Finite Group Databasebeta

← #10687354

Subgroups — #10687354

Order 768.

682 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 682 conjugacy classes of subgroups — too many to draw. Only its 111 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#85819C211yesmaximal, characteristic
2C3×#1985C211yesmaximal, characteristic, complemented
3#71625C211yesmaximal, characteristic, complemented
4#70439C211yesmaximal, characteristic, complemented
5#70433C211yesmaximal, characteristic
6#69801C211yesmaximal, characteristic, complemented
7#69798C211yesmaximal, characteristic
8#8461—31nomaximal, Sylow
9C2×C3×C4×Q8C2211yescharacteristic
10#5816C412yes—
11#5585—21no—
12#5584—21no—
13#5581—22no—
14#5579—21no—
15#5568—22no—
16#5567—21no—
17#5515—22no—
18#5504C412yes—
19#5479—22no—
20#5477C2211yescharacteristic
21#5470C2211yescharacteristic
22C2×C3×C4⋊C8C2211yescharacteristic
23C2×C3×#97C2211yescharacteristic
24#4926C2211yescharacteristic
25#4926C2211yescharacteristic
26C3×#336—22no—
27#4528—22no—
28#4486—21no—
29#4485—21no—
30#4482—21no—
31#4481—21no—
32#4465—22no—
33#4464—22no—
34#3507—31no—
35#1985D311yescharacteristic, complemented
36#1552—31no—
37#1551—31no—
38#1531—31no—
39#1530—31no—
40#1467—31no—
41C22×C3×Q8C2×C411yescharacteristic
42C2×#235C2×C412yes—
43C22×C3×C8—21no—
44C22×C3×C8—21no—
45C3×C4×Q8—21no—
46C3×C4×Q8—21no—
47C3×C4×Q8—21no—
48C3×C4×Q8—21no—
49C2×C3×C4⋊C4C2×C411yescharacteristic
50C2×C3×C4⋊C4D412yes—
51C2×C3×C42D412yes—
52C2×C3×C42C2311yescharacteristic
53#947—22no—
54#946—22no—
55#946—22no—
56#944—22no—
57#930—21no—
58#930—21no—
59#929—24no—
60C4×#67—24no—
61#926C2×C412yes—
62#924—21no—
63#924—21no—
64#898—24no—
65#896—24no—
66#894—24no—
67#893—22no—
68#892—22no—
69D3×C4⋊C4—22no—
70#889—24no—
71#887—22no—
72#876—24no—
73#875—22no—
74#875D412yes—
75#874D411yescharacteristic
76C4×D12—22no—
77C4×D12—21no—
78C4×D12—21no—
79#872—24no—
80#870—22no—
81#870D412yes—
82#869D411yescharacteristic
83#869D411yescharacteristic
84#869D411yescharacteristic
85C4×Dic6—22no—
86C4×Dic6—21no—
87C4×Dic6—21no—
88C3×C4⋊C8—21no—
89C3×C4⋊C8—21no—
90C3×#97—21no—
91C3×#97—21no—
92#830—21no—
93#830—21no—
94C4⋊(C3⋊C8)—21no—
95C4⋊(C3⋊C8)—21no—
96#803—21no—
97#803—21no—
98#559—61no—
99#557—61no—
100#556—62no—
101#556—61no—
102#554—61no—
103#552—62no—
104#539—62no—
105#536—32no—
106#533—31no—
107#532—31no—
108#529—62no—
109#520—32no—
110C2×C4×Q8C2×D311yescharacteristic
111#432—31no—
112#432—31no—
113C2×C4⋊C8C2×D311yescharacteristic
114C2×#97C2×D311yescharacteristic
115#336—22no—
116#333—61no—
117#332—61no—
118#332—61no—
119#331—61no—
120#330—62no—
121#329—62no—
122#324—62no—
123C2×C3×Q8—21no—
124C2×C3×Q8—21no—
125C2×C3×Q8—42no—
126C22×C3×C4C22⋊C412yes—
127C22×C3×C4C22⋊C412yes—
128C22×C3×C4C22×C411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C2×#67—22no—
132#235—44no—
133C2×D12—21no—
134C2×D12—21no—
135C2×D12C22⋊C412yes—
136C2×C4×D3—22no—
137C2×C4×D3—24no—
138C2×Dic6—21no—
139C2×Dic6—21no—
140C2×Dic6C22⋊C412yes—
141C2×C3×C8—22no—
142C2×C3×C8—41no—
143C2×C3×C8—41no—
144C2×C3×C8—42no—
145C2×C3×C8—41no—
146C3×C4⋊C4—21no—
147C3×C4⋊C4—21no—
148C3×C4⋊C4—22no—
149C3×C4⋊C4—22no—
150C3×C42—22no—
151C3×C42—22no—
152C3×C42C2×D411yescharacteristic
153C3×C42C2×D411yescharacteristic
154C3×C42C2×D411yescharacteristic
155C3×C42C2×D411yescharacteristic
156#217—22no—
157(C2×D3)⋊C4—24no—
158(C2×D3)⋊C4—24no—
159(C2×D3)⋊C4—24no—
160(C2×D3)⋊C4—22no—
161(C2×D3)⋊C4—22no—
162C4⋊Dic3—21no—
163C4⋊Dic3—21no—
164C4⋊Dic3—21no—
165C4⋊Dic3—21no—
166C4⋊Dic3C22⋊C412yes—
167C4⋊Dic3C22⋊C412yes—
168Dic3⋊C4—24no—
169Dic3⋊C4—24no—
170Dic3⋊C4—24no—
171Dic3⋊C4—22no—
172Dic3⋊C4—22no—
173C4×Dic3—24no—
174C4×Dic3—22no—
175C2×C3⋊C8—41no—
176C2×C3⋊C8—41no—
177C2×#41—32no—
178C22×Q8C4×D311yescharacteristic
179C22×C8—21no—
180C22×C8—21no—
181C4⋊Q8—31no—
182C4⋊Q8—31no—
183C4⋊Q8—31no—
184C4⋊Q8—62no—
185#127—31no—
186#126—64no—
187#126—64no—
188#125—32no—
189#125—62no—
190#125—64no—
191#124—32no—
192#124—62no—
193#124—62no—
194#123—64no—
195#123—64no—
196C22⋊Q8—64no—
197C22⋊Q8—62no—
198C22⋊Q8—62no—
199C22⋊Q8—61no—
200C22⋊Q8—61no—
201#121—61no—
202#121—61no—
203C4×Q8—21no—
204C4×Q8—21no—
205C4×Q8—21no—
206C4×Q8—21no—
207C4×Q8—62no—
208C4×Q8—62no—
209C4×Q8—61no—
210C4×Q8—61no—
211C4×D4—62no—
212C4×D4—62no—
213C4×D4—64no—
214C4×D4—61no—
215C4×D4—61no—
216#117—32no—
217#117—64no—
218#117—62no—
219C2×C4⋊C4—62no—
220C2×C4⋊C4C4×D311yescharacteristic
221C2×C4⋊C4D1212yes—
222C2×C42#6712yes—
223C2×C42C22×D311yescharacteristic
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—61no—
227C4⋊C8—61no—
228C22⋊C8—61no—
229C22⋊C8—61no—
230#97—21no—
231#97—21no—
232#97—61no—
233#97—61no—
234C23×C3C2×C22⋊C411yescharacteristic
235C22×D3—22no—
236C3×Q8—42no—
237C3×Q8—42no—
238C2×C3×C4—22no—
239C2×C3×C4—21no—
240C2×C3×C4—21no—
241C2×C3×C4—22no—
242C2×C3×C4—22no—
243C2×C3×C4—24no—
244C2×C3×C4—41no—
245C2×C3×C4—42no—
246C2×C3×C4C2×C22⋊C411yescharacteristic
247C2×C3×C4C2×C22⋊C411yescharacteristic
248C2×C3×C4#12011yescharacteristic
249C2×C3×C4#12011yescharacteristic
250C2×C3×C4#12011yescharacteristic
251C2×C3×C4#12011yescharacteristic
252#67—44no—
253C2×Dic3—22no—
254C2×Dic3—22no—
255C2×Dic3—22no—
256D12—42no—
257D12—41no—
258D12—41no—
259D12—81no—
260C4×D3—44no—
261C4×D3—84no—
262Dic6—41no—
263Dic6—41no—
264Dic6—42no—
265Dic6—81no—
266C3×C8—42no—
267C3×C8—81no—
268C3⋊C8—81no—
269C3⋊C8—81no—
270#41—124no—
271C2×Q8—21no—
272C2×Q8—21no—
273C2×Q8—32no—
274C2×Q8—42no—
275C2×Q8—61no—
276C2×Q8—61no—
277C2×D4—32no—
278C2×D4—62no—
279C2×D4—61no—
280C2×D4—61no—
281C22×C4—64no—
282C22×C4—62no—
283C22×C4(C2×D3)⋊C412yes—
284C22×C4(C2×D3)⋊C412yes—
285C22×C4C2×C4×D311yescharacteristic
286C22×C4C2×D1211yescharacteristic
287C22×C4C2×#6711yescharacteristic
288C2×C8—22no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—42no—
292C2×C8—41no—
293C2×C8—121no—
294C2×C8—121no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—32no—
300C4⋊C4—32no—
301C4⋊C4—64no—
302C4⋊C4—64no—
303C4⋊C4—64no—
304C4⋊C4—61no—
305C4⋊C4—62no—
306C4⋊C4—61no—
307C4⋊C4—62no—
308C4⋊C4—61no—
309C4⋊C4—61no—
310C22⋊C4—64no—
311C22⋊C4—64no—
312C22⋊C4—64no—
313C22⋊C4—62no—
314C22⋊C4—62no—
315C22⋊C4—62no—
316C42—22no—
317C42—22no—
318C42—64no—
319C42—62no—
320C42D3×D411yescharacteristic
321C42D3×D411yescharacteristic
322C42D3×D411yescharacteristic
323C42D3×D411yescharacteristic
324C22×C3—21no—
325C22×C3—21no—
326C22×C3#37911yescharacteristic
327C22×C3#41312yes—
328C2×D3—42no—
329C2×D3—42no—
330C2×D3—42no—
331C3×C4—21no—
332C3×C4—21no—
333C3×C4—21no—
334C3×C4—21no—
335C3×C4—21no—
336C3×C4—42no—
337C3×C4—42no—
338C3×C4#45411yescharacteristic
339C3×C4#45611yescharacteristic
340Dic3—42no—
341Dic3—42no—
342Dic3—42no—
343C23—62no—
344C23C2×(C2×D3)⋊C411yescharacteristic
345Q8—42no—
346Q8—42no—
347Q8—122no—
348Q8—121no—
349Q8—121no—
350Q8—241no—
351D4—122no—
352D4—124no—
353D4—121no—
354D4—121no—
355D4—241no—
356C2×C4—22no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—24no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—62no—
365C2×C4—62no—
366C2×C4—62no—
367C2×C4—124no—
368C2×C4—244no—
369C2×C4D3×C22⋊C411yescharacteristic
370C2×C4D3×C22⋊C411yescharacteristic
371C2×C4#88211yescharacteristic
372C2×C4#88211yescharacteristic
373C2×C4#93711yescharacteristic
374C2×C4#93711yescharacteristic
375C8—42no—
376C8—81no—
377C8—241no—
378C8—241no—
379C2×C3—22no—
380C2×C3#178511yescharacteristic
381C2×C3#179311yescharacteristic
382C2×C3#179411yescharacteristic
383D3—82no—
384C22—21no—
385C22—21no—
386C22—122no—
387C22—122no—
388C22—122no—
389C22#495511yescharacteristic
390C22#513212yes—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—21no—
395C4—21no—
396C4—42no—
397C4—42no—
398C4—122no—
399C4—122no—
400C4—122no—
401C4#506211yescharacteristic
402C4#509511yescharacteristic
403C3#846111yescharacteristic, complemented, Sylow
404C2—22no—
405C2—242no—
406C2#6969211yescharacteristic
407C2#6998911yescharacteristic
408C2#7001011yescharacteristic
subgroupG/H
centerC22#4955
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1985C2
socleC22×C3#379
prime radical#5477C22
cosocleC2×C3×C42C23
hypercenter#1985D3
hypocenterC3#8461
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1985

Complements

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

NH ≅ G/Ntype
2. C3×#1985C2semidirect
3. #71625C2semidirect
4. #70439C2semidirect
6. #69801C2semidirect
35. #1985D3semidirect
403. C3#8461semidirect

Maximal subgroups

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

subgroupindex
#858192
C3×#19852
#716252
#704392
#704332
#698012
#697982
#84613