Finite Group Databasebeta

← #10687355

Subgroups — #10687355

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#86260C211yesmaximal, characteristic
2C3×#1531C211yesmaximal, characteristic, complemented
3#71831C211yesmaximal, characteristic, complemented
4#70830C211yesmaximal, characteristic, complemented
5#70816C211yesmaximal, characteristic
6#69838C211yesmaximal, characteristic, complemented
7#69798C211yesmaximal, characteristic
8#8461—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10#5732—21no—
11#5730—21no—
12#5694—22no—
13#5645—22no—
14#5624—22no—
15#5585—21no—
16#5584—22no—
17#5570—21no—
18#5526C412yes—
19#5523C412yes—
20#5477C2211yescharacteristic
21C2×C4×Dic6C2211yescharacteristic
22C3×#432C2211yescharacteristic
23C2×C3×#97C2211yescharacteristic
24#4926C2211yescharacteristic
25C2×C4⋊(C3⋊C8)C2211yescharacteristic
26C3×#330—22no—
27#4550—22no—
28#4547—21no—
29#4547—21no—
30#4485—21no—
31#4481—21no—
32#4469—22no—
33#4464—22no—
34#3507—31no—
35#1985—31no—
36#1552—31no—
37#1551—31no—
38#1531D311yescharacteristic, complemented
39#1530—31no—
40#1467—31no—
41C2×#235C2×C412yes—
42C22×Dic6C2×C411yescharacteristic
43C22×C3×C8—21no—
44C3×C4⋊Q8D411yescharacteristic
45C3×C4⋊Q8D411yescharacteristic
46C3×#125D412yes—
47C3×C22⋊Q8—21no—
48C3×C22⋊Q8—21no—
49C3×C4×Q8—21no—
50C3×C4×Q8—21no—
51C3×#117C2×C412yes—
52C2×C3×C42C2311yescharacteristic
53#947—22no—
54#946—24no—
55Q8×Dic3—22no—
56#930—21no—
57#930—21no—
58C2×C4⋊Dic3C2×C411yescharacteristic
59C2×Dic3⋊C4D412yes—
60C2×C4×Dic3D412yes—
61C22×C3⋊C8—21no—
62#898—24no—
63#898—24no—
64#897—22no—
65#897—22no—
66#894—24no—
67#893—22no—
68#892—24no—
69#892—22no—
70D3×C4⋊C4—22no—
71#890—24no—
72#890—22no—
73#886—24no—
74#67⋊C4—24no—
75#875—22no—
76#875D412yes—
77#874D411yescharacteristic
78C4×D12—22no—
79C4×D12—21no—
80C4×D12—21no—
81#869—22no—
82#869D411yescharacteristic
83C4×Dic6—22no—
84C4×Dic6—21no—
85C4×Dic6—21no—
86C4×Dic6—21no—
87C4×Dic6—21no—
88C3×C4⋊C8—21no—
89C3×C22⋊C8—21no—
90C3×#97—21no—
91C3×#97—21no—
92C3×#97—21no—
93#830—21no—
94C4⋊(C3⋊C8)—21no—
95C4⋊(C3⋊C8)—21no—
96C4⋊(C3⋊C8)—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#533C2×D311yescharacteristic
107#532—31no—
108#529—62no—
109#520—32no—
110C2×C4×Q8—31no—
111#432—31no—
112#432C2×D311yescharacteristic
113C2×C4⋊C8—31no—
114C2×#97C2×D311yescharacteristic
115#336—62no—
116#333—61no—
117#332—61no—
118#332—61no—
119#331—61no—
120#330—22no—
121#329—62no—
122#324—62no—
123C2×C3×Q8—21no—
124C2×C3×Q8—21no—
125C22×C3×C4C22×C411yescharacteristic
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C2×#67—22no—
129C22×Dic3C22⋊C412yes—
130C22×Dic3C22⋊C412yes—
131#235—44no—
132C2×D12—21no—
133C2×D12—21no—
134C2×D12C22⋊C412yes—
135C2×C4×D3—22no—
136C2×C4×D3—24no—
137C2×Dic6—21no—
138C2×Dic6—21no—
139C2×Dic6—42no—
140C2×Dic6C22⋊C412yes—
141C2×C3×C8—22no—
142C2×C3×C8—41no—
143C2×C3×C8—41no—
144C2×C3×C8—41no—
145C3×C4⋊C4—21no—
146C3×C4⋊C4—22no—
147C3×C4⋊C4—22no—
148C3×C4⋊C4—21no—
149C3×C4⋊C4C22⋊C412yes—
150C3×C4⋊C4C22⋊C412yes—
151C3×C22⋊C4—22no—
152C3×C42—22no—
153C3×C42C2×D411yescharacteristic
154C3×C42C2×D411yescharacteristic
155C3×C42C2×D411yescharacteristic
156C3×C42C2×D411yescharacteristic
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—24no—
163C4⋊Dic3—24no—
164C4⋊Dic3—21no—
165C4⋊Dic3—21no—
166C4⋊Dic3—21no—
167C4⋊Dic3—21no—
168Dic3⋊C4—24no—
169Dic3⋊C4—22no—
170Dic3⋊C4—22no—
171C4×Dic3—24no—
172C4×Dic3—22no—
173C4×Dic3—22no—
174C2×C3⋊C8—42no—
175C2×C3⋊C8—41no—
176C2×C3⋊C8—41no—
177C2×#41—32no—
178C22×Q8—31no—
179C22×C8—21no—
180C22×C8—61no—
181C4⋊Q8—31no—
182C4⋊Q8—62no—
183C4⋊Q8D1211yescharacteristic
184C4⋊Q8D1211yescharacteristic
185#127—31no—
186#126—64no—
187#126—64no—
188#125—62no—
189#125—64no—
190#125#6712yes—
191#124—32no—
192#124—62no—
193#124—62no—
194#123—64no—
195#123—64no—
196C22⋊Q8—21no—
197C22⋊Q8—21no—
198C22⋊Q8—64no—
199C22⋊Q8—62no—
200C22⋊Q8—62no—
201#121—61no—
202#121—61no—
203C4×Q8—21no—
204C4×Q8—21no—
205C4×Q8—62no—
206C4×Q8—62no—
207C4×Q8—61no—
208C4×Q8—61no—
209C4×Q8—61no—
210C4×Q8—61no—
211C4×D4—62no—
212C4×D4—64no—
213C4×D4—62no—
214C4×D4—61no—
215C4×D4—61no—
216#117—64no—
217#117—62no—
218#117C4×D312yes—
219C2×C4⋊C4—31no—
220C2×C4⋊C4—32no—
221C2×C4⋊C4—62no—
222C2×C42—32no—
223C2×C42C22×D311yescharacteristic
224C4⋊C8—21no—
225C4⋊C8—61no—
226C4⋊C8—61no—
227C4⋊C8—61no—
228C22⋊C8—21no—
229C22⋊C8—61no—
230#97—21no—
231#97—21no—
232#97—21no—
233#97—61no—
234C23×C3C2×C22⋊C411yescharacteristic
235C22×D3—22no—
236C3×Q8—41no—
237C3×Q8—41no—
238C3×Q8—81no—
239C2×C3×C4—22no—
240C2×C3×C4—21no—
241C2×C3×C4—21no—
242C2×C3×C4—22no—
243C2×C3×C4—22no—
244C2×C3×C4—41no—
245C2×C3×C4C2×C22⋊C411yescharacteristic
246C2×C3×C4C2×C22⋊C411yescharacteristic
247C2×C3×C4#12011yescharacteristic
248C2×C3×C4#12011yescharacteristic
249C2×C3×C4#12011yescharacteristic
250C2×C3×C4#12011yescharacteristic
251#67—44no—
252C2×Dic3—22no—
253C2×Dic3—22no—
254C2×Dic3—22no—
255C2×Dic3—24no—
256C2×Dic3—42no—
257D12—42no—
258D12—41no—
259D12—41no—
260D12—81no—
261C4×D3—44no—
262C4×D3—84no—
263Dic6—42no—
264Dic6—42no—
265Dic6—42no—
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—61no—
275C2×Q8—61no—
276C2×Q8—122no—
277C2×D4—32no—
278C2×D4—62no—
279C2×D4—61no—
280C2×D4—61no—
281C22×C4—32no—
282C22×C4—32no—
283C22×C4—64no—
284C22×C4—62no—
285C22×C4C2×C4×D311yescharacteristic
286C22×C4D3×D411yescharacteristic
287C22×C4D3×D411yescharacteristic
288C2×C8—22no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—122no—
293C2×C8—121no—
294C2×C8—121no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—64no—
300C4⋊C4—64no—
301C4⋊C4—64no—
302C4⋊C4—61no—
303C4⋊C4—62no—
304C4⋊C4—61no—
305C4⋊C4—61no—
306C4⋊C4—62no—
307C4⋊C4—61no—
308C4⋊C4(C2×D3)⋊C412yes—
309C4⋊C4(C2×D3)⋊C412yes—
310C22⋊C4—22no—
311C22⋊C4—64no—
312C22⋊C4—64no—
313C22⋊C4—64no—
314C22⋊C4—62no—
315C22⋊C4—62no—
316C42—22no—
317C42—64no—
318C42—62no—
319C42—62no—
320C42C2×D1211yescharacteristic
321C42D3×D411yescharacteristic
322C42D3×D411yescharacteristic
323C42C2×#6711yescharacteristic
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—
344C23D3×C22⋊C411yescharacteristic
345Q8—41no—
346Q8—41no—
347Q8—81no—
348Q8—122no—
349Q8—122no—
350Q8—122no—
351D4—122no—
352D4—124no—
353D4—121no—
354D4—121no—
355D4—241no—
356C2×C4—22no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—41no—
362C2×C4—62no—
363C2×C4—62no—
364C2×C4—62no—
365C2×C4—64no—
366C2×C4—124no—
367C2×C4—122no—
368C2×C4—244no—
369C2×C4D3×C22⋊C411yescharacteristic
370C2×C4#88211yescharacteristic
371C2×C4#88211yescharacteristic
372C2×C4C2×(C2×D3)⋊C411yescharacteristic
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#475112yes—
390C22#495511yescharacteristic
391C4—21no—
392C4—21no—
393C4—21no—
394C4—21no—
395C4—21no—
396C4—42no—
397C4—42no—
398C4—122no—
399C4—122no—
400C4—122no—
401C4#482611yescharacteristic
402C4#482711yescharacteristic
403C3#846111yescharacteristic, complemented, Sylow
404C2—22no—
405C2—242no—
406C2#7011411yescharacteristic
407C2#7038711yescharacteristic
408C2#7041311yescharacteristic
subgroupG/H
centerC22#4955
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1531C2
socleC22×C3#379
prime radical#5477C22
cosocleC2×C3×C42C23
hypercenter#1531D3
hypocenterC3#8461
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1531

Complements

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

NH ≅ G/Ntype
2. C3×#1531C2semidirect
3. #71831C2semidirect
4. #70830C2semidirect
6. #69838C2semidirect
38. #1531D3semidirect
403. C3#8461semidirect

Maximal subgroups

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

subgroupindex
#862602
C3×#15312
#718312
#708302
#708162
#698382
#697982
#84613