Finite Group Databasebeta

← #10696787

Subgroups — #10696787

Order 768.

458 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 79 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83338C211yesmaximal, characteristic
2#83327C211yesmaximal, characteristic, complemented
3#79351C211yesmaximal, characteristic
4C3×#1497C211yesmaximal, characteristic, complemented
5#70784C211yesmaximal, characteristic
6#70769C211yesmaximal, characteristic, complemented
7#67753C211yesmaximal, characteristic, complemented
8#10125—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×#875C2211yescharacteristic
11#5470C2211yescharacteristic
12C3×#432C2211yescharacteristic
13C3×#431C2211yescharacteristic
14#5089—21no—
15#5089—21no—
16#5082—21no—
17#5081—21no—
18#5068—22no—
19#5045—22no—
20#5021—21no—
21#5020—21no—
22#5004—21no—
23#5001—21no—
24#4926C2211yescharacteristic
25C4×#208C2211yescharacteristic
26#4845—22no—
27#4739—21no—
28#4731—21no—
29#4715—21no—
30#4714—21no—
31C3×Q8⋊C8—22no—
32#4484—22no—
33#4468—22no—
34#3237—31no—
35#3215—31no—
36#3140—31no—
37#1497D311yescharacteristic, complemented
38#1485—31no—
39#1469—31no—
40#1291—31no—
41C22×D12D411yescharacteristic
42C22×Dic6D411yescharacteristic
43C3×C4⋊Q8D411yescharacteristic
44C3×C4⋊Q8D411yescharacteristic
45C3×#125—21no—
46C3×C22⋊Q8—21no—
47C3×C22⋊Q8—21no—
48C3×C4×Q8—21no—
49C3×C4×Q8—21no—
50C3×#117—21no—
51C2×C3×C42C2311yescharacteristic
52C2×(C2×D3)⋊C4—21no—
53C2×(C2×D3)⋊C4—21no—
54#926—21no—
55#924—21no—
56#924—21no—
57C2×#208—21no—
58C2×Dic12—22no—
59C2×#204—22no—
60#875—21no—
61#875—21no—
62#875—21no—
63#875—21no—
64#870—21no—
65#869D411yescharacteristic
66#869D411yescharacteristic
67C4×Dic6—21no—
68C4×Dic6—21no—
69C3×C4⋊C8—21no—
70C3×C4⋊C8D412yes—
71C3×C22⋊C8—21no—
72C3×C22⋊C8—21no—
73C3×#97—21no—
74C3×C4×C8—21no—
75#835—22no—
76#835—22no—
77#830D412yes—
78#821—22no—
79#816—22no—
80#810—22no—
81#810—22no—
82#809—22no—
83#809—22no—
84#808—22no—
85#807—22no—
86C4⋊(C3⋊C8)—21no—
87#803—21no—
88#803—21no—
89C4×C3⋊C8D412yes—
90#533—31no—
91#533C2×D311yescharacteristic
92C2×#124—31no—
93#501—61no—
94#497—61no—
95#491—61no—
96#489—61no—
97#489—61no—
98#489—61no—
99#488—62no—
100#485—62no—
101#477—61no—
102#475—61no—
103#464—62no—
104#451—61no—
105#450—61no—
106#432—31no—
107#432C2×D311yescharacteristic
108#431C2×D311yescharacteristic
109#419—61no—
110#418—61no—
111C4×M16—31no—
112Q8⋊C8—22no—
113Q8⋊C8—62no—
114#323—62no—
115C23×D3—21no—
116C2×C3×Q8—21no—
117C2×C3×Q8—21no—
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4#4111yescharacteristic
121C22×Dic3—21no—
122C2×D12—21no—
123C2×D12—21no—
124C2×D12—42no—
125C2×Dic6—21no—
126C2×Dic6—21no—
127C2×Dic6—21no—
128C2×Dic6—21no—
129C2×Dic6—42no—
130C2×C3×C8—22no—
131C2×C3×C8—41no—
132C3×C4⋊C4—21no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4—41no—
137C3×C4⋊C4—41no—
138C3×C22⋊C4—41no—
139C3×C42—41no—
140C3×C42C2×D411yescharacteristic
141C3×C42C2×D411yescharacteristic
142C3×C42C2×D411yescharacteristic
143C3×C42C2×D411yescharacteristic
144#217—41no—
145(C2×D3)⋊C4—41no—
146(C2×D3)⋊C4—41no—
147(C2×D3)⋊C4—41no—
148(C2×D3)⋊C4—41no—
149C4⋊Dic3—21no—
150C4⋊Dic3—21no—
151C4⋊Dic3—21no—
152C4⋊Dic3—21no—
153Dic3⋊C4—41no—
154Dic3⋊C4—41no—
155C4×Dic3—41no—
156#208—42no—
157C2×C3⋊C8—22no—
158C2×C3⋊C8—22no—
159Dic12—82no—
160#204—82no—
161C22×Q8—31no—
162C22×D4—31no—
163C2×Q16—62no—
164C2×SD16—62no—
165C2×M16—61no—
166C4⋊Q8—31no—
167C4⋊Q8—31no—
168C4⋊Q8D1211yescharacteristic
169C4⋊Q8D1211yescharacteristic
170#125—21no—
171#125—61no—
172#124—61no—
173#124—61no—
174#124—61no—
175#124—61no—
176C22⋊Q8—21no—
177C22⋊Q8—21no—
178C22⋊Q8—61no—
179C22⋊Q8—61no—
180C4×Q8—21no—
181C4×Q8—21no—
182C4×Q8—61no—
183C4×Q8—61no—
184#117—21no—
185#117—61no—
186C2×C22⋊C4—61no—
187C2×C22⋊C4—61no—
188C2×C42C22×D311yescharacteristic
189#107—62no—
190#106—62no—
191C4⋊C8—21no—
192C4⋊C8—61no—
193C4⋊C8#6712yes—
194#103—62no—
195#103—62no—
196#103—62no—
197#103—62no—
198#103—62no—
199#102—62no—
200#102—62no—
201#102—62no—
202C22⋊C8—21no—
203C22⋊C8—21no—
204C22⋊C8—32no—
205#97—21no—
206#97—61no—
207#97—61no—
208C4×C8—21no—
209C4×C8—32no—
210C23×C3#12111yescharacteristic
211C22×D3—41no—
212C22×D3—42no—
213C22×D3—81no—
214C22×D3—81no—
215C3×Q8—42no—
216C3×Q8—42no—
217C2×C3×C4—22no—
218C2×C3×C4—41no—
219C2×C3×C4—41no—
220C2×C3×C4—41no—
221C2×C3×C4—41no—
222C2×C3×C4#12011yescharacteristic
223C2×C3×C4#12011yescharacteristic
224C2×C3×C4#12011yescharacteristic
225C2×C3×C4#12111yescharacteristic
226C2×C3×C4#12111yescharacteristic
227C2×C3×C4#12711yescharacteristic
228C2×Dic3—41no—
229C2×Dic3—41no—
230C2×Dic3—41no—
231C2×Dic3—81no—
232D12—42no—
233D12—42no—
234Dic6—42no—
235Dic6—42no—
236Dic6—42no—
237Dic6—42no—
238C3×C8—42no—
239C3×C8—81no—
240C3⋊C8—42no—
241C3⋊C8—42no—
242C24—61no—
243C2×Q8—21no—
244C2×Q8—21no—
245C2×Q8—61no—
246C2×Q8—61no—
247C2×Q8—61no—
248C2×Q8—61no—
249C2×Q8—122no—
250C2×D4—61no—
251C2×D4—61no—
252C2×D4—122no—
253C22×C4—61no—
254C22×C4#23511yescharacteristic
255C22×C4D3×D411yescharacteristic
256C22×C4D3×D411yescharacteristic
257Q16—242no—
258SD16—242no—
259M16—122no—
260C2×C8—22no—
261C2×C8—41no—
262C2×C8—62no—
263C2×C8—62no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—61no—
271C4⋊C4—61no—
272C4⋊C4—61no—
273C4⋊C4—61no—
274C4⋊C4—121no—
275C4⋊C4—121no—
276C22⋊C4—41no—
277C22⋊C4—121no—
278C22⋊C4—121no—
279C22⋊C4—121no—
280C22⋊C4—121no—
281C22⋊C4—121no—
282C42—41no—
283C42—121no—
284C42C2×D1211yescharacteristic
285C42D3×D411yescharacteristic
286C42D3×D411yescharacteristic
287C42C2×#6711yescharacteristic
288C22×C3—21no—
289C22×C3—21no—
290C22×C3—21no—
291C22×C3#39211yescharacteristic
292C2×D3—81no—
293C2×D3—81no—
294C2×D3—81no—
295C2×D3—162no—
296C3×C4—21no—
297C3×C4—21no—
298C3×C4—21no—
299C3×C4—21no—
300C3×C4—21no—
301C3×C4—81no—
302C3×C4—81no—
303C3×C4#45512yes—
304Dic3—81no—
305Dic3—81no—
306Dic3—81no—
307C23—122no—
308C23—121no—
309C23—241no—
310C23—241no—
311C23#88411yescharacteristic
312Q8—42no—
313Q8—42no—
314Q8—122no—
315Q8—122no—
316Q8—122no—
317Q8—122no—
318D4—122no—
319D4—122no—
320C2×C4—22no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—121no—
326C2×C4—121no—
327C2×C4—121no—
328C2×C4—241no—
329C2×C4#88211yescharacteristic
330C2×C4#88211yescharacteristic
331C2×C4#88411yescharacteristic
332C2×C4#93011yescharacteristic
333C2×C4#93711yescharacteristic
334C2×C4#94011yescharacteristic
335C8—42no—
336C8—81no—
337C8—122no—
338C8—122no—
339C2×C3—41no—
340C2×C3#200411yescharacteristic
341C2×C3#200911yescharacteristic
342C2×C3#201711yescharacteristic
343D3—161no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—241no—
348C22—241no—
349C22—241no—
350C22—482no—
351C22#496011yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—81no—
358C4—81no—
359C4—241no—
360C4—241no—
361C4—241no—
362C4#520212yes—
363C3#1012511yescharacteristic, complemented, Sylow
364C2—41no—
365C2—481no—
366C2#7001311yescharacteristic
367C2#7034411yescharacteristic
368C2#7041211yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1497C2
socleC22×C3#392
prime radicalC22×D12D4
hypercenter#1497D3
hypocenterC3#10125
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1497

Complements

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

NH ≅ G/Ntype
2. #83327C2semidirect
4. C3×#1497C2semidirect
6. #70769C2semidirect
7. #67753C2semidirect
37. #1497D3semidirect
363. C3#10125semidirect

Maximal subgroups

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

subgroupindex
#833382
#833272
#793512
C3×#14972
#707842
#707692
#677532
#101253