Finite Group Databasebeta

← #10696664

Subgroups — #10696664

Order 768.

552 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 87 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#83370C211yesmaximal, characteristic, complemented
2#83366C211yesmaximal, characteristic
3#78538C211yesmaximal, characteristic
4C3×#1526C211yesmaximal, characteristic, complemented
5#69879C211yesmaximal, characteristic, complemented
6#69853C211yesmaximal, characteristic
7#67669C211yesmaximal, characteristic, complemented
8#10107—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×#909—22no—
11#5477C2211yescharacteristic
12C2×#869C2211yescharacteristic
13C3×C4×M16C2211yescharacteristic
14C3×C4×M16C2211yescharacteristic
15#5093—21no—
16#5092—21no—
17#5087—21no—
18#5083—21no—
19#5050—22no—
20#5049—22no—
21#5031—22no—
22#5027—21no—
23#5021—21no—
24#4925C2211yescharacteristic
25#4924C2211yescharacteristic
26#4715—22no—
27#4699—21no—
28#4693—21no—
29C3×#330—22no—
30#4554—22no—
31#4459—22no—
32#3254—31no—
33#3209—31no—
34#3152—31no—
35#1997—31no—
36#1526D311yescharacteristic, complemented
37#1484—31no—
38#1299—31no—
39C2×#235—21no—
40C22×Dic6D412yes—
41C2×C3×M16—21no—
42C2×C3×M16—21no—
43C3×C4⋊Q8D411yescharacteristic
44C3×C4⋊Q8D411yescharacteristic
45C3×#125—21no—
46C3×C22⋊Q8—21no—
47C3×C22⋊Q8—21no—
48C3×C4×Q8—21no—
49C3×C4×Q8—21no—
50C3×#117—21no—
51C2×C3×C42C2311yescharacteristic
52C2×#216—24no—
53C2×#215—22no—
54#930—21no—
55#930—21no—
56C2×C4⋊Dic3—21no—
57C2×C4⋊Dic3—21no—
58C2×#208D412yes—
59#909—48no—
60C2×Dic12—24no—
61C2×#204—24no—
62#875—21no—
63#874D411yescharacteristic
64C4×D12—21no—
65C4×D12—21no—
66#869—21no—
67#869—21no—
68#869—21no—
69#869—21no—
70#869D411yescharacteristic
71C3×#97—21no—
72C3×#97D412yes—
73C3×C4×C8—21no—
74C3×C4×C8—21no—
75#835—22no—
76#830—21no—
77#810—24no—
78#809—22no—
79#807—22no—
80C4⋊(C3⋊C8)—21no—
81C4⋊(C3⋊C8)—21no—
82C4⋊(C3⋊C8)D412yes—
83C4×C3⋊C8—21no—
84C2×#137—62no—
85#533C2×D311yescharacteristic
86#532—31no—
87C2×C4⋊Q8—31no—
88#497—62no—
89#494—61no—
90#494—61no—
91#492—61no—
92#487—61no—
93#477—61no—
94#474—61no—
95#468—62no—
96#462—62no—
97#461—61no—
98#460—61no—
99#451—62no—
100#431—31no—
101#423—31no—
102C4×M16C2×D311yescharacteristic
103C4×M16C2×D311yescharacteristic
104#343—62no—
105#330—22no—
106D4⋊C8—62no—
107C2×C3×Q8—21no—
108C2×C3×Q8—21no—
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4#4111yescharacteristic
112C2×#67—41no—
113C22×Dic3—22no—
114#235—44no—
115C2×D12—21no—
116C2×D12—21no—
117C2×D12—21no—
118C2×C4×D3—41no—
119C2×Dic6—22no—
120C2×Dic6—22no—
121C2×Dic6—21no—
122C2×Dic6—44no—
123C3×M16—24no—
124C3×M16—22no—
125C3×M16—41no—
126C2×C3×C8—22no—
127C2×C3×C8—41no—
128C3×C4⋊C4—21no—
129C3×C4⋊C4—21no—
130C3×C4⋊C4—21no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—41no—
133C3×C4⋊C4—41no—
134C3×C22⋊C4—41no—
135C3×C42—41no—
136C3×C42C2×D411yescharacteristic
137C3×C42C2×D411yescharacteristic
138C3×C42C2×D411yescharacteristic
139C3×C42C2×D411yescharacteristic
140#216—84no—
141#215—82no—
142(C2×D3)⋊C4—41no—
143(C2×D3)⋊C4—41no—
144C4⋊Dic3—21no—
145C4⋊Dic3—21no—
146C4⋊Dic3—41no—
147C4⋊Dic3—41no—
148C4⋊Dic3—41no—
149C4⋊Dic3—41no—
150#208—42no—
151C2×C3⋊C8—22no—
152C2×C3⋊C8—22no—
153Dic12—48no—
154#204—48no—
155C2×#41—61no—
156C22×Q8—32no—
157#137—128no—
158C2×Q16—64no—
159C2×Q16—64no—
160C2×SD16—64no—
161C2×SD16—62no—
162C2×M16—21no—
163C2×M16—21no—
164C2×M16—32no—
165C4⋊Q8—31no—
166C4⋊Q8—61no—
167C4⋊Q8—61no—
168C4⋊Q8—61no—
169C4⋊Q8—61no—
170C4⋊Q8D1211yescharacteristic
171C4⋊Q8D1211yescharacteristic
172#127—31no—
173#125—21no—
174#124—61no—
175C22⋊Q8—21no—
176C22⋊Q8—21no—
177#121—61no—
178#121—61no—
179C4×Q8—21no—
180C4×Q8—21no—
181C4×D4—61no—
182C4×D4—61no—
183#117—21no—
184C2×C4⋊C4—61no—
185C2×C4⋊C4—61no—
186C2×C42C22×D311yescharacteristic
187#107—62no—
188C4⋊C8—32no—
189C4⋊C8—61no—
190C4⋊C8—61no—
191#103—64no—
192#103—62no—
193#102—62no—
194C22⋊C8—61no—
195#97—21no—
196#97#6712yes—
197C4×C8—21no—
198C4×C8—21no—
199C4×C8—61no—
200C23×C3#12111yescharacteristic
201C22×D3—41no—
202C3×Q8—42no—
203C3×Q8—41no—
204C3×Q8—41no—
205C2×C3×C4—22no—
206C2×C3×C4—21no—
207C2×C3×C4—21no—
208C2×C3×C4—22no—
209C2×C3×C4—41no—
210C2×C3×C4—41no—
211C2×C3×C4#12011yescharacteristic
212C2×C3×C4#12011yescharacteristic
213C2×C3×C4#12011yescharacteristic
214C2×C3×C4#12111yescharacteristic
215C2×C3×C4#12111yescharacteristic
216C2×C3×C4#12711yescharacteristic
217#67—82no—
218C2×Dic3—41no—
219C2×Dic3—42no—
220C2×Dic3—82no—
221D12—42no—
222D12—41no—
223D12—41no—
224D12—42no—
225C4×D3—82no—
226Dic6—44no—
227Dic6—42no—
228Dic6—44no—
229C3×C8—24no—
230C3×C8—42no—
231C3⋊C8—42no—
232C3⋊C8—42no—
233#41—124no—
234C2×Q8—21no—
235C2×Q8—21no—
236C2×Q8—62no—
237C2×Q8—61no—
238C2×Q8—62no—
239C2×Q8—124no—
240C2×D4—61no—
241C2×D4—61no—
242C2×D4—61no—
243C2×D4—121no—
244C22×C4—62no—
245C22×C4—121no—
246C22×C4#23511yescharacteristic
247C22×C4D3×D411yescharacteristic
248C22×C4D3×D411yescharacteristic
249Q16—128no—
250Q16—244no—
251SD16—128no—
252SD16—242no—
253M16—24no—
254M16—22no—
255M16—41no—
256M16—122no—
257C2×C8—22no—
258C2×C8—41no—
259C2×C8—62no—
260C2×C8—62no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—61no—
268C4⋊C4—61no—
269C4⋊C4—121no—
270C4⋊C4—121no—
271C4⋊C4—121no—
272C4⋊C4—121no—
273C22⋊C4—41no—
274C22⋊C4—121no—
275C22⋊C4—121no—
276C42—41no—
277C42C2×D1211yescharacteristic
278C42D3×D411yescharacteristic
279C42D3×D411yescharacteristic
280C42C2×#6711yescharacteristic
281C22×C3—21no—
282C22×C3—21no—
283C22×C3—21no—
284C22×C3#39211yescharacteristic
285C2×D3—81no—
286C2×D3—81no—
287C2×D3—81no—
288C3×C4—21no—
289C3×C4—21no—
290C3×C4—21no—
291C3×C4—81no—
292C3×C4—81no—
293C3×C4#45312yes—
294C3×C4#45511yescharacteristic
295C3×C4#45611yescharacteristic
296C3×C4#47312yes—
297Dic3—82no—
298Dic3—81no—
299C23—121no—
300C23#88411yescharacteristic
301Q8—42no—
302Q8—41no—
303Q8—41no—
304Q8—124no—
305Q8—122no—
306Q8—124no—
307D4—122no—
308D4—122no—
309D4—121no—
310D4—121no—
311D4—242no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—121no—
319C2×C4—122no—
320C2×C4—242no—
321C2×C4—242no—
322C2×C4#88211yescharacteristic
323C2×C4#88211yescharacteristic
324C2×C4#88411yescharacteristic
325C2×C4#93011yescharacteristic
326C2×C4#93711yescharacteristic
327C2×C4#94011yescharacteristic
328C8—24no—
329C8—42no—
330C8—122no—
331C8—122no—
332C2×C3—41no—
333C2×C3#200211yescharacteristic
334C2×C3#200711yescharacteristic
335C2×C3#202011yescharacteristic
336D3—161no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—241no—
341C22—241no—
342C22—241no—
343C22#496011yescharacteristic
344C4—21no—
345C4—21no—
346C4—21no—
347C4—81no—
348C4—81no—
349C4—241no—
350C4—242no—
351C4#475512yes—
352C4#482212yes—
353C4#482511yescharacteristic
354C4#482611yescharacteristic
355C3#1010711yescharacteristic, complemented, Sylow
356C2—41no—
357C2—481no—
358C2#7012511yescharacteristic
359C2#7038511yescharacteristic
360C2#7039311yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1526C2
socleC22×C3#392
prime radical#5477C22
hypercenter#1526D3
hypocenterC3#10107
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1526

Complements

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

NH ≅ G/Ntype
1. #83370C2semidirect
4. C3×#1526C2semidirect
5. #69879C2semidirect
7. #67669C2semidirect
36. #1526D3semidirect
355. C3#10107semidirect

Maximal subgroups

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

subgroupindex
#833702
#833662
#785382
C3×#15262
#698792
#698532
#676692
#101073