Finite Group Databasebeta

← #10697125

Subgroups — #10697125

Order 768.

476 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 103 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#83224C211yesmaximal, characteristic
2C2×#5080C211yesmaximal, characteristic
3#78537C211yesmaximal, characteristic
4C3×#1997C211yesmaximal, characteristic
5#70506C211yesmaximal, characteristic
6#70501C211yesmaximal, characteristic
7#67667C211yesmaximal, characteristic
8#10173—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10#5470C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#423C2211yescharacteristic
13C3×C4×M16C2211yescharacteristic
14#5120—22no—
15#5093—21no—
16#5091—22no—
17#5080—22no—
18#5080—22no—
19#5078—21no—
20#4970—22no—
21C2×#808—22no—
22#4925C2211yescharacteristic
23C2×C4×C3⋊C8C2211yescharacteristic
24#4703—22no—
25#4681—21no—
26#4681—21no—
27C3×#343—22no—
28#4484—22no—
29#4482—22no—
30#3230—31no—
31C2×#498—31no—
32#3139—31no—
33#1997D311yescharacteristic
34#1546—31no—
35#1493—31no—
36#1278—31no—
37C22×C3×Q8—21no—
38C22×Dic6—21no—
39C2×C3×M16—21no—
40C2×C3×M16D412yes—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4D412yes—
47C2×C3×C42C2311yescharacteristic
48#926—21no—
49C2×C4⋊Dic3—21no—
50C2×C4⋊Dic3—21no—
51#924—21no—
52#924—21no—
53C22×C3⋊C8—21no—
54#870—21no—
55#869—21no—
56#869—21no—
57#869—21no—
58#869D411yescharacteristic
59#869D411yescharacteristic
60#869D412yes—
61C4×Dic6—21no—
62C4×Dic6—21no—
63C3×C4⋊C8—21no—
64C3×C4⋊C8—21no—
65C3×#97—21no—
66C3×C4×C8—21no—
67#830—21no—
68#816—24no—
69#816—24no—
70#810—24no—
71#808—28no—
72#807—24no—
73C4⋊(C3⋊C8)D412yes—
74C4×C3⋊C8—21no—
75C4×C3⋊C8—21no—
76C4×C3⋊C8D412yes—
77#533—31no—
78C2×C4⋊Q8—31no—
79C2×C4⋊Q8C2×D311yescharacteristic
80#500—61no—
81#498—62no—
82#498—62no—
83#491—62no—
84#487—61no—
85#487—61no—
86#487—61no—
87#484—62no—
88#477—62no—
89#431—31no—
90C2×#106—62no—
91#423C2×D311yescharacteristic
92#419—62no—
93C4×M16C2×D311yescharacteristic
94C2×C4×C8—31no—
95#343—22no—
96#332—62no—
97Q8⋊C8—62no—
98C2×C3×Q8—41no—
99C2×C3×Q8—41no—
100C2×C3×Q8—82no—
101C22×C3×C4—22no—
102C22×C3×C4C2×D411yescharacteristic
103C22×C3×C4C2×D411yescharacteristic
104C22×C3×C4#4111yescharacteristic
105C22×Dic3—41no—
106C2×Dic6—21no—
107C2×Dic6—22no—
108C2×Dic6—21no—
109C2×Dic6—42no—
110C2×Dic6—42no—
111C2×Dic6—41no—
112C3×M16—22no—
113C3×M16—22no—
114C3×M16—42no—
115C2×C3×C8—22no—
116C2×C3×C8—41no—
117C3×C4⋊C4—22no—
118C3×C4⋊C4—22no—
119C3×C4⋊C4—41no—
120C3×C4⋊C4—41no—
121C3×C42C2×D411yescharacteristic
122C3×C42C2×D411yescharacteristic
123C3×C42C2×D411yescharacteristic
124C3×C42C2×D411yescharacteristic
125#217—41no—
126C4⋊Dic3—21no—
127C4⋊Dic3—22no—
128C4⋊Dic3—21no—
129C4⋊Dic3—21no—
130C4⋊Dic3—21no—
131C4⋊Dic3—22no—
132C4⋊Dic3—41no—
133C4⋊Dic3—41no—
134Dic3⋊C4—41no—
135Dic3⋊C4—41no—
136C4×Dic3—41no—
137C2×C3⋊C8—24no—
138C2×C3⋊C8—22no—
139C2×C3⋊C8—22no—
140C22×Q8—21no—
141C22×Q8—61no—
142C2×M16—21no—
143C2×M16#6712yes—
144C22×C8—61no—
145C4⋊Q8—21no—
146C4⋊Q8—21no—
147C4⋊Q8—21no—
148C4⋊Q8—21no—
149C4⋊Q8—32no—
150C4⋊Q8—31no—
151C4⋊Q8—31no—
152C4⋊Q8—61no—
153C4⋊Q8—61no—
154C4⋊Q8—61no—
155#125—61no—
156C22⋊Q8—61no—
157C22⋊Q8—61no—
158C4×Q8—61no—
159C4×Q8—61no—
160#117—61no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—61no—
163C2×C4⋊C4—61no—
164C2×C4⋊C4D1212yes—
165C2×C42C22×D311yescharacteristic
166#107—64no—
167#106—68no—
168C4⋊C8—21no—
169C4⋊C8—21no—
170C4⋊C8—32no—
171#103—64no—
172#103—64no—
173#103—64no—
174C22⋊C8—61no—
175#97—21no—
176C4×C8—21no—
177C4×C8—32no—
178C4×C8—61no—
179C4×C8—61no—
180C23×C3#12111yescharacteristic
181C3×Q8—82no—
182C3×Q8—82no—
183C2×C3×C4—22no—
184C2×C3×C4—22no—
185C2×C3×C4—42no—
186C2×C3×C4—82no—
187C2×C3×C4#12011yescharacteristic
188C2×C3×C4#12011yescharacteristic
189C2×C3×C4#12011yescharacteristic
190C2×C3×C4#12111yescharacteristic
191C2×C3×C4#12111yescharacteristic
192C2×C3×C4#12711yescharacteristic
193C2×C3×C4#13512yes—
194C2×C3×C4#13712yes—
195C2×Dic3—41no—
196C2×Dic3—41no—
197C2×Dic3—42no—
198C2×Dic3—82no—
199Dic6—44no—
200Dic6—42no—
201Dic6—42no—
202Dic6—81no—
203Dic6—81no—
204C3×C8—42no—
205C3×C8—81no—
206C3⋊C8—24no—
207C3⋊C8—42no—
208C2×Q8—41no—
209C2×Q8—41no—
210C2×Q8—61no—
211C2×Q8—61no—
212C2×Q8—62no—
213C2×Q8—82no—
214C2×Q8—121no—
215C2×Q8—122no—
216C2×Q8—122no—
217C22×C4—22no—
218C22×C4—121no—
219C22×C4C2×D1211yescharacteristic
220C22×C4#23511yescharacteristic
221C22×C4C2×#6711yescharacteristic
222M16—22no—
223M16—22no—
224M16—42no—
225C2×C8—22no—
226C2×C8—41no—
227C2×C8—64no—
228C2×C8—62no—
229C2×C8—62no—
230C4⋊C4—22no—
231C4⋊C4—22no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4—62no—
235C4⋊C4—61no—
236C4⋊C4—62no—
237C4⋊C4—61no—
238C4⋊C4—61no—
239C4⋊C4—61no—
240C4⋊C4—121no—
241C4⋊C4—121no—
242C4⋊C4—121no—
243C4⋊C4—121no—
244C22⋊C4—121no—
245C42—121no—
246C42D3×D411yescharacteristic
247C42D3×D411yescharacteristic
248C42D3×D411yescharacteristic
249C42D3×D411yescharacteristic
250C22×C3—21no—
251C22×C3—21no—
252C22×C3#39211yescharacteristic
253C22×C3#46412yes—
254C3×C4—21no—
255C3×C4—21no—
256C3×C4—82no—
257C3×C4#45212yes—
258C3×C4#45612yes—
259C3×C4#47112yes—
260C3×C4#49511yescharacteristic
261C3×C4#49711yescharacteristic
262Dic3—81no—
263Dic3—82no—
264Dic3—81no—
265C23#93011yescharacteristic
266Q8—82no—
267Q8—82no—
268Q8—122no—
269Q8—122no—
270Q8—124no—
271Q8—241no—
272Q8—241no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—42no—
276C2×C4—82no—
277C2×C4—121no—
278C2×C4—121no—
279C2×C4—122no—
280C2×C4—242no—
281C2×C4#88211yescharacteristic
282C2×C4#88211yescharacteristic
283C2×C4#88411yescharacteristic
284C2×C4#88411yescharacteristic
285C2×C4#93711yescharacteristic
286C2×C4#94011yescharacteristic
287C2×C4#94212yes—
288C2×C4#95012yes—
289C8—42no—
290C8—64no—
291C8—81no—
292C8—122no—
293C2×C3—22no—
294C2×C3#200511yescharacteristic
295C2×C3#201511yescharacteristic
296C2×C3#201911yescharacteristic
297C22—21no—
298C22—21no—
299C22#496011yescharacteristic
300C22#502812yes—
301C4—21no—
302C4—21no—
303C4—82no—
304C4—242no—
305C4—241no—
306C4—241no—
307C4#474812yes—
308C4#475212yes—
309C4#504712yes—
310C4#505911yescharacteristic
311C4#508611yescharacteristic
312C3#1017311yescharacteristic, complemented, Sylow
313C2—22no—
314C2#6968911yescharacteristic
315C2#7009111yescharacteristic
316C2#7011911yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1997C2
socleC22×C3#392
prime radicalC23×C3#121
hypercenter#1997D3
hypocenterC3#10173
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1997

Complements

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

NH ≅ G/Ntype
312. C3#10173semidirect

Maximal subgroups

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

subgroupindex
#832242
C2×#50802
#785372
C3×#19972
#705062
#705012
#676672
#101733