Finite Group Databasebeta

← #10883490

Subgroups — #10883490

Order 768.

822 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2931C211yesmaximal, characteristic, complemented
2#83314C211yesmaximal, characteristic, complemented
3#83297C211yesmaximal, characteristic
4C2×C4×#204C211yesmaximal, characteristic, complemented
5#72019C212yesmaximal, complemented
6#72018C212yesmaximal, complemented
7#70528C211yesmaximal, characteristic, complemented
8#70527C211yesmaximal, characteristic, complemented
9#70007C212yesmaximal, complemented
10#69995C212yesmaximal, complemented
11#69791C211yesmaximal, characteristic, complemented
12#28175—31nomaximal, Sylow
13C2×C3×#131C2212yes—
14C3×#532C2211yescharacteristic
15C2×#951C2212yes—
16C2×#949C2212yes—
17C22×#204C2211yescharacteristic
18C2×C4×D12C2211yescharacteristic
19C2×C4×Dic6C2211yescharacteristic
20C3×#436—21no—
21C3×#436—21no—
22C3×C8×D4—21no—
23C3×C8×D4—21no—
24C3×#423C2211yescharacteristic
25C3×#408C2212yes—
26C2×C3×C4×C8C2211yescharacteristic
27#5232C2212yes—
28#5232C2212yes—
29C2×#824C2212yes—
30C2×#823C2212yes—
31C2×#821C2211yescharacteristic
32C2×#817C2211yescharacteristic
33C2×#816C2211yescharacteristic
34#5090—21no—
35#5083—21no—
36#5075—21no—
37#5072—21no—
38#5057—22no—
39#5054—22no—
40#4924C2211yescharacteristic
41#4924C2211yescharacteristic
42C4×#204—21no—
43C4×#204—21no—
44C4×#204—21no—
45C4×#204—21no—
46#4554C2212yes—
47#4554C2212yes—
48#4552C2212yes—
49#4552C2212yes—
50#3037—31no—
51#3036—31no—
52C2×C4×SD16—31no—
53#2931D311yescharacteristic, complemented
54#2079—32no—
55#2019—32no—
56#1908—32no—
57#1907—32no—
58#1837—31no—
59#1801—31no—
60#1800—31no—
61C2×C3×#41C2312yes—
62C22×D12C2311yescharacteristic
63C22×C4×D3—21no—
64C22×Dic6C2311yescharacteristic
65C3×#131—42no—
66C3×#131—42no—
67C2×C3×M16—22no—
68C2×C3×M16C2312yes—
69C22×C3×C8—22no—
70C22×C3×C8C2311yescharacteristic
71C22×C3×C8C2311yescharacteristic
72C3×C4⋊Q8D411yescharacteristic
73C3×#127D411yescharacteristic
74C3×#124D412yes—
75C3×#121—21no—
76C3×#121—21no—
77C3×C4×D4—21no—
78C3×C4×D4—21no—
79C2×C3×C42C2311yescharacteristic
80#951—24no—
81#951—42no—
82#949—24no—
83#949—42no—
84C2×#216—22no—
85C2×#215—22no—
86C2×#214—22no—
87C2×#213—22no—
88C2×(C2×D3)⋊C4—21no—
89C2×C4⋊Dic3C2311yescharacteristic
90C2×C4⋊Dic3C2311yescharacteristic
91C2×Dic3⋊C4—21no—
92C2×C4×Dic3—21no—
93C2×#208C2312yes—
94C2×#208C2312yes—
95C2×#204—22no—
96C2×#204—22no—
97C2×#204—21no—
98C2×#204—21no—
99C4×D12—21no—
100C4×D12—21no—
101C4×D12—21no—
102C4×D12—21no—
103C4×Dic6—21no—
104C4×Dic6—21no—
105C4×Dic6—21no—
106C4×Dic6—21no—
107C3×C4⋊C8—21no—
108C3×C4⋊C8—21no—
109C3×C22⋊C8—22no—
110C3×C22⋊C8—22no—
111C3×C4×C8D411yescharacteristic
112C3×C4×C8D411yescharacteristic
113C3×C4×C8D411yescharacteristic
114C3×C4×C8D411yescharacteristic
115#835—22no—
116#835—22no—
117#832—22no—
118#832—22no—
119#824—42no—
120#823—42no—
121#821—21no—
122#821—21no—
123#817—21no—
124#817—21no—
125#816—21no—
126#816—21no—
127C4⋊(C3⋊C8)—21no—
128C4⋊(C3⋊C8)—21no—
129C4⋊(C3⋊C8)—21no—
130C4⋊(C3⋊C8)—21no—
131C2×#137—32no—
132C2×AGL(1,ℤ/8)—32no—
133C22×SD16—31no—
134C2×#131C2×D312yes—
135#532C2×D311yescharacteristic
136C2×C4×Q8—31no—
137C2×C4×D4—31no—
138#464—62no—
139#463—62no—
140#462—61no—
141#461—61no—
142#460—61no—
143#459—61no—
144C4×SD16—61no—
145C4×SD16—61no—
146C4×SD16—61no—
147C4×SD16—61no—
148#436—21no—
149#436—21no—
150C8×D4—21no—
151C8×D4—21no—
152C2×#106—31no—
153#423—31no—
154#423—31no—
155#423C2×D311yescharacteristic
156#417—32no—
157#417—32no—
158C2×#103—31no—
159C2×#102—31no—
160C2×#101—32no—
161C2×#100—32no—
162#408C2×D312yes—
163C2×C4×C8C2×D311yescharacteristic
164#343—32no—
165#343—32no—
166#341—32no—
167#341—32no—
168C23×D3—21no—
169C3×#41—22no—
170C3×#41—22no—
171C3×#41—22no—
172C3×#41—22no—
173C2×C3×Q8C2×D412yes—
174C2×C3×D4—22no—
175C2×C3×D4—21no—
176C2×C3×D4—21no—
177C2×C3×D4C2×D412yes—
178C22×C3×C4—22no—
179C22×C3×C4#4111yescharacteristic
180C22×C3×C4#4111yescharacteristic
181C22×C3×C4C2411yescharacteristic
182C22×Dic3—21no—
183C22×Dic3—21no—
184C22×Dic3—21no—
185C2×D12—21no—
186C2×D12—21no—
187C2×D12—21no—
188C2×D12—21no—
189C2×D12—21no—
190C2×D12#4112yes—
191C2×C4×D3—41no—
192C2×C4×D3—41no—
193C2×C4×D3—82no—
194C2×Dic6—21no—
195C2×Dic6—21no—
196C2×Dic6—21no—
197C2×Dic6—21no—
198C2×Dic6—21no—
199C2×Dic6#4112yes—
200C3×M16—22no—
201C3×M16—22no—
202C3×M16—44no—
203C2×C3×C8—21no—
204C2×C3×C8—21no—
205C2×C3×C8—21no—
206C2×C3×C8—21no—
207C2×C3×C8—22no—
208C2×C3×C8—44no—
209C2×C3×C8C2×D411yescharacteristic
210C2×C3×C8C2×D411yescharacteristic
211C2×C3×C8C2×D411yescharacteristic
212C2×C3×C8C2×D411yescharacteristic
213C3×C4⋊C4—21no—
214C3×C4⋊C4—21no—
215C3×C22⋊C4—22no—
216C3×C22⋊C4—22no—
217C3×C42C2×D411yescharacteristic
218C3×C42C2×D411yescharacteristic
219C3×C42C2×D411yescharacteristic
220C3×C42C2×D411yescharacteristic
221#216—44no—
222#215—44no—
223#214—44no—
224#213—44no—
225(C2×D3)⋊C4—41no—
226(C2×D3)⋊C4—41no—
227C4⋊Dic3—21no—
228C4⋊Dic3—21no—
229C4⋊Dic3—21no—
230C4⋊Dic3—21no—
231Dic3⋊C4—41no—
232Dic3⋊C4—41no—
233C4×Dic3—41no—
234C4×Dic3—41no—
235#208—22no—
236#208—22no—
237#208—22no—
238#208—22no—
239C2×C3⋊C8—22no—
240C2×C3⋊C8—22no—
241#204—24no—
242#204—24no—
243C2×#41C22×D312yes—
244C22×Q8—31no—
245C22×D4—31no—
246C23×C4—61no—
247#137—64no—
248#137—122no—
249AGL(1,ℤ/8)—64no—
250AGL(1,ℤ/8)—122no—
251C2×Q16—62no—
252C2×SD16—62no—
253C2×SD16—62no—
254C2×SD16—62no—
255C2×SD16—62no—
256C2×SD16—61no—
257C2×SD16—61no—
258C2×D8—62no—
259#131—42no—
260#131—42no—
261C2×M16—22no—
262C2×M16—32no—
263C2×M16—32no—
264C2×M16C22×D312yes—
265C22×C8—22no—
266C22×C8C22×D311yescharacteristic
267C22×C8C22×D311yescharacteristic
268C4⋊Q8D1211yescharacteristic
269#127D1211yescharacteristic
270#124D1212yes—
271#121—21no—
272#121—21no—
273C4×Q8—61no—
274C4×Q8—61no—
275C4×Q8—61no—
276C4×Q8—61no—
277C4×D4—21no—
278C4×D4—21no—
279C4×D4—61no—
280C4×D4—61no—
281C4×D4—61no—
282C4×D4—61no—
283C2×C4⋊C4—31no—
284C2×C4⋊C4—31no—
285C2×C4⋊C4—61no—
286C2×C22⋊C4—61no—
287C2×C42—61no—
288C2×C42C22×D311yescharacteristic
289#106—61no—
290#106—61no—
291C4⋊C8—21no—
292C4⋊C8—21no—
293C4⋊C8—61no—
294C4⋊C8—61no—
295C4⋊C8—61no—
296C4⋊C8—61no—
297#103—62no—
298#103—62no—
299#103—61no—
300#103—61no—
301#102—62no—
302#102—62no—
303#102—61no—
304#102—61no—
305#101—122no—
306#100—122no—
307C22⋊C8—22no—
308C22⋊C8—22no—
309C4×C8#6711yescharacteristic
310C4×C8#6711yescharacteristic
311C4×C8#6711yescharacteristic
312C4×C8#6711yescharacteristic
313C23×C3—22no—
314C23×C3C2×#4111yescharacteristic
315C22×D3—22no—
316C22×D3—41no—
317C22×D3—41no—
318C22×D3—81no—
319C22×D3—81no—
320C3×Q8—22no—
321C3×Q8—22no—
322C3×D4—22no—
323C3×D4—22no—
324C3×D4—42no—
325C3×D4—42no—
326C3×D4—42no—
327C3×D4—42no—
328C2×C3×C4—22no—
329C2×C3×C4—22no—
330C2×C3×C4—41no—
331C2×C3×C4—42no—
332C2×C3×C4—42no—
333C2×C3×C4#12111yescharacteristic
334C2×C3×C4#12111yescharacteristic
335C2×C3×C4#12111yescharacteristic
336C2×C3×C4#12111yescharacteristic
337C2×C3×C4C22×D411yescharacteristic
338C2×C3×C4C22×D411yescharacteristic
339C2×C3×C4C2×#4111yescharacteristic
340C2×C3×C4C2×#4111yescharacteristic
341C2×C3×C4H(5,𝔽2)11yescharacteristic
342C2×C3×C4H(5,𝔽2)11yescharacteristic
343C2×Dic3—22no—
344C2×Dic3—41no—
345C2×Dic3—41no—
346C2×Dic3—41no—
347C2×Dic3—41no—
348C2×Dic3—81no—
349C2×Dic3—81no—
350D12—22no—
351D12—22no—
352D12—41no—
353D12—41no—
354C4×D3—82no—
355C4×D3—82no—
356Dic6—22no—
357Dic6—22no—
358Dic6—41no—
359Dic6—41no—
360C3×C8—21no—
361C3×C8—21no—
362C3×C8—21no—
363C3×C8—21no—
364C3×C8—42no—
365C3⋊C8—42no—
366C3⋊C8—42no—
367C24—61no—
368#41—22no—
369#41—22no—
370#41—22no—
371#41—22no—
372C2×Q8—32no—
373C2×Q8—61no—
374C2×Q8—61no—
375C2×Q8—61no—
376C2×Q8—61no—
377C2×Q8—61no—
378C2×Q8C2×D1212yes—
379C2×D4—22no—
380C2×D4—21no—
381C2×D4—21no—
382C2×D4—32no—
383C2×D4—61no—
384C2×D4—61no—
385C2×D4—61no—
386C2×D4—61no—
387C2×D4—61no—
388C2×D4C2×D1212yes—
389C22×C4—22no—
390C22×C4—61no—
391C22×C4—61no—
392C22×C4—61no—
393C22×C4—121no—
394C22×C4—121no—
395C22×C4—242no—
396C22×C4#23511yescharacteristic
397C22×C4#23511yescharacteristic
398C22×C4C23×D311yescharacteristic
399Q16—124no—
400SD16—64no—
401SD16—64no—
402SD16—124no—
403SD16—124no—
404D8—124no—
405M16—22no—
406M16—22no—
407M16—44no—
408M16—62no—
409M16—62no—
410M16—62no—
411M16—62no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—22no—
417C2×C8—44no—
418C2×C8—62no—
419C2×C8—62no—
420C2×C8C2×#6711yescharacteristic
421C2×C8C2×#6711yescharacteristic
422C2×C8C2×#6711yescharacteristic
423C2×C8C2×#6711yescharacteristic
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—61no—
427C4⋊C4—61no—
428C4⋊C4—61no—
429C4⋊C4—61no—
430C4⋊C4—121no—
431C4⋊C4—121no—
432C22⋊C4—22no—
433C22⋊C4—22no—
434C22⋊C4—121no—
435C22⋊C4—121no—
436C42—121no—
437C42—121no—
438C42C2×D1211yescharacteristic
439C42C2×D1211yescharacteristic
440C42C2×#6711yescharacteristic
441C42C2×#6711yescharacteristic
442C22×C3—21no—
443C22×C3—21no—
444C22×C3—42no—
445C22×C3—42no—
446C22×C3—42no—
447C22×C3C2×#12111yescharacteristic
448C22×C3#55311yescharacteristic
449C22×C3#55311yescharacteristic
450C2×D3—42no—
451C2×D3—82no—
452C2×D3—81no—
453C2×D3—81no—
454C2×D3—82no—
455C3×C4—21no—
456C3×C4—21no—
457C3×C4—21no—
458C3×C4—42no—
459C3×C4#47112yes—
460C3×C4#52511yescharacteristic
461C3×C4#53511yescharacteristic
462C3×C4#54611yescharacteristic
463C3×C4#54611yescharacteristic
464Dic3—42no—
465Dic3—81no—
466Dic3—81no—
467C23—22no—
468C23—62no—
469C23—121no—
470C23—121no—
471C23—241no—
472C23—241no—
473C23C2×#23511yescharacteristic
474Q8—22no—
475Q8—22no—
476Q8—62no—
477Q8—62no—
478Q8—121no—
479Q8—121no—
480D4—22no—
481D4—22no—
482D4—42no—
483D4—42no—
484D4—42no—
485D4—42no—
486D4—62no—
487D4—62no—
488D4—121no—
489D4—121no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—62no—
496C2×C4—121no—
497C2×C4—121no—
498C2×C4—121no—
499C2×C4—121no—
500C2×C4—242no—
501C2×C4—242no—
502C2×C4—241no—
503C2×C4—241no—
504C2×C4#93011yescharacteristic
505C2×C4#93011yescharacteristic
506C2×C4#93011yescharacteristic
507C2×C4#93011yescharacteristic
508C2×C4C22×D1211yescharacteristic
509C2×C4#100411yescharacteristic
510C2×C4#100411yescharacteristic
511C2×C4D3×#4111yescharacteristic
512C2×C4D3×#4111yescharacteristic
513C2×C4C22×#6711yescharacteristic
514C8—21no—
515C8—21no—
516C8—21no—
517C8—21no—
518C8—42no—
519C8—122no—
520C8—122no—
521C2×C3—21no—
522C2×C3—21no—
523C2×C3—82no—
524C2×C3#253511yescharacteristic
525C2×C3C2×#47111yescharacteristic
526C2×C3#306711yescharacteristic
527D3—82no—
528C22—21no—
529C22—21no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22—122no—
534C22—242no—
535C22—242no—
536C22—241no—
537C22—241no—
538C22#555811yescharacteristic
539C22#555811yescharacteristic
540C22C2×#93011yescharacteristic
541C4—21no—
542C4—21no—
543C4—21no—
544C4—42no—
545C4—122no—
546C4—241no—
547C4—241no—
548C4#514112yes—
549C4#555411yescharacteristic
550C4#555411yescharacteristic
551C4#580511yescharacteristic
552C4#582811yescharacteristic
553C3#2817511yescharacteristic, complemented, Sylow
554C2—21no—
555C2—21no—
556C2—82no—
557C2—242no—
558C2#8241611yescharacteristic
559C2C2×#514111yescharacteristic
560C2#8380511yescharacteristic
subgroupG/H
centerC22C2×#930
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2931C2
socleC22×C3C2×#121
prime radical#83297C2
hypercenter#2931D3
hypocenterC3#28175
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2931

Complements

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

NH ≅ G/Ntype
1. C3×#2931C2semidirect
2. #83314C2semidirect
4. C2×C4×#204C2semidirect
5. #72019C2semidirect
6. #72018C2semidirect
7. #70528C2semidirect
8. #70527C2semidirect
9. #70007C2semidirect
10. #69995C2semidirect
11. #69791C2semidirect
53. #2931D3semidirect
553. C3#28175semidirect

Maximal subgroups

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

subgroupindex
C3×#29312
#833142
#832972
C2×C4×#2042
#720192
#720182
#705282
#705272
#700072
#699952
#697912
#281753