Finite Group Databasebeta

← #10696628

Subgroups — #10696628

Order 768.

568 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 83 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#83264C211yesmaximal, characteristic
2#83207C211yesmaximal, characteristic, complemented
3#79351C211yesmaximal, characteristic
4C3×#1556C211yesmaximal, characteristic, complemented
5#70836C211yesmaximal, characteristic, complemented
6#70804C211yesmaximal, characteristic
7#67748C211yesmaximal, characteristic, complemented
8#10102—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×#947—21no—
11C2×#944—21no—
12C2×#875C2211yescharacteristic
13#5470C2211yescharacteristic
14C3×#432C2211yescharacteristic
15C3×#432C2211yescharacteristic
16#5081—22no—
17#5080—21no—
18#5078—21no—
19#4993—22no—
20#4993—22no—
21#4964—22no—
22#4932C2211yescharacteristic
23C4×#208C2211yescharacteristic
24#4845—22no—
25#4739—21no—
26#4731—21no—
27#4715—21no—
28#4714—21no—
29C3×#332—22no—
30#4467—22no—
31#4464—22no—
32#3215—31no—
33#3165—31no—
34#2658—31no—
35#1556D311yescharacteristic, complemented
36#1527—31no—
37#1506—31no—
38#1300—31no—
39C22×C3×Q8—21no—
40C22×D12D411yescharacteristic
41C22×Dic6D411yescharacteristic
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4D412yes—
48C2×C3×C42C2311yescharacteristic
49#947—44no—
50#944—44no—
51C2×(C2×D3)⋊C4—24no—
52C2×(C2×D3)⋊C4—21no—
53C2×(C2×D3)⋊C4—21no—
54#926—21no—
55#924—21no—
56#924—21no—
57C2×Dic3⋊C4—24no—
58C2×C4×Dic3—21no—
59C2×#208—21no—
60C2×Dic12—22no—
61C2×#204—22no—
62#875—21no—
63#875—21no—
64#875—21no—
65#875—21no—
66#870—21no—
67#869D411yescharacteristic
68#869D411yescharacteristic
69C4×Dic6—21no—
70C4×Dic6—21no—
71C3×C4⋊C8—21no—
72C3×C4⋊C8D412yes—
73C3×C22⋊C8—21no—
74C3×C22⋊C8—21no—
75C3×#97—21no—
76C3×#97—21no—
77C3⋊#95—21no—
78C3⋊#95D412yes—
79#821—22no—
80#816—22no—
81#808—24no—
82#807—24no—
83#803D412yes—
84C4×C3⋊C8—21no—
85#533—31no—
86C2×C4⋊Q8—61no—
87C2×C4⋊Q8C2×D311yescharacteristic
88C2×#124—31no—
89C2×#124—61no—
90#501—62no—
91#500—61no—
92#498—61no—
93#497—61no—
94#489—61no—
95#464—62no—
96#451—61no—
97#450—61no—
98#432C2×D311yescharacteristic
99#432C2×D311yescharacteristic
100#428—62no—
101C4×M16—31no—
102#399—62no—
103#399—62no—
104#376—31no—
105#332—22no—
106#329—62no—
107#328—62no—
108C23×D3—21no—
109C2×C3×Q8—42no—
110C2×C3×Q8—42no—
111C2×C3×Q8—41no—
112C2×C3×Q8—41no—
113C22×C3×C4—22no—
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C22×Dic3—22no—
118C22×Dic3—21no—
119C2×D12—21no—
120C2×D12—21no—
121C2×D12—42no—
122C2×Dic6—21no—
123C2×Dic6—21no—
124C2×Dic6—21no—
125C2×Dic6—21no—
126C2×Dic6—42no—
127C2×C3×C8—22no—
128C2×C3×C8—41no—
129C3×C4⋊C4—22no—
130C3×C4⋊C4—22no—
131C3×C4⋊C4—41no—
132C3×C4⋊C4—41no—
133C3×C42C2×D411yescharacteristic
134C3×C42C2×D411yescharacteristic
135C3×C42C2×D411yescharacteristic
136C3×C42#4111yescharacteristic
137#217—41no—
138(C2×D3)⋊C4—48no—
139(C2×D3)⋊C4—41no—
140(C2×D3)⋊C4—41no—
141(C2×D3)⋊C4—41no—
142(C2×D3)⋊C4—41no—
143C4⋊Dic3—21no—
144C4⋊Dic3—21no—
145C4⋊Dic3—21no—
146C4⋊Dic3—21no—
147Dic3⋊C4—48no—
148Dic3⋊C4—41no—
149Dic3⋊C4—41no—
150C4×Dic3—24no—
151C4×Dic3—41no—
152#208—24no—
153C2×C3⋊C8—22no—
154Dic12—82no—
155#204—82no—
156C22×Q8—21no—
157C22×Q8—31no—
158C22×D4—31no—
159C2×Q16—62no—
160C2×SD16—62no—
161C2×M16—61no—
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164C4⋊Q8—21no—
165C4⋊Q8—21no—
166C4⋊Q8—31no—
167C4⋊Q8—31no—
168C4⋊Q8—124no—
169#125—61no—
170#124—61no—
171#124—61no—
172#124—61no—
173#124—61no—
174#124—124no—
175C22⋊Q8—61no—
176C22⋊Q8—61no—
177C4×Q8—61no—
178C4×Q8—61no—
179#117—61no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—64no—
182C2×C4⋊C4D1212yes—
183C2×C22⋊C4—64no—
184C2×C22⋊C4—61no—
185C2×C22⋊C4—61no—
186C2×C42—61no—
187C2×C42C22×D311yescharacteristic
188#107—64no—
189#106—64no—
190C4⋊C8—21no—
191C4⋊C8#6712yes—
192#103—62no—
193#102—62no—
194C22⋊C8—21no—
195C22⋊C8—21no—
196#97—21no—
197#97—21no—
198#97—32no—
199C4×C8—61no—
200#95—32no—
201#95—61no—
202C23×C3#12011yescharacteristic
203C22×D3—42no—
204C22×D3—42no—
205C22×D3—41no—
206C22×D3—42no—
207C3×Q8—82no—
208C3×Q8—82no—
209C2×C3×C4—22no—
210C2×C3×C4—22no—
211C2×C3×C4—41no—
212C2×C3×C4—42no—
213C2×C3×C4—42no—
214C2×C3×C4—42no—
215C2×C3×C4#12011yescharacteristic
216C2×C3×C4#12011yescharacteristic
217C2×C3×C4#12111yescharacteristic
218C2×C3×C4#12111yescharacteristic
219C2×C3×C4#12111yescharacteristic
220C2×C3×C4#12711yescharacteristic
221C2×Dic3—24no—
222C2×Dic3—42no—
223C2×Dic3—42no—
224C2×Dic3—42no—
225C2×Dic3—41no—
226C2×Dic3—41no—
227C2×Dic3—41no—
228D12—42no—
229D12—81no—
230Dic6—42no—
231Dic6—42no—
232Dic6—81no—
233Dic6—81no—
234C3×C8—42no—
235C3×C8—81no—
236C3⋊C8—24no—
237C24—61no—
238C2×Q8—42no—
239C2×Q8—42no—
240C2×Q8—41no—
241C2×Q8—41no—
242C2×Q8—61no—
243C2×Q8—61no—
244C2×Q8—61no—
245C2×Q8—61no—
246C2×Q8—122no—
247C2×D4—61no—
248C2×D4—61no—
249C2×D4—122no—
250C22×C4—22no—
251C22×C4—62no—
252C22×C4—61no—
253C22×C4C2×D1211yescharacteristic
254C22×C4D3×D411yescharacteristic
255C22×C4D3×D411yescharacteristic
256Q16—242no—
257SD16—242no—
258M16—64no—
259C2×C8—22no—
260C2×C8—41no—
261C2×C8—62no—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—61no—
267C4⋊C4—61no—
268C4⋊C4—61no—
269C4⋊C4—61no—
270C4⋊C4—128no—
271C4⋊C4—121no—
272C4⋊C4—121no—
273C22⋊C4—128no—
274C22⋊C4—121no—
275C22⋊C4—121no—
276C22⋊C4—121no—
277C22⋊C4—121no—
278C22⋊C4—121no—
279C42—64no—
280C42—121no—
281C42#23511yescharacteristic
282C42D3×D411yescharacteristic
283C42D3×D411yescharacteristic
284C42C2×#6711yescharacteristic
285C22×C3—21no—
286C22×C3—21no—
287C22×C3#39211yescharacteristic
288C22×C3#45612yes—
289C2×D3—84no—
290C2×D3—81no—
291C2×D3—81no—
292C2×D3—81no—
293C3×C4—21no—
294C3×C4—21no—
295C3×C4—21no—
296C3×C4—21no—
297C3×C4—21no—
298C3×C4—82no—
299C3×C4#45312yes—
300Dic3—44no—
301Dic3—81no—
302Dic3—81no—
303Dic3—81no—
304C23—122no—
305C23—122no—
306C23—121no—
307C23—122no—
308C23#88211yescharacteristic
309Q8—82no—
310Q8—82no—
311Q8—122no—
312Q8—122no—
313Q8—241no—
314Q8—241no—
315D4—122no—
316D4—241no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—41no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—64no—
324C2×C4—122no—
325C2×C4—122no—
326C2×C4—122no—
327C2×C4—121no—
328C2×C4—121no—
329C2×C4—121no—
330C2×C4#88211yescharacteristic
331C2×C4#88411yescharacteristic
332C2×C4#88411yescharacteristic
333C2×C4#93011yescharacteristic
334C2×C4#93711yescharacteristic
335C2×C4#94011yescharacteristic
336C8—42no—
337C8—64no—
338C8—81no—
339C2×C3—22no—
340C2×C3#200211yescharacteristic
341C2×C3#200511yescharacteristic
342C2×C3#200911yescharacteristic
343D3—161no—
344C22—21no—
345C22—21no—
346C22—244no—
347C22—241no—
348C22—241no—
349C22—241no—
350C22#475212yes—
351C22#496011yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—82no—
358C4—124no—
359C4—241no—
360C4—241no—
361C4—241no—
362C4#519812yes—
363C3#1010211yescharacteristic, complemented, Sylow
364C2—22no—
365C2—481no—
366C2#7011911yescharacteristic
367C2#7032411yescharacteristic
368C2#7035111yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1556C2
socleC22×C3#392
prime radicalC22×D12D4
hypercenter#1556D3
hypocenterC3#10102
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1556

Complements

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

NH ≅ G/Ntype
2. #83207C2semidirect
4. C3×#1556C2semidirect
5. #70836C2semidirect
7. #67748C2semidirect
35. #1556D3semidirect
363. C3#10102semidirect

Maximal subgroups

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

subgroupindex
#832642
#832072
#793512
C3×#15562
#708362
#708042
#677482
#101023