Finite Group Databasebeta

← #10695099

Subgroups — #10695099

Order 768.

492 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 117 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#83290C211yesmaximal, characteristic
2C3×#1962C211yesmaximal, characteristic
3#71802C211yesmaximal, characteristic
4#70501C211yesmaximal, characteristic, complemented
5#70491C211yesmaximal, characteristic, complemented
6#69396C211yesmaximal, characteristic, complemented
7C4×#803C211yesmaximal, characteristic, complemented
8#9851—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10#5470C2211yescharacteristic
11C3×#423C2211yescharacteristic
12C2×C3×#104—22no—
13C3×C43C2211yescharacteristic
14#5093C411yescharacteristic, complemented
15#5080C411yescharacteristic, complemented
16#5076C411yescharacteristic, complemented
17#5070C411yescharacteristic, complemented
18#5041—22no—
19C4×#214—22no—
20#4924C2211yescharacteristic
21C2×#803—22no—
22C2×C4×C3⋊C8C2211yescharacteristic
23C2×C4×C3⋊C8C2211yescharacteristic
24D4⋊(C3⋊C8)—22no—
25#4523—22no—
26#4484—22no—
27#4479—22no—
28#3167—31no—
29#1994—31no—
30#1962D311yescharacteristic
31C4×#97—31no—
32#1567—31no—
33#1493—31no—
34#1492—31no—
35C2×C3×#41—21no—
36C2×C3×M16—21no—
37C3×C4⋊Q8C2×C411yescharacteristic
38C3×#127C2×C411yescharacteristic
39C3×#124—21no—
40C3×#121—21no—
41C3×#121—21no—
42C3×C4×D4—21no—
43C3×C4×D4—21no—
44C2×C3×C42—21no—
45C2×C3×C42—21no—
46C2×C3×C42D412yes—
47C2×C3×C42C2311yescharacteristic
48C2×#214—22no—
49#926—21no—
50#924—21no—
51#924—21no—
52C2×#208—21no—
53C22×C3⋊C8—21no—
54C22×C3⋊C8D412yes—
55#870—21no—
56#869C2×C411yescharacteristic
57#869C2×C411yescharacteristic
58C4×Dic6—21no—
59C4×Dic6—21no—
60C3×C4⋊C8—21no—
61C3×C4⋊C8—21no—
62C3×#104—44no—
63#832—22no—
64#810—22no—
65#808—22no—
66C4⋊(C3⋊C8)—21no—
67C4⋊(C3⋊C8)—21no—
68#803—28no—
69C4×C3⋊C8—21no—
70C4×C3⋊C8C2×C411yescharacteristic
71C4×C3⋊C8C2×C411yescharacteristic
72C4×C3⋊C8D412yes—
73C4×C3⋊C8D412yes—
74#533—31no—
75#532C2×D311yescharacteristic
76#498—31no—
77#492—31no—
78#487—31no—
79#486—31no—
80#465—62no—
81C4×SD16—62no—
82#423—31no—
83#423C2×D311yescharacteristic
84C2×#104—22no—
85C2×#97—62no—
86C2×C4×C8—31no—
87C2×C4×C8—31no—
88C43C2×D311yescharacteristic
89#339—62no—
90#335—62no—
91Q8⋊C8—62no—
92D4⋊C8—62no—
93C3×#41—42no—
94C3×#41—42no—
95C2×C3×Q8—21no—
96C2×C3×D4—21no—
97C2×C3×D4—21no—
98C2×C3×D4—21no—
99C2×C3×D4—41no—
100C22×C3×C4—22no—
101C22×C3×C4—41no—
102C22×C3×C4C2×D411yescharacteristic
103C22×C3×C4C2×D411yescharacteristic
104C22×C3×C4#4111yescharacteristic
105C2×Dic6—21no—
106C2×Dic6—21no—
107C3×M16—42no—
108C2×C3×C8—41no—
109C3×C4⋊C4—21no—
110C3×C4⋊C4—21no—
111C3×C22⋊C4—41no—
112C3×C22⋊C4—41no—
113C3×C42—24no—
114C3×C42—22no—
115C3×C42—22no—
116C3×C42—42no—
117C3×C42C22×C411yescharacteristic
118C3×C42C2×D411yescharacteristic
119C3×C42C2×D411yescharacteristic
120C3×C42#4111yescharacteristic
121#217—41no—
122#214—44no—
123C4⋊Dic3—21no—
124C4⋊Dic3—21no—
125C4⋊Dic3—21no—
126C4⋊Dic3—21no—
127Dic3⋊C4—41no—
128Dic3⋊C4—41no—
129C4×Dic3—41no—
130#208—42no—
131C2×C3⋊C8—24no—
132C2×C3⋊C8—22no—
133C2×C3⋊C8—24no—
134C2×C3⋊C8—41no—
135C2×C3⋊C8C22⋊C412yes—
136C2×C3⋊C8C22⋊C412yes—
137C2×#41—21no—
138C2×SD16—62no—
139C2×M16—21no—
140C2×M16—61no—
141C22×C8—32no—
142C22×C8—61no—
143C4⋊Q8—31no—
144C4⋊Q8—31no—
145C4⋊Q8C4×D311yescharacteristic
146#127C4×D311yescharacteristic
147#125—61no—
148#124—21no—
149C22⋊Q8—61no—
150C22⋊Q8—61no—
151#121—21no—
152#121—21no—
153C4×Q8—61no—
154C4×Q8—61no—
155C4×D4—21no—
156C4×D4—21no—
157#117—61no—
158C2×C42—21no—
159C2×C42—21no—
160C2×C42#6712yes—
161C2×C42C22×D311yescharacteristic
162#106—62no—
163C4⋊C8—21no—
164C4⋊C8—21no—
165C4⋊C8—61no—
166C4⋊C8—61no—
167#104—44no—
168#103—62no—
169#102—62no—
170#97—68no—
171C4×C8—31no—
172C4×C8—31no—
173C4×C8—32no—
174C4×C8—32no—
175C4×C8—61no—
176C23×C3—41no—
177C23×C3#12111yescharacteristic
178C3×Q8—42no—
179C3×D4—42no—
180C3×D4—42no—
181C3×D4—42no—
182C3×D4—82no—
183C2×C3×C4—22no—
184C2×C3×C4—22no—
185C2×C3×C4—24no—
186C2×C3×C4—42no—
187C2×C3×C4—42no—
188C2×C3×C4—41no—
189C2×C3×C4—81no—
190C2×C3×C4—81no—
191C2×C3×C4C2×C22⋊C411yescharacteristic
192C2×C3×C4C4×D411yescharacteristic
193C2×C3×C4C4×D411yescharacteristic
194C2×C3×C4#12111yescharacteristic
195C2×C3×C4#12411yescharacteristic
196C2×C3×C4#12711yescharacteristic
197C2×C3×C4AGL(1,ℤ/8)12yes—
198C2×C3×C4#13712yes—
199C2×Dic3—41no—
200C2×Dic3—41no—
201Dic6—42no—
202Dic6—42no—
203C3×C8—81no—
204C3⋊C8—24no—
205C3⋊C8—22no—
206C3⋊C8—81no—
207C3⋊C8#10414yescomplemented
208#41—42no—
209#41—42no—
210C2×Q8—21no—
211C2×Q8—61no—
212C2×Q8—61no—
213C2×D4—21no—
214C2×D4—21no—
215C2×D4—21no—
216C2×D4—41no—
217C22×C4—22no—
218C22×C4—41no—
219C22×C4D3×D411yescharacteristic
220C22×C4#23711yescharacteristic
221C22×C4C2×#6711yescharacteristic
222SD16—124no—
223M16—42no—
224M16—122no—
225C2×C8—32no—
226C2×C8—32no—
227C2×C8—41no—
228C2×C8—64no—
229C2×C8—64no—
230C2×C8—62no—
231C2×C8—121no—
232C4⋊C4—21no—
233C4⋊C4—21no—
234C4⋊C4—61no—
235C4⋊C4—61no—
236C4⋊C4—61no—
237C4⋊C4—61no—
238C4⋊C4—121no—
239C4⋊C4—121no—
240C22⋊C4—41no—
241C22⋊C4—41no—
242C22⋊C4—121no—
243C42—24no—
244C42—22no—
245C42—22no—
246C42—42no—
247C42—121no—
248C42C2×C4×D311yescharacteristic
249C42#23511yescharacteristic
250C42D3×D411yescharacteristic
251C42D3×D411yescharacteristic
252C22×C3—21no—
253C22×C3—21no—
254C22×C3—81no—
255C22×C3—81no—
256C22×C3—81no—
257C22×C3#39011yescharacteristic
258C22×C3#46812yes—
259C3×C4—21no—
260C3×C4—21no—
261C3×C4—44no—
262C3×C4—81no—
263C3×C4C2×#10412yes—
264C3×C4#44012yes—
265C3×C4#44412yes—
266C3×C4#49611yescharacteristic
267C3×C4#49711yescharacteristic
268Dic3—81no—
269Dic3—81no—
270C23—41no—
271C23#93911yescharacteristic
272Q8—42no—
273Q8—122no—
274Q8—122no—
275D4—42no—
276D4—42no—
277D4—42no—
278D4—82no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—24no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—41no—
285C2×C4—81no—
286C2×C4—81no—
287C2×C4—121no—
288C2×C4—121no—
289C2×C4D3×C22⋊C411yescharacteristic
290C2×C4#67⋊C411yescharacteristic
291C2×C4#88411yescharacteristic
292C2×C4#88511yescharacteristic
293C2×C4C4×#6711yescharacteristic
294C2×C4#93212yes—
295C2×C4#94011yescharacteristic
296C2×C4#95112yes—
297C8—34no—
298C8—64no—
299C8—62no—
300C8—81no—
301C8—241no—
302C2×C3—22no—
303C2×C3—161no—
304C2×C3#196211yescharacteristic
305C2×C3#197311yescharacteristic
306C2×C3#197511yescharacteristic
307C22—21no—
308C22—21no—
309C22—81no—
310C22—81no—
311C22—81no—
312C22#495711yescharacteristic
313C22#504212yes—
314C4—21no—
315C4—21no—
316C4—44no—
317C4—81no—
318C4—241no—
319C4—241no—
320C4D3×#10412yes—
321C4#482312yes—
322C4#501812yes—
323C4#506011yescharacteristic
324C4#507411yescharacteristic
325C3#985111yescharacteristic, complemented, Sylow
326C2—22no—
327C2—161no—
328C2#6964911yescharacteristic
329C2#7035611yescharacteristic
330C2#7037411yescharacteristic
subgroupG/H
centerC2×C4#940
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1962C2
socleC22×C3#390
prime radicalC3×#532C22
cosocleC2×C3×C42C23
hypercenter#1962D3
hypocenterC3#9851
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1962

Complements

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

NH ≅ G/Ntype
4. #70501C2semidirect
5. #70491C2semidirect
6. #69396C2semidirect
7. C4×#803C2semidirect
14. #5093C4semidirect
15. #5080C4semidirect
16. #5076C4semidirect
17. #5070C4semidirect
207. C3⋊C8#104semidirect
325. C3#9851semidirect

Maximal subgroups

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

subgroupindex
#832902
C3×#19622
#718022
#705012
#704912
#693962
C4×#8032
#98513