Finite Group Databasebeta

← #10697294

Subgroups — #10697294

Order 768.

524 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 103 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#83363C211yesmaximal, characteristic
2#83341C211yesmaximal, characteristic, complemented
3#79312C211yesmaximal, characteristic
4C3×#1503C211yesmaximal, characteristic, complemented
5#69492C211yesmaximal, characteristic
6#69489C211yesmaximal, characteristic, complemented
7#67744C211yesmaximal, characteristic, complemented
8#10197—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10#5477C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×C4×C4⋊C4C2211yescharacteristic
14#5084—21no—
15#5083—21no—
16#5080—21no—
17#5078—21no—
18#5064—22no—
19#5048—22no—
20#5027—21no—
21#5020—21no—
22#4995—22no—
23C2×#805—22no—
24C2×C4⋊(C3⋊C8)C2211yescharacteristic
25C4×#208C2211yescharacteristic
26#4841—22no—
27#4738—22no—
28#4693—21no—
29#4681—21no—
30C3×#328—22no—
31#4529—22no—
32#3246—31no—
33#3206—31no—
34#3165—31no—
35#1987—31no—
36#1980—31no—
37#1503D311yescharacteristic, complemented
38#1283—31no—
39C2×#235—21no—
40C22×Dic6—21no—
41C3×C4⋊Q8D411yescharacteristic
42C3×C4⋊Q8D411yescharacteristic
43C3×#125—21no—
44C3×C22⋊Q8—21no—
45C3×C22⋊Q8—21no—
46C3×C4×Q8—21no—
47C3×C4×Q8—21no—
48C3×#117—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C42—21no—
51C2×C3×C42C2311yescharacteristic
52C2×#215—22no—
53#930—21no—
54#930—21no—
55C2×C4⋊Dic3—21no—
56C2×C4⋊Dic3—21no—
57C2×#208D412yes—
58C22×C3⋊C8D412yes—
59C2×#204—24no—
60#875—21no—
61#874D411yescharacteristic
62C4×D12—21no—
63C4×D12—21no—
64#869—21no—
65#869—21no—
66#869—21no—
67#869D411yescharacteristic
68#869D412yes—
69C3×C4⋊C8—21no—
70C3×C22⋊C8D412yes—
71C3×C4×C8—21no—
72C3×#95—21no—
73#835—22no—
74#816—24no—
75#809—22no—
76#808—24no—
77#808—22no—
78#807—24no—
79#805—44no—
80C4⋊(C3⋊C8)—21no—
81C4⋊(C3⋊C8)—21no—
82#803—21no—
83C4×C3⋊C8—21no—
84#533C2×D311yescharacteristic
85#532—31no—
86C2×C4⋊Q8—31no—
87#501—62no—
88#500—61no—
89#498—61no—
90#492—61no—
91#487—61no—
92#476—61no—
93#475—61no—
94#465—62no—
95#461—62no—
96#461—61no—
97#460—61no—
98#448—62no—
99#431C2×D311yescharacteristic
100#428—62no—
101C2×C4⋊C8—31no—
102C2×#104—62no—
103C4×M16—31no—
104C4×C4⋊C4C2×D311yescharacteristic
105#340—62no—
106#328—22no—
107C2×C3×Q8—21no—
108C2×C3×Q8—21no—
109C22×C3×C4—41no—
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4C2×D411yescharacteristic
113C2×#67—41no—
114C22×Dic3—41no—
115#235—42no—
116#235—42no—
117C2×D12—21no—
118C2×D12—21no—
119C2×D12—21no—
120C2×C4×D3—41no—
121C2×Dic6—22no—
122C2×Dic6—21no—
123C2×Dic6—41no—
124C2×Dic6—82no—
125C2×C3×C8—22no—
126C3×C4⋊C4—22no—
127C3×C4⋊C4—21no—
128C3×C4⋊C4—21no—
129C3×C4⋊C4—21no—
130C3×C4⋊C4—21no—
131C3×C4⋊C4—41no—
132C3×C4⋊C4—41no—
133C3×C4⋊C4—41no—
134C3×C22⋊C4—41no—
135C3×C42—41no—
136C3×C42—42no—
137C3×C42C2×D411yescharacteristic
138C3×C42C2×D411yescharacteristic
139C3×C42C2×D411yescharacteristic
140C3×C42#4111yescharacteristic
141#215—82no—
142(C2×D3)⋊C4—41no—
143(C2×D3)⋊C4—41no—
144C4⋊Dic3—22no—
145C4⋊Dic3—21no—
146C4⋊Dic3—21no—
147C4⋊Dic3—22no—
148C4⋊Dic3—41no—
149C4⋊Dic3—41no—
150#208—22no—
151#208SD1614yescomplemented
152C2×C3⋊C8—22no—
153C2×C3⋊C8—22no—
154C2×C3⋊C8—42no—
155#204—84no—
156C2×#41—61no—
157C22×Q8—61no—
158C2×SD16—64no—
159C2×SD16—62no—
160C2×M16—32no—
161C22×C8—32no—
162C4⋊Q8—32no—
163C4⋊Q8—31no—
164C4⋊Q8—61no—
165C4⋊Q8—61no—
166C4⋊Q8—61no—
167C4⋊Q8D1211yescharacteristic
168C4⋊Q8D1211yescharacteristic
169#127—31no—
170#125—21no—
171#124—61no—
172C22⋊Q8—21no—
173C22⋊Q8—21no—
174#121—61no—
175#121—61no—
176C4×Q8—21no—
177C4×Q8—21no—
178C4×D4—61no—
179C4×D4—61no—
180#117—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—61no—
183C2×C4⋊C4—61no—
184C2×C42—21no—
185C2×C42C22×D311yescharacteristic
186#107—64no—
187#106—64no—
188#106—62no—
189C4⋊C8—21no—
190C4⋊C8—61no—
191C4⋊C8—61no—
192#104—124no—
193#103—64no—
194#103—62no—
195#102—62no—
196C22⋊C8#6712yes—
197#97—61no—
198C4×C8—21no—
199C4×C8—61no—
200#95—21no—
201C23×C3#12711yescharacteristic
202C22×D3—41no—
203C3×Q8—41no—
204C3×Q8—41no—
205C3×Q8—81no—
206C2×C3×C4—21no—
207C2×C3×C4—21no—
208C2×C3×C4—41no—
209C2×C3×C4—42no—
210C2×C3×C4—41no—
211C2×C3×C4—41no—
212C2×C3×C4—81no—
213C2×C3×C4—81no—
214C2×C3×C4#12011yescharacteristic
215C2×C3×C4#12011yescharacteristic
216C2×C3×C4#12011yescharacteristic
217C2×C3×C4#12111yescharacteristic
218C2×C3×C4#12111yescharacteristic
219C2×C3×C4#12111yescharacteristic
220C2×C3×C4C2×SD1612yes—
221C2×C3×C4#13712yes—
222#67—81no—
223#67—81no—
224C2×Dic3—42no—
225C2×Dic3—41no—
226C2×Dic3—82no—
227D12—42no—
228D12—41no—
229D12—41no—
230D12—42no—
231C4×D3—82no—
232Dic6—44no—
233Dic6—42no—
234Dic6—82no—
235C3×C8—42no—
236C3⋊C8—24no—
237C3⋊C8—42no—
238#41—122no—
239#41—122no—
240C2×Q8—21no—
241C2×Q8—21no—
242C2×Q8—61no—
243C2×Q8—62no—
244C2×Q8—121no—
245C2×Q8—242no—
246C2×D4—61no—
247C2×D4—61no—
248C2×D4—61no—
249C2×D4—121no—
250C22×C4—41no—
251C22×C4—121no—
252C22×C4—121no—
253C22×C4D3×D411yescharacteristic
254C22×C4D3×D411yescharacteristic
255C22×C4C2×#6711yescharacteristic
256SD16—244no—
257SD16—242no—
258M16—34no—
259M16—62no—
260C2×C8—22no—
261C2×C8—62no—
262C2×C8—62no—
263C2×C8—122no—
264C4⋊C4—22no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—61no—
273C4⋊C4—62no—
274C4⋊C4—62no—
275C4⋊C4—61no—
276C4⋊C4—121no—
277C4⋊C4—121no—
278C22⋊C4—41no—
279C22⋊C4—121no—
280C22⋊C4—121no—
281C42—41no—
282C42—42no—
283C42C2×D1211yescharacteristic
284C42#23511yescharacteristic
285C42D3×D411yescharacteristic
286C42D3×D411yescharacteristic
287C22×C3—21no—
288C22×C3—21no—
289C22×C3#39211yescharacteristic
290C22×C3#49211yescharacteristic
291C22×C3#49711yescharacteristic
292C2×D3—81no—
293C2×D3—81no—
294C2×D3—81no—
295C3×C4—21no—
296C3×C4—21no—
297C3×C4—21no—
298C3×C4—82no—
299C3×C4—81no—
300C3×C4—81no—
301C3×C4#44812yes—
302C3×C4#45511yescharacteristic
303C3×C4#45611yescharacteristic
304C3×C4#46112yes—
305Dic3—82no—
306Dic3—81no—
307C23—121no—
308C23#94011yescharacteristic
309Q8—41no—
310Q8—41no—
311Q8—81no—
312Q8—124no—
313Q8—122no—
314Q8—242no—
315D4—122no—
316D4—122no—
317D4—121no—
318D4—121no—
319D4—241no—
320D4—241no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—41no—
324C2×C4—42no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—121no—
330C2×C4—122no—
331C2×C4—242no—
332C2×C4—242no—
333C2×C4#88211yescharacteristic
334C2×C4#88211yescharacteristic
335C2×C4#88411yescharacteristic
336C2×C4#88411yescharacteristic
337C2×C4D3×SD1612yes—
338C2×C4#91812yes—
339C2×C4#93011yescharacteristic
340C2×C4#93711yescharacteristic
341C8—42no—
342C8—64no—
343C8—122no—
344C2×C3—21no—
345C2×C3—21no—
346C2×C3#200711yescharacteristic
347C2×C3#200811yescharacteristic
348C2×C3#201311yescharacteristic
349D3—161no—
350C22—21no—
351C22—21no—
352C22—241no—
353C22—241no—
354C22—241no—
355C22#496011yescharacteristic
356C22#517511yescharacteristic
357C22#519011yescharacteristic
358C4—21no—
359C4—21no—
360C4—21no—
361C4—82no—
362C4—81no—
363C4—81no—
364C4—241no—
365C4—242no—
366C4#480612yes—
367C4#481812yes—
368C4#482511yescharacteristic
369C4#482611yescharacteristic
370C3#1019711yescharacteristic, complemented, Sylow
371C2—21no—
372C2—21no—
373C2—481no—
374C2#7005011yescharacteristic
375C2#7028211yescharacteristic
376C2#7039311yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1503C2
socleC22×C3#392
prime radical#5477C22
hypercenter#1503D3
hypocenterC3#10197
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1503

Complements

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

NH ≅ G/Ntype
2. #83341C2semidirect
4. C3×#1503C2semidirect
6. #69489C2semidirect
7. #67744C2semidirect
37. #1503D3semidirect
151. #208SD16semidirect
370. C3#10197semidirect

Maximal subgroups

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

subgroupindex
#833632
#833412
#793122
C3×#15032
#694922
#694892
#677442
#101973