Finite Group Databasebeta

← #10692130

Subgroups — #10692130

Order 768.

482 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 93 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#79355C211yesmaximal, characteristic
2#73625C211yesmaximal, characteristic, complemented
3#70832C211yesmaximal, characteristic
4#70802C211yesmaximal, characteristic, complemented
5#69492C211yesmaximal, characteristic
6C3×#1285C211yesmaximal, characteristic, complemented
7#67748C211yesmaximal, characteristic, complemented
8#9330—31nomaximal, Sylow
9#5477C2211yescharacteristic
10C2×C4×Dic6C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C3×#432C2211yescharacteristic
13C3×C4×C4⋊C4C2211yescharacteristic
14#4973—21no—
15#4972—21no—
16#4933—22no—
17#4932C2211yescharacteristic
18C2×#805—22no—
19C4×#208C2211yescharacteristic
20#4850C412yescomplemented
21#4848C412yescomplemented
22#4738—21no—
23#4736—21no—
24#4710—22no—
25#4651—21no—
26#4650—21no—
27#4484—22no—
28#4470—21no—
29#4470—21no—
30#3112—31no—
31#2320—31no—
32#1980—31no—
33#1505—31no—
34#1491—31no—
35#1300—31no—
36#1285D311yescharacteristic, complemented
37C2×#235—21no—
38C22×Dic6D411yescharacteristic
39C2×C3×C4⋊C4—21no—
40C2×C3×C42D412yes—
41C2×C3×C42C2311yescharacteristic
42#930—21no—
43#930—21no—
44C2×C4⋊Dic3D411yescharacteristic
45C2×Dic3⋊C4—21no—
46C2×Dic3⋊C4—22no—
47C2×C4×Dic3—21no—
48C2×C4×Dic3—21no—
49C2×#208—21no—
50C2×#204—22no—
51#875—21no—
52#874C2×C411yescharacteristic
53C4×D12—21no—
54C4×D12—21no—
55#869C2×C411yescharacteristic
56C4×Dic6—21no—
57C4×Dic6—21no—
58C4×Dic6C2×C412yes—
59C4×Dic6D412yes—
60C3×C4⋊C8—21no—
61C3×C4⋊C8C2×C412yes—
62C3×C22⋊C8—21no—
63C3×C22⋊C8—21no—
64C3×#97—21no—
65C3×#97—21no—
66C3×#95—21no—
67C3⋊#95—22no—
68C3⋊#95—22no—
69C3⋊#95—22no—
70C3⋊#95—21no—
71C3⋊#95D411yescharacteristic
72C3⋊#95D411yescharacteristic
73#821—22no—
74#818—22no—
75#816—22no—
76#805—44no—
77#803—21no—
78C4×C3⋊C8—21no—
79#532—31no—
80C2×C4×Q8—31no—
81#480—61no—
82#477—32no—
83#460—32no—
84#448—61no—
85#440—62no—
86#432C2×D311yescharacteristic
87#432C2×D311yescharacteristic
88C2×#104—62no—
89C4×M16—31no—
90#391—61no—
91#389—61no—
92#387—62no—
93#383—61no—
94#382—61no—
95C4×C4⋊C4C2×D311yescharacteristic
96#376—31no—
97#327—61no—
98#327—61no—
99Q8⋊C8—62no—
100C22×C3×C4—22no—
101C22×C3×C4C2×D411yescharacteristic
102C22×C3×C4C2×D411yescharacteristic
103C22×C3×C4C2×D411yescharacteristic
104C2×#67—41no—
105C22×Dic3—21no—
106C22×Dic3—21no—
107C22×Dic3—21no—
108C22×Dic3—21no—
109#235—42no—
110#235—42no—
111C2×D12—21no—
112C2×D12—21no—
113C2×D12—21no—
114C2×C4×D3—41no—
115C2×Dic6—21no—
116C2×Dic6—21no—
117C2×Dic6—42no—
118C2×Dic6C22⋊C412yes—
119C2×C3×C8—22no—
120C2×C3×C8—41no—
121C3×C4⋊C4—42no—
122C3×C42—42no—
123C3×C42C22×C411yescharacteristic
124C3×C42C2×D411yescharacteristic
125C3×C42#4111yescharacteristic
126C3×C42#4111yescharacteristic
127(C2×D3)⋊C4—41no—
128(C2×D3)⋊C4—41no—
129C4⋊Dic3—21no—
130C4⋊Dic3—21no—
131C4⋊Dic3—21no—
132C4⋊Dic3C22⋊C412yes—
133Dic3⋊C4—22no—
134Dic3⋊C4—22no—
135Dic3⋊C4—82no—
136C4×Dic3—22no—
137C4×Dic3—22no—
138C4×Dic3—42no—
139#208—22no—
140#208—41no—
141C2×C3⋊C8—41no—
142#204—44no—
143C2×#41—61no—
144C22×Q8—31no—
145C2×SD16—62no—
146C2×M16—61no—
147C4⋊Q8—31no—
148#127—31no—
149#124—61no—
150#121—61no—
151#121—61no—
152C4×Q8—32no—
153C4×Q8—32no—
154C4×Q8—61no—
155C4×Q8—61no—
156C4×D4—61no—
157C4×D4—61no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—31no—
160C2×C4⋊C4—61no—
161C2×C4⋊C4—62no—
162C2×C42—61no—
163C2×C42—61no—
164C2×C42D1212yes—
165C2×C42C22×D311yescharacteristic
166#107—62no—
167C4⋊C8—21no—
168C4⋊C8C4×D312yes—
169#104—124no—
170#103—62no—
171#102—62no—
172C22⋊C8—21no—
173C22⋊C8—21no—
174#97—21no—
175#97—21no—
176#97—61no—
177C4×C8—61no—
178#95—21no—
179#95—31no—
180#95—31no—
181#95—62no—
182#95—62no—
183#95—62no—
184#95—61no—
185C23×C3#12011yescharacteristic
186C22×D3—41no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—22no—
190C2×C3×C4—41no—
191C2×C3×C4—42no—
192C2×C3×C4—42no—
193C2×C3×C4—42no—
194C2×C3×C4C2×C22⋊C411yescharacteristic
195C2×C3×C4C4×D411yescharacteristic
196C2×C3×C4C4×D411yescharacteristic
197C2×C3×C4#12111yescharacteristic
198C2×C3×C4#12111yescharacteristic
199C2×C3×C4#12311yescharacteristic
200#67—81no—
201#67—81no—
202C2×Dic3—22no—
203C2×Dic3—22no—
204C2×Dic3—42no—
205C2×Dic3—41no—
206C2×Dic3—41no—
207C2×Dic3—41no—
208C2×Dic3—41no—
209C2×Dic3—41no—
210C2×Dic3—41no—
211C2×Dic3—41no—
212C2×Dic3—41no—
213C2×Dic3—41no—
214D12—42no—
215D12—42no—
216D12—81no—
217C4×D3—82no—
218Dic6—22no—
219Dic6—22no—
220Dic6—42no—
221Dic6—42no—
222C3×C8—42no—
223C3×C8—81no—
224C3⋊C8—42no—
225#41—122no—
226#41—122no—
227C2×Q8—32no—
228C2×Q8—61no—
229C2×Q8—61no—
230C2×Q8—122no—
231C2×D4—61no—
232C2×D4—61no—
233C2×D4—61no—
234C2×D4—121no—
235C22×C4—22no—
236C22×C4—61no—
237C22×C4—61no—
238C22×C4—61no—
239C22×C4—61no—
240C22×C4—121no—
241C22×C4C2×D1211yescharacteristic
242C22×C4D3×D411yescharacteristic
243C22×C4D3×D411yescharacteristic
244SD16—124no—
245M16—62no—
246M16—121no—
247C2×C8—22no—
248C2×C8—41no—
249C2×C8—121no—
250C4⋊C4—32no—
251C4⋊C4—42no—
252C4⋊C4—62no—
253C4⋊C4—61no—
254C4⋊C4—62no—
255C4⋊C4—61no—
256C4⋊C4—61no—
257C4⋊C4—242no—
258C22⋊C4—121no—
259C22⋊C4—121no—
260C42—42no—
261C42—62no—
262C42—62no—
263C42—122no—
264C42C2×C4×D311yescharacteristic
265C42#23511yescharacteristic
266C42D3×D411yescharacteristic
267C42#23911yescharacteristic
268C22×C3—21no—
269C22×C3—21no—
270C22×C3#38611yescharacteristic
271C22×C3#45311yescharacteristic
272C22×C3#45411yescharacteristic
273C2×D3—81no—
274C2×D3—81no—
275C2×D3—81no—
276C3×C4—21no—
277C3×C4—21no—
278C3×C4—21no—
279C3×C4—21no—
280C3×C4—21no—
281C3×C4—82no—
282C3×C4#42112yes—
283Dic3—42no—
284Dic3—42no—
285Dic3—81no—
286Dic3—81no—
287Dic3—81no—
288C23—121no—
289C23#88211yescharacteristic
290Q8—62no—
291Q8—62no—
292Q8—122no—
293Q8—122no—
294D4—122no—
295D4—122no—
296D4—241no—
297D4—241no—
298D4—241no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—22no—
302C2×C4—41no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—62no—
307C2×C4—62no—
308C2×C4—122no—
309C2×C4—121no—
310C2×C4—121no—
311C2×C4—121no—
312C2×C4—121no—
313C2×C4—121no—
314C2×C4—121no—
315C2×C4—121no—
316C2×C4—121no—
317C2×C4—121no—
318C2×C4—242no—
319C2×C4C4×D1211yescharacteristic
320C2×C4D3×C22⋊C411yescharacteristic
321C2×C4#88411yescharacteristic
322C2×C4#89311yescharacteristic
323C2×C4#89411yescharacteristic
324C2×C4#89511yescharacteristic
325C8—42no—
326C8—81no—
327C8—122no—
328C2×C3—21no—
329C2×C3—21no—
330C2×C3#188311yescharacteristic
331C2×C3#189711yescharacteristic
332C2×C3#190511yescharacteristic
333D3—161no—
334C22—21no—
335C22—21no—
336C22—241no—
337C22—241no—
338C22—241no—
339C22#466511yescharacteristic
340C22#474711yescharacteristic
341C22#475311yescharacteristic
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—82no—
348C4—122no—
349C4—122no—
350C4—241no—
351C4—241no—
352C4—241no—
353C4#482112yes—
354C3#933011yescharacteristic, complemented, Sylow
355C2—21no—
356C2—21no—
357C2—481no—
358C2#6870711yescharacteristic
359C2#6875711yescharacteristic
360C2#6900511yescharacteristic
subgroupG/H
centerC22#4665
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1285C2
socleC22×C3#386
prime radical#5477C22
cosocleC2×C3×C42C23
hypercenter#1285D3
hypocenterC3#9330
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1285

Complements

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

NH ≅ G/Ntype
2. #73625C2semidirect
4. #70802C2semidirect
6. C3×#1285C2semidirect
7. #67748C2semidirect
20. #4850C4semidirect
21. #4848C4semidirect
36. #1285D3semidirect
354. C3#9330semidirect

Maximal subgroups

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

subgroupindex
#793552
#736252
#708322
#708022
#694922
C3×#12852
#677482
#93303