Finite Group Databasebeta

← #10849686

Subgroups — #10849686

Order 768.

1038 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#85788C211yesmaximal, characteristic
2C3×#2563C211yesmaximal, characteristic, complemented
3#83207C211yesmaximal, characteristic, complemented
4#83199C211yesmaximal, characteristic, complemented
5C4×#833C211yesmaximal, characteristic, complemented
6#69665C214yesmaximal, complemented
7#69562C214yesmaximal, complemented
8#69493C211yesmaximal, characteristic, complemented
9#69492C211yesmaximal, characteristic, complemented
10#25693—31nomaximal, Sylow
11C2×C3×C4⋊Q8C2211yescharacteristic
12C2×C3×C4×D4C2211yescharacteristic
13#5805C2212yes—
14#5791—22no—
15#5563—21no—
16#5559—22no—
17#5558—22no—
18#5553—21no—
19#5551—21no—
20#5542—21no—
21#5504C2212yes—
22#5494—22no—
23#5477C2211yescharacteristic
24#5470C2211yescharacteristic
25C3×#393—22no—
26C3×#390C2212yes—
27C3×#382C2212yes—
28C3×#380—22no—
29C3×C4×C4⋊C4C2211yescharacteristic
30C2×#833C2212yes—
31#5093—21no—
32#5089—22no—
33#5088—21no—
34#5081—22no—
35#5080—21no—
36#5078—21no—
37#4966C2214yes—
38#4964C2214yes—
39C2×#805C2214yes—
40C4×#208C2211yescharacteristic
41C4×#208C2211yescharacteristic
42#4530C2212yes—
43#4523C2214yes—
44#3506—31no—
45#3165—31no—
46#3141—31no—
47#2563D311yescharacteristic, complemented
48#2037—34no—
49#1986—31no—
50#1980—31no—
51#1860—34no—
52C4×#100—31no—
53C22×C3×Q8—21no—
54C22×C3×D4C2311yescharacteristic
55C23×C3×C4D412yes—
56C2×#235—22no—
57C2×#235C2312yes—
58C3×C4⋊Q8—21no—
59C3×C4⋊Q8—21no—
60C3×C4⋊Q8—21no—
61C3×C4⋊Q8—21no—
62C3×C4×D4—21no—
63C3×C4×D4—21no—
64C3×C4×D4—21no—
65C3×C4×D4—21no—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4C2312yes—
69C2×C3×C4⋊C4C2311yescharacteristic
70C2×C3×C22⋊C4—24no—
71C2×C3×C22⋊C4D412yes—
72C2×C3×C42C2312yes—
73C2×C3×C42C2311yescharacteristic
74#953—24no—
75#940—22no—
76#938—22no—
77#938—22no—
78#936—22no—
79#930—22no—
80#930—21no—
81#930—21no—
82#929—24no—
83C4×#67—24no—
84C4×#67—24no—
85#926—22no—
86#926C2312yes—
87#924—22no—
88#924—21no—
89#924—21no—
90C2×#208C2312yes—
91C2×#208C2312yes—
92#893—22no—
93#892—22no—
94D3×C4⋊C4—22no—
95#887—22no—
96#885—24no—
97#884—24no—
98#883—24no—
99#879—24no—
100#875D412yes—
101#874D411yescharacteristic
102C4×D12—21no—
103C4×D12—21no—
104#870D412yes—
105#869D411yescharacteristic
106#869D411yescharacteristic
107#869D411yescharacteristic
108C4×Dic6—21no—
109C4×Dic6—21no—
110C3×#95—21no—
111C3×#95—24no—
112#833—24no—
113#810—24no—
114#809—24no—
115#808—24no—
116#807—24no—
117#805—28no—
118#803—21no—
119#803—21no—
120C4×C3⋊C8—21no—
121C4×C3⋊C8—21no—
122#559—62no—
123#556—61no—
124#555—61no—
125#553—62no—
126#550—61no—
127#547—61no—
128#538—62no—
129#535—32no—
130#533—31no—
131#532—31no—
132C2×C4⋊Q8C2×D311yescharacteristic
133#525—62no—
134#520—32no—
135C2×C4×D4C2×D311yescharacteristic
136#501—62no—
137#500—61no—
138#498—61no—
139#489—62no—
140#487—61no—
141#486—61no—
142#428—34no—
143C2×#104—34no—
144#417—34no—
145C2×#100—32no—
146C4×M16—31no—
147C4×M16—31no—
148#393—22no—
149#390C2×D312yes—
150#382C2×D312yes—
151#380—22no—
152C4×C4⋊C4C2×D311yescharacteristic
153#343—32no—
154#339—34no—
155C24×C3C2×D412yes—
156C2×C3×Q8—41no—
157C2×C3×Q8—41no—
158C2×C3×Q8—82no—
159C2×C3×D4—21no—
160C2×C3×D4—21no—
161C2×C3×D4—22no—
162C2×C3×D4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—44no—
168C22×C3×C4C2×D411yescharacteristic
169C22×C3×C4C2×D412yes—
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2411yescharacteristic
172C2×#67—22no—
173C2×#67—24no—
174C2×#67—24no—
175#235—24no—
176#235—24no—
177#235—84no—
178C2×D12—21no—
179C2×D12—21no—
180C2×D12C2×D412yes—
181C2×C4×D3—22no—
182C2×C4×D3—24no—
183C2×Dic6—21no—
184C2×Dic6—21no—
185C2×Dic6C2×D412yes—
186C3×C4⋊C4—21no—
187C3×C4⋊C4—21no—
188C3×C4⋊C4—22no—
189C3×C4⋊C4—22no—
190C3×C4⋊C4—42no—
191C3×C4⋊C4—41no—
192C3×C4⋊C4—41no—
193C3×C22⋊C4—22no—
194C3×C22⋊C4—22no—
195C3×C22⋊C4—84no—
196C3×C42—24no—
197C3×C42C2×D411yescharacteristic
198C3×C42C2×D411yescharacteristic
199C3×C42C2×D411yescharacteristic
200C3×C42C2×D411yescharacteristic
201#217—24no—
202#217—24no—
203#217—22no—
204(C2×D3)⋊C4—24no—
205(C2×D3)⋊C4—22no—
206(C2×D3)⋊C4—22no—
207C4⋊Dic3—21no—
208C4⋊Dic3—21no—
209C4⋊Dic3—21no—
210C4⋊Dic3—21no—
211C4⋊Dic3C2×D412yes—
212C4⋊Dic3C2×D412yes—
213Dic3⋊C4—24no—
214Dic3⋊C4—22no—
215Dic3⋊C4—22no—
216C4×Dic3—24no—
217C4×Dic3—22no—
218#208—22no—
219#208—22no—
220#208—22no—
221#208#4114yescomplemented
222C2×C3⋊C8—22no—
223C2×C3⋊C8—22no—
224C2×#41—32no—
225C2×#41—62no—
226C22×Q8—21no—
227C22×D4C22×D311yescharacteristic
228C23×C4#6712yes—
229C2×M16—32no—
230C2×M16—32no—
231C4⋊Q8—21no—
232C4⋊Q8—21no—
233C4⋊Q8—21no—
234C4⋊Q8—21no—
235C4⋊Q8—31no—
236C4⋊Q8—31no—
237C4⋊Q8—31no—
238#127—31no—
239#127—62no—
240#126—64no—
241#125—32no—
242#124—32no—
243#124—64no—
244#124—62no—
245#123—64no—
246#123—64no—
247C22⋊Q8—62no—
248C22⋊Q8—61no—
249C22⋊Q8—61no—
250#121—64no—
251#121—62no—
252#121—62no—
253#121—62no—
254#121—61no—
255#121—61no—
256#120—64no—
257C4×Q8—62no—
258C4×Q8—61no—
259C4×Q8—61no—
260C4×D4—21no—
261C4×D4—21no—
262C4×D4—21no—
263C4×D4—21no—
264C4×D4—62no—
265C4×D4—64no—
266C4×D4—64no—
267C4×D4—61no—
268C4×D4—61no—
269#117—32no—
270#117—62no—
271#117—62no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—62no—
275C2×C4⋊C4C22×D312yes—
276C2×C4⋊C4C22×D311yescharacteristic
277C2×C22⋊C4—24no—
278C2×C22⋊C4#6712yes—
279C2×C42C22×D312yes—
280C2×C42C22×D311yescharacteristic
281#107—64no—
282#106—64no—
283#104—68no—
284#103—64no—
285#102—64no—
286#100—64no—
287#97—61no—
288#97—61no—
289C4×C8—61no—
290C4×C8—61no—
291#95—21no—
292#95—24no—
293C23×C3—22no—
294C23×C3—22no—
295C23×C3—22no—
296C23×C3—42no—
297C23×C3—42no—
298C23×C3C22×D411yescharacteristic
299C22×D3—22no—
300C3×Q8—82no—
301C3×Q8—82no—
302C3×D4—42no—
303C3×D4—42no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—41no—
307C2×C3×C4—44no—
308C2×C3×C4—42no—
309C2×C3×C4—42no—
310C2×C3×C4—44no—
311C2×C3×C4—42no—
312C2×C3×C4—42no—
313C2×C3×C4—42no—
314C2×C3×C4—82no—
315C2×C3×C4#12011yescharacteristic
316C2×C3×C4#12011yescharacteristic
317C2×C3×C4#12011yescharacteristic
318C2×C3×C4#12011yescharacteristic
319C2×C3×C4C22×D411yescharacteristic
320C2×C3×C4C22×D411yescharacteristic
321C2×C3×C4C2×#4112yes—
322C2×C3×C42−1+412yes—
323#67—42no—
324#67—42no—
325#67—84no—
326#67—84no—
327C2×Dic3—22no—
328C2×Dic3—22no—
329C2×Dic3—22no—
330D12—24no—
331D12—42no—
332D12—81no—
333C4×D3—44no—
334C4×D3—84no—
335Dic6—24no—
336Dic6—42no—
337Dic6—81no—
338C3⋊C8—24no—
339C3⋊C8—42no—
340C24C2×#6712yes—
341#41—64no—
342#41—64no—
343#41—244no—
344C2×Q8—32no—
345C2×Q8—41no—
346C2×Q8—41no—
347C2×Q8—61no—
348C2×Q8—61no—
349C2×Q8—82no—
350C2×D4—21no—
351C2×D4—21no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—32no—
355C2×D4—62no—
356C2×D4—64no—
357C2×D4—64no—
358C2×D4—61no—
359C2×D4—61no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—44no—
365C22×C4—64no—
366C22×C4—62no—
367C22×C4C2×#6711yescharacteristic
368C22×C4C2×#6712yes—
369C22×C4C2×#6711yescharacteristic
370C22×C4C23×D311yescharacteristic
371M16—34no—
372M16—62no—
373M16—62no—
374M16—62no—
375C2×C8—62no—
376C2×C8—62no—
377C4⋊C4—21no—
378C4⋊C4—21no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—32no—
382C4⋊C4—32no—
383C4⋊C4—42no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—64no—
387C4⋊C4—61no—
388C4⋊C4—61no—
389C4⋊C4—62no—
390C4⋊C4—61no—
391C4⋊C4—62no—
392C4⋊C4—61no—
393C22⋊C4—22no—
394C22⋊C4—22no—
395C22⋊C4—64no—
396C22⋊C4—64no—
397C22⋊C4—64no—
398C22⋊C4—62no—
399C22⋊C4—62no—
400C22⋊C4—62no—
401C22⋊C4—84no—
402C42—24no—
403C42—64no—
404C42—62no—
405C42D3×D411yescharacteristic
406C42D3×D411yescharacteristic
407C42D3×D411yescharacteristic
408C42D3×D411yescharacteristic
409C22×C3—21no—
410C22×C3—21no—
411C22×C3—44no—
412C22×C3—42no—
413C22×C3—42no—
414C22×C3—42no—
415C22×C3—82no—
416C22×C3C2×#12011yescharacteristic
417C22×C3#53211yescharacteristic
418C22×C3#53611yescharacteristic
419C2×D3—42no—
420C2×D3—42no—
421C2×D3—42no—
422C3×C4—21no—
423C3×C4—21no—
424C3×C4—21no—
425C3×C4—42no—
426C3×C4—82no—
427C3×C4—82no—
428C3×C4#45512yes—
429C3×C4#54712yes—
430C3×C4#54812yes—
431Dic3—42no—
432Dic3—42no—
433Dic3—42no—
434C23—22no—
435C23—22no—
436C23—22no—
437C23—42no—
438C23—42no—
439C23—62no—
440C23C22×#6711yescharacteristic
441Q8—64no—
442Q8—82no—
443Q8—82no—
444Q8—122no—
445Q8—241no—
446D4—42no—
447D4—42no—
448D4—64no—
449D4—122no—
450D4—122no—
451D4—122no—
452D4—241no—
453D4—244no—
454D4—244no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—41no—
458C2×C4—44no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—44no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—62no—
466C2×C4—62no—
467C2×C4—62no—
468C2×C4—82no—
469C2×C4—124no—
470C2×C4—244no—
471C2×C4#93711yescharacteristic
472C2×C4#93711yescharacteristic
473C2×C4#93711yescharacteristic
474C2×C4#93711yescharacteristic
475C2×C4C2×D3×D411yescharacteristic
476C2×C4C2×D3×D411yescharacteristic
477C2×C4#100712yes—
478C2×C4D3×#4112yes—
479C8—64no—
480C8—122no—
481C2×C3—21no—
482C2×C3—21no—
483C2×C3—82no—
484C2×C3#243211yescharacteristic
485C2×C3C2×#45511yescharacteristic
486C2×C3#302111yescharacteristic
487D3—82no—
488C22—21no—
489C22—21no—
490C22—44no—
491C22—42no—
492C22—42no—
493C22—42no—
494C22—82no—
495C22—122no—
496C22—122no—
497C22—122no—
498C22C2×#93711yescharacteristic
499C22#581611yescharacteristic
500C22#583211yescharacteristic
501C4—21no—
502C4—21no—
503C4—21no—
504C4—42no—
505C4—82no—
506C4—82no—
507C4—122no—
508C4—122no—
509C4—122no—
510C4#506112yes—
511C4#563112yes—
512C4#563312yes—
513C3#2569311yescharacteristic, complemented, Sylow
514C2—21no—
515C2—21no—
516C2—82no—
517C2—242no—
518C2#8275911yescharacteristic
519C2C2×#506111yescharacteristic
520C2#8339511yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2563C2
socleC22×C3C2×#120
prime radical#85788C2
hypercenter#2563D3
hypocenterC3#25693
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2563

Complements

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

NH ≅ G/Ntype
2. C3×#2563C2semidirect
3. #83207C2semidirect
4. #83199C2semidirect
5. C4×#833C2semidirect
6. #69665C2semidirect
7. #69562C2semidirect
8. #69493C2semidirect
9. #69492C2semidirect
47. #2563D3semidirect
221. #208#41semidirect
513. C3#25693semidirect

Maximal subgroups

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

subgroupindex
#857882
C3×#25632
#832072
#831992
C4×#8332
#696652
#695622
#694932
#694922
#256933