Finite Group Databasebeta

← #10849639

Subgroups — #10849639

Order 768.

964 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#86204C211yesmaximal, characteristic
2C3×#2862C211yesmaximal, characteristic, complemented
3#83212C211yesmaximal, characteristic
4#83207C211yesmaximal, characteristic, complemented
5#71789C211yesmaximal, characteristic, complemented
6#69695C214yesmaximal
7#69672C214yesmaximal, complemented
8#69396C211yesmaximal, characteristic, complemented
9C4×#805C211yesmaximal, characteristic
10#25688—31nomaximal, Sylow
11C2×C3×C4⋊Q8C2211yescharacteristic
12C2×C3×C4⋊Q8C2211yescharacteristic
13#5816C2212yes—
14#5812C2212yes—
15#5732—21no—
16#5732—21no—
17#5731—22no—
18#5728—21no—
19#5728—21no—
20#5721—21no—
21#5721—21no—
22#5513—22no—
23#5513—22no—
24C4×#235C2211yescharacteristic
25#5470C2211yescharacteristic
26C3×#391—22no—
27C3×#391C2214yes—
28C3×#382—22no—
29C3×C43C2211yescharacteristic
30C2×#836C2214yes—
31#5094—21no—
32#5091—21no—
33#5087—21no—
34#5084—21no—
35#5081—22no—
36#5080—21no—
37#5078—21no—
38#4966C2212yes—
39#4964C2214yes—
40#4964C2212yes—
41C2×#805C2212yes—
42#4924C2211yescharacteristic
43C4×#208C2211yescharacteristic
44#4523C2212yes—
45#4523C2214yes—
46#3509—31no—
47#3165—31no—
48#3079—31no—
49#2862D311yescharacteristic, complemented
50#2038—34no—
51#1994—31no—
52#1966—31no—
53#1857—34no—
54C4×#104—31no—
55C22×C3×Q8—21no—
56C22×C3×Q8C2312yes—
57C2×#235C2311yescharacteristic
58C3×C4⋊Q8—21no—
59C3×C4⋊Q8—21no—
60C3×C4⋊Q8—21no—
61C3×C4⋊Q8—21no—
62C3×C4⋊Q8—21no—
63C3×C4⋊Q8—21no—
64C3×C4⋊Q8—21no—
65C3×C4⋊Q8—21no—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4D412yes—
68C2×C3×C4⋊C4D412yes—
69C2×C3×C4⋊C4C2312yes—
70C2×C3×C42—21no—
71C2×C3×C42—21no—
72C2×C3×C42C2312yes—
73C2×C3×C42C2311yescharacteristic
74#947—22no—
75#946—22no—
76#946—22no—
77Q8×Dic3—22no—
78Q8×Dic3—22no—
79Q8×Dic3—22no—
80Q8×Dic3—22no—
81#944—22no—
82#929—24no—
83C4×#67—21no—
84C4×#67—21no—
85#926—24no—
86#926C2312yes—
87#926C2311yescharacteristic
88#924—21no—
89#924—21no—
90C2×#208C2312yes—
91C2×#208C2312yes—
92#898—24no—
93#898—24no—
94#897—22no—
95#897—22no—
96#892—22no—
97#892—22no—
98#890—22no—
99#890—22no—
100#889—24no—
101#889—24no—
102#887—22no—
103#887—22no—
104C4×D12D411yescharacteristic
105C4×D12D411yescharacteristic
106#872—21no—
107C42×D3—21no—
108#870D412yes—
109#869D411yescharacteristic
110#869D411yescharacteristic
111C4×Dic6—21no—
112C4×Dic6—21no—
113C4×Dic6D411yescharacteristic
114C4×Dic6D411yescharacteristic
115C3×#95—28no—
116#836—28no—
117#810—22no—
118#809—22no—
119#808—24no—
120#808—22no—
121#807—24no—
122#807—22no—
123#805—24no—
124C4⋊(C3⋊C8)—21no—
125C4⋊(C3⋊C8)—21no—
126#803—21no—
127C4×C3⋊C8—21no—
128#557—61no—
129#557—61no—
130#556—61no—
131#556—61no—
132#552—62no—
133#552—62no—
134#541—62no—
135#536—32no—
136#533—31no—
137#533—61no—
138#533—61no—
139C2×C4⋊Q8C2×D311yescharacteristic
140C2×C4⋊Q8C2×D311yescharacteristic
141#519—32no—
142C4×#41—31no—
143#501—62no—
144#500—61no—
145#498—61no—
146#477—61no—
147#476—61no—
148#475—61no—
149#474—61no—
150#428—32no—
151#428—34no—
152#423—31no—
153C2×#104—32no—
154#417—32no—
155C2×#101—34no—
156C4×M16—31no—
157#391—22no—
158#391C2×D314yes—
159#382—22no—
160C43C2×D311yescharacteristic
161#339—32no—
162#339—34no—
163C2×C3×Q8—22no—
164C2×C3×Q8—22no—
165C2×C3×Q8—24no—
166C2×C3×Q8—24no—
167C2×C3×Q8—44no—
168C2×C3×Q8—41no—
169C2×C3×Q8—41no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4C2×D414yes—
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2×D411yescharacteristic
175C22×C3×C4C2411yescharacteristic
176C2×#67—21no—
177#235—22no—
178#235—22no—
179C2×D12C2×D411yescharacteristic
180C2×C4×D3—21no—
181C2×C4×D3—21no—
182C2×C4×D3—21no—
183C2×Dic6—21no—
184C2×Dic6—21no—
185C2×Dic6C2×D411yescharacteristic
186C3×C4⋊C4—22no—
187C3×C4⋊C4—22no—
188C3×C4⋊C4—22no—
189C3×C4⋊C4—22no—
190C3×C4⋊C4—22no—
191C3×C4⋊C4—22no—
192C3×C4⋊C4—41no—
193C3×C4⋊C4—41no—
194C3×C42—24no—
195C3×C42—22no—
196C3×C42—22no—
197C3×C42—42no—
198C3×C42C2×D411yescharacteristic
199C3×C42C2×D411yescharacteristic
200C3×C42C2×D411yescharacteristic
201C3×C42C2×D411yescharacteristic
202#217—22no—
203#217—21no—
204(C2×D3)⋊C4—24no—
205(C2×D3)⋊C4—24no—
206(C2×D3)⋊C4—21no—
207(C2×D3)⋊C4—21no—
208C4⋊Dic3—24no—
209C4⋊Dic3—24no—
210C4⋊Dic3—21no—
211C4⋊Dic3—21no—
212C4⋊Dic3C2×D411yescharacteristic
213C4⋊Dic3C2×D412yes—
214C4⋊Dic3C2×D411yescharacteristic
215C4⋊Dic3C2×D412yes—
216Dic3⋊C4—24no—
217Dic3⋊C4—24no—
218Dic3⋊C4—22no—
219Dic3⋊C4—22no—
220Dic3⋊C4—21no—
221Dic3⋊C4—21no—
222C4×Dic3—24no—
223C4×Dic3—24no—
224C4×Dic3—21no—
225C4×Dic3—21no—
226C4×Dic3—22no—
227C4×Dic3—21no—
228#208—22no—
229#208—24no—
230#208#4114yes—
231C2×C3⋊C8—22no—
232C2×C3⋊C8—22no—
233C2×#41—31no—
234C22×Q8—21no—
235C22×Q8C22×D312yes—
236C2×M16—32no—
237C2×M16—32no—
238C4⋊Q8—21no—
239C4⋊Q8—21no—
240C4⋊Q8—21no—
241C4⋊Q8—21no—
242C4⋊Q8—21no—
243C4⋊Q8—21no—
244C4⋊Q8—21no—
245C4⋊Q8—21no—
246C4⋊Q8—31no—
247C4⋊Q8—31no—
248C4⋊Q8—62no—
249#126—64no—
250#126—64no—
251#125—32no—
252#125—62no—
253#125—62no—
254#125—64no—
255#125—64no—
256#124—62no—
257#123—64no—
258C22⋊Q8—62no—
259C22⋊Q8—62no—
260C22⋊Q8—62no—
261C22⋊Q8—62no—
262C22⋊Q8—61no—
263C22⋊Q8—61no—
264C4×Q8—31no—
265C4×Q8—31no—
266C4×Q8—62no—
267C4×Q8—62no—
268C4×Q8—62no—
269C4×Q8—62no—
270C4×Q8—62no—
271C4×Q8—62no—
272C4×Q8—61no—
273C4×Q8—61no—
274C4×D4—31no—
275C4×D4—31no—
276C4×D4—61no—
277C4×D4—61no—
278#117—31no—
279#117—32no—
280#117—62no—
281#117—62no—
282#117—64no—
283#117—61no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4#6712yes—
286C2×C4⋊C4#6712yes—
287C2×C4⋊C4C22×D312yes—
288C2×C42—21no—
289C2×C42—21no—
290C2×C42—61no—
291C2×C42C22×D312yes—
292C2×C42C22×D311yescharacteristic
293#107—62no—
294#107—64no—
295#106—62no—
296#106—64no—
297C4⋊C8—61no—
298C4⋊C8—61no—
299#104—64no—
300#103—62no—
301#102—62no—
302#101—68no—
303#97—61no—
304C4×C8—61no—
305#95—28no—
306C23×C3C22×D411yescharacteristic
307C22×D3—21no—
308C3×Q8—42no—
309C3×Q8—42no—
310C3×Q8—44no—
311C3×Q8—82no—
312C3×Q8—82no—
313C2×C3×C4—22no—
314C2×C3×C4—22no—
315C2×C3×C4—24no—
316C2×C3×C4—24no—
317C2×C3×C4—28no—
318C2×C3×C4—42no—
319C2×C3×C4—42no—
320C2×C3×C4—44no—
321C2×C3×C4—42no—
322C2×C3×C4#12011yescharacteristic
323C2×C3×C4#12011yescharacteristic
324C2×C3×C4#12011yescharacteristic
325C2×C3×C4#12011yescharacteristic
326C2×C3×C4C22×D411yescharacteristic
327C2×C3×C4C22×D411yescharacteristic
328C2×C3×C4C2×#4112yes—
329C2×C3×C42−1+412yes—
330#67—42no—
331C2×Dic3—21no—
332C2×Dic3—21no—
333C2×Dic3—22no—
334C2×Dic3—22no—
335C2×Dic3—21no—
336D12—21no—
337D12—21no—
338C4×D3—42no—
339C4×D3—42no—
340C4×D3—42no—
341Dic6—21no—
342Dic6—21no—
343Dic6—42no—
344Dic6—42no—
345C3⋊C8—24no—
346C3⋊C8—42no—
347#41—62no—
348#41—62no—
349C2×Q8—22no—
350C2×Q8—22no—
351C2×Q8—24no—
352C2×Q8—24no—
353C2×Q8—31no—
354C2×Q8—44no—
355C2×Q8—41no—
356C2×Q8—41no—
357C2×Q8—61no—
358C2×Q8—61no—
359C2×D4—31no—
360C2×D4—61no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—61no—
364C22×C4—61no—
365C22×C4—61no—
366C22×C4C2×#6714yes—
367C22×C4C2×#6711yescharacteristic
368C22×C4C2×#6711yescharacteristic
369C22×C4C23×D311yescharacteristic
370M16—34no—
371M16—62no—
372M16—64no—
373C2×C8—62no—
374C2×C8—62no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—31no—
382C4⋊C4—32no—
383C4⋊C4—31no—
384C4⋊C4—32no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—64no—
388C4⋊C4—64no—
389C4⋊C4—64no—
390C4⋊C4—64no—
391C4⋊C4—61no—
392C4⋊C4—61no—
393C4⋊C4—61no—
394C4⋊C4—61no—
395C4⋊C4—62no—
396C4⋊C4—62no—
397C22⋊C4—64no—
398C22⋊C4—64no—
399C22⋊C4—61no—
400C22⋊C4—62no—
401C22⋊C4—61no—
402C22⋊C4—61no—
403C42—24no—
404C42—22no—
405C42—22no—
406C42—42no—
407C42—64no—
408C42—64no—
409C42—61no—
410C42—61no—
411C42—61no—
412C42—62no—
413C42D3×D411yescharacteristic
414C42D3×D411yescharacteristic
415C42D3×D411yescharacteristic
416C42D3×D411yescharacteristic
417C22×C3—21no—
418C22×C3—21no—
419C22×C3C2×#12011yescharacteristic
420C22×C3#54712yes—
421C2×D3—41no—
422C2×D3—41no—
423C2×D3—41no—
424C3×C4—21no—
425C3×C4—21no—
426C3×C4—44no—
427C3×C4—44no—
428C3×C4—82no—
429C3×C4#45412yes—
430C3×C4#45612yes—
431C3×C4#53211yescharacteristic
432C3×C4#53611yescharacteristic
433C3×C4#54812yes—
434Dic3—41no—
435Dic3—42no—
436Dic3—41no—
437Dic3—42no—
438Dic3—41no—
439C23—61no—
440C23C22×#6711yescharacteristic
441Q8—42no—
442Q8—42no—
443Q8—44no—
444Q8—61no—
445Q8—61no—
446Q8—82no—
447Q8—82no—
448Q8—122no—
449Q8—122no—
450D4—61no—
451D4—61no—
452D4—122no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—28no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—44no—
461C2×C4—42no—
462C2×C4—61no—
463C2×C4—61no—
464C2×C4—62no—
465C2×C4—62no—
466C2×C4—61no—
467C2×C4—122no—
468C2×C4—122no—
469C2×C4—122no—
470C2×C4#93711yescharacteristic
471C2×C4#93711yescharacteristic
472C2×C4#93711yescharacteristic
473C2×C4#93711yescharacteristic
474C2×C4C2×D3×D411yescharacteristic
475C2×C4C2×D3×D411yescharacteristic
476C2×C4D3×#4112yes—
477C2×C4#101012yes—
478C8—64no—
479C8—122no—
480C2×C3—22no—
481C2×C3#243211yescharacteristic
482C2×C3C2×#45411yescharacteristic
483C2×C3C2×#45611yescharacteristic
484D3—81no—
485C22—21no—
486C22—21no—
487C22—121no—
488C22—121no—
489C22—121no—
490C22C2×#93711yescharacteristic
491C22#582412yes—
492C4—21no—
493C4—21no—
494C4—44no—
495C4—44no—
496C4—82no—
497C4—121no—
498C4—122no—
499C4—121no—
500C4—121no—
501C4—122no—
502C4#506212yes—
503C4#509512yes—
504C4#566811yescharacteristic
505C4#567511yescharacteristic
506C4#567712yes—
507C3#2568811yescharacteristic, complemented, Sylow
508C2—22no—
509C2—241no—
510C2C2×#506211yescharacteristic
511C2#8295911yescharacteristic
512C2C2×#509511yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2862C2
socleC22×C3C2×#120
prime radicalC2×#235C23
hypercenter#2862D3
hypocenterC3#25688
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2862

Complements

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

NH ≅ G/Ntype
2. C3×#2862C2semidirect
4. #83207C2semidirect
5. #71789C2semidirect
7. #69672C2semidirect
8. #69396C2semidirect
49. #2862D3semidirect
507. C3#25688semidirect

Maximal subgroups

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

subgroupindex
#862042
C3×#28622
#832122
#832072
#717892
#696952
#696722
#693962
C4×#8052
#256883