Finite Group Databasebeta

← #10697361

Subgroups — #10697361

Order 768.

474 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 99 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#83223C211yesmaximal, characteristic
2#83198C211yesmaximal, characteristic
3#79037C211yesmaximal, characteristic
4C3×#1988C211yesmaximal, characteristic
5#69493C211yesmaximal, characteristic
6#69471C211yesmaximal, characteristic
7#67706C211yesmaximal, characteristic
8#10207—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10#5470C2211yescharacteristic
11#5470C2211yescharacteristic
12C2×C3×C4⋊C8C2211yescharacteristic
13C3×C4×C4⋊C4C2211yescharacteristic
14#5225—22no—
15#5116—22no—
16#5094—22no—
17#5093—21no—
18#5082—22no—
19#5080—21no—
20#5079—21no—
21#5079—21no—
22#4971—22no—
23#4925C2211yescharacteristic
24C4×#208C2211yescharacteristic
25#4850—21no—
26#4850—21no—
27#4849—21no—
28#4849—21no—
29C3×#340—22no—
30#4523—22no—
31#4452—22no—
32#3249—31no—
33#3229—31no—
34#3139—31no—
35#1988D311yescharacteristic
36#1986—31no—
37#1503—31no—
38#1283—31no—
39C22×C3×Q8—21no—
40C22×C3×C8D412yes—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8D412yes—
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C42—21no—
49C2×C3×C42C2311yescharacteristic
50#926—21no—
51#926—21no—
52#924—21no—
53#924—21no—
54#924—21no—
55#924—21no—
56C2×#208D412yes—
57C2×Dic12—22no—
58#870—21no—
59#870—21no—
60#869D411yescharacteristic
61#869D411yescharacteristic
62#869D411yescharacteristic
63#869D411yescharacteristic
64C4×Dic6—21no—
65C4×Dic6—21no—
66C4×Dic6—21no—
67C4×Dic6—21no—
68C3×C4⋊C8—21no—
69C3×C4⋊C8—21no—
70C3×#95—21no—
71#835—24no—
72#835—24no—
73#830D412yes—
74#818—22no—
75#816—22no—
76#816—22no—
77#810—24no—
78#808—24no—
79C4⋊(C3⋊C8)—21no—
80#803—21no—
81C4×C3⋊C8—21no—
82C4×C3⋊C8—21no—
83#533—31no—
84#533—31no—
85C2×C4⋊Q8C2×D311yescharacteristic
86#498—61no—
87#491—62no—
88#487—61no—
89#487—61no—
90#487—61no—
91#483—62no—
92#477—61no—
93#477—61no—
94#475—62no—
95#467—62no—
96#462—61no—
97#462—61no—
98#431—31no—
99C2×C4⋊C8C2×D311yescharacteristic
100#419—62no—
101C4×M16—31no—
102C4×C4⋊C4C2×D311yescharacteristic
103#340—22no—
104#339—62no—
105#328—62no—
106C2×C3×Q8—22no—
107C2×C3×Q8—41no—
108C2×C3×Q8—42no—
109C2×C3×Q8—42no—
110C22×C3×C4—41no—
111C22×C3×C4—41no—
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C2×Dic6—21no—
116C2×Dic6—21no—
117C2×Dic6—21no—
118C2×Dic6—21no—
119C2×C3×C8—22no—
120C2×C3×C8—42no—
121C3×C4⋊C4—22no—
122C3×C4⋊C4—22no—
123C3×C4⋊C4—22no—
124C3×C4⋊C4—41no—
125C3×C4⋊C4—41no—
126C3×C4⋊C4—41no—
127C3×C42—42no—
128C3×C42C2×D411yescharacteristic
129C3×C42C2×D411yescharacteristic
130C3×C42C2×D411yescharacteristic
131C3×C42#4111yescharacteristic
132#217—41no—
133#217—41no—
134C4⋊Dic3—21no—
135C4⋊Dic3—21no—
136C4⋊Dic3—21no—
137C4⋊Dic3—21no—
138C4⋊Dic3—21no—
139C4⋊Dic3—21no—
140C4⋊Dic3—21no—
141C4⋊Dic3—21no—
142Dic3⋊C4—41no—
143Dic3⋊C4—41no—
144Dic3⋊C4—41no—
145Dic3⋊C4—41no—
146C4×Dic3—41no—
147C4×Dic3—41no—
148#208—22no—
149#208—22no—
150C2×C3⋊C8—22no—
151C2×C3⋊C8—22no—
152Dic12—82no—
153C22×Q8—21no—
154C2×Q16—62no—
155C2×M16—32no—
156C22×C8#6712yes—
157C4⋊Q8—21no—
158C4⋊Q8—21no—
159C4⋊Q8—21no—
160C4⋊Q8—31no—
161C4⋊Q8—31no—
162C4⋊Q8—31no—
163C4⋊Q8—31no—
164C4⋊Q8D1212yes—
165#125—61no—
166#125—61no—
167C22⋊Q8—61no—
168C22⋊Q8—61no—
169C22⋊Q8—61no—
170C22⋊Q8—61no—
171C4×Q8—61no—
172C4×Q8—61no—
173C4×Q8—61no—
174C4×Q8—61no—
175#117—61no—
176#117—61no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C42—21no—
181C2×C42C22×D311yescharacteristic
182#107—62no—
183#106—64no—
184C4⋊C8—21no—
185C4⋊C8—21no—
186C4⋊C8—61no—
187#103—64no—
188#103—62no—
189#103—62no—
190#103—64no—
191#103—64no—
192C22⋊C8—32no—
193#97—61no—
194C4×C8—61no—
195C4×C8—61no—
196#95—21no—
197C23×C3#12711yescharacteristic
198C3×Q8—44no—
199C3×Q8—82no—
200C2×C3×C4—21no—
201C2×C3×C4—21no—
202C2×C3×C4—41no—
203C2×C3×C4—42no—
204C2×C3×C4—42no—
205C2×C3×C4—81no—
206C2×C3×C4—81no—
207C2×C3×C4—81no—
208C2×C3×C4—81no—
209C2×C3×C4#12011yescharacteristic
210C2×C3×C4#12011yescharacteristic
211C2×C3×C4#12011yescharacteristic
212C2×C3×C4#12111yescharacteristic
213C2×C3×C4#12111yescharacteristic
214C2×C3×C4#12111yescharacteristic
215C2×C3×C4#13512yes—
216C2×C3×C4#13712yes—
217C2×Dic3—41no—
218C2×Dic3—41no—
219C2×Dic3—41no—
220C2×Dic3—41no—
221Dic6—42no—
222Dic6—41no—
223Dic6—41no—
224Dic6—41no—
225Dic6—41no—
226Dic6—81no—
227C3×C8—42no—
228C3⋊C8—42no—
229C3⋊C8—42no—
230C2×Q8—22no—
231C2×Q8—41no—
232C2×Q8—42no—
233C2×Q8—42no—
234C2×Q8—61no—
235C2×Q8—61no—
236C2×Q8—61no—
237C2×Q8—61no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4D3×D411yescharacteristic
241C22×C4D3×D411yescharacteristic
242C22×C4C2×#6711yescharacteristic
243Q16—242no—
244M16—62no—
245M16—62no—
246C2×C8—22no—
247C2×C8—42no—
248C2×C8—62no—
249C2×C8—62no—
250C4⋊C4—22no—
251C4⋊C4—22no—
252C4⋊C4—22no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—61no—
257C4⋊C4—61no—
258C4⋊C4—61no—
259C4⋊C4—61no—
260C4⋊C4—61no—
261C4⋊C4—61no—
262C4⋊C4—61no—
263C4⋊C4—61no—
264C4⋊C4—121no—
265C4⋊C4—121no—
266C4⋊C4—121no—
267C4⋊C4—121no—
268C22⋊C4—121no—
269C22⋊C4—121no—
270C42—42no—
271C42—121no—
272C42—121no—
273C42C2×D1211yescharacteristic
274C42#23511yescharacteristic
275C42D3×D411yescharacteristic
276C42D3×D411yescharacteristic
277C22×C3—21no—
278C22×C3—21no—
279C22×C3#39211yescharacteristic
280C22×C3#49511yescharacteristic
281C22×C3#49711yescharacteristic
282C3×C4—21no—
283C3×C4—21no—
284C3×C4—21no—
285C3×C4—82no—
286C3×C4—82no—
287C3×C4#45212yes—
288C3×C4#45511yescharacteristic
289C3×C4#45611yescharacteristic
290C3×C4#46412yes—
291Dic3—81no—
292Dic3—81no—
293Dic3—81no—
294Dic3—81no—
295C23#94011yescharacteristic
296Q8—44no—
297Q8—82no—
298Q8—122no—
299Q8—121no—
300Q8—121no—
301Q8—121no—
302Q8—121no—
303Q8—241no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—41no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—121no—
314C2×C4—121no—
315C2×C4—121no—
316C2×C4—121no—
317C2×C4#88211yescharacteristic
318C2×C4#88211yescharacteristic
319C2×C4#88411yescharacteristic
320C2×C4#88411yescharacteristic
321C2×C4#91212yes—
322C2×C4#91512yes—
323C2×C4#93011yescharacteristic
324C2×C4#93711yescharacteristic
325C8—42no—
326C8—122no—
327C8—122no—
328C2×C3—21no—
329C2×C3—21no—
330C2×C3#200711yescharacteristic
331C2×C3#200911yescharacteristic
332C2×C3#201511yescharacteristic
333C22—21no—
334C22—21no—
335C22#496011yescharacteristic
336C22#515511yescharacteristic
337C22#516711yescharacteristic
338C4—21no—
339C4—21no—
340C4—21no—
341C4—82no—
342C4—82no—
343C4—241no—
344C4—241no—
345C4—241no—
346C4—241no—
347C4#476712yes—
348C4#478312yes—
349C4#506111yescharacteristic
350C4#509511yescharacteristic
351C3#1020711yescharacteristic, complemented, Sylow
352C2—21no—
353C2—21no—
354C2#6969411yescharacteristic
355C2#7021511yescharacteristic
356C2#7035111yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1988C2
socleC22×C3#392
prime radicalC23×C3#127
hypercenter#1988D3
hypocenterC3#10207
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1988

Complements

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

NH ≅ G/Ntype
351. C3#10207semidirect

Maximal subgroups

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

subgroupindex
#832232
#831982
#790372
C3×#19882
#694932
#694712
#677062
#102073