Finite Group Databasebeta

← #10691862

Subgroups — #10691862

Order 768.

426 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 107 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#79045C211yesmaximal, characteristic
3C3×#1546C211yesmaximal, characteristic
4#71786C211yesmaximal, characteristic
5#70793C211yesmaximal, characteristic
6#67740C211yesmaximal, characteristic
7#67713C211yesmaximal, characteristic
8#9292—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10#5470C2211yescharacteristic
11C3×#431C2211yescharacteristic
12C3×#423C2211yescharacteristic
13C2×#835—21no—
14C2×#835—21no—
15#5122—21no—
16#5119—21no—
17#5094—22no—
18#5093C412yescomplemented
19#5080C412yescomplemented
20#4971—22no—
21#4936C2211yescharacteristic
22#4925C2211yescharacteristic
23C2×#803C2211yescharacteristic
24#4828—22no—
25#4828—22no—
26C3×#332—22no—
27#4467—22no—
28#3229—31no—
29#2961—31no—
30#1960—31no—
31#1546D311yescharacteristic
32#1518—31no—
33#1286—31no—
34#1275—31no—
35C22×C3×Q8—21no—
36C2×C3×M16—21no—
37C3×C4⋊Q8—21no—
38C3×C4⋊Q8—21no—
39C3×C4⋊Q8—21no—
40C3×C4⋊Q8C2×C412yes—
41C2×C3×C4⋊C4—21no—
42C2×C3×C4⋊C4—21no—
43C2×C3×C42C2311yescharacteristic
44#926—21no—
45C2×C4⋊Dic3—21no—
46#924—21no—
47#924D412yes—
48C22×C3⋊C8D411yescharacteristic
49C22×C3⋊C8D411yescharacteristic
50#870—21no—
51#869C2×C411yescharacteristic
52#869C2×C411yescharacteristic
53C4×Dic6—21no—
54C4×Dic6D412yes—
55C3×C4⋊C8—21no—
56C3×C4⋊C8—21no—
57C3×C4⋊C8D412yes—
58C3×C22⋊C8—21no—
59C3×C4×C8—21no—
60#835—42no—
61#835—42no—
62C3⋊#95—21no—
63C3⋊#95—21no—
64#830—21no—
65#815—22no—
66Dic3⋊C8—24no—
67C8×Dic3—22no—
68#810—24no—
69#808—24no—
70C4⋊(C3⋊C8)—21no—
71#803—21no—
72#803—21no—
73C4×C3⋊C8C2×C412yes—
74#533—31no—
75C2×C4⋊Q8C2×D311yescharacteristic
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#423C2×D311yescharacteristic
86C2×#103—61no—
87C2×#103—61no—
88#405—61no—
89C2×#97—31no—
90#382—31no—
91#332—22no—
92#328—62no—
93C2×C3×Q8—22no—
94C2×C3×Q8—41no—
95C2×C3×Q8—41no—
96C2×C3×Q8—41no—
97C2×C3×Q8—41no—
98C2×C3×Q8—41no—
99C22×C3×C4—41no—
100C22×C3×C4C2×D411yescharacteristic
101C22×C3×C4C2×D411yescharacteristic
102C22×C3×C4#4111yescharacteristic
103C22×Dic3—41no—
104C2×Dic6—21no—
105C2×Dic6C22⋊C412yes—
106C3×M16—41no—
107C3×M16—41no—
108C2×C3×C8—22no—
109C2×C3×C8—41no—
110C3×C4⋊C4—22no—
111C3×C4⋊C4—22no—
112C3×C4⋊C4—41no—
113C3×C4⋊C4—41no—
114C3×C42C22×C411yescharacteristic
115C3×C42C2×D411yescharacteristic
116C3×C42C2×D411yescharacteristic
117C3×C42#4111yescharacteristic
118#217—22no—
119C4⋊Dic3—21no—
120C4⋊Dic3—21no—
121C4⋊Dic3—21no—
122C4⋊Dic3—41no—
123C4⋊Dic3—41no—
124C4⋊Dic3C22⋊C412yes—
125Dic3⋊C4—22no—
126Dic3⋊C4—22no—
127C4×Dic3—22no—
128C2×C3⋊C8—21no—
129C2×C3⋊C8—21no—
130C2×C3⋊C8—22no—
131C2×C3⋊C8—21no—
132C2×C3⋊C8—21no—
133C2×C3⋊C8—21no—
134C2×C3⋊C8SD1612yes—
135C22×Q8—21no—
136C2×M16—21no—
137C22×C8—31no—
138C22×C8—31no—
139C4⋊Q8—21no—
140C4⋊Q8—21no—
141C4⋊Q8—21no—
142C4⋊Q8—31no—
143C4⋊Q8—31no—
144C4⋊Q8C4×D312yes—
145#125—61no—
146C22⋊Q8—32no—
147C22⋊Q8—61no—
148C4×Q8—32no—
149C4×Q8—61no—
150#117—61no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—61no—
154C2×C42C22×D311yescharacteristic
155#106—64no—
156C4⋊C8—21no—
157C4⋊C8—21no—
158C4⋊C8—64no—
159C4⋊C8—61no—
160C4⋊C8#6712yes—
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—61no—
173#95—61no—
174C23×C3#12111yescharacteristic
175C3×Q8—42no—
176C3×Q8—42no—
177C3×Q8—81no—
178C3×Q8—81no—
179C2×C3×C4—41no—
180C2×C3×C4—41no—
181C2×C3×C4—42no—
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—22no—
195C2×Dic3—22no—
196C2×Dic3—81no—
197C2×Dic3—81no—
198C2×Dic3—81no—
199Dic6—42no—
200Dic6—81no—
201C3×C8—42no—
202C3×C8—81no—
203C3⋊C8—22no—
204C3⋊C8—42no—
205C3⋊C8—41no—
206C2×Q8—22no—
207C2×Q8—32no—
208C2×Q8—41no—
209C2×Q8—41no—
210C2×Q8—41no—
211C2×Q8—41no—
212C2×Q8—41no—
213C2×Q8—61no—
214C22×C4—41no—
215C22×C4—121no—
216C22×C4#23511yescharacteristic
217C22×C4D3×D411yescharacteristic
218C22×C4D3×D411yescharacteristic
219M16—41no—
220M16—41no—
221C2×C8—22no—
222C2×C8—32no—
223C2×C8—41no—
224C2×C8—61no—
225C2×C8—61no—
226C2×C8—62no—
227C2×C8—61no—
228C2×C8—61no—
229C2×C8—61no—
230C4⋊C4—22no—
231C4⋊C4—22no—
232C4⋊C4—32no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—61no—
236C4⋊C4—61no—
237C4⋊C4—62no—
238C4⋊C4—61no—
239C4⋊C4—62no—
240C4⋊C4—121no—
241C4⋊C4—121no—
242C22⋊C4—62no—
243C42—62no—
244C42C2×C4×D311yescharacteristic
245C42D3×D411yescharacteristic
246C42#23711yescharacteristic
247C42C2×#6711yescharacteristic
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—82no—
258C3×C4C4×SD1611yescharacteristic
259C3×C4#44011yescharacteristic
260C3×C4#44811yescharacteristic
261C3×C4#44911yescharacteristic
262Dic3—42no—
263Dic3—42no—
264Dic3—161no—
265C23#88411yescharacteristic
266Q8—42no—
267Q8—42no—
268Q8—81no—
269Q8—81no—
270Q8—122no—
271Q8—241no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—62no—
276C2×C4—62no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4—241no—
280C2×C4—241no—
281C2×C4—241no—
282C2×C4D3×C22⋊C411yescharacteristic
283C2×C4#67⋊C411yescharacteristic
284C2×C4#88311yescharacteristic
285C2×C4#91111yescharacteristic
286C2×C4#91211yescharacteristic
287C2×C4D3×SD1611yescharacteristic
288C2×C4#91511yescharacteristic
289C2×C4C4×#6711yescharacteristic
290C2×C4#93711yescharacteristic
291C2×C4#93811yescharacteristic
292C8—42no—
293C8—62no—
294C8—81no—
295C8—122no—
296C8—121no—
297C2×C3—21no—
298C2×C3—21no—
299C2×C3#188011yescharacteristic
300C2×C3#190211yescharacteristic
301C2×C3#190911yescharacteristic
302C22—21no—
303C22—21no—
304C22#476211yescharacteristic
305C22#476711yescharacteristic
306C22#495611yescharacteristic
307C4—21no—
308C4—21no—
309C4—21no—
310C4—21no—
311C4—82no—
312C4—122no—
313C4—122no—
314C4—481no—
315C4#475711yescharacteristic
316C4#475811yescharacteristic
317C4#515811yescharacteristic
318C4#517011yescharacteristic
319C3#929211yescharacteristic, complemented, Sylow
320C2—21no—
321C2—21no—
322C2#7012211yescharacteristic
323C2#7021311yescharacteristic
324C2#7035011yescharacteristic
subgroupG/H
centerC22#4956
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1546C2
socleC22×C3#386
prime radicalC23×C3#121
cosocleC2×C3×C42C23
hypercenter#1546D3
hypocenterC3#9292
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1546

Complements

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

NH ≅ G/Ntype
18. #5093C4semidirect
19. #5080C4semidirect
319. C3#9292semidirect

Maximal subgroups

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

subgroupindex
#832232
#790452
C3×#15462
#717862
#707932
#677402
#677132
#92923