Finite Group Databasebeta

← #10823666

Subgroups — #10823666

Order 768.

1118 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3460C211yesmaximal, characteristic
2C4×#951C212yesmaximal, complemented
3#82328C211yesmaximal, characteristic, complemented
4#71905C211yesmaximal, characteristic, complemented
5#71900C211yesmaximal, characteristic, complemented
6#71881C211yesmaximal, characteristic, complemented
7#71876C211yesmaximal, characteristic, complemented
8#71857C212yesmaximal, complemented
9#71852C212yesmaximal, complemented
10#71643C212yesmaximal, complemented
11C4×#833C211yesmaximal, characteristic, complemented
12#23593—31nomaximal, Sylow
13C3×#544C2211yescharacteristic
14C3×#542—22no—
15C3×#540—22no—
16C3×#538—22no—
17C3×#537C2211yescharacteristic
18C3×#535C2212yes—
19C3×#532—22no—
20C3×#529—22no—
21C3×#525—22no—
22C3×C4×#41C2212yes—
23C2×C3×C4×D4C2211yescharacteristic
24C2×#951C2212yes—
25C2×#936C2211yescharacteristic
26C2×C4×Dic6C2211yescharacteristic
27C2×#837C2212yes—
28C2×#837C2212yes—
29#5232C2212yes—
30#5221C2211yescharacteristic
31C2×#833C2212yes—
32#5210—22no—
33#5030—22no—
34C4×#216—22no—
35#5018—22no—
36C4×#214—22no—
37#5004C2212yes—
38#4995C2212yes—
39#4975C2211yescharacteristic
40#4973C2211yescharacteristic
41#4944—22no—
42#4932C2211yescharacteristic
43C4×#208C2212yes—
44#4532C2212yes—
45#4532C2212yes—
46#4530C2211yescharacteristic
47#4530C2211yescharacteristic
48#3460D311yescharacteristic
49C4×#137—32no—
50#2326—31no—
51#2060—31no—
52#2039—31no—
53#2037—32no—
54#2028—31no—
55#2007—31no—
56#2004—32no—
57#1759—32no—
58C4×#100—31no—
59C2×C3×#41—22no—
60C2×C3×#41C2312yes—
61C22×C3×D4C2311yescharacteristic
62C23×C3×C4D412yes—
63C22×Dic6C2311yescharacteristic
64C3×C4⋊Q8—22no—
65C3×#127—22no—
66C3×#126—24no—
67C3×#125—22no—
68C3×#124—24no—
69C3×#124—22no—
70C3×#123—24no—
71C3×#123—24no—
72C3×C22⋊Q8—22no—
73C3×C22⋊Q8—22no—
74C3×C22⋊Q8—22no—
75C3×#121—22no—
76C3×#121—24no—
77C3×#121—22no—
78C3×#121—22no—
79C3×#121—22no—
80C3×#121—22no—
81C3×#120—24no—
82C3×C4×Q8D412yes—
83C3×C4×Q8D412yes—
84C3×C4×D4—24no—
85C3×C4×D4—22no—
86C3×C4×D4—22no—
87C3×C4×D4—22no—
88C3×C4×D4—22no—
89C3×C4×D4D412yes—
90C3×C4×D4D412yes—
91C3×#117—22no—
92C3×#117—22no—
93C3×#117C2312yes—
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C22⋊C4D412yes—
96C2×C3×C42—22no—
97C2×C3×C42C2311yescharacteristic
98C2×#217—22no—
99C2×#217—22no—
100#951—24no—
101#951—24no—
102C2×#216—22no—
103#936—22no—
104#936—22no—
105C2×#214—22no—
106C2×C4⋊Dic3C2311yescharacteristic
107C2×Dic3⋊C4—21no—
108C2×Dic3⋊C4—22no—
109C2×C4×Dic3—21no—
110C2×C4×Dic3C2311yescharacteristic
111C2×C4×Dic3C2311yescharacteristic
112C2×#208C2312yes—
113C2×#208C2312yes—
114C4×Dic6—22no—
115C4×Dic6—22no—
116#837—24no—
117#837—24no—
118#835—22no—
119#833—24no—
120#832—22no—
121C3⋊#95—21no—
122C3⋊#95—21no—
123C3⋊#95—22no—
124#810—22no—
125#810—22no—
126#808—22no—
127#807—22no—
128#803—22no—
129C4×C3⋊C8—22no—
130C2×#137—32no—
131#544C2×D311yescharacteristic
132#542—22no—
133#540—22no—
134#538—22no—
135#537C2×D311yescharacteristic
136#535C2×D312yes—
137#532—22no—
138#529—22no—
139C2×#124—31no—
140#525—22no—
141C4×#41C2×D312yes—
142C2×C4×Q8—31no—
143C2×C4×D4C2×D311yescharacteristic
144#441—62no—
145#440—62no—
146C4×Q16—62no—
147C4×SD16—62no—
148#428—32no—
149C2×#104—32no—
150C2×#104—32no—
151#419—32no—
152#417—32no—
153C2×#100—32no—
154C4×M16—32no—
155#391—31no—
156#390—31no—
157#388—62no—
158#385—62no—
159#384—31no—
160#376—31no—
161#343—31no—
162#343—31no—
163#339—32no—
164#339—32no—
165C24×C3C2×D412yes—
166C3×#41—22no—
167C3×#41—22no—
168C3×#41—22no—
169C3×#41—22no—
170C3×#41—84no—
171C2×C3×Q8—22no—
172C2×C3×Q8C2×D412yes—
173C2×C3×D4—22no—
174C2×C3×D4—24no—
175C2×C3×D4—24no—
176C2×C3×D4—22no—
177C2×C3×D4—22no—
178C2×C3×D4—22no—
179C2×C3×D4—21no—
180C2×C3×D4—21no—
181C2×C3×D4C2×D412yes—
182C22×C3×C4—22no—
183C22×C3×C4—22no—
184C22×C3×C4—22no—
185C22×C3×C4—22no—
186C22×C3×C4—22no—
187C22×C3×C4—82no—
188C22×C3×C4C2×D412yes—
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2411yescharacteristic
192C22×Dic3—21no—
193C22×Dic3—21no—
194C22×Dic3—21no—
195C22×Dic3—21no—
196C2×Dic6—22no—
197C2×Dic6—22no—
198C2×Dic6—21no—
199C2×Dic6#4112yescomplemented
200C3×C4⋊C4—22no—
201C3×C4⋊C4—22no—
202C3×C4⋊C4—24no—
203C3×C4⋊C4—22no—
204C3×C4⋊C4—22no—
205C3×C4⋊C4—22no—
206C3×C4⋊C4C2×D412yes—
207C3×C4⋊C4C2×D412yes—
208C3×C22⋊C4—24no—
209C3×C22⋊C4—24no—
210C3×C22⋊C4—24no—
211C3×C22⋊C4—22no—
212C3×C22⋊C4—24no—
213C3×C22⋊C4—22no—
214C3×C22⋊C4—22no—
215C3×C42—22no—
216C3×C42—22no—
217C3×C42—22no—
218C3×C42C2×D412yes—
219C3×C42C2×D412yes—
220#217—28no—
221#217—82no—
222#216—28no—
223#214—28no—
224C4⋊Dic3—22no—
225Dic3⋊C4—41no—
226Dic3⋊C4—41no—
227Dic3⋊C4—82no—
228C4×Dic3—21no—
229C4×Dic3—22no—
230C4×Dic3—42no—
231C4×Dic3#4112yescomplemented
232#208—22no—
233#208—22no—
234#208—22no—
235#208#4114yescomplemented
236C2×C3⋊C8—22no—
237C2×C3⋊C8—22no—
238C2×#41—22no—
239C2×#41C22×D312yes—
240C22×Q8—31no—
241C22×D4C22×D311yescharacteristic
242C23×C4#6712yes—
243#137—64no—
244#137—64no—
245C2×Q16—62no—
246C2×SD16—62no—
247C2×M16—32no—
248C2×M16—32no—
249C4⋊Q8—22no—
250#127—22no—
251#126—24no—
252#125—22no—
253#124—24no—
254#124—22no—
255#124—62no—
256#124—62no—
257#123—24no—
258#123—24no—
259C22⋊Q8—22no—
260C22⋊Q8—22no—
261C22⋊Q8—22no—
262#121—22no—
263#121—24no—
264#121—22no—
265#121—22no—
266#121—22no—
267#121—22no—
268#120—24no—
269C4×Q8—62no—
270C4×Q8—62no—
271C4×Q8#6712yes—
272C4×Q8#6712yes—
273C4×D4—24no—
274C4×D4—22no—
275C4×D4—22no—
276C4×D4—22no—
277C4×D4—22no—
278C4×D4#6712yes—
279C4×D4#6712yes—
280#117—22no—
281#117—22no—
282#117C22×D312yes—
283C2×C4⋊C4—31no—
284C2×C4⋊C4—61no—
285C2×C4⋊C4—62no—
286C2×C4⋊C4C22×D311yescharacteristic
287C2×C22⋊C4—62no—
288C2×C22⋊C4—62no—
289C2×C22⋊C4#6712yes—
290C2×C42—22no—
291C2×C42—31no—
292C2×C42—31no—
293C2×C42—61no—
294C2×C42C22×D311yescharacteristic
295#107—62no—
296#106—62no—
297#104—64no—
298#104—64no—
299#103—62no—
300#103—62no—
301#103—62no—
302#102—62no—
303#100—64no—
304#97—62no—
305C4×C8—62no—
306#95—61no—
307#95—61no—
308#95—62no—
309C23×C3—22no—
310C23×C3—22no—
311C23×C3—22no—
312C23×C3—22no—
313C23×C3—42no—
314C23×C3—42no—
315C23×C3C22×D411yescharacteristic
316C3×Q8—22no—
317C3×Q8—22no—
318C3×Q8—82no—
319C3×D4—22no—
320C3×D4—22no—
321C3×D4—42no—
322C3×D4—42no—
323C3×D4—42no—
324C3×D4—42no—
325C3×D4—84no—
326C3×D4—82no—
327C3×D4—84no—
328C2×C3×C4—22no—
329C2×C3×C4—22no—
330C2×C3×C4—22no—
331C2×C3×C4—22no—
332C2×C3×C4—41no—
333C2×C3×C4—41no—
334C2×C3×C4—44no—
335C2×C3×C4—44no—
336C2×C3×C4—42no—
337C2×C3×C4—42no—
338C2×C3×C4—42no—
339C2×C3×C4—82no—
340C2×C3×C4—82no—
341C2×C3×C4#12012yes—
342C2×C3×C4#12011yescharacteristic
343C2×C3×C4#12011yescharacteristic
344C2×C3×C4C22×D412yes—
345C2×C3×C4C2×#4111yescharacteristic
346C2×C3×C4C2×#4112yes—
347C2×C3×C4C2×#4111yescharacteristic
348C2×Dic3—22no—
349C2×Dic3—22no—
350C2×Dic3—41no—
351C2×Dic3—41no—
352C2×Dic3—41no—
353C2×Dic3—41no—
354C2×Dic3—42no—
355C2×Dic3—41no—
356C2×Dic3—41no—
357C2×Dic3—41no—
358C2×Dic3—41no—
359Dic6—24no—
360Dic6—42no—
361C3⋊C8—24no—
362C3⋊C8—42no—
363C24C2×#6712yes—
364#41—22no—
365#41—22no—
366#41—22no—
367#41—22no—
368#41—84no—
369C2×Q8—22no—
370C2×Q8—32no—
371C2×Q8—61no—
372C2×Q8—62no—
373C2×Q8—62no—
374C2×Q8C2×#6712yes—
375C2×D4—22no—
376C2×D4—24no—
377C2×D4—24no—
378C2×D4—22no—
379C2×D4—22no—
380C2×D4—21no—
381C2×D4—21no—
382C2×D4—22no—
383C2×D4C2×#6712yes—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—61no—
390C22×C4—61no—
391C22×C4—61no—
392C22×C4—61no—
393C22×C4—82no—
394C22×C4C2×#6712yes—
395C22×C4C2×#6711yescharacteristic
396C22×C4C2×#6711yescharacteristic
397C22×C4C23×D311yescharacteristic
398Q16—68no—
399SD16—68no—
400M16—34no—
401M16—62no—
402M16—62no—
403M16—62no—
404C2×C8—62no—
405C2×C8—62no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—24no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—62no—
413C4⋊C4—121no—
414C4⋊C4—121no—
415C4⋊C4—242no—
416C4⋊C4C2×#6712yes—
417C4⋊C4C2×#6712yes—
418C22⋊C4—24no—
419C22⋊C4—24no—
420C22⋊C4—24no—
421C22⋊C4—22no—
422C22⋊C4—24no—
423C22⋊C4—22no—
424C22⋊C4—22no—
425C22⋊C4—68no—
426C22⋊C4—242no—
427C42—22no—
428C42—22no—
429C42—22no—
430C42—32no—
431C42—61no—
432C42—62no—
433C42—122no—
434C42C2×#6712yes—
435C42C2×#6712yes—
436C22×C3—21no—
437C22×C3—21no—
438C22×C3—44no—
439C22×C3—42no—
440C22×C3—42no—
441C22×C3—42no—
442C22×C3—42no—
443C22×C3—42no—
444C22×C3—42no—
445C22×C3—82no—
446C22×C3C2×#12011yescharacteristic
447C22×C3#52511yescharacteristic
448C22×C3#53211yescharacteristic
449C3×C4—21no—
450C3×C4—21no—
451C3×C4—22no—
452C3×C4—42no—
453C3×C4—42no—
454C3×C4—42no—
455C3×C4—42no—
456C3×C4#52512yes—
457C3×C4#53212yes—
458Dic3—42no—
459Dic3—42no—
460Dic3—81no—
461Dic3—81no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—42no—
467C23—42no—
468C23C22×#6711yescharacteristic
469Q8—22no—
470Q8—22no—
471Q8—64no—
472Q8—82no—
473Q8—122no—
474D4—22no—
475D4—22no—
476D4—42no—
477D4—42no—
478D4—42no—
479D4—42no—
480D4—82no—
481D4—84no—
482D4—84no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—62no—
495C2×C4—62no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—121no—
499C2×C4—121no—
500C2×C4—121no—
501C2×C4—121no—
502C2×C4—121no—
503C2×C4—122no—
504C2×C4—121no—
505C2×C4—121no—
506C2×C4—121no—
507C2×C4#95311yescharacteristic
508C2×C4#95312yes—
509C2×C4#95311yescharacteristic
510C2×C4D3×#4111yescharacteristic
511C2×C4D3×#4112yes—
512C2×C4D3×#4111yescharacteristic
513C2×C4C22×#6712yes—
514C8—64no—
515C8—122no—
516C2×C3—21no—
517C2×C3—21no—
518C2×C3—82no—
519C2×C3—82no—
520C2×C3#238811yescharacteristic
521C2×C3#301911yescharacteristic
522C2×C3#302211yescharacteristic
523C22—21no—
524C22—21no—
525C22—44no—
526C22—42no—
527C22—42no—
528C22—42no—
529C22—42no—
530C22—42no—
531C22—42no—
532C22—82no—
533C22#582811yescharacteristic
534C22#583211yescharacteristic
535C22C2×#95311yescharacteristic
536C4—21no—
537C4—21no—
538C4—22no—
539C4—42no—
540C4—42no—
541C4—42no—
542C4—42no—
543C4—122no—
544C4—122no—
545C4—241no—
546C4—241no—
547C4#582812yes—
548C4#583212yes—
549C3#2359311yescharacteristic, complemented, Sylow
550C2—21no—
551C2—21no—
552C2—82no—
553C2—82no—
554C2#8409511yescharacteristic
555C2#8410411yescharacteristic
556C2#8412311yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC3×#3460C2
socleC22×C3C2×#120
hypercenter#3460D3
hypocenterC3#23593
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3460

Complements

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

NH ≅ G/Ntype
2. C4×#951C2semidirect
3. #82328C2semidirect
4. #71905C2semidirect
5. #71900C2semidirect
6. #71881C2semidirect
7. #71876C2semidirect
8. #71857C2semidirect
9. #71852C2semidirect
10. #71643C2semidirect
11. C4×#833C2semidirect
199. C2×Dic6#41semidirect
231. C4×Dic3#41semidirect
235. #208#41semidirect
549. C3#23593semidirect

Maximal subgroups

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

subgroupindex
C3×#34602
C4×#9512
#823282
#719052
#719002
#718812
#718762
#718572
#718522
#716432
C4×#8332
#235933