Finite Group Databasebeta

← #10696093

Subgroups — #10696093

Order 768.

576 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 79 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#83002C211yesmaximal, characteristic
2#82939C211yesmaximal, characteristic, complemented
3#79344C211yesmaximal, characteristic, complemented
4C3×#1533C211yesmaximal, characteristic, complemented
5#70837C211yesmaximal, characteristic, complemented
6#70799C211yesmaximal, characteristic, complemented
7#67752C211yesmaximal, characteristic, complemented
8#10022—31nomaximal, Sylow
9C2×C3×#124C2211yescharacteristic
10C2×C4×D12C2211yescharacteristic
11#5470C2211yescharacteristic
12C3×#432C2211yescharacteristic
13C3×#432C2211yescharacteristic
14#5231—21no—
15#5225—21no—
16#5223—21no—
17#5215—21no—
18#5056—22no—
19#5055—21no—
20#5055—21no—
21#4960—21no—
22#4956—22no—
23#4953—21no—
24#4937C2211yescharacteristic
25C4×#208C2211yescharacteristic
26#4846—21no—
27#4841—21no—
28#4731—21no—
29#4722—21no—
30#4710—22no—
31C3×#329—22no—
32#4483—22no—
33#4467—22no—
34#3140—31no—
35#3110—31no—
36#2631—31no—
37#1533D311yescharacteristic, complemented
38#1515—31no—
39#1490—31no—
40#1288—31no—
41C22×C3×Q8D411yescharacteristic
42C22×C3×D4D411yescharacteristic
43C22×#67—22no—
44C22×D12D411yescharacteristic
45C22×C4×D3—21no—
46C3×#124—21no—
47C3×#124—21no—
48C3×#124—21no—
49C3×#124—21no—
50C2×C3×C22⋊C4—21no—
51C2×C3×C22⋊C4—21no—
52C2×C3×C42C2311yescharacteristic
53C2×#217—22no—
54C2×(C2×D3)⋊C4—21no—
55C2×(C2×D3)⋊C4—22no—
56#926—21no—
57C2×C4⋊Dic3D411yescharacteristic
58#924—21no—
59#924—21no—
60C2×#208—21no—
61C2×#204—22no—
62C4×D12—21no—
63C4×D12—21no—
64C4×D12—21no—
65C4×D12—21no—
66#870—21no—
67#869D411yescharacteristic, complemented
68#869D411yescharacteristic, complemented
69C4×Dic6—21no—
70C4×Dic6—21no—
71C3×C4⋊C8—21no—
72C3×C4⋊C8—21no—
73C3×C22⋊C8—21no—
74C3×C22⋊C8—21no—
75C3×#97—21no—
76C3×#97D412yescomplemented
77#835—22no—
78#835—22no—
79#832—22no—
80#832—22no—
81C3⋊#95—22no—
82C3⋊#95—21no—
83C3⋊#95—21no—
84C3⋊#95D412yes—
85#821—22no—
86#818—22no—
87#816—22no—
88#803—21no—
89C4×C3⋊C8D412yescomplemented
90#533—31no—
91C2×#124C2×D311yescharacteristic
92C2×C4×D4—31no—
93#489—61no—
94#489—61no—
95#488—62no—
96#484—61no—
97#474—61no—
98#461—61no—
99#450—61no—
100#440—62no—
101#432C2×D311yescharacteristic
102#432C2×D311yescharacteristic
103#419—61no—
104#418—61no—
105C4×M16—31no—
106#397—61no—
107#393—61no—
108#392—61no—
109#392—61no—
110#386—62no—
111#383—31no—
112#329—22no—
113#328—62no—
114D4⋊C8—62no—
115C24×C3—21no—
116C23×D3—21no—
117C2×C3×Q8—21no—
118C2×C3×Q8—21no—
119C2×C3×Q8—42no—
120C2×C3×D4—21no—
121C2×C3×D4—21no—
122C2×C3×D4—42no—
123C22×C3×C4—21no—
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4C2×D411yescharacteristic
127C2×#67—44no—
128C2×#67—82no—
129C2×#67—82no—
130C22×Dic3—22no—
131C22×Dic3—21no—
132C2×D12—21no—
133C2×D12—21no—
134C2×D12—42no—
135C2×C4×D3—41no—
136C2×C4×D3—41no—
137C2×C4×D3—82no—
138C2×Dic6—21no—
139C2×Dic6—21no—
140C2×C3×C8—22no—
141C2×C3×C8—41no—
142C3×C22⋊C4—41no—
143C3×C22⋊C4—41no—
144C3×C22⋊C4—41no—
145C3×C22⋊C4—41no—
146C3×C42C2×D411yescharacteristic
147C3×C42C2×D411yescharacteristic
148C3×C42C2×D411yescharacteristic
149C3×C42#4111yescharacteristic
150#217—41no—
151#217—82no—
152(C2×D3)⋊C4—41no—
153(C2×D3)⋊C4—41no—
154(C2×D3)⋊C4—82no—
155C4⋊Dic3—21no—
156C4⋊Dic3—21no—
157C4⋊Dic3—21no—
158C4⋊Dic3—21no—
159C4⋊Dic3—21no—
160C4⋊Dic3—21no—
161Dic3⋊C4—41no—
162Dic3⋊C4—41no—
163C4×Dic3—41no—
164#208—42no—
165C2×C3⋊C8—22no—
166#204—44no—
167C22×Q8D1211yescharacteristic
168C22×D4—31no—
169C22×D4—62no—
170C22×D4D1211yescharacteristic
171C23×C4—61no—
172C2×SD16—62no—
173C2×M16—61no—
174C4⋊Q8—31no—
175C4⋊Q8—31no—
176#125—61no—
177#124—21no—
178#124—21no—
179#124—21no—
180#124—21no—
181C22⋊Q8—61no—
182C22⋊Q8—61no—
183C4×Q8—61no—
184C4×Q8—61no—
185C4×D4—61no—
186C4×D4—61no—
187C4×D4—61no—
188C4×D4—61no—
189#117—61no—
190C2×C4⋊C4—31no—
191C2×C22⋊C4—21no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—61no—
194C2×C22⋊C4—62no—
195C2×C22⋊C4—62no—
196C2×C42C22×D311yescharacteristic
197#107—62no—
198C4⋊C8—21no—
199C4⋊C8—21no—
200#103—62no—
201#103—62no—
202#103—62no—
203#102—62no—
204#102—62no—
205#102—62no—
206C22⋊C8—21no—
207C22⋊C8—21no—
208#97—21no—
209#97—61no—
210#97#6712yescomplemented
211C4×C8—32no—
212#95—32no—
213#95—62no—
214#95—61no—
215#95—61no—
216C23×C3—41no—
217C23×C3—42no—
218C23×C3—41no—
219C23×C3—41no—
220C23×C3—42no—
221C23×C3#12011yescharacteristic
222C22×D3—22no—
223C22×D3—42no—
224C22×D3—41no—
225C22×D3—41no—
226C22×D3—42no—
227C3×Q8—42no—
228C3×Q8—81no—
229C3×D4—42no—
230C3×D4—81no—
231C2×C3×C4—22no—
232C2×C3×C4—41no—
233C2×C3×C4—41no—
234C2×C3×C4—41no—
235C2×C3×C4—42no—
236C2×C3×C4#12011yescharacteristic
237C2×C3×C4#12011yescharacteristic
238C2×C3×C4#12111yescharacteristic
239C2×C3×C4#12111yescharacteristic
240C2×C3×C4#12111yescharacteristic
241C2×C3×C4#12711yescharacteristic
242#67—88no—
243C2×Dic3—42no—
244C2×Dic3—42no—
245C2×Dic3—42no—
246C2×Dic3—41no—
247C2×Dic3—41no—
248C2×Dic3—41no—
249C2×Dic3—42no—
250D12—22no—
251D12—22no—
252D12—41no—
253D12—41no—
254C4×D3—82no—
255C4×D3—82no—
256Dic6—42no—
257Dic6—81no—
258C3×C8—42no—
259C3×C8—81no—
260C3⋊C8—42no—
261C24—21no—
262C24—61no—
263C2×Q8—21no—
264C2×Q8—21no—
265C2×Q8—42no—
266C2×Q8—61no—
267C2×Q8—61no—
268C2×D4—21no—
269C2×D4—21no—
270C2×D4—42no—
271C2×D4—61no—
272C2×D4—61no—
273C2×D4—122no—
274C2×D4—124no—
275C2×D4—242no—
276C2×D4—242no—
277C22×C4—21no—
278C22×C4—62no—
279C22×C4—61no—
280C22×C4—121no—
281C22×C4—121no—
282C22×C4—242no—
283C22×C4C2×D1211yescharacteristic
284C22×C4D3×D411yescharacteristic
285C22×C4D3×D411yescharacteristic
286SD16—124no—
287M16—122no—
288C2×C8—22no—
289C2×C8—41no—
290C2×C8—62no—
291C4⋊C4—61no—
292C4⋊C4—61no—
293C4⋊C4—61no—
294C4⋊C4—61no—
295C4⋊C4—61no—
296C4⋊C4—61no—
297C4⋊C4—121no—
298C4⋊C4—121no—
299C22⋊C4—41no—
300C22⋊C4—41no—
301C22⋊C4—41no—
302C22⋊C4—41no—
303C22⋊C4—121no—
304C22⋊C4—121no—
305C22⋊C4—121no—
306C22⋊C4—242no—
307C22⋊C4—242no—
308C42—121no—
309C42#23511yescharacteristic
310C42D3×D411yescharacteristic
311C42D3×D411yescharacteristic
312C42C2×#6711yescharacteristic
313C22×C3—21no—
314C22×C3—21no—
315C22×C3—82no—
316C22×C3—82no—
317C22×C3—81no—
318C22×C3—81no—
319C22×C3—81no—
320C22×C3#39211yescharacteristic
321C22×C3#45512yes—
322C2×D3—44no—
323C2×D3—42no—
324C2×D3—82no—
325C2×D3—81no—
326C2×D3—81no—
327C3×C4—21no—
328C3×C4—21no—
329C3×C4—21no—
330C3×C4—21no—
331C3×C4—21no—
332C3×C4—21no—
333C3×C4—81no—
334Dic3—82no—
335Dic3—81no—
336Dic3—81no—
337Dic3—81no—
338C23—41no—
339C23—42no—
340C23—41no—
341C23—41no—
342C23—42no—
343C23—62no—
344C23—122no—
345C23—121no—
346C23—121no—
347C23—122no—
348C23#88211yescharacteristic
349Q8—42no—
350Q8—81no—
351Q8—122no—
352Q8—241no—
353D4—42no—
354D4—62no—
355D4—62no—
356D4—81no—
357D4—121no—
358D4—121no—
359D4—248no—
360C2×C4—22no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—42no—
365C2×C4—122no—
366C2×C4—122no—
367C2×C4—122no—
368C2×C4—121no—
369C2×C4—121no—
370C2×C4—121no—
371C2×C4—122no—
372C2×C4—242no—
373C2×C4—242no—
374C2×C4#88211yescharacteristic
375C2×C4#88411yescharacteristic
376C2×C4#88411yescharacteristic
377C2×C4#93011yescharacteristic
378C2×C4#93711yescharacteristic
379C2×C4#94011yescharacteristic
380C8—42no—
381C8—81no—
382C8—122no—
383C2×C3—22no—
384C2×C3—161no—
385C2×C3#200411yescharacteristic
386C2×C3#200811yescharacteristic
387C2×C3#201011yescharacteristic
388D3—82no—
389C22—21no—
390C22—21no—
391C22—82no—
392C22—82no—
393C22—81no—
394C22—81no—
395C22—81no—
396C22—124no—
397C22—122no—
398C22—242no—
399C22—241no—
400C22—241no—
401C22#474612yes—
402C22#496011yescharacteristic
403C4—21no—
404C4—21no—
405C4—21no—
406C4—21no—
407C4—21no—
408C4—21no—
409C4—81no—
410C4—242no—
411C4—241no—
412C4—241no—
413C4—241no—
414C3#1002211yescharacteristic, complemented, Sylow
415C2—22no—
416C2—161no—
417C2—242no—
418C2#7008411yescharacteristic
419C2#7040611yescharacteristic
420C2#7040911yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1533C2
socleC22×C3#392
prime radical#83002C2
hypercenter#1533D3
hypocenterC3#10022
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1533

Complements

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

NH ≅ G/Ntype
2. #82939C2semidirect
3. #79344C2semidirect
4. C3×#1533C2semidirect
5. #70837C2semidirect
6. #70799C2semidirect
7. #67752C2semidirect
37. #1533D3semidirect
67. #869D4semidirect
68. #869D4semidirect
76. C3×#97D4semidirect
89. C4×C3⋊C8D4semidirect
210. #97#67semidirect
414. C3#10022semidirect

Maximal subgroups

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

subgroupindex
#830022
#829392
#793442
C3×#15332
#708372
#707992
#677522
#100223