Finite Group Databasebeta

← #10690374

Subgroups — #10690374

Order 768.

424 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 107 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#79310C211yesmaximal, characteristic
2#79045C211yesmaximal, characteristic
3#70500C211yesmaximal, characteristic
4#70473C211yesmaximal, characteristic
5#70437C211yesmaximal, characteristic
6C3×#1301C211yesmaximal, characteristic
7#67741C211yesmaximal, characteristic
8#9048—31nomaximal, Sylow
9#5470C2211yescharacteristic
10C2×C4×Dic6C2211yescharacteristic
11C3×#431C2211yescharacteristic
12C3×#423C2211yescharacteristic
13C2×C3×C4⋊C8C2211yescharacteristic
14#5122—21no—
15#5119—21no—
16#4925C2211yescharacteristic
17C2×#803C2211yescharacteristic
18#4850—21no—
19#4849—21no—
20#4828—22no—
21#4828—22no—
22#4739C412yes—
23#4722C412yes—
24C4×Dic12—22no—
25#4550—21no—
26#4550—21no—
27#4486—21no—
28#4484—22no—
29#4481—21no—
30#3104—31no—
31#2961—31no—
32#1985—31no—
33#1542—31no—
34#1487—31no—
35#1301D311yescharacteristic
36#1281—31no—
37C22×Dic6C2×C411yescharacteristic
38C2×C3×M16—21no—
39C22×C3×C8—21no—
40C2×C3×C42C2311yescharacteristic
41#926—21no—
42C2×C4⋊Dic3C2×C411yescharacteristic
43#924—21no—
44#924C2×C412yes—
45C2×Dic3⋊C4—21no—
46C2×C4×Dic3—21no—
47C22×C3⋊C8D411yescharacteristic
48C22×C3⋊C8D411yescharacteristic
49C2×Dic12—22no—
50#870—21no—
51#869D411yescharacteristic
52#869D411yescharacteristic
53C4×Dic6—21no—
54C4×Dic6—21no—
55C4×Dic6—21no—
56C4×Dic6—21no—
57C4×Dic6—21no—
58C4×Dic6D412yes—
59C3×C4⋊C8—21no—
60C3×C4⋊C8—21no—
61C3×C4⋊C8—21no—
62C3×C4⋊C8—21no—
63C3×C4⋊C8D412yes—
64C3×C22⋊C8C2×C412yes—
65C3×C4×C8—21no—
66#830—21no—
67#818—22no—
68#816—22no—
69#816—22no—
70#815—22no—
71Dic3⋊C8—24no—
72C8×Dic3—22no—
73C4⋊(C3⋊C8)—21no—
74#803—21no—
75#803—21no—
76C4×C3⋊C8—21no—
77#533—31no—
78C2×C4×Q8—31no—
79#484—32no—
80#477—61no—
81#462—61no—
82#451—32no—
83#446—62no—
84#446—62no—
85C4×Q16—62no—
86#431—31no—
87#431C2×D311yescharacteristic
88#424—61no—
89#423C2×D311yescharacteristic
90C2×C4⋊C8C2×D311yescharacteristic
91#405—61no—
92C2×#97—31no—
93#336—61no—
94#336—61no—
95#333—61no—
96#332—61no—
97Q8⋊C8—62no—
98C22×C3×C4C22×C411yescharacteristic
99C22×C3×C4C2×D411yescharacteristic
100C22×C3×C4#4111yescharacteristic
101C22×Dic3—21no—
102C22×Dic3—21no—
103C2×Dic6—21no—
104C2×Dic6—21no—
105C2×Dic6—21no—
106C2×Dic6—42no—
107C2×Dic6C22⋊C412yes—
108C3×M16—41no—
109C3×M16—41no—
110C2×C3×C8—22no—
111C2×C3×C8—22no—
112C2×C3×C8—41no—
113C2×C3×C8—41no—
114C2×C3×C8—41no—
115C3×C42C2×D411yescharacteristic
116C3×C42C2×D411yescharacteristic
117C3×C42C2×D411yescharacteristic
118C3×C42#4111yescharacteristic
119#217—22no—
120C4⋊Dic3—21no—
121C4⋊Dic3—21no—
122C4⋊Dic3—21no—
123C4⋊Dic3—21no—
124C4⋊Dic3—21no—
125C4⋊Dic3C22⋊C412yes—
126Dic3⋊C4—22no—
127Dic3⋊C4—22no—
128Dic3⋊C4—41no—
129Dic3⋊C4—41no—
130C4×Dic3—22no—
131C4×Dic3—41no—
132C4×Dic3—41no—
133C2×C3⋊C8—21no—
134C2×C3⋊C8—21no—
135C2×C3⋊C8—21no—
136C2×C3⋊C8—21no—
137C2×C3⋊C8—21no—
138C2×C3⋊C8—41no—
139C2×C3⋊C8Q1612yescomplemented
140Dic12—44no—
141C22×Q8—31no—
142C2×Q16—62no—
143C2×M16—21no—
144C22×C8—21no—
145C22×C8—31no—
146C22×C8—31no—
147C4⋊Q8—31no—
148C4⋊Q8—31no—
149#125—61no—
150C22⋊Q8—32no—
151C22⋊Q8—61no—
152C4×Q8—32no—
153C4×Q8—61no—
154C4×Q8—61no—
155C4×Q8—61no—
156C4×Q8—61no—
157C4×Q8—61no—
158#117—61no—
159C2×C4⋊C4—31no—
160C2×C4⋊C4—61no—
161C2×C42—61no—
162C2×C42C22×D311yescharacteristic
163#107—62no—
164C4⋊C8—21no—
165C4⋊C8—21no—
166C4⋊C8—21no—
167C4⋊C8—21no—
168C4⋊C8—64no—
169C4⋊C8—61no—
170C4⋊C8D1212yes—
171#103—62no—
172#103—62no—
173C22⋊C8—61no—
174C22⋊C8C4×D312yes—
175#97—62no—
176#97—61no—
177#97—61no—
178C4×C8—21no—
179C4×C8—62no—
180C4×C8—61no—
181C23×C3C4×D411yescharacteristic
182C2×C3×C4—41no—
183C2×C3×C4—41no—
184C2×C3×C4C2×C22⋊C411yescharacteristic
185C2×C3×C4C4×D411yescharacteristic
186C2×C3×C4#12011yescharacteristic
187C2×C3×C4#12111yescharacteristic
188C2×C3×C4#12111yescharacteristic
189C2×C3×C4#12311yescharacteristic
190C2×C3×C4C2×Q1611yescharacteristic
191C2×C3×C4#13511yescharacteristic
192C2×C3×C4#13711yescharacteristic
193C2×C3×C4#13711yescharacteristic
194C2×Dic3—22no—
195C2×Dic3—22no—
196C2×Dic3—22no—
197C2×Dic3—41no—
198C2×Dic3—41no—
199C2×Dic3—42no—
200C2×Dic3—41no—
201Dic6—22no—
202Dic6—22no—
203Dic6—41no—
204Dic6—41no—
205Dic6—42no—
206Dic6—81no—
207C3×C8—42no—
208C3×C8—42no—
209C3×C8—81no—
210C3⋊C8—22no—
211C3⋊C8—41no—
212C3⋊C8—81no—
213C2×Q8—32no—
214C2×Q8—61no—
215C2×Q8—61no—
216C2×Q8—61no—
217C2×Q8—122no—
218C22×C4—61no—
219C22×C4—61no—
220C22×C4C2×C4×D311yescharacteristic
221C22×C4D3×D411yescharacteristic
222C22×C4#23911yescharacteristic
223Q16—124no—
224M16—41no—
225M16—41no—
226C2×C8—22no—
227C2×C8—22no—
228C2×C8—32no—
229C2×C8—41no—
230C2×C8—41no—
231C2×C8—41no—
232C2×C8—61no—
233C2×C8—61no—
234C2×C8—61no—
235C2×C8—61no—
236C2×C8—61no—
237C2×C8—121no—
238C4⋊C4—32no—
239C4⋊C4—62no—
240C4⋊C4—61no—
241C4⋊C4—61no—
242C4⋊C4—62no—
243C4⋊C4—61no—
244C4⋊C4—61no—
245C4⋊C4—61no—
246C4⋊C4—121no—
247C4⋊C4—121no—
248C22⋊C4—62no—
249C42—62no—
250C42—121no—
251C42—121no—
252C42C2×D1211yescharacteristic
253C42#23511yescharacteristic
254C42D3×D411yescharacteristic
255C42D3×D411yescharacteristic
256C22×C3—21no—
257C22×C3—21no—
258C22×C3#38611yescharacteristic
259C22×C3C4×Q1611yescharacteristic
260C22×C3#44111yescharacteristic
261C3×C4—21no—
262C3×C4—21no—
263C3×C4—21no—
264C3×C4—21no—
265C3×C4#45111yescharacteristic
266C3×C4#45211yescharacteristic
267C3×C4#46211yescharacteristic
268C3×C4#46411yescharacteristic
269Dic3—42no—
270Dic3—42no—
271Dic3—42no—
272Dic3—81no—
273C23#89311yescharacteristic
274Q8—62no—
275Q8—62no—
276Q8—122no—
277Q8—121no—
278Q8—121no—
279Q8—241no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—62no—
283C2×C4—62no—
284C2×C4—62no—
285C2×C4—121no—
286C2×C4—121no—
287C2×C4—122no—
288C2×C4—121no—
289C2×C4C4×D1211yescharacteristic
290C2×C4D3×C22⋊C411yescharacteristic
291C2×C4#88211yescharacteristic
292C2×C4#88411yescharacteristic
293C2×C4#89411yescharacteristic
294C2×C4#89511yescharacteristic
295C2×C4#91211yescharacteristic
296C2×C4#91511yescharacteristic
297C2×C4#91511yescharacteristic
298C2×C4D3×Q1611yescharacteristic
299C8—42no—
300C8—42no—
301C8—62no—
302C8—81no—
303C8—121no—
304C8—241no—
305C2×C3—21no—
306C2×C3—21no—
307C2×C3#187811yescharacteristic
308C2×C3#188111yescharacteristic
309C2×C3#189111yescharacteristic
310C22—21no—
311C22—21no—
312C22#466511yescharacteristic
313C22#485311yescharacteristic
314C22#487311yescharacteristic
315C4—21no—
316C4—21no—
317C4—21no—
318C4—21no—
319C4—122no—
320C4—122no—
321C4—122no—
322C4—241no—
323C4#476711yescharacteristic
324C4#478311yescharacteristic
325C4#479511yescharacteristic
326C4#480911yescharacteristic
327C3#904811yescharacteristic, complemented, Sylow
328C2—21no—
329C2—21no—
330C2#6890811yescharacteristic
331C2#6894711yescharacteristic
332C2#6903611yescharacteristic
subgroupG/H
centerC2×C4#882
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1301C2
socleC22×C3#386
prime radicalC23×C3C4×D4
cosocleC2×C3×C42C23
hypercenter#1301D3
hypocenterC3#9048
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1301

Complements

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

NH ≅ G/Ntype
139. C2×C3⋊C8Q16semidirect
327. C3#9048semidirect

Maximal subgroups

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

subgroupindex
#793102
#790452
#705002
#704732
#704372
C3×#13012
#677412
#90483