Finite Group Databasebeta

← #10696773

Subgroups — #10696773

Order 768.

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

normal subgroups only

This group has 442 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#83284C211yesmaximal, characteristic
2#83274C211yesmaximal, characteristic
3#79320C211yesmaximal, characteristic, complemented
4C3×#1496C211yesmaximal, characteristic
5#70787C211yesmaximal, characteristic, complemented
6#70769C211yesmaximal, characteristic, complemented
7#67753C211yesmaximal, characteristic, complemented
8#10123—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10#5470C2211yescharacteristic
11C2×#870C2211yescharacteristic
12C3×#432C2211yescharacteristic
13C3×#431C2211yescharacteristic
14#5232—21no—
15#5232—21no—
16#5082—21no—
17#5081—21no—
18#5071—21no—
19#5071—21no—
20#5056—22no—
21#5035—22no—
22#5008—21no—
23#5007—21no—
24#4926C2211yescharacteristic
25C4×#208C2211yescharacteristic
26#4829—22no—
27#4722—21no—
28#4721—21no—
29#4703—21no—
30#4702—21no—
31C3×D4⋊C8—22no—
32#4484—22no—
33#4469—22no—
34#3236—31no—
35#3235—31no—
36#3142—31no—
37#1496D311yescharacteristic
38#1485—31no—
39#1470—31no—
40#1291—31no—
41C2×C3×#41—21no—
42C3×C4⋊Q8D411yescharacteristic
43C3×#127D411yescharacteristic
44C3×#124—21no—
45C3×#121—21no—
46C3×#121—21no—
47C3×C4×D4—21no—
48C3×C4×D4—21no—
49C2×C3×C42C2311yescharacteristic
50#926—21no—
51C2×C4⋊Dic3D411yescharacteristic
52C2×C4⋊Dic3D411yescharacteristic
53#924—21no—
54#924—21no—
55C2×Dic3⋊C4—21no—
56C2×Dic3⋊C4—21no—
57C2×#208—21no—
58#870—21no—
59#870—21no—
60#870—21no—
61#870—21no—
62#870—21no—
63#869D411yescharacteristic
64#869D411yescharacteristic
65C4×Dic6—21no—
66C4×Dic6—21no—
67C3×C4⋊C8—21no—
68C3×C4⋊C8D412yes—
69C3×C22⋊C8—21no—
70C3×C22⋊C8—21no—
71C3×#97—21no—
72C3×C4×C8—21no—
73#835—22no—
74#835—22no—
75#832—22no—
76#832—22no—
77#832—22no—
78#832—22no—
79#830D412yes—
80#818—22no—
81#817—22no—
82#816—22no—
83#816—22no—
84#808—22no—
85#807—22no—
86C4⋊(C3⋊C8)—21no—
87#803—21no—
88#803—21no—
89C4×C3⋊C8D412yes—
90#533—31no—
91#532C2×D311yescharacteristic
92C2×#125—31no—
93#501—61no—
94#501—61no—
95#491—61no—
96#491—61no—
97#490—61no—
98#490—61no—
99#488—62no—
100#484—61no—
101#483—61no—
102#482—62no—
103#479—62no—
104#476—61no—
105#474—61no—
106#432—31no—
107#432C2×D311yescharacteristic
108#431C2×D311yescharacteristic
109#417—61no—
110#417—61no—
111C4×M16—31no—
112Q8⋊C8—62no—
113D4⋊C8—22no—
114#324—62no—
115C3×#41—44no—
116C2×C3×Q8—21no—
117C2×C3×D4—21no—
118C2×C3×D4—21no—
119C2×C3×D4—21no—
120C2×C3×D4—41no—
121C22×C3×C4—41no—
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4#4111yescharacteristic
125C22×Dic3—21no—
126C22×Dic3—21no—
127C2×Dic6—21no—
128C2×Dic6—21no—
129C2×C3×C8—22no—
130C2×C3×C8—41no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—21no—
133C3×C22⋊C4—41no—
134C3×C22⋊C4—41no—
135C3×C42C2×D411yescharacteristic
136C3×C42C2×D411yescharacteristic
137C3×C42C2×D411yescharacteristic
138C3×C42C2×D411yescharacteristic
139#217—41no—
140C4⋊Dic3—21no—
141C4⋊Dic3—21no—
142C4⋊Dic3—21no—
143C4⋊Dic3—21no—
144C4⋊Dic3—21no—
145C4⋊Dic3—21no—
146C4⋊Dic3—21no—
147C4⋊Dic3—21no—
148Dic3⋊C4—41no—
149Dic3⋊C4—41no—
150Dic3⋊C4—41no—
151Dic3⋊C4—41no—
152Dic3⋊C4—41no—
153Dic3⋊C4—41no—
154C4×Dic3—41no—
155#208—42no—
156C2×C3⋊C8—22no—
157C2×C3⋊C8—22no—
158C2×#41—21no—
159C2×M16—61no—
160C4⋊Q8—31no—
161C4⋊Q8—31no—
162C4⋊Q8D1211yescharacteristic
163#127D1211yescharacteristic
164#125—61no—
165#125—61no—
166#125—61no—
167#125—61no—
168#125—61no—
169#124—21no—
170C22⋊Q8—61no—
171C22⋊Q8—61no—
172#121—21no—
173#121—21no—
174C4×Q8—61no—
175C4×Q8—61no—
176C4×D4—21no—
177C4×D4—21no—
178#117—61no—
179C2×C4⋊C4—31no—
180C2×C4⋊C4—31no—
181C2×C4⋊C4—61no—
182C2×C4⋊C4—61no—
183C2×C42C22×D311yescharacteristic
184#107—62no—
185#107—62no—
186#106—62no—
187#106—62no—
188C4⋊C8—21no—
189C4⋊C8—61no—
190C4⋊C8#6712yes—
191#103—62no—
192#103—62no—
193#103—62no—
194#103—62no—
195#102—62no—
196#102—62no—
197#102—62no—
198#102—62no—
199C22⋊C8—21no—
200C22⋊C8—21no—
201C22⋊C8—32no—
202#97—21no—
203#97—61no—
204#97—61no—
205C4×C8—21no—
206C4×C8—32no—
207C23×C3—41no—
208C23×C3#12111yescharacteristic
209C3×Q8—42no—
210C3×D4—42no—
211C3×D4—42no—
212C3×D4—42no—
213C3×D4—82no—
214C2×C3×C4—22no—
215C2×C3×C4—41no—
216C2×C3×C4—41no—
217C2×C3×C4—41no—
218C2×C3×C4—82no—
219C2×C3×C4#12011yescharacteristic
220C2×C3×C4#12011yescharacteristic
221C2×C3×C4#12011yescharacteristic
222C2×C3×C4#12111yescharacteristic
223C2×C3×C4#12111yescharacteristic
224C2×C3×C4#12711yescharacteristic
225C2×Dic3—42no—
226C2×Dic3—41no—
227C2×Dic3—41no—
228C2×Dic3—41no—
229C2×Dic3—41no—
230C2×Dic3—81no—
231Dic6—42no—
232Dic6—42no—
233C3×C8—42no—
234C3×C8—81no—
235C3⋊C8—42no—
236C3⋊C8—42no—
237#41—44no—
238C2×Q8—21no—
239C2×Q8—61no—
240C2×Q8—61no—
241C2×D4—21no—
242C2×D4—21no—
243C2×D4—21no—
244C2×D4—41no—
245C22×C4—41no—
246C22×C4—61no—
247C22×C4—61no—
248C22×C4#23511yescharacteristic
249C22×C4D3×D411yescharacteristic
250C22×C4D3×D411yescharacteristic
251M16—122no—
252C2×C8—22no—
253C2×C8—41no—
254C2×C8—62no—
255C2×C8—62no—
256C4⋊C4—21no—
257C4⋊C4—21no—
258C4⋊C4—61no—
259C4⋊C4—61no—
260C4⋊C4—61no—
261C4⋊C4—61no—
262C4⋊C4—61no—
263C4⋊C4—61no—
264C4⋊C4—61no—
265C4⋊C4—61no—
266C4⋊C4—121no—
267C4⋊C4—121no—
268C4⋊C4—121no—
269C4⋊C4—121no—
270C4⋊C4—121no—
271C4⋊C4—121no—
272C22⋊C4—41no—
273C22⋊C4—41no—
274C22⋊C4—121no—
275C42—121no—
276C42C2×D1211yescharacteristic
277C42D3×D411yescharacteristic
278C42D3×D411yescharacteristic
279C42C2×#6711yescharacteristic
280C22×C3—21no—
281C22×C3—21no—
282C22×C3—21no—
283C22×C3—81no—
284C22×C3—81no—
285C22×C3—81no—
286C22×C3#39211yescharacteristic
287C3×C4—21no—
288C3×C4—21no—
289C3×C4—21no—
290C3×C4—21no—
291C3×C4—21no—
292C3×C4—81no—
293C3×C4#45412yes—
294Dic3—81no—
295Dic3—81no—
296Dic3—81no—
297Dic3—81no—
298C23—41no—
299C23#88411yescharacteristic
300Q8—42no—
301Q8—122no—
302Q8—122no—
303D4—42no—
304D4—42no—
305D4—42no—
306D4—82no—
307C2×C4—22no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—82no—
312C2×C4—122no—
313C2×C4—121no—
314C2×C4—121no—
315C2×C4—121no—
316C2×C4—121no—
317C2×C4—241no—
318C2×C4#88211yescharacteristic
319C2×C4#88211yescharacteristic
320C2×C4#88411yescharacteristic
321C2×C4#93011yescharacteristic
322C2×C4#93711yescharacteristic
323C2×C4#94011yescharacteristic
324C8—42no—
325C8—81no—
326C8—122no—
327C8—122no—
328C2×C3—41no—
329C2×C3—161no—
330C2×C3#200311yescharacteristic
331C2×C3#200911yescharacteristic
332C2×C3#201711yescharacteristic
333C22—21no—
334C22—21no—
335C22—21no—
336C22—81no—
337C22—81no—
338C22—81no—
339C22#496011yescharacteristic
340C4—21no—
341C4—21no—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—81no—
346C4—241no—
347C4—241no—
348C4—241no—
349C4—241no—
350C4#520112yes—
351C3#1012311yescharacteristic, complemented, Sylow
352C2—41no—
353C2—161no—
354C2#7001311yescharacteristic
355C2#7032611yescharacteristic
356C2#7041211yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1496C2
socleC22×C3#392
prime radicalC3×#532C22
hypercenter#1496D3
hypocenterC3#10123
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1496

Complements

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

NH ≅ G/Ntype
3. #79320C2semidirect
5. #70787C2semidirect
6. #70769C2semidirect
7. #67753C2semidirect
351. C3#10123semidirect

Maximal subgroups

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

subgroupindex
#832842
#832742
#793202
C3×#14962
#707872
#707692
#677532
#101233