683C2semidirect
150. C2×#216Q8semidirect
694. C3#61379semidirect

Maximal subgroups

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

subgroupindex
C3×#35022
#862582
#837092
#836842
#836832
#834802
#834722
#833672
#833402
#833292
#825502
#825492
#824312
C2×#50302
#613793