Finite Group Databasebeta

← #10697007

Subgroups — #10697007

Order 768.

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

normal subgroups only

This group has 498 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#83283C211yesmaximal, characteristic
2#83271C211yesmaximal, characteristic
3C2×#4681C211yesmaximal, characteristic, complemented
4C3×#1492C211yesmaximal, characteristic
5#70524C211yesmaximal, characteristic, complemented
6#70508C211yesmaximal, characteristic, complemented
7#67656C211yesmaximal, characteristic, complemented
8#10156—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×#869C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#423C2211yescharacteristic
13C2×C3×C4×C8C2211yescharacteristic
14#5232—22no—
15C2×#816—22no—
16#5080—21no—
17#5079—21no—
18#5070—21no—
19#5070—21no—
20#5055—22no—
21#5034—22no—
22#5008—22no—
23#4925C2211yescharacteristic
24C4×#208C2211yescharacteristic
25#4681—22no—
26#4681—22no—
27C3×D4⋊C8—22no—
28#4554—22no—
29#4486—22no—
30#3227—31no—
31#3141—31no—
32C2×#487—31no—
33#1997—31no—
34#1549—31no—
35#1492D311yescharacteristic
36#1278—31no—
37C2×C3×#41—21no—
38C22×Dic6—21no—
39C22×Dic6—21no—
40C2×C3×M16—21no—
41C22×C3×C8—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
50C2×C4⋊Dic3—21no—
51C2×C4⋊Dic3—21no—
52C2×C4⋊Dic3—21no—
53C2×C4⋊Dic3D412yes—
54C2×#208D412yes—
55#869—21no—
56#869—21no—
57#869—21no—
58#869—21no—
59#869—21no—
60#869—21no—
61#869—21no—
62#869D412yes—
63C3×C4⋊C8—21no—
64C3×C4⋊C8—21no—
65C3×C4×C8—21no—
66C3×C4×C8D412yes—
67#835—24no—
68#832—24no—
69#832—24no—
70#830—21no—
71#816—28no—
72#808—24no—
73C4⋊(C3⋊C8)D412yes—
74#803—21no—
75C4×C3⋊C8—21no—
76C4×C3⋊C8—21no—
77#532C2×D311yescharacteristic
78C2×C4⋊Q8—31no—
79C2×C4⋊Q8—31no—
80#498—61no—
81#489—62no—
82#487—62no—
83#487—61no—
84#487—62no—
85#486—61no—
86#486—61no—
87#481—62no—
88#476—62no—
89#431—31no—
90#423C2×D311yescharacteristic
91#417—62no—
92C2×#103—62no—
93C4×M16—31no—
94C2×C4×C8C2×D311yescharacteristic
95#343—62no—
96#332—62no—
97D4⋊C8—22no—
98C3×#41—42no—
99C3×#41—42no—
100C2×C3×Q8—21no—
101C2×C3×D4—21no—
102C2×C3×D4—21no—
103C2×C3×D4—21no—
104C2×C3×D4—41no—
105C22×C3×C4—41no—
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4#4111yescharacteristic
109C22×Dic3—22no—
110C22×Dic3—41no—
111C2×Dic6—22no—
112C2×Dic6—44no—
113C2×Dic6—41no—
114C2×Dic6—41no—
115C2×Dic6—41no—
116C2×Dic6—82no—
117C3×M16—42no—
118C2×C3×C8—22no—
119C2×C3×C8—42no—
120C2×C3×C8—41no—
121C3×C4⋊C4—21no—
122C3×C4⋊C4—21no—
123C3×C22⋊C4—41no—
124C3×C22⋊C4—41no—
125C3×C42C2×D411yescharacteristic
126C3×C42C2×D411yescharacteristic
127C3×C42C2×D411yescharacteristic
128C3×C42C2×D411yescharacteristic
129C4⋊Dic3—22no—
130C4⋊Dic3—22no—
131C4⋊Dic3—22no—
132C4⋊Dic3—22no—
133C4⋊Dic3—41no—
134C4⋊Dic3—41no—
135C4⋊Dic3—41no—
136C4⋊Dic3—41no—
137#208—22no—
138#208—22no—
139C2×C3⋊C8—22no—
140C2×C3⋊C8—22no—
141C2×#41—21no—
142C22×Q8—61no—
143C22×Q8—61no—
144C2×M16—21no—
145C2×M16—32no—
146C22×C8—21no—
147C4⋊Q8—32no—
148C4⋊Q8—61no—
149C4⋊Q8—61no—
150C4⋊Q8—61no—
151C4⋊Q8—61no—
152C4⋊Q8—61no—
153C4⋊Q8—61no—
154C4⋊Q8—61no—
155C4⋊Q8D1211yescharacteristic
156#127D1211yescharacteristic
157#124—21no—
158#121—21no—
159#121—21no—
160C4×D4—21no—
161C4×D4—21no—
162C2×C4⋊C4—32no—
163C2×C4⋊C4—61no—
164C2×C4⋊C4—61no—
165C2×C4⋊C4—61no—
166C2×C42C22×D311yescharacteristic
167#106—64no—
168C4⋊C8—21no—
169C4⋊C8—21no—
170C4⋊C8—32no—
171#103—68no—
172#103—64no—
173#102—64no—
174#102—64no—
175C22⋊C8—61no—
176#97—61no—
177C4×C8—21no—
178C4×C8—61no—
179C4×C8—61no—
180C4×C8#6712yes—
181C23×C3—41no—
182C23×C3#12111yescharacteristic
183C3×Q8—42no—
184C3×D4—42no—
185C3×D4—42no—
186C3×D4—42no—
187C3×D4—82no—
188C2×C3×C4—22no—
189C2×C3×C4—22no—
190C2×C3×C4—41no—
191C2×C3×C4—81no—
192C2×C3×C4—81no—
193C2×C3×C4#12011yescharacteristic
194C2×C3×C4#12011yescharacteristic
195C2×C3×C4#12011yescharacteristic
196C2×C3×C4#12111yescharacteristic
197C2×C3×C4#12111yescharacteristic
198C2×C3×C4#12711yescharacteristic
199C2×C3×C4#13512yes—
200C2×C3×C4#13712yes—
201C2×Dic3—42no—
202C2×Dic3—42no—
203C2×Dic3—82no—
204C2×Dic3—82no—
205Dic6—44no—
206Dic6—82no—
207Dic6—82no—
208Dic6—82no—
209C3×C8—42no—
210C3×C8—81no—
211C3⋊C8—42no—
212C3⋊C8—42no—
213#41—42no—
214#41—42no—
215C2×Q8—21no—
216C2×Q8—62no—
217C2×Q8—124no—
218C2×Q8—121no—
219C2×Q8—121no—
220C2×Q8—121no—
221C2×Q8—242no—
222C2×D4—21no—
223C2×D4—21no—
224C2×D4—21no—
225C2×D4—41no—
226C22×C4—41no—
227C22×C4—62no—
228C22×C4—121no—
229C22×C4#23511yescharacteristic
230C22×C4D3×D411yescharacteristic
231C22×C4D3×D411yescharacteristic
232M16—42no—
233M16—62no—
234M16—62no—
235C2×C8—22no—
236C2×C8—42no—
237C2×C8—41no—
238C2×C8—62no—
239C2×C8—62no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—62no—
243C4⋊C4—62no—
244C4⋊C4—62no—
245C4⋊C4—62no—
246C4⋊C4—121no—
247C4⋊C4—121no—
248C4⋊C4—121no—
249C4⋊C4—121no—
250C22⋊C4—41no—
251C22⋊C4—41no—
252C42C2×D1211yescharacteristic
253C42D3×D411yescharacteristic
254C42D3×D411yescharacteristic
255C42C2×#6711yescharacteristic
256C22×C3—21no—
257C22×C3—21no—
258C22×C3—81no—
259C22×C3—81no—
260C22×C3—81no—
261C22×C3#39211yescharacteristic
262C22×C3#46412yes—
263C3×C4—21no—
264C3×C4—21no—
265C3×C4—81no—
266C3×C4#45212yes—
267C3×C4#45412yes—
268C3×C4#47112yes—
269C3×C4#49511yescharacteristic
270C3×C4#49711yescharacteristic
271Dic3—82no—
272Dic3—82no—
273C23—41no—
274C23#88411yescharacteristic
275Q8—42no—
276Q8—124no—
277Q8—242no—
278Q8—242no—
279Q8—242no—
280D4—42no—
281D4—42no—
282D4—42no—
283D4—82no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—41no—
287C2×C4—81no—
288C2×C4—81no—
289C2×C4—122no—
290C2×C4—122no—
291C2×C4—242no—
292C2×C4—242no—
293C2×C4#88211yescharacteristic
294C2×C4#88211yescharacteristic
295C2×C4#88411yescharacteristic
296C2×C4#91212yes—
297C2×C4#91512yes—
298C2×C4#93011yescharacteristic
299C2×C4#93711yescharacteristic
300C2×C4#94011yescharacteristic
301C8—42no—
302C8—81no—
303C8—122no—
304C8—122no—
305C2×C3—22no—
306C2×C3—161no—
307C2×C3#200311yescharacteristic
308C2×C3#201511yescharacteristic
309C2×C3#201911yescharacteristic
310C22—21no—
311C22—21no—
312C22—81no—
313C22—81no—
314C22—81no—
315C22#476712yes—
316C22#496011yescharacteristic
317C4—21no—
318C4—21no—
319C4—81no—
320C4—242no—
321C4—242no—
322C4#478312yes—
323C4#482712yes—
324C4#514112yes—
325C4#515511yescharacteristic
326C4#516711yescharacteristic
327C3#1015611yescharacteristic, complemented, Sylow
328C2—22no—
329C2—161no—
330C2#7000711yescharacteristic
331C2#7021511yescharacteristic
332C2#7038811yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1492C2
socleC22×C3#392
prime radicalC3×#532C22
hypercenter#1492D3
hypocenterC3#10156
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1492

Complements

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

NH ≅ G/Ntype
3. C2×#4681C2semidirect
5. #70524C2semidirect
6. #70508C2semidirect
7. #67656C2semidirect
327. C3#10156semidirect

Maximal subgroups

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

subgroupindex
#832832
#832712
C2×#46812
C3×#14922
#705242
#705082
#676562
#101563