Finite Group Databasebeta

← #10696662

Subgroups — #10696662

Order 768.

552 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 89 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
1C3×#3209C211yesmaximal, characteristic, complemented
2#83366C211yesmaximal, characteristic
3#83203C211yesmaximal, characteristic
4#71821C211yesmaximal, characteristic, complemented
5#71790C211yesmaximal, characteristic, complemented
6#70761C211yesmaximal, characteristic
7#67743C211yesmaximal, characteristic, complemented
8#10107—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×C4⋊Q8C2211yescharacteristic
11C2×#942—22no—
12#5477C2211yescharacteristic
13C3×#494—21no—
14C3×#487—21no—
15C3×#462—22no—
16C3×#451—22no—
17C3×#431C2211yescharacteristic
18#5092—21no—
19#5087—21no—
20#5086—22no—
21#5085—21no—
22#5083—21no—
23#5049—22no—
24#5027—21no—
25#5021—21no—
26#4924C2211yescharacteristic
27C4×#208C2211yescharacteristic
28C4×#208C2211yescharacteristic
29#4543—22no—
30#4530—22no—
31#4483—22no—
32#3254—31no—
33#3209D311yescharacteristic, complemented
34#3152—31no—
35#1997—31no—
36#1526—31no—
37#1484—31no—
38#1299—31no—
39C22×C3×Q8D412yes—
40C2×#235—21no—
41C2×C3×Q16—24no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8D411yescharacteristic
47C3×C4⋊Q8D411yescharacteristic
48C3×#125—21no—
49C3×C22⋊Q8—21no—
50C3×C22⋊Q8—21no—
51C3×C4×Q8—21no—
52C3×C4×Q8—21no—
53C3×#117—21no—
54C2×C3×C4⋊C4—21no—
55C2×C3×C4⋊C4—21no—
56C2×C3×C42C2311yescharacteristic
57C2×#216—24no—
58#942—48no—
59C2×#215—24no—
60C2×#215—22no—
61#930—21no—
62#930—21no—
63C2×#208—21no—
64C2×#208—21no—
65C2×#208D412yes—
66#875—21no—
67#874D411yescharacteristic
68C4×D12—21no—
69C4×D12—21no—
70#869D411yescharacteristic
71C3×C4⋊C8D412yes—
72C3×#103—24no—
73C3×C22⋊C8—21no—
74C3×C4×C8—21no—
75#835—22no—
76#809—22no—
77#807—22no—
78C4⋊(C3⋊C8)—21no—
79C4⋊(C3⋊C8)—21no—
80#803—21no—
81#803D412yes—
82C4×C3⋊C8—21no—
83C4×C3⋊C8—21no—
84C2×#137—62no—
85#533C2×D311yescharacteristic
86#532—31no—
87C2×C4⋊Q8C2×D311yescharacteristic
88#497—62no—
89#494—21no—
90#494—61no—
91#492—61no—
92#487—21no—
93#477—61no—
94#474—61no—
95#468—62no—
96#462—22no—
97#461—61no—
98#460—61no—
99#451—22no—
100#431C2×D311yescharacteristic
101#423—31no—
102C4×M16—31no—
103C4×M16—31no—
104#343—62no—
105#330—62no—
106D4⋊C8—62no—
107C2×C3×Q8—21no—
108C2×C3×Q8—22no—
109C2×C3×Q8—22no—
110C2×C3×Q8—21no—
111C2×C3×Q8—44no—
112C22×C3×C4—22no—
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4#4111yescharacteristic
116C2×#67—41no—
117#235—44no—
118C2×D12—21no—
119C2×D12—21no—
120C2×D12—21no—
121C2×C4×D3—41no—
122C2×Dic6—21no—
123C3×Q16—84no—
124C2×C3×C8—22no—
125C3×C4⋊C4—21no—
126C3×C4⋊C4—21no—
127C3×C4⋊C4—21no—
128C3×C4⋊C4—21no—
129C3×C4⋊C4—41no—
130C3×C4⋊C4—41no—
131C3×C4⋊C4—41no—
132C3×C4⋊C4—41no—
133C3×C4⋊C4—41no—
134C3×C4⋊C4—41no—
135C3×C22⋊C4—41no—
136C3×C42—41no—
137C3×C42C2×D411yescharacteristic
138C3×C42C2×D411yescharacteristic
139C3×C42C2×D411yescharacteristic
140C3×C42C2×D411yescharacteristic
141#216—48no—
142#215—48no—
143#215—82no—
144(C2×D3)⋊C4—41no—
145(C2×D3)⋊C4—41no—
146C4⋊Dic3—21no—
147C4⋊Dic3—21no—
148#208—24no—
149#208—22no—
150#208—41no—
151#208—42no—
152C2×C3⋊C8—22no—
153C2×C3⋊C8—22no—
154C2×C3⋊C8—41no—
155C2×#41—61no—
156C22×Q8#6712yes—
157#137—128no—
158C2×Q16—24no—
159C2×Q16—64no—
160C2×SD16—64no—
161C2×SD16—62no—
162C2×M16—32no—
163C2×M16—61no—
164C2×M16—61no—
165C4⋊Q8—21no—
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169C4⋊Q8—31no—
170C4⋊Q8#6711yescharacteristic
171C4⋊Q8#6711yescharacteristic
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—21no—
185C2×C4⋊C4—21no—
186C2×C42C22×D311yescharacteristic
187#107—62no—
188C4⋊C8—61no—
189C4⋊C8—61no—
190C4⋊C8#6712yes—
191#103—24no—
192#103—62no—
193#102—62no—
194C22⋊C8—21no—
195#97—32no—
196#97—61no—
197C4×C8—21no—
198C4×C8—61no—
199C4×C8—61no—
200C23×C3#12111yescharacteristic
201C22×D3—41no—
202C3×Q8—44no—
203C3×Q8—42no—
204C3×Q8—44no—
205C3×Q8—41no—
206C3×Q8—41no—
207C2×C3×C4—22no—
208C2×C3×C4—21no—
209C2×C3×C4—21no—
210C2×C3×C4—22no—
211C2×C3×C4—41no—
212C2×C3×C4—41no—
213C2×C3×C4—42no—
214C2×C3×C4—82no—
215C2×C3×C4#12011yescharacteristic
216C2×C3×C4#12011yescharacteristic
217C2×C3×C4#12011yescharacteristic
218C2×C3×C4#12111yescharacteristic
219C2×C3×C4#12111yescharacteristic
220C2×C3×C4#12711yescharacteristic
221#67—82no—
222C2×Dic3—41no—
223D12—42no—
224D12—41no—
225D12—41no—
226D12—42no—
227C4×D3—82no—
228Dic6—42no—
229C3×C8—42no—
230C3⋊C8—24no—
231C3⋊C8—42no—
232C3⋊C8—42no—
233#41—124no—
234C2×Q8—21no—
235C2×Q8—22no—
236C2×Q8—22no—
237C2×Q8—21no—
238C2×Q8—44no—
239C2×Q8—61no—
240C2×D4—61no—
241C2×D4—61no—
242C2×D4—61no—
243C2×D4—121no—
244C22×C4—22no—
245C22×C4—121no—
246C22×C4D3×D411yescharacteristic
247C22×C4#23711yescharacteristic
248C22×C4C2×#6711yescharacteristic
249Q16—84no—
250Q16—128no—
251SD16—128no—
252SD16—242no—
253M16—64no—
254M16—62no—
255M16—121no—
256M16—122no—
257C2×C8—22no—
258C2×C8—62no—
259C2×C8—62no—
260C2×C8—121no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—61no—
272C4⋊C4—61no—
273C22⋊C4—41no—
274C22⋊C4—121no—
275C22⋊C4—121no—
276C42—41no—
277C42D3×D411yescharacteristic
278C42D3×D411yescharacteristic
279C42C2×#6711yescharacteristic
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—82no—
292C3×C4—81no—
293C3×C4—81no—
294C3×C4#45312yes—
295C3×C4#45511yescharacteristic
296C3×C4#45611yescharacteristic
297C3×C4#47312yes—
298Dic3—81no—
299C23—121no—
300C23#93911yescharacteristic
301Q8—44no—
302Q8—41no—
303Q8—41no—
304Q8—42no—
305Q8—44no—
306Q8—122no—
307D4—122no—
308D4—122no—
309D4—121no—
310D4—121no—
311D4—242no—
312C2×C4—22no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—22no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—42no—
319C2×C4—82no—
320C2×C4—121no—
321C2×C4—242no—
322C2×C4#93711yescharacteristic
323C2×C4#93711yescharacteristic
324C2×C4#93811yescharacteristic
325C2×C4#93911yescharacteristic
326C2×C4#94011yescharacteristic
327C2×C4#95311yescharacteristic
328C8—42no—
329C8—64no—
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#522311yescharacteristic
344C4—21no—
345C4—21no—
346C4—21no—
347C4—81no—
348C4—82no—
349C4—81no—
350C4—241no—
351C4#519212yes—
352C4#520211yescharacteristic
353C4#520511yescharacteristic
354C4#524112yes—
355C3#1010711yescharacteristic, complemented, Sylow
356C2—41no—
357C2—481no—
358C2#7185611yescharacteristic
359C2#7207411yescharacteristic
360C2#7211011yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3209C2
socleC22×C3#392
prime radical#5477C22
hypercenter#3209D3
hypocenterC3#10107
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3209

Complements

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

NH ≅ G/Ntype
1. C3×#3209C2semidirect
4. #71821C2semidirect
5. #71790C2semidirect
7. #67743C2semidirect
33. #3209D3semidirect
355. C3#10107semidirect

Maximal subgroups

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

subgroupindex
C3×#32092
#833662
#832032
#718212
#717902
#707612
#677432
#101073