Finite Group Databasebeta

← #10687351

Subgroups — #10687351

Order 768.

658 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 658 conjugacy classes of subgroups — too many to draw. Only its 115 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#86258C211yesmaximal, characteristic
2C3×#1552C211yesmaximal, characteristic
3#71822C211yesmaximal, characteristic
4#70830C211yesmaximal, characteristic
5#70813C211yesmaximal, characteristic
6#70473C211yesmaximal, characteristic
7#70439C211yesmaximal, characteristic
8#8460—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10#5722—21no—
11#5721—21no—
12#5685—22no—
13#5623—22no—
14#5622—22no—
15#5567—22no—
16#5567—21no—
17Q8×Dic6—21no—
18#5519C412yes—
19#5516C412yes—
20#5470C2211yescharacteristic
21C2×C4×Dic6C2211yescharacteristic
22C3×#432C2211yescharacteristic
23C2×C3×C4⋊C8C2211yescharacteristic
24#4926C2211yescharacteristic
25C2×#803C2211yescharacteristic
26C3×#332—21no—
27C3×#332—21no—
28#4550—22no—
29#4543—22no—
30#4486—21no—
31#4482—21no—
32#4469—22no—
33#4465—22no—
34#3502—31no—
35#1985—31no—
36#1552—31no—
37#1552D311yescharacteristic
38#1531—31no—
39#1531—31no—
40#1467—31no—
41C22×Dic6C2×C411yescharacteristic
42C22×C3×C8—21no—
43C3×C4⋊Q8D411yescharacteristic
44C3×C4⋊Q8D411yescharacteristic
45C3×#125D412yes—
46C3×C22⋊Q8—21no—
47C3×C22⋊Q8—21no—
48C3×C4×Q8—21no—
49C3×C4×Q8—21no—
50C3×#117C2×C412yes—
51C2×C3×C42C2311yescharacteristic
52Q8×Dic3—22no—
53#944—22no—
54#944—24no—
55#926C2×C412yes—
56C2×C4⋊Dic3C2×C411yescharacteristic
57#924—21no—
58#924—21no—
59C2×Dic3⋊C4D412yes—
60C2×C4×Dic3D412yes—
61C22×C3⋊C8—21no—
62#890—22no—
63#889—24no—
64#889—24no—
65#889—24no—
66#888—24no—
67#888—22no—
68#888—22no—
69#887—24no—
70#887—22no—
71#887—22no—
72#887—22no—
73#878—24no—
74#877—24no—
75#870—22no—
76#870D412yes—
77#869—22no—
78#869D411yescharacteristic
79#869D411yescharacteristic
80C4×Dic6—21no—
81C4×Dic6—21no—
82C4×Dic6—21no—
83C4×Dic6—21no—
84C4×Dic6—21no—
85C4×Dic6—21no—
86C4×Dic6—22no—
87C4×Dic6—22no—
88C3×C4⋊C8—21no—
89C3×C4⋊C8—21no—
90C3×C4⋊C8—21no—
91C3×C22⋊C8—21no—
92C3×#97—21no—
93#830—21no—
94C4⋊(C3⋊C8)—21no—
95#803—21no—
96#803—21no—
97#803—21no—
98Q82—61no—
99Q82—61no—
100#557—62no—
101#557—61no—
102#557—61no—
103#557—62no—
104#544—32no—
105#544—62no—
106#533—31no—
107#533—62no—
108#533C2×D311yescharacteristic
109#519—32no—
110C2×C4×Q8—31no—
111#432—31no—
112#432C2×D311yescharacteristic
113C2×C4⋊C8C2×D311yescharacteristic
114C2×#97—31no—
115#336—62no—
116#332—21no—
117#332—21no—
118#332—61no—
119#332—61no—
120#330—62no—
121#330—62no—
122#324—62no—
123C2×C3×Q8—21no—
124C2×C3×Q8—21no—
125C22×C3×C4C22×C411yescharacteristic
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C22×Dic3C22⋊C412yes—
129C22×Dic3C22⋊C412yes—
130C2×Dic6—21no—
131C2×Dic6—21no—
132C2×Dic6—24no—
133C2×Dic6—21no—
134C2×Dic6—24no—
135C2×Dic6—21no—
136C2×C3×C8—42no—
137C2×C3×C8—41no—
138C2×C3×C8—41no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—22no—
141C3×C4⋊C4—22no—
142C3×C4⋊C4—21no—
143C3×C4⋊C4C22⋊C412yes—
144C3×C4⋊C4C22⋊C412yes—
145C3×C22⋊C4—22no—
146C3×C42—22no—
147C3×C42C2×D411yescharacteristic
148C3×C42C2×D411yescharacteristic
149C3×C42C2×D411yescharacteristic
150C3×C42C2×D411yescharacteristic
151#217—22no—
152C4⋊Dic3—24no—
153C4⋊Dic3—21no—
154C4⋊Dic3—21no—
155C4⋊Dic3—21no—
156C4⋊Dic3—21no—
157C4⋊Dic3C22⋊C412yes—
158C4⋊Dic3C22⋊C412yes—
159Dic3⋊C4—24no—
160Dic3⋊C4—24no—
161Dic3⋊C4—24no—
162Dic3⋊C4—24no—
163Dic3⋊C4—22no—
164Dic3⋊C4—22no—
165Dic3⋊C4—22no—
166Dic3⋊C4—22no—
167C4×Dic3—24no—
168C4×Dic3—24no—
169C4×Dic3—22no—
170C4×Dic3—22no—
171C4×Dic3—22no—
172C2×C3⋊C8—22no—
173C2×C3⋊C8—41no—
174C2×C3⋊C8—41no—
175C2×C3⋊C8—41no—
176C22×Q8—31no—
177C22×C8—21no—
178C22×C8—61no—
179C4⋊Q8—31no—
180C4⋊Q8—31no—
181C4⋊Q8—62no—
182C4⋊Q8—62no—
183C4⋊Q8—64no—
184C4⋊Q8—62no—
185C4⋊Q8—64no—
186C4⋊Q8—62no—
187C4⋊Q8#6711yescharacteristic
188C4⋊Q8#6711yescharacteristic
189#125—32no—
190#125—62no—
191#125—64no—
192#125—64no—
193#125—64no—
194#125—62no—
195#125D1212yes—
196C22⋊Q8—21no—
197C22⋊Q8—21no—
198C22⋊Q8—64no—
199C22⋊Q8—61no—
200C22⋊Q8—61no—
201C4×Q8—21no—
202C4×Q8—21no—
203C4×Q8—62no—
204C4×Q8—62no—
205C4×Q8—64no—
206C4×Q8—62no—
207C4×Q8—62no—
208C4×Q8—62no—
209C4×Q8—62no—
210C4×Q8—61no—
211C4×Q8—61no—
212C4×Q8—61no—
213C4×Q8—61no—
214C4×Q8—61no—
215C4×Q8—61no—
216#117—32no—
217#117—64no—
218#117C4×D312yes—
219C2×C4⋊C4—31no—
220C2×C4⋊C4—32no—
221C2×C42—32no—
222C2×C42C22×D311yescharacteristic
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—61no—
227C22⋊C8—21no—
228C22⋊C8—61no—
229#97—21no—
230#97—61no—
231#97—61no—
232#97—61no—
233C23×C3C2×C22⋊C411yescharacteristic
234C3×Q8—41no—
235C3×Q8—41no—
236C3×Q8—81no—
237C2×C3×C4—22no—
238C2×C3×C4—21no—
239C2×C3×C4—21no—
240C2×C3×C4—22no—
241C2×C3×C4—22no—
242C2×C3×C4—41no—
243C2×C3×C4C2×C22⋊C411yescharacteristic
244C2×C3×C4C2×C22⋊C411yescharacteristic
245C2×C3×C4#12011yescharacteristic
246C2×C3×C4#12011yescharacteristic
247C2×C3×C4#12011yescharacteristic
248C2×C3×C4#12011yescharacteristic
249C2×Dic3—22no—
250C2×Dic3—22no—
251C2×Dic3—22no—
252C2×Dic3—22no—
253C2×Dic3—22no—
254C2×Dic3—42no—
255C2×Dic3#10114yes—
256Dic6—24no—
257Dic6—24no—
258Dic6—41no—
259Dic6—41no—
260Dic6—81no—
261Dic6—84no—
262C3×C8—81no—
263C3×C8—81no—
264C3⋊C8—42no—
265C3⋊C8—81no—
266C2×Q8—21no—
267C2×Q8—21no—
268C2×Q8—64no—
269C2×Q8—64no—
270C2×Q8—61no—
271C2×Q8—61no—
272C2×Q8—61no—
273C2×Q8—61no—
274C22×C4—32no—
275C22×C4—32no—
276C22×C4C2×C4×D311yescharacteristic
277C22×C4D3×D411yescharacteristic
278C22×C4D3×D411yescharacteristic
279C2×C8—42no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—62no—
283C2×C8—121no—
284C2×C8—121no—
285C2×C8—121no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—32no—
291C4⋊C4—32no—
292C4⋊C4—64no—
293C4⋊C4—64no—
294C4⋊C4—64no—
295C4⋊C4—64no—
296C4⋊C4—64no—
297C4⋊C4—61no—
298C4⋊C4—62no—
299C4⋊C4—62no—
300C4⋊C4—61no—
301C4⋊C4—61no—
302C4⋊C4—62no—
303C4⋊C4—61no—
304C4⋊C4—62no—
305C4⋊C4(C2×D3)⋊C412yes—
306C4⋊C4(C2×D3)⋊C412yes—
307C22⋊C4—22no—
308C22⋊C4—62no—
309C42—22no—
310C42—64no—
311C42—64no—
312C42—62no—
313C42—62no—
314C42—62no—
315C42C2×D1211yescharacteristic
316C42D3×D411yescharacteristic
317C42D3×D411yescharacteristic
318C42C2×#6711yescharacteristic
319C22×C3—21no—
320C22×C3—21no—
321C22×C3#37911yescharacteristic
322C22×C3C2×#10112yes—
323C3×C4—21no—
324C3×C4—21no—
325C3×C4—21no—
326C3×C4—21no—
327C3×C4—21no—
328C3×C4—42no—
329C3×C4—42no—
330C3×C4#45411yescharacteristic
331C3×C4#45611yescharacteristic
332Dic3—24no—
333Dic3—42no—
334Dic3—42no—
335Dic3—42no—
336C23D3×C22⋊C411yescharacteristic
337Q8—41no—
338Q8—41no—
339Q8—64no—
340Q8—64no—
341Q8—81no—
342Q8—121no—
343Q8—121no—
344Q8—241no—
345Q8—244no—
346C2×C4—22no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—34no—
352C2×C4—41no—
353C2×C4—62no—
354C2×C4—62no—
355C2×C4—62no—
356C2×C4—62no—
357C2×C4—62no—
358C2×C4—122no—
359C2×C4D3×C22⋊C411yescharacteristic
360C2×C4#88211yescharacteristic
361C2×C4#88211yescharacteristic
362C2×C4C2×(C2×D3)⋊C411yescharacteristic
363C2×C4#93711yescharacteristic
364C2×C4#93711yescharacteristic
365C8—81no—
366C8—81no—
367C8—122no—
368C8—241no—
369C2×C3—22no—
370C2×C3#178411yescharacteristic
371C2×C3#179311yescharacteristic
372C2×C3#179411yescharacteristic
373C22—21no—
374C22—21no—
375C22D3×#10112yes—
376C22#495511yescharacteristic
377C4—21no—
378C4—21no—
379C4—21no—
380C4—21no—
381C4—21no—
382C4—42no—
383C4—42no—
384C4—64no—
385C4—122no—
386C4—122no—
387C4—122no—
388C4#520111yescharacteristic
389C4#520511yescharacteristic
390C3#846011yescharacteristic, complemented, Sylow
391C2—22no—
392C2#7011311yescharacteristic
393C2#7029511yescharacteristic
394C2#7030811yescharacteristic
subgroupG/H
centerC22#4955
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1552C2
socleC22×C3#379
prime radicalC23×C3C2×C22⋊C4
cosocleC2×C3×C42C23
hypercenter#1552D3
hypocenterC3#8460
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1552

Complements

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

NH ≅ G/Ntype
390. C3#8460semidirect

Maximal subgroups

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

subgroupindex
#862582
C3×#15522
#718222
#708302
#708132
#704732
#704392
#84603