Finite Group Databasebeta

← #10691863

Subgroups — #10691863

Order 768.

426 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 113 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#83329C211yesmaximal, characteristic
2#83323C211yesmaximal, characteristic
3C3×#1518C211yesmaximal, characteristic
4#70778C211yesmaximal, characteristic
5#69552C211yesmaximal, characteristic
6#67740C211yesmaximal, characteristic
7#67631C211yesmaximal, characteristic
8#9292—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×#869C2211yescharacteristic
11C3×#431C2211yescharacteristic
12C3×#382C2211yescharacteristic
13#5080C412yescomplemented
14#5079C412yescomplemented
15#5043—22no—
16#5024—22no—
17#5024—22no—
18#5020—22no—
19#4999—21no—
20#4997—21no—
21C2×#810—21no—
22C2×#810—21no—
23#4925C2211yescharacteristic
24#4924C2211yescharacteristic
25C2×#803C2211yescharacteristic
26C3×#328—22no—
27#4486—22no—
28#3229—31no—
29#2961—31no—
30#1960—31no—
31#1546—31no—
32#1518D311yescharacteristic
33#1286—31no—
34#1275—31no—
35C22×Dic6—21no—
36C3×C4⋊Q8C2×C411yescharacteristic
37C3×C4⋊Q8C2×C411yescharacteristic
38C3×#125—21no—
39C3×C22⋊Q8—21no—
40C3×C22⋊Q8D412yes—
41C3×C4×Q8—21no—
42C3×C4×Q8D412yes—
43C3×#117—21no—
44C2×C3×C4⋊C4—21no—
45C2×C3×C42C2311yescharacteristic
46C2×C4⋊Dic3—21no—
47C2×C4⋊Dic3—21no—
48C2×#208—21no—
49C22×C3⋊C8D411yescharacteristic
50C22×C3⋊C8D411yescharacteristic
51#869—21no—
52#869—21no—
53#869—21no—
54#869C2×C412yes—
55C3×C4⋊C8—21no—
56C3×C22⋊C8—21no—
57C3×C4×C8—21no—
58C3×#95—21no—
59C3×#95—21no—
60#835—24no—
61#830—21no—
62#810—42no—
63#810—42no—
64#808—24no—
65C4⋊(C3⋊C8)—24no—
66C4⋊(C3⋊C8)—21no—
67C4⋊(C3⋊C8)—21no—
68C4⋊(C3⋊C8)D412yes—
69#803—22no—
70#803—21no—
71#803—21no—
72C4×C3⋊C8—22no—
73C4×C3⋊C8C2×C412yes—
74#533C2×D311yescharacteristic
75C2×C4⋊Q8—31no—
76#498—32no—
77#487—32no—
78#483—62no—
79#475—62no—
80#446—62no—
81#446—62no—
82#431—31no—
83#431C2×D311yescharacteristic
84#424—61no—
85#423—31no—
86C2×#103—61no—
87C2×#103—61no—
88#405—61no—
89C2×#97—31no—
90#382C2×D311yescharacteristic
91#332—62no—
92#328—22no—
93C2×C3×Q8—21no—
94C2×C3×Q8C22⋊C412yes—
95C22×C3×C4—41no—
96C22×C3×C4C2×D411yescharacteristic
97C22×C3×C4C2×D411yescharacteristic
98C22×C3×C4#4111yescharacteristic
99C22×Dic3—41no—
100C2×Dic6—22no—
101C2×Dic6—41no—
102C2×Dic6—41no—
103C2×Dic6—41no—
104C2×Dic6—41no—
105C2×Dic6—41no—
106C2×C3×C8—41no—
107C3×C4⋊C4—21no—
108C3×C4⋊C4—21no—
109C3×C4⋊C4—21no—
110C3×C4⋊C4—22no—
111C3×C4⋊C4—22no—
112C3×C4⋊C4—41no—
113C3×C4⋊C4—41no—
114C3×C4⋊C4C22⋊C412yes—
115C3×C22⋊C4—22no—
116C3×C42—22no—
117C3×C42C22×C411yescharacteristic
118C3×C42C2×D411yescharacteristic
119C3×C42C2×D411yescharacteristic
120C3×C42#4111yescharacteristic
121C4⋊Dic3—22no—
122C4⋊Dic3—22no—
123C4⋊Dic3—41no—
124C4⋊Dic3—41no—
125#208—41no—
126#208—41no—
127C2×C3⋊C8—21no—
128C2×C3⋊C8—21no—
129C2×C3⋊C8—22no—
130C2×C3⋊C8—22no—
131C2×C3⋊C8—21no—
132C2×C3⋊C8—21no—
133C2×C3⋊C8—21no—
134C2×C3⋊C8SD1612yes—
135C22×Q8—61no—
136C2×M16—61no—
137C22×C8—31no—
138C22×C8—31no—
139C4⋊Q8—32no—
140C4⋊Q8—61no—
141C4⋊Q8—61no—
142C4⋊Q8—61no—
143C4⋊Q8C4×D311yescharacteristic
144C4⋊Q8C4×D311yescharacteristic
145#125—21no—
146C22⋊Q8—21no—
147C22⋊Q8#6712yes—
148C4×Q8—21no—
149C4×Q8D1212yes—
150#117—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—61no—
153C2×C4⋊C4—61no—
154C2×C42C22×D311yescharacteristic
155#106—64no—
156C4⋊C8—21no—
157C4⋊C8—32no—
158C4⋊C8—64no—
159C4⋊C8—61no—
160C4⋊C8—61no—
161#103—64no—
162#103—122no—
163#103—122no—
164C22⋊C8—21no—
165C22⋊C8—61no—
166#97—62no—
167#97—61no—
168#97—61no—
169C4×C8—21no—
170C4×C8—32no—
171C4×C8—62no—
172#95—21no—
173#95—21no—
174C23×C3#12111yescharacteristic
175C3×Q8—42no—
176C3×Q8—81no—
177C2×C3×C4—22no—
178C2×C3×C4—22no—
179C2×C3×C4—41no—
180C2×C3×C4—41no—
181C2×C3×C4—81no—
182C2×C3×C4—81no—
183C2×C3×C4—81no—
184C2×C3×C4C2×C22⋊C411yescharacteristic
185C2×C3×C4C4×D411yescharacteristic
186C2×C3×C4C4×D411yescharacteristic
187C2×C3×C4#12011yescharacteristic
188C2×C3×C4#12111yescharacteristic
189C2×C3×C4#12311yescharacteristic
190C2×C3×C4C2×SD1611yescharacteristic
191C2×C3×C4#13511yescharacteristic
192C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
193C2×C3×C4#13711yescharacteristic
194C2×Dic3—42no—
195C2×Dic3—81no—
196C2×Dic3—81no—
197Dic6—42no—
198Dic6—42no—
199Dic6—81no—
200Dic6—81no—
201C3×C8—81no—
202C3⋊C8—22no—
203C3⋊C8—42no—
204C3⋊C8—41no—
205C3⋊C8—42no—
206C2×Q8—21no—
207C2×Q8—62no—
208C2×Q8—121no—
209C2×Q8—121no—
210C2×Q8—121no—
211C2×Q8—121no—
212C2×Q8—121no—
213C2×Q8(C2×D3)⋊C412yes—
214C22×C4—41no—
215C22×C4—121no—
216C22×C4D3×D411yescharacteristic
217C22×C4#23711yescharacteristic
218C22×C4C2×#6711yescharacteristic
219M16—121no—
220M16—121no—
221C2×C8—32no—
222C2×C8—41no—
223C2×C8—61no—
224C2×C8—61no—
225C2×C8—62no—
226C2×C8—61no—
227C2×C8—61no—
228C2×C8—61no—
229C2×C8—62no—
230C4⋊C4—21no—
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4—22no—
234C4⋊C4—22no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—62no—
238C4⋊C4—62no—
239C4⋊C4—121no—
240C4⋊C4—121no—
241C4⋊C4(C2×D3)⋊C412yes—
242C22⋊C4—22no—
243C42—22no—
244C42C2×C4×D311yescharacteristic
245C42C2×D1211yescharacteristic
246C42D3×D411yescharacteristic
247C42#23711yescharacteristic
248C22×C3—21no—
249C22×C3—21no—
250C22×C3#38611yescharacteristic
251C22×C3#46011yescharacteristic
252C22×C3#46411yescharacteristic
253C3×C4—21no—
254C3×C4—21no—
255C3×C4—21no—
256C3×C4—21no—
257C3×C4—42no—
258C3×C4—42no—
259C3×C4—161no—
260C3×C4C4×SD1611yescharacteristic
261C3×C4#44011yescharacteristic
262C3×C4#44811yescharacteristic
263C3×C4#44911yescharacteristic
264Dic3—82no—
265C23#93911yescharacteristic
266Q8—42no—
267Q8—81no—
268Q8—122no—
269Q8—122no—
270Q8—241no—
271Q8—241no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4—122no—
280C2×C4—241no—
281C2×C4—241no—
282C2×C4#67⋊C411yescharacteristic
283C2×C4#67⋊C411yescharacteristic
284C2×C4#88211yescharacteristic
285C2×C4#88611yescharacteristic
286C2×C4D3×SD1611yescharacteristic
287C2×C4#91411yescharacteristic
288C2×C4#91811yescharacteristic
289C2×C4#91911yescharacteristic
290C2×C4C2×(C2×D3)⋊C411yescharacteristic
291C2×C4#93911yescharacteristic
292C8—62no—
293C8—81no—
294C8—122no—
295C8—121no—
296C8—122no—
297C2×C3—21no—
298C2×C3—21no—
299C2×C3#188011yescharacteristic
300C2×C3#190211yescharacteristic
301C2×C3#190911yescharacteristic
302C22—21no—
303C22—21no—
304C22#495811yescharacteristic
305C22#516311yescharacteristic
306C22#518511yescharacteristic
307C4—21no—
308C4—21no—
309C4—21no—
310C4—21no—
311C4—42no—
312C4—42no—
313C4—161no—
314C4—242no—
315C4#478811yescharacteristic
316C4#480611yescharacteristic
317C4#481011yescharacteristic
318C4#481711yescharacteristic
319C3#929211yescharacteristic, complemented, Sylow
320C2—21no—
321C2—21no—
322C2#7004611yescharacteristic
323C2#7024011yescharacteristic
324C2#7039911yescharacteristic
subgroupG/H
centerC22#4958
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1518C2
socleC22×C3#386
prime radicalC23×C3#121
cosocleC2×C3×C42C23
hypercenter#1518D3
hypocenterC3#9292
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1518

Complements

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

NH ≅ G/Ntype
13. #5080C4semidirect
14. #5079C4semidirect
319. C3#9292semidirect

Maximal subgroups

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

subgroupindex
#833292
#833232
C3×#15182
#707782
#695522
#677402
#676312
#92923