Finite Group Databasebeta

← #10687353

Subgroups — #10687353

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#86224C211yesmaximal, characteristic
2C3×#1530C211yesmaximal, characteristic
3#71807C211yesmaximal, characteristic
4#70830C211yesmaximal, characteristic, complemented
5#70820C211yesmaximal, characteristic, complemented
6#69838C211yesmaximal, characteristic, complemented
7#69801C211yesmaximal, characteristic, complemented
8#8461—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10#5827C412yes—
11#5825C412yes—
12#5721—21no—
13#5713—21no—
14#5671—22no—
15#5602—22no—
16#5600—22no—
17#5543—21no—
18#5542—21no—
19#5541—22no—
20#5470C2211yescharacteristic
21C2×C4×Dic6C2211yescharacteristic
22C3×#432C2211yescharacteristic
23C2×C3×#97C2211yescharacteristic
24#4926C2211yescharacteristic
25C2×C4⋊(C3⋊C8)C2211yescharacteristic
26C3×#329—22no—
27#4550—22no—
28#4546—21no—
29#4544—21no—
30#4486—21no—
31#4482—21no—
32#4469—22no—
33#4465—22no—
34#3507—31no—
35#1985—31no—
36#1552—31no—
37#1551—31no—
38#1531—31no—
39#1530D311yescharacteristic
40#1467—31no—
41C2×C3×#41C2×C412yes—
42C22×Dic6C2×C411yescharacteristic
43C22×C3×C8—21no—
44C3×C4⋊Q8D411yescharacteristic
45C3×#127D411yescharacteristic
46C3×#124D412yes—
47C3×#121—21no—
48C3×#121—21no—
49C3×C4×D4—21no—
50C3×C4×D4—21no—
51C2×C3×C42C2311yescharacteristic
52Q8×Dic3—22no—
53#944—22no—
54#936—22no—
55#936—22no—
56#935—24no—
57#935—24no—
58D4×Dic3—22no—
59D4×Dic3—22no—
60D4×Dic3—24no—
61#926—22no—
62#926C2×C412yes—
63C2×C4⋊Dic3—22no—
64C2×C4⋊Dic3C2×C411yescharacteristic
65#924—21no—
66#924—21no—
67#924—22no—
68#924—22no—
69C2×Dic3⋊C4D412yes—
70C2×C4×Dic3D412yes—
71C22×C3⋊C8—21no—
72#890—22no—
73#889—24no—
74#887—22no—
75#879—24no—
76#879—24no—
77#878—24no—
78#877—24no—
79#870D412yes—
80#869D411yescharacteristic
81#869D411yescharacteristic
82C4×Dic6—21no—
83C4×Dic6—21no—
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—61no—
101#556—62no—
102#554—61no—
103#552—62no—
104#539—62no—
105#536—32no—
106#533—31no—
107#532C2×D311yescharacteristic
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—62no—
121#329—22no—
122#324—62no—
123C3×#41—44no—
124C2×C3×Q8C22⋊C412yes—
125C2×C3×D4—22no—
126C2×C3×D4—21no—
127C2×C3×D4—21no—
128C2×C3×D4C22⋊C412yes—
129C22×C3×C4—22no—
130C22×C3×C4C22×C411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D411yescharacteristic
133C22×Dic3—24no—
134C22×Dic3C22⋊C412yes—
135C22×Dic3C22⋊C412yes—
136C2×Dic6—21no—
137C2×Dic6—21no—
138C2×Dic6—21no—
139C2×Dic6—21no—
140C2×Dic6—42no—
141C2×C3×C8—22no—
142C2×C3×C8—41no—
143C2×C3×C8—41no—
144C2×C3×C8—41no—
145C3×C4⋊C4—21no—
146C3×C4⋊C4—21no—
147C3×C22⋊C4—22no—
148C3×C22⋊C4—22no—
149C3×C42C2×D411yescharacteristic
150C3×C42C2×D411yescharacteristic
151C3×C42C2×D411yescharacteristic
152C3×C42C2×D411yescharacteristic
153#217—24no—
154#217—24no—
155#217—24no—
156#217—22no—
157C4⋊Dic3—24no—
158C4⋊Dic3—21no—
159C4⋊Dic3—21no—
160C4⋊Dic3—21no—
161C4⋊Dic3—21no—
162C4⋊Dic3C22⋊C412yes—
163C4⋊Dic3C22⋊C412yes—
164Dic3⋊C4—24no—
165Dic3⋊C4—24no—
166Dic3⋊C4—22no—
167Dic3⋊C4—22no—
168Dic3⋊C4—22no—
169Dic3⋊C4—22no—
170C4×Dic3—24no—
171C4×Dic3—22no—
172C4×Dic3—22no—
173C4×Dic3—22no—
174C2×C3⋊C8—42no—
175C2×C3⋊C8—41no—
176C2×C3⋊C8—41no—
177C2×#41C4×D312yes—
178C22×Q8—31no—
179C22×C8—21no—
180C22×C8—61no—
181C4⋊Q8—31no—
182C4⋊Q8—31no—
183C4⋊Q8—62no—
184C4⋊Q8D1211yescharacteristic
185#127D1211yescharacteristic
186#126—64no—
187#126—64no—
188#125—32no—
189#125—62no—
190#125—64no—
191#124—62no—
192#124—62no—
193#124#6712yes—
194#123—64no—
195#123—64no—
196C22⋊Q8—64no—
197C22⋊Q8—62no—
198C22⋊Q8—62no—
199C22⋊Q8—61no—
200C22⋊Q8—61no—
201#121—21no—
202#121—21no—
203C4×Q8—62no—
204C4×Q8—62no—
205C4×Q8—61no—
206C4×Q8—61no—
207C4×Q8—61no—
208C4×Q8—61no—
209C4×Q8—61no—
210C4×Q8—61no—
211C4×D4—21no—
212C4×D4—21no—
213C4×D4—64no—
214C4×D4—62no—
215C4×D4—62no—
216#117—32no—
217#117—64no—
218#117—62no—
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×C3—22no—
235C23×C3C2×C22⋊C411yescharacteristic
236C3×Q8—42no—
237C3×D4—42no—
238C3×D4—44no—
239C3×D4—41no—
240C3×D4—41no—
241C3×D4—81no—
242C2×C3×C4—22no—
243C2×C3×C4—21no—
244C2×C3×C4—21no—
245C2×C3×C4—22no—
246C2×C3×C4—41no—
247C2×C3×C4—44no—
248C2×C3×C4C2×C22⋊C411yescharacteristic
249C2×C3×C4C2×C22⋊C411yescharacteristic
250C2×C3×C4#12011yescharacteristic
251C2×C3×C4#12011yescharacteristic
252C2×C3×C4#12011yescharacteristic
253C2×C3×C4#12011yescharacteristic
254C2×Dic3—22no—
255C2×Dic3—22no—
256C2×Dic3—22no—
257C2×Dic3—22no—
258C2×Dic3—24no—
259C2×Dic3—42no—
260C2×Dic3—84no—
261Dic6—42no—
262Dic6—42no—
263Dic6—41no—
264Dic6—41no—
265Dic6—81no—
266C3×C8—42no—
267C3×C8—81no—
268C3⋊C8—81no—
269C3⋊C8—81no—
270#41—44no—
271C2×Q8—61no—
272C2×Q8—61no—
273C2×Q8—61no—
274C2×Q8—61no—
275C2×Q8—122no—
276C2×Q8(C2×D3)⋊C412yes—
277C2×D4—22no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4(C2×D3)⋊C412yes—
281C22×C4—22no—
282C22×C4—32no—
283C22×C4—32no—
284C22×C4—64no—
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—32no—
298C4⋊C4—32no—
299C4⋊C4—64no—
300C4⋊C4—64no—
301C4⋊C4—64no—
302C4⋊C4—61no—
303C4⋊C4—62no—
304C4⋊C4—61no—
305C4⋊C4—62no—
306C4⋊C4—61no—
307C4⋊C4—62no—
308C4⋊C4—62no—
309C4⋊C4—61no—
310C22⋊C4—22no—
311C22⋊C4—22no—
312C22⋊C4—64no—
313C22⋊C4—64no—
314C22⋊C4—64no—
315C22⋊C4—62no—
316C42—64no—
317C42—62no—
318C42—62no—
319C42—62no—
320C42C2×D1211yescharacteristic
321C42D3×D411yescharacteristic
322C42D3×D411yescharacteristic
323C42C2×#6711yescharacteristic
324C22×C3—21no—
325C22×C3—21no—
326C22×C3—42no—
327C22×C3—42no—
328C22×C3—42no—
329C22×C3#37911yescharacteristic
330C22×C3#41312yes—
331C3×C4—21no—
332C3×C4—21no—
333C3×C4—21no—
334C3×C4—21no—
335C3×C4—21no—
336C3×C4—42no—
337C3×C4#45411yescharacteristic
338C3×C4#45611yescharacteristic
339Dic3—42no—
340Dic3—42no—
341Dic3—42no—
342Dic3—42no—
343C23—22no—
344C23D3×C22⋊C411yescharacteristic
345Q8—42no—
346Q8—122no—
347Q8—122no—
348Q8—121no—
349Q8—121no—
350Q8—241no—
351D4—42no—
352D4—44no—
353D4—41no—
354D4—41no—
355D4—81no—
356C2×C4—22no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—22no—
360C2×C4—41no—
361C2×C4—44no—
362C2×C4—62no—
363C2×C4—62no—
364C2×C4—62no—
365C2×C4—62no—
366C2×C4—64no—
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—82no—
381C2×C3#178511yescharacteristic
382C2×C3#179311yescharacteristic
383C2×C3#179411yescharacteristic
384C22—21no—
385C22—21no—
386C22—42no—
387C22—42no—
388C22—42no—
389C22#474512yes—
390C22#495511yescharacteristic
391C4—21no—
392C4—21no—
393C4—21no—
394C4—21no—
395C4—21no—
396C4—42no—
397C4—122no—
398C4—122no—
399C4—122no—
400C4—122no—
401C4#482611yescharacteristic
402C4#482711yescharacteristic
403C3#846111yescharacteristic, complemented, Sylow
404C2—22no—
405C2—82no—
406C2#7008011yescharacteristic
407C2#7038711yescharacteristic
408C2#7041311yescharacteristic
subgroupG/H
centerC22#4955
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1530C2
socleC22×C3#379
prime radicalC3×#532C22
cosocleC2×C3×C42C23
hypercenter#1530D3
hypocenterC3#8461
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1530

Complements

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

NH ≅ G/Ntype
4. #70830C2semidirect
5. #70820C2semidirect
6. #69838C2semidirect
7. #69801C2semidirect
403. C3#8461semidirect

Maximal subgroups

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

subgroupindex
#862242
C3×#15302
#718072
#708302
#708202
#698382
#698012
#84613