Finite Group Databasebeta

← #10824873

Subgroups — #10824873

Order 768.

752 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Dic3×Q16C211yesmaximal, characteristic, complemented
2#79070C211yesmaximal, characteristic
3#78946C212yesmaximal
4C3×#2047C211yesmaximal, characteristic, complemented
5#72074C211yesmaximal, characteristic
6#70532C211yesmaximal, characteristic, complemented
7#70337C212yesmaximal, complemented
8#70126C212yesmaximal
9Dic3×#101C212yesmaximal, complemented
10#68921C211yesmaximal, characteristic
11#68916C211yesmaximal, characteristic, complemented
12#23717—31nomaximal, Sylow
13C22×C3×Q16C2211yescharacteristic
14C2×Q8×Dic3C2212yes—
15C2×#942C2212yes—
16#5522C2211yescharacteristic
17C3×#424C2211yescharacteristic
18C3×#419C2212yes—
19C2×C3×#101C2212yes—
20C2×#836C2212yes—
21C2×#835C2212yes—
22Dic3×Q16—22no—
23Dic3×Q16—22no—
24#5133C2212yes—
25#5129C2211yescharacteristic
26C2×#822C2212yes—
27Dic3×M16C2212yes—
28Dic3×M16C2211yescharacteristic
29#5110C2211yescharacteristic
30C2×#818C2211yescharacteristic
31C2×C8×Dic3C2211yescharacteristic
32#4999C2211yescharacteristic
33#4995C2212yes—
34#4840—22no—
35#4833—22no—
36#4817—22no—
37#4790—22no—
38#4789—22no—
39#4788—22no—
40C3×#341C2211yescharacteristic
41#4554C2212yes—
42#4554C2212yes—
43#4552C2211yescharacteristic
44C4×#137—32no—
45C2×C4×Q16—31no—
46#2922—31no—
47#2089—31no—
48#2078—31no—
49#2064—32no—
50#2052—32no—
51#2047D311yescharacteristic, complemented
52#2020—31no—
53C4×#108—31no—
54C4×#101—32no—
55C22×C3×Q8C2312yes—
56C2×#235C2311yescharacteristic
57C2×C3×Q16—24no—
58C2×C3×Q16—22no—
59C2×C3×M16C2312yes—
60C2×C3×M16C2311yescharacteristic
61C22×C3×C8C2311yescharacteristic
62C3×#117C2311yescharacteristic
63Q8×Dic3—22no—
64Q8×Dic3—22no—
65Q8×Dic3—22no—
66Q8×Dic3—22no—
67C2×#216—22no—
68#942—24no—
69#942—24no—
70C2×#215—22no—
71C2×C4⋊Dic3—22no—
72C2×C4⋊Dic3C2312yes—
73C2×C4×Dic3—22no—
74C2×C4×Dic3C2311yescharacteristic
75C2×#208C2311yescharacteristic
76C2×#208C2312yes—
77C22×C3⋊C8C2311yescharacteristic
78#893D411yescharacteristic
79#893D411yescharacteristic
80#887D411yescharacteristic
81#887D411yescharacteristic
82#67⋊C4—21no—
83#67⋊C4—21no—
84#872—21no—
85C42×D3—21no—
86C3×C4⋊C8—22no—
87C3×#103—22no—
88C3×#103—22no—
89C3×#101—24no—
90#836—24no—
91#835—24no—
92#822—24no—
93#821—22no—
94(C2×D3)⋊C8—22no—
95(C2×D3)⋊C8—22no—
96#818—22no—
97#816—22no—
98#815—22no—
99#815—21no—
100C8×Dic3—22no—
101C8×Dic3—21no—
102C8×Dic3D412yes—
103C8×Dic3D412yes—
104#808—22no—
105#807—22no—
106C4⋊(C3⋊C8)—22no—
107C2×#137—32no—
108C22×Q16C2×D311yescharacteristic
109C4×#41—31no—
110C2×C4×Q8—32no—
111#441—62no—
112#440—62no—
113C4×Q16—62no—
114C4×Q16—62no—
115C4×Q16—62no—
116C4×SD16—62no—
117#433—62no—
118#431—62no—
119C2×#108—32no—
120#428—32no—
121C2×#107—31no—
122#424—31no—
123#424C2×D311yescharacteristic
124#419C2×D312yes—
125#417—32no—
126C2×#103—32no—
127C2×#101—32no—
128C2×#101C2×D312yes—
129#408—31no—
130#408—31no—
131C4×M16—31no—
132C4×M16—32no—
133C2×C4×C8—31no—
134#343—32no—
135#343—32no—
136#341—31no—
137#341C2×D311yescharacteristic
138C2×C3×Q8—22no—
139C2×C3×Q8—22no—
140C2×C3×Q8—22no—
141C2×C3×Q8—22no—
142C2×C3×Q8—22no—
143C2×C3×Q8#4114yes—
144C22×C3×C4—22no—
145C22×C3×C4C2411yescharacteristic
146C2×#67—21no—
147C22×Dic3—22no—
148C22×Dic3#4111yescharacteristic
149C22×Dic3#4111yescharacteristic
150#235—22no—
151#235—22no—
152C2×D12C2×D411yescharacteristic
153C2×C4×D3—21no—
154C2×C4×D3—21no—
155C2×C4×D3—21no—
156C2×Dic6C2×D411yescharacteristic
157C3×Q16—28no—
158C3×M16—22no—
159C3×M16—22no—
160C3×M16—22no—
161C2×C3×C8—22no—
162C2×C3×C8—22no—
163C2×C3×C8—21no—
164C2×C3×C8C2×D412yes—
165C3×C4⋊C4C2×D411yescharacteristic
166C3×C4⋊C4C2×D411yescharacteristic
167C3×C22⋊C4—21no—
168C3×C42—21no—
169#216—28no—
170#215—28no—
171(C2×D3)⋊C4—21no—
172(C2×D3)⋊C4—21no—
173C4⋊Dic3—22no—
174C4⋊Dic3—22no—
175C4⋊Dic3—42no—
176C4⋊Dic3—42no—
177Dic3⋊C4—21no—
178Dic3⋊C4—21no—
179C4×Dic3—21no—
180C4×Dic3—21no—
181C4×Dic3—42no—
182C4×Dic3—42no—
183C4×Dic3C2×D411yescharacteristic
184C4×Dic3C2×D411yescharacteristic
185C4×Dic3C2×D411yescharacteristic
186C4×Dic3C2×D411yescharacteristic
187#208—22no—
188#208—22no—
189#208#4114yes—
190C2×C3⋊C8—22no—
191C2×C3⋊C8—22no—
192C2×C3⋊C8—21no—
193C2×C3⋊C8C2×D412yes—
194C2×#41—31no—
195C22×Q8C22×D312yes—
196#137—64no—
197#137—64no—
198C2×Q16—24no—
199C2×Q16—22no—
200C2×Q16—62no—
201C2×SD16—62no—
202C2×M16—32no—
203C2×M16—31no—
204C2×M16C22×D312yes—
205C2×M16C22×D311yescharacteristic
206C22×C8—31no—
207C22×C8C22×D311yescharacteristic
208C4×Q8—31no—
209C4×Q8—31no—
210C4×Q8—62no—
211C4×Q8—62no—
212C4×Q8—62no—
213C4×Q8—62no—
214C4×D4—31no—
215C4×D4—31no—
216C4×D4—61no—
217C4×D4—61no—
218#117—61no—
219#117C22×D311yescharacteristic
220C2×C4⋊C4—32no—
221C2×C4⋊C4—62no—
222C2×C42—31no—
223C2×C42—61no—
224C2×C42—62no—
225#108—64no—
226#107—62no—
227#107—62no—
228#106—62no—
229C4⋊C8—22no—
230C4⋊C8—62no—
231#103—22no—
232#103—22no—
233#103—62no—
234#103—64no—
235#102—62no—
236#101—24no—
237#101—64no—
238C22⋊C8—62no—
239C22⋊C8—62no—
240#97—61no—
241#97—62no—
242C4×C8—32no—
243C4×C8—32no—
244C4×C8—61no—
245C4×C8—62no—
246C23×C3C2×#4111yescharacteristic
247C22×D3—21no—
248C3×Q8—24no—
249C3×Q8—24no—
250C3×Q8—42no—
251C3×Q8—42no—
252C2×C3×C4—21no—
253C2×C3×C4—21no—
254C2×C3×C4—24no—
255C2×C3×C4—42no—
256C2×C3×C4—42no—
257C2×C3×C4C22×D411yescharacteristic
258C2×C3×C4C22×D411yescharacteristic
259C2×C3×C4C2×#4112yes—
260C2×C3×C4C2×#4112yes—
261#67—42no—
262C2×Dic3—22no—
263C2×Dic3—21no—
264C2×Dic3—41no—
265C2×Dic3—42no—
266C2×Dic3—82no—
267C2×Dic3#12111yescharacteristic
268C2×Dic3#12111yescharacteristic
269C2×Dic3#12111yescharacteristic
270C2×Dic3#12111yescharacteristic
271D12—22no—
272C4×D3—42no—
273C4×D3—41no—
274C4×D3—41no—
275C4×D3—42no—
276Dic6—22no—
277C3×C8—22no—
278C3×C8—41no—
279C3×C8—42no—
280C3⋊C8—24no—
281C3⋊C8—22no—
282C3⋊C8—41no—
283#41—62no—
284#41—62no—
285C2×Q8—22no—
286C2×Q8—22no—
287C2×Q8—22no—
288C2×Q8—22no—
289C2×Q8—22no—
290C2×Q8—31no—
291C2×Q8#23714yes—
292C2×D4—31no—
293C2×D4—61no—
294C22×C4—22no—
295C22×C4—31no—
296C22×C4—31no—
297C22×C4—61no—
298C22×C4—61no—
299C22×C4—61no—
300C22×C4—62no—
301C22×C4C23×D311yescharacteristic
302Q16—28no—
303Q16—68no—
304SD16—68no—
305M16—22no—
306M16—22no—
307M16—22no—
308M16—34no—
309M16—62no—
310M16—62no—
311C2×C8—22no—
312C2×C8—21no—
313C2×C8—22no—
314C2×C8—32no—
315C2×C8—62no—
316C2×C8—62no—
317C2×C8—61no—
318C2×C8D3×D412yes—
319C4⋊C4—61no—
320C4⋊C4—61no—
321C4⋊C4—62no—
322C4⋊C4—62no—
323C4⋊C4—122no—
324C4⋊C4—122no—
325C4⋊C4D3×D411yescharacteristic
326C4⋊C4D3×D411yescharacteristic
327C22⋊C4—21no—
328C22⋊C4—61no—
329C22⋊C4—61no—
330C42—21no—
331C42—31no—
332C42—31no—
333C42—31no—
334C42—31no—
335C42—61no—
336C42—61no—
337C42—122no—
338C42—122no—
339C22×C3—21no—
340C22×C3—21no—
341C22×C3C2×#12111yescharacteristic
342C22×C3#52912yes—
343C2×D3—41no—
344C2×D3—41no—
345C2×D3—41no—
346C3×C4—44no—
347C3×C4—41no—
348C3×C4—41no—
349C3×C4#52511yescharacteristic
350C3×C4#52512yes—
351C3×C4#53211yescharacteristic
352Dic3—21no—
353Dic3—21no—
354Dic3—21no—
355Dic3—41no—
356Dic3—82no—
357Dic3#47312yes—
358C23—61no—
359C23D3×#4111yescharacteristic
360Q8—24no—
361Q8—24no—
362Q8—42no—
363Q8—42no—
364Q8—62no—
365D4—62no—
366D4—122no—
367C2×C4—24no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—31no—
371C2×C4—31no—
372C2×C4—31no—
373C2×C4—31no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—62no—
377C2×C4—61no—
378C2×C4—122no—
379C2×C4—122no—
380C2×C4—121no—
381C2×C4—121no—
382C2×C4—121no—
383C2×C4—122no—
384C2×C4—242no—
385C2×C4C2×D3×D411yescharacteristic
386C2×C4C2×D3×D411yescharacteristic
387C2×C4C2×#23712yes—
388C2×C4D3×#4112yes—
389C8—22no—
390C8—41no—
391C8—42no—
392C8—62no—
393C8—64no—
394C8—121no—
395C2×C3—22no—
396C2×C3#239211yescharacteristic
397C2×C3C2×#47311yescharacteristic
398C2×C3#306711yescharacteristic
399D3—81no—
400C22—21no—
401C22—21no—
402C22—121no—
403C22—121no—
404C22—121no—
405C22D3×#12111yescharacteristic
406C22#564912yes—
407C4—32no—
408C4—41no—
409C4—41no—
410C4—44no—
411C4—61no—
412C4—61no—
413C4—61no—
414C4—121no—
415C4—242no—
416C4#558611yescharacteristic
417C4#560612yes—
418C4#566811yescharacteristic
419C3#2371711yescharacteristic, complemented, Sylow
420C2—22no—
421C2—241no—
422C2#7689111yescharacteristic
423C2D3×#47311yescharacteristic
424C2#8094311yescharacteristic
subgroupG/H
centerC22D3×#121
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2047C2
socleC22×C3C2×#121
prime radicalC2×#235C23
hypercenter#2047D3
hypocenterC3#23717
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #2047

Complements

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

NH ≅ G/Ntype
1. C2×Dic3×Q16C2semidirect
4. C3×#2047C2semidirect
6. #70532C2semidirect
7. #70337C2semidirect
9. Dic3×#101C2semidirect
11. #68916C2semidirect
51. #2047D3semidirect
419. C3#23717semidirect

Maximal subgroups

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

subgroupindex
C2×Dic3×Q162
#790702
#789462
C3×#20472
#720742
#705322
#703372
#701262
Dic3×#1012
#689212
#689162
#237173