Finite Group Databasebeta

← #10784386

Subgroups — #10784386

Order 768.

704 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3209C211yesmaximal, characteristic, complemented
2#83359C211yesmaximal, characteristic, complemented
3#83356C211yesmaximal, characteristic, complemented
4#83216C211yesmaximal, characteristic, complemented
5#83215C211yesmaximal, characteristic, complemented
6#79304C211yesmaximal, characteristic
7#71302C211yesmaximal, characteristic, complemented
8#71301C211yesmaximal, characteristic, complemented
9#71300C211yesmaximal, characteristic, complemented
10#71297C211yesmaximal, characteristic, complemented
11#71225C211yesmaximal, characteristic, complemented
12#71218C211yesmaximal, characteristic, complemented
13#71216C211yesmaximal, characteristic, complemented
14#71212C211yesmaximal, characteristic, complemented
15#70744C211yesmaximal, characteristic, complemented
16#19741—31nomaximal, Sylow
17C3×#533C2211yescharacteristic
18C2×C3×C4⋊Q8C2211yescharacteristic
19#5477C2211yescharacteristic
20#5477C2211yescharacteristic
21C3×#494C2211yescharacteristic
22C3×#494C2211yescharacteristic
23C3×#487C2211yescharacteristic
24C3×#487C2211yescharacteristic
25C3×#462—22no—
26C3×#451—22no—
27C3×#431C2211yescharacteristic
28#5093C2211yescharacteristic
29#5092C2211yescharacteristic
30#5088C2211yescharacteristic
31#5088C2211yescharacteristic
32#5088C2211yescharacteristic
33#5088C2211yescharacteristic
34#5088C2211yescharacteristic
35#5088C2211yescharacteristic
36#5088C2211yescharacteristic
37#5087—22no—
38#5085C2211yescharacteristic
39#5084—22no—
40#5080C2211yescharacteristic
41#5080C2211yescharacteristic
42#5080C2211yescharacteristic
43#5078C2211yescharacteristic
44#5078C2211yescharacteristic
45#5078C2211yescharacteristic
46#5066—22no—
47#5047—22no—
48#5028—22no—
49#5003—22no—
50#4968—22no—
51#4967—22no—
52#4925C2211yescharacteristic
53#4925C2211yescharacteristic
54#4844—22no—
55#4734—22no—
56#4695C2211yescharacteristic
57#4694C2211yescharacteristic
58#4693C2211yescharacteristic
59#4681C2211yescharacteristic
60#4455C2211yescharacteristic
61#4454C2211yescharacteristic
62#4454C2211yescharacteristic
63C8×C3⋊C8C2211yescharacteristic
64#3228—31no—
65#3227—31no—
66#3226—31no—
67#3210—31no—
68#3209D311yescharacteristic, complemented
69#3207—31no—
70#1723—31no—
71#1722—31no—
72#1720—31no—
73#1720—31no—
74#1717—31no—
75#1714—31no—
76#1709—31no—
77#1707—31no—
78#1454—31no—
79C22×C3×Q8—21no—
80C2×#235—21no—
81C2×#235—21no—
82C2×C3×Q16—22no—
83C2×C3×Q16—22no—
84C3×C4⋊Q8—21no—
85C3×C4⋊Q8—21no—
86C3×C4⋊Q8—21no—
87C3×C4⋊Q8C2311yescharacteristic
88C3×C4⋊Q8C2311yescharacteristic
89C3×C4⋊Q8C2311yescharacteristic
90C3×C4⋊Q8C2311yescharacteristic
91C3×#125—21no—
92C3×C22⋊Q8—21no—
93C3×C22⋊Q8—21no—
94C3×C4×Q8—21no—
95C3×C4×Q8—21no—
96C3×#117—21no—
97C2×C3×C4⋊C4—21no—
98C2×C3×C4⋊C4—21no—
99C2×C3×C42C2311yescharacteristic
100C2×#216—22no—
101C2×#215—22no—
102#930—21no—
103#930—21no—
104#930—21no—
105#930—21no—
106C2×D24—22no—
107C2×#204—22no—
108#875—21no—
109#875—21no—
110#874C2311yescharacteristic
111#874C2311yescharacteristic
112C4×D12—21no—
113C4×D12—21no—
114C4×D12—21no—
115C4×D12—21no—
116#869C2311yescharacteristic
117#869C2311yescharacteristic
118C3×C4⋊C8D412yes—
119C3×#103—22no—
120C3×#103—22no—
121C3×C22⋊C8D412yes—
122C3×C4×C8C2311yescharacteristic
123C3×C4×C8C2311yescharacteristic
124#830D412yes—
125#830D412yes—
126#821—22no—
127#816—22no—
128#810—22no—
129#809—22no—
130#809—22no—
131#809—22no—
132#809—22no—
133#809—22no—
134#809—22no—
135#809—22no—
136#808—22no—
137#808—22no—
138#808—22no—
139#807—22no—
140#807—22no—
141#807—22no—
142C4⋊(C3⋊C8)D412yes—
143C4⋊(C3⋊C8)D412yes—
144C4×C3⋊C8C2311yescharacteristic
145C4×C3⋊C8C2311yescharacteristic
146C4×C3⋊C8C2311yescharacteristic
147C4×C3⋊C8C2311yescharacteristic
148#533C2×D311yescharacteristic
149#532—31no—
150#532—31no—
151C2×C4⋊Q8C2×D311yescharacteristic
152#500—31no—
153#500—31no—
154#500—31no—
155#498—31no—
156#498—31no—
157#498—31no—
158#494—31no—
159#494C2×D311yescharacteristic
160#494C2×D311yescharacteristic
161#493—31no—
162#492—31no—
163#492—31no—
164#487—31no—
165#487—31no—
166#487C2×D311yescharacteristic
167#487C2×D311yescharacteristic
168#486—31no—
169#486—31no—
170#486—31no—
171#486—31no—
172#486—31no—
173#486—31no—
174#486—31no—
175#486—31no—
176#482—62no—
177#481—62no—
178#480—62no—
179#478—62no—
180#476—62no—
181#474—62no—
182#464—62no—
183#463—62no—
184#462—22no—
185#452—62no—
186#451—22no—
187#449—62no—
188#431—31no—
189#431—31no—
190#431C2×D311yescharacteristic
191#322—31no—
192#322—31no—
193#322—31no—
194C82—31no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—41no—
200C2×C3×Q8—82no—
201C22×C3×C4—41no—
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D411yescharacteristic
205C2×#67—41no—
206C2×#67—41no—
207#235—82no—
208#235—82no—
209C2×D12—21no—
210C2×D12—21no—
211C2×D12—21no—
212C2×D12—21no—
213C2×D12—21no—
214C2×D12—21no—
215C2×C4×D3—41no—
216C2×C4×D3—41no—
217C2×Dic6—21no—
218C2×Dic6—21no—
219C3×Q16—82no—
220C3×Q16—82no—
221C2×C3×C8C2×D412yes—
222C2×C3×C8C2×D412yes—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—41no—
232C3×C4⋊C4—41no—
233C3×C4⋊C4—41no—
234C3×C4⋊C4—41no—
235C3×C22⋊C4—41no—
236C3×C42—41no—
237C3×C42C2×D411yescharacteristic
238C3×C42C2×D411yescharacteristic
239C3×C42C2×D411yescharacteristic
240C3×C42C2411yescharacteristic
241#216—82no—
242#215—82no—
243(C2×D3)⋊C4—41no—
244(C2×D3)⋊C4—41no—
245(C2×D3)⋊C4—41no—
246(C2×D3)⋊C4—41no—
247C4⋊Dic3—21no—
248C4⋊Dic3—21no—
249C4⋊Dic3—21no—
250C4⋊Dic3—21no—
251C2×C3⋊C8C2×D412yes—
252C2×C3⋊C8C2×D412yes—
253C2×C3⋊C8C2×D412yes—
254C2×C3⋊C8C2×D412yes—
255D24—82no—
256#204—82no—
257C2×#41—61no—
258C2×#41—61no—
259C22×Q8—21no—
260C2×Q16—22no—
261C2×Q16—22no—
262C2×Q16—62no—
263C2×SD16—62no—
264C2×SD16—62no—
265C2×D8—62no—
266C4⋊Q8—21no—
267C4⋊Q8—21no—
268C4⋊Q8—21no—
269C4⋊Q8—31no—
270C4⋊Q8—31no—
271C4⋊Q8C22×D311yescharacteristic
272C4⋊Q8C22×D311yescharacteristic
273C4⋊Q8C22×D311yescharacteristic
274C4⋊Q8C22×D311yescharacteristic
275#127—31no—
276#127—31no—
277#125—21no—
278#124—61no—
279#124—61no—
280C22⋊Q8—21no—
281C22⋊Q8—21no—
282#121—61no—
283#121—61no—
284#121—61no—
285#121—61no—
286C4×Q8—21no—
287C4×Q8—21no—
288C4×D4—61no—
289C4×D4—61no—
290C4×D4—61no—
291C4×D4—61no—
292#117—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C42C22×D311yescharacteristic
296#107—62no—
297#107—62no—
298#107—62no—
299#106—62no—
300#106—62no—
301#106—62no—
302C4⋊C8—32no—
303C4⋊C8—32no—
304C4⋊C8#6712yes—
305#103—22no—
306#103—22no—
307#103—62no—
308#103—62no—
309#102—62no—
310#102—62no—
311#102—62no—
312#102—62no—
313#102—62no—
314#102—62no—
315#102—62no—
316#102—62no—
317C22⋊C8—32no—
318C22⋊C8—32no—
319C22⋊C8#6712yes—
320C4×C8—31no—
321C4×C8—31no—
322C4×C8—31no—
323C4×C8—31no—
324C4×C8C22×D311yescharacteristic
325C4×C8C22×D311yescharacteristic
326C23×C3#12711yescharacteristic
327C22×D3—41no—
328C22×D3—41no—
329C3×Q8—42no—
330C3×Q8—42no—
331C3×Q8—42no—
332C3×Q8—42no—
333C3×Q8—82no—
334C2×C3×C4—22no—
335C2×C3×C4—22no—
336C2×C3×C4—22no—
337C2×C3×C4—41no—
338C2×C3×C4—41no—
339C2×C3×C4—41no—
340C2×C3×C4—41no—
341C2×C3×C4—82no—
342C2×C3×C4#12711yescharacteristic
343C2×C3×C4#12711yescharacteristic
344C2×C3×C4#12711yescharacteristic
345C2×C3×C4C22×D411yescharacteristic
346C2×C3×C4C22×D411yescharacteristic
347C2×C3×C4C22×D411yescharacteristic
348#67—82no—
349#67—82no—
350C2×Dic3—41no—
351C2×Dic3—41no—
352D12—42no—
353D12—42no—
354D12—42no—
355D12—42no—
356D12—42no—
357D12—42no—
358C4×D3—82no—
359C4×D3—82no—
360Dic6—42no—
361Dic6—42no—
362C3×C8—22no—
363C3×C8—22no—
364C3⋊C8—22no—
365C3⋊C8—22no—
366C3⋊C8—22no—
367C3⋊C8—22no—
368#41—242no—
369#41—242no—
370C2×Q8—21no—
371C2×Q8—21no—
372C2×Q8—21no—
373C2×Q8—21no—
374C2×Q8—41no—
375C2×Q8—61no—
376C2×Q8—61no—
377C2×Q8—82no—
378C2×D4—61no—
379C2×D4—61no—
380C2×D4—61no—
381C2×D4—61no—
382C2×D4—61no—
383C2×D4—61no—
384C2×D4—121no—
385C2×D4—121no—
386C22×C4—41no—
387C22×C4—121no—
388C22×C4—121no—
389C22×C4D3×D411yescharacteristic
390C22×C4D3×D411yescharacteristic
391C22×C4C2×#6711yescharacteristic
392Q16—82no—
393Q16—82no—
394Q16—242no—
395SD16—242no—
396SD16—242no—
397D8—242no—
398C2×C8—32no—
399C2×C8—32no—
400C2×C8—32no—
401C2×C8—32no—
402C2×C8C2×#6712yes—
403C2×C8C2×#6712yes—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—21no—
407C4⋊C4—21no—
408C4⋊C4—21no—
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—61no—
417C4⋊C4—61no—
418C4⋊C4—61no—
419C4⋊C4—61no—
420C22⋊C4—41no—
421C22⋊C4—121no—
422C22⋊C4—121no—
423C22⋊C4—121no—
424C22⋊C4—121no—
425C42—41no—
426C42D3×D411yescharacteristic
427C42D3×D411yescharacteristic
428C42C2×#6711yescharacteristic
429C42C23×D311yescharacteristic
430C22×C3—21no—
431C22×C3—21no—
432C22×C3—21no—
433C22×C3C2×#12711yescharacteristic
434C2×D3—81no—
435C2×D3—81no—
436C2×D3—81no—
437C2×D3—81no—
438C2×D3—81no—
439C2×D3—81no—
440C3×C4—21no—
441C3×C4—21no—
442C3×C4—21no—
443C3×C4—81no—
444C3×C4—81no—
445C3×C4—81no—
446C3×C4—81no—
447C3×C4#57612yes—
448C3×C4#57712yes—
449C3×C4#57812yes—
450Dic3—81no—
451Dic3—81no—
452C23—121no—
453C23—121no—
454C23#94011yescharacteristic
455Q8—42no—
456Q8—42no—
457Q8—42no—
458Q8—42no—
459Q8—82no—
460Q8—122no—
461Q8—122no—
462D4—122no—
463D4—122no—
464D4—122no—
465D4—122no—
466D4—122no—
467D4—122no—
468D4—242no—
469D4—242no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—82no—
478C2×C4—121no—
479C2×C4—121no—
480C2×C4—242no—
481C2×C4—242no—
482C2×C4#94011yescharacteristic
483C2×C4#94011yescharacteristic
484C2×C4#94011yescharacteristic
485C2×C4C2×D3×D411yescharacteristic
486C2×C4C2×D3×D411yescharacteristic
487C2×C4C22×#6711yescharacteristic
488C8—22no—
489C8—22no—
490C8—62no—
491C8—62no—
492C8—62no—
493C8—62no—
494C2×C3—41no—
495C2×C3#315411yescharacteristic
496C2×C3#315511yescharacteristic
497C2×C3#315611yescharacteristic
498D3—161no—
499D3—161no—
500C22—21no—
501C22—21no—
502C22—21no—
503C22—241no—
504C22—241no—
505C22—241no—
506C22—241no—
507C22—241no—
508C22—241no—
509C22C2×#94011yescharacteristic
510C4—21no—
511C4—21no—
512C4—21no—
513C4—81no—
514C4—81no—
515C4—81no—
516C4—81no—
517C4—241no—
518C4—241no—
519C4#576612yes—
520C4#577812yes—
521C4#583512yes—
522C3#1974111yescharacteristic, complemented, Sylow
523C2—41no—
524C2—481no—
525C2—481no—
526C2#8338211yescharacteristic
527C2#8388111yescharacteristic
528C2#8397411yescharacteristic
subgroupG/H
centerC22C2×#940
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#3209C2
socleC22×C3C2×#127
prime radical#79304C2
hypercenter#3209D3
hypocenterC3#19741
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3209

Complements

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

NH ≅ G/Ntype
1. C3×#3209C2semidirect
2. #83359C2semidirect
3. #83356C2semidirect
4. #83216C2semidirect
5. #83215C2semidirect
7. #71302C2semidirect
8. #71301C2semidirect
9. #71300C2semidirect
10. #71297C2semidirect
11. #71225C2semidirect
12. #71218C2semidirect
13. #71216C2semidirect
14. #71212C2semidirect
15. #70744C2semidirect
68. #3209D3semidirect
522. C3#19741semidirect

Maximal subgroups

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

subgroupindex
C3×#32092
#833592
#833562
#832162
#832152
#793042
#713022
#713012
#713002
#712972
#712252
#712182
#712162
#712122
#707442
#197413