Finite Group Databasebeta

← #10696224

Subgroups — #10696224

Order 768.

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

normal subgroups only

This group has 476 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#83351C211yesmaximal, characteristic
2#83327C211yesmaximal, characteristic, complemented
3#79344C211yesmaximal, characteristic
4C3×#1537C211yesmaximal, characteristic, complemented
5#70827C211yesmaximal, characteristic
6#70803C211yesmaximal, characteristic, complemented
7#67756C211yesmaximal, characteristic, complemented
8#10043—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C4×D12C2211yescharacteristic
11#5470C2211yescharacteristic
12C3×#432C2211yescharacteristic
13C3×#432C2211yescharacteristic
14#5084—21no—
15#5083—21no—
16#5082—21no—
17#5081—21no—
18#5046—21no—
19#5045—22no—
20#5043—21no—
21C4×#215—22no—
22#5021—21no—
23#5020—21no—
24#4926C2211yescharacteristic
25#4925C2211yescharacteristic
26#4846—21no—
27#4841—21no—
28#4731—21no—
29#4722—21no—
30#4710—22no—
31C3×#330—22no—
32#4484—22no—
33#4468—22no—
34#3237—31no—
35#3110—31no—
36#3101—31no—
37#1537D311yescharacteristic, complemented
38#1497—31no—
39#1466—31no—
40#1287—31no—
41C22×D12D411yescharacteristic
42C22×C4×D3—21no—
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×#215—22no—
53C2×(C2×D3)⋊C4—21no—
54#926—21no—
55C2×C4⋊Dic3D411yescharacteristic
56#924—21no—
57#924—21no—
58C2×#204—22no—
59C4×D12—21no—
60C4×D12—21no—
61C4×D12—21no—
62C4×D12—21no—
63#870—21no—
64#869D411yescharacteristic
65#869D411yescharacteristic
66C4×Dic6—21no—
67C4×Dic6—21no—
68C3×C4⋊C8—21no—
69C3×C4⋊C8—21no—
70C3×C22⋊C8—21no—
71C3×C22⋊C8—21no—
72C3×#97—21no—
73C3×#97D412yes—
74#835—22no—
75#835—22no—
76#835—22no—
77#830—21no—
78#830D412yes—
79#821—22no—
80#818—22no—
81#816—22no—
82#809—22no—
83#808—22no—
84#808—22no—
85#807—22no—
86C4⋊(C3⋊C8)—21no—
87C4⋊(C3⋊C8)—21no—
88#803—21no—
89C4×C3⋊C8D412yes—
90#533—31no—
91#533C2×D311yescharacteristic
92C2×C4×D4—31no—
93#501—61no—
94#491—61no—
95#485—62no—
96#484—61no—
97#483—61no—
98#477—61no—
99#476—61no—
100#475—61no—
101#474—61no—
102#461—61no—
103#461—61no—
104#450—61no—
105#450—61no—
106#440—62no—
107C4×SD16—62no—
108#432—31no—
109#432C2×D311yescharacteristic
110#432C2×D311yescharacteristic
111#431—31no—
112#330—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×C4×D3—41no—
126C2×C4×D3—41no—
127C2×C4×D3—82no—
128C2×Dic6—21no—
129C2×Dic6—21no—
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#215—44no—
146(C2×D3)⋊C4—41no—
147(C2×D3)⋊C4—41no—
148C4⋊Dic3—21no—
149C4⋊Dic3—21no—
150C4⋊Dic3—21no—
151C4⋊Dic3—21no—
152C4⋊Dic3—21no—
153C4⋊Dic3—21no—
154Dic3⋊C4—41no—
155Dic3⋊C4—41no—
156C4×Dic3—41no—
157C2×C3⋊C8—22no—
158C2×C3⋊C8—22no—
159#204—44no—
160C22×D4—31no—
161C23×C4—61no—
162C2×SD16—62no—
163C2×SD16—62no—
164C4⋊Q8—31no—
165C4⋊Q8—31no—
166C4⋊Q8D1211yescharacteristic
167C4⋊Q8D1211yescharacteristic
168#125—21no—
169#125—61no—
170C22⋊Q8—21no—
171C22⋊Q8—21no—
172C22⋊Q8—61no—
173C22⋊Q8—61no—
174C4×Q8—21no—
175C4×Q8—21no—
176C4×Q8—61no—
177C4×Q8—61no—
178C4×D4—61no—
179C4×D4—61no—
180C4×D4—61no—
181C4×D4—61no—
182#117—21no—
183#117—61no—
184C2×C4⋊C4—31no—
185C2×C22⋊C4—61no—
186C2×C42C22×D311yescharacteristic
187#107—62no—
188#107—62no—
189#106—62no—
190#106—62no—
191C4⋊C8—21no—
192C4⋊C8—21no—
193C4⋊C8—61no—
194C4⋊C8—61no—
195#103—62no—
196#103—62no—
197#103—62no—
198#103—62no—
199#102—62no—
200#102—62no—
201C22⋊C8—21no—
202C22⋊C8—21no—
203C22⋊C8—32no—
204C22⋊C8—61no—
205#97—21no—
206#97—61no—
207#97#6712yes—
208C4×C8—32no—
209C23×C3#12111yescharacteristic
210C22×D3—22no—
211C22×D3—41no—
212C22×D3—41no—
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—22no—
233D12—22no—
234D12—41no—
235D12—41no—
236C4×D3—82no—
237C4×D3—82no—
238Dic6—42no—
239Dic6—42no—
240C3×C8—42no—
241C3×C8—81no—
242C3⋊C8—42no—
243C3⋊C8—42no—
244C24—61no—
245C2×Q8—21no—
246C2×Q8—21no—
247C2×Q8—61no—
248C2×Q8—61no—
249C2×D4—61no—
250C2×D4—61no—
251C2×D4—122no—
252C22×C4—61no—
253C22×C4—121no—
254C22×C4—121no—
255C22×C4—242no—
256C22×C4#23511yescharacteristic
257C22×C4D3×D411yescharacteristic
258C22×C4D3×D411yescharacteristic
259SD16—124no—
260SD16—124no—
261C2×C8—22no—
262C2×C8—41no—
263C2×C8—62no—
264C2×C8—62no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—61no—
272C4⋊C4—61no—
273C4⋊C4—61no—
274C4⋊C4—61no—
275C4⋊C4—61no—
276C4⋊C4—61no—
277C4⋊C4—121no—
278C4⋊C4—121no—
279C22⋊C4—41no—
280C22⋊C4—121no—
281C22⋊C4—121no—
282C22⋊C4—121no—
283C42—41no—
284C42—121no—
285C42C2×D1211yescharacteristic
286C42D3×D411yescharacteristic
287C42D3×D411yescharacteristic
288C42C2×#6711yescharacteristic
289C22×C3—21no—
290C22×C3—21no—
291C22×C3—21no—
292C22×C3#39211yescharacteristic
293C2×D3—42no—
294C2×D3—82no—
295C2×D3—81no—
296C2×D3—81no—
297C2×D3—82no—
298C3×C4—21no—
299C3×C4—21no—
300C3×C4—21no—
301C3×C4—21no—
302C3×C4—21no—
303C3×C4—81no—
304C3×C4—81no—
305C3×C4#47112yes—
306Dic3—81no—
307Dic3—81no—
308Dic3—81no—
309C23—62no—
310C23—121no—
311C23—121no—
312C23—241no—
313C23—241no—
314C23#88411yescharacteristic
315Q8—42no—
316Q8—42no—
317Q8—122no—
318Q8—122no—
319D4—62no—
320D4—62no—
321D4—121no—
322D4—121no—
323C2×C4—22no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—121no—
329C2×C4—121no—
330C2×C4—121no—
331C2×C4—242no—
332C2×C4—242no—
333C2×C4—241no—
334C2×C4#88211yescharacteristic
335C2×C4#88211yescharacteristic
336C2×C4#88411yescharacteristic
337C2×C4#93011yescharacteristic
338C2×C4#93711yescharacteristic
339C2×C4#94011yescharacteristic
340C8—42no—
341C8—81no—
342C8—122no—
343C8—122no—
344C2×C3—41no—
345C2×C3#200911yescharacteristic
346C2×C3#201011yescharacteristic
347C2×C3#201911yescharacteristic
348D3—82no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—122no—
353C22—242no—
354C22—242no—
355C22—241no—
356C22—241no—
357C22#496011yescharacteristic
358C4—21no—
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—81no—
364C4—81no—
365C4—241no—
366C4—241no—
367C4—241no—
368C4#475412yes—
369C3#1004311yescharacteristic, complemented, Sylow
370C2—41no—
371C2—242no—
372C2#7011711yescharacteristic
373C2#7040611yescharacteristic
374C2#7041211yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1537C2
socleC22×C3#392
prime radicalC22×D12D4
hypercenter#1537D3
hypocenterC3#10043
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1537

Complements

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

NH ≅ G/Ntype
2. #83327C2semidirect
4. C3×#1537C2semidirect
6. #70803C2semidirect
7. #67756C2semidirect
37. #1537D3semidirect
369. C3#10043semidirect

Maximal subgroups

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

subgroupindex
#833512
#833272
#793442
C3×#15372
#708272
#708032
#677562
#100433