Finite Group Databasebeta

← #10690107

Subgroups — #10690107

Order 768.

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

normal subgroups only

This group has 474 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#83027C211yesmaximal, characteristic
2C2×#5064C211yesmaximal, characteristic
3C2×C3×#330C211yesmaximal, characteristic
4#69828C211yesmaximal, characteristic
5#69822C211yesmaximal, characteristic
6#67659C211yesmaximal, characteristic
7#67659C211yesmaximal, characteristic
8#9000—31nomaximal, Sylow
9C2×C3×#125C2211yescharacteristic
10C2×#869C2211yescharacteristic
11C2×C3×#97C2211yescharacteristic
12C2×C3×#97C2211yescharacteristic
13C2×#816—21no—
14C2×#816—21no—
15#5064—22no—
16#5064C414yescomplemented
17#4984—21no—
18#4984—21no—
19#4980—21no—
20#4980—21no—
21#4980—21no—
22#4980—21no—
23C2×#808—21no—
24C2×#807—21no—
25#4936C2211yescharacteristic
26#4936C2211yescharacteristic
27C2×#803C2211yescharacteristic
28C3×#330—22no—
29#4551—21no—
30#4551—21no—
31C2×#501—31no—
32#2382—31no—
33#1993—31no—
34#1960—31no—
35C2×#330D311yescharacteristic
36#1295—31no—
37#1295—31no—
38C22×Dic6—21no—
39C22×C3×C8—21no—
40C22×C3×C8—21no—
41C3×#125—21no—
42C3×#125—21no—
43C3×#125C2×C412yes—
44C3×#125D412yes—
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4D411yescharacteristic
48C2×C3×C4⋊C4D411yescharacteristic
49C2×C3×C42C2311yescharacteristic
50C2×C4⋊Dic3—21no—
51C2×C4⋊Dic3—21no—
52C2×C4⋊Dic3—21no—
53C2×C4⋊Dic3—21no—
54C2×C4⋊Dic3D411yescharacteristic
55C2×C4⋊Dic3D411yescharacteristic
56C2×C4⋊Dic3D411yescharacteristic
57C2×C4⋊Dic3D411yescharacteristic
58C2×C4×Dic3—21no—
59C2×C4×Dic3—21no—
60C22×C3⋊C8—21no—
61#869—21no—
62#869—21no—
63#869—21no—
64#869C2×C412yes—
65C3×#97—21no—
66C3×#97—21no—
67C3×#97—21no—
68C3×#97—21no—
69C3⋊#95—21no—
70C3⋊#95—21no—
71C3⋊#95—21no—
72C3⋊#95—21no—
73C3⋊#95—21no—
74C3⋊#95—21no—
75C3⋊#95—21no—
76C3⋊#95—21no—
77#816—42no—
78#816—42no—
79#808—24no—
80#807—24no—
81#803—21no—
82#803C2×C412yes—
83C2×C4⋊Q8—31no—
84C2×#125C2×D311yescharacteristic
85#501—34no—
86#501—62no—
87C2×#107—61no—
88C2×#106—61no—
89C2×#103—61no—
90C2×#103—61no—
91C2×#97—31no—
92C2×#97C2×D311yescharacteristic
93C2×#97C2×D311yescharacteristic
94#389—61no—
95#389—61no—
96#387—61no—
97#387—61no—
98#387—61no—
99#387—61no—
100#382—31no—
101#382—31no—
102#340—61no—
103#340—61no—
104#330—22no—
105C22×C3×C4—21no—
106C22×C3×C4—21no—
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4C2×D411yescharacteristic
110C22×Dic3—21no—
111C22×Dic3—21no—
112C22×Dic3—21no—
113C22×Dic3—21no—
114C22×Dic3—41no—
115C2×Dic6—22no—
116C2×Dic6—41no—
117C2×Dic6—41no—
118C2×Dic6—41no—
119C2×Dic6—41no—
120C2×Dic6—41no—
121C2×C3×C8—22no—
122C2×C3×C8—22no—
123C2×C3×C8—41no—
124C2×C3×C8—41no—
125C2×C3×C8—41no—
126C2×C3×C8—41no—
127C3×C4⋊C4—22no—
128C3×C4⋊C4—22no—
129C3×C4⋊C4—22no—
130C3×C4⋊C4—22no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—21no—
133C3×C4⋊C4C22⋊C412yes—
134C3×C4⋊C4C22⋊C412yes—
135C3×C42C22×C411yescharacteristic
136C3×C42C2×D411yescharacteristic
137C3×C42#4111yescharacteristic
138C3×C42#4111yescharacteristic
139C4⋊Dic3—22no—
140C4⋊Dic3—22no—
141C4⋊Dic3—41no—
142C4⋊Dic3—41no—
143C4⋊Dic3—41no—
144C4⋊Dic3—41no—
145C4⋊Dic3—41no—
146C4⋊Dic3—41no—
147C4⋊Dic3—81no—
148C4⋊Dic3—81no—
149C4×Dic3—81no—
150C4×Dic3—81no—
151C2×C3⋊C8—22no—
152C2×C3⋊C8—22no—
153C2×C3⋊C8—22no—
154C22×Q8—61no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—61no—
158C4⋊Q8—32no—
159C4⋊Q8—61no—
160C4⋊Q8—61no—
161C4⋊Q8—61no—
162#125—21no—
163#125—21no—
164#125C4×D312yes—
165#125#6712yes—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—31no—
169C2×C4⋊C4—31no—
170C2×C4⋊C4—31no—
171C2×C4⋊C4—31no—
172C2×C4⋊C4—61no—
173C2×C4⋊C4—61no—
174C2×C4⋊C4—61no—
175C2×C4⋊C4—61no—
176C2×C4⋊C4D1211yescharacteristic
177C2×C4⋊C4D1211yescharacteristic
178C2×C42—61no—
179C2×C42—61no—
180C2×C42C22×D311yescharacteristic
181#107—64no—
182#106—64no—
183#103—122no—
184#103—122no—
185#97—21no—
186#97—21no—
187#97—21no—
188#97—21no—
189#97—32no—
190#97—61no—
191#95—61no—
192#95—61no—
193#95—61no—
194#95—61no—
195#95—61no—
196#95—61no—
197#95—61no—
198#95—61no—
199C23×C3#12011yescharacteristic
200C2×C3×C4—21no—
201C2×C3×C4—21no—
202C2×C3×C4—21no—
203C2×C3×C4—21no—
204C2×C3×C4—21no—
205C2×C3×C4—21no—
206C2×C3×C4—22no—
207C2×C3×C4—22no—
208C2×C3×C4—41no—
209C2×C3×C4—41no—
210C2×C3×C4—41no—
211C2×C3×C4—41no—
212C2×C3×C4C2×C22⋊C411yescharacteristic
213C2×C3×C4C4×D411yescharacteristic
214C2×C3×C4C4×D411yescharacteristic
215C2×C3×C4#12111yescharacteristic
216C2×C3×C4#12111yescharacteristic
217C2×C3×C4#12311yescharacteristic
218C2×Dic3—42no—
219C2×Dic3—41no—
220C2×Dic3—41no—
221C2×Dic3—41no—
222C2×Dic3—41no—
223C2×Dic3—41no—
224C2×Dic3—41no—
225C2×Dic3—41no—
226C2×Dic3—41no—
227C2×Dic3—41no—
228C2×Dic3—41no—
229C2×Dic3—41no—
230C2×Dic3—41no—
231C2×Dic3—81no—
232C2×Dic3—81no—
233Dic6—42no—
234Dic6—42no—
235Dic6—81no—
236Dic6—81no—
237C3×C8—42no—
238C3×C8—42no—
239C3⋊C8—24no—
240C2×Q8—62no—
241C2×Q8—121no—
242C2×Q8—121no—
243C2×Q8—121no—
244C2×Q8—121no—
245C2×Q8—121no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—61no—
249C22×C4—61no—
250C22×C4—61no—
251C22×C4—61no—
252C22×C4—121no—
253C22×C4C2×D1211yescharacteristic
254C22×C4D3×D411yescharacteristic
255C22×C4D3×D411yescharacteristic
256C2×C8—22no—
257C2×C8—22no—
258C2×C8—41no—
259C2×C8—41no—
260C2×C8—41no—
261C2×C8—41no—
262C2×C8—62no—
263C2×C8—62no—
264C2×C8—62no—
265C4⋊C4—22no—
266C4⋊C4—22no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—22no—
270C4⋊C4—22no—
271C4⋊C4—62no—
272C4⋊C4—62no—
273C4⋊C4—121no—
274C4⋊C4—121no—
275C4⋊C4—121no—
276C4⋊C4—121no—
277C4⋊C4—121no—
278C4⋊C4—121no—
279C4⋊C4—241no—
280C4⋊C4—241no—
281C4⋊C4(C2×D3)⋊C412yes—
282C4⋊C4(C2×D3)⋊C412yes—
283C42—241no—
284C42—241no—
285C42C2×C4×D311yescharacteristic
286C42#23711yescharacteristic
287C42#23711yescharacteristic
288C42C2×#6711yescharacteristic
289C22×C3#38611yescharacteristic
290C22×C3#45311yescharacteristic
291C22×C3#45411yescharacteristic
292C22×C3#45511yescharacteristic
293C22×C3#45611yescharacteristic
294C22×C3#45611yescharacteristic
295C22×C3#45611yescharacteristic
296C3×C4—21no—
297C3×C4—21no—
298C3×C4—21no—
299C3×C4—21no—
300C3×C4—21no—
301C3×C4—21no—
302C3×C4—42no—
303C3×C4—42no—
304Dic3—82no—
305Dic3—81no—
306Dic3—81no—
307Dic3—81no—
308Dic3—81no—
309C23#88211yescharacteristic
310Q8—122no—
311Q8—122no—
312Q8—241no—
313Q8—241no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—122no—
327C2×C4—121no—
328C2×C4—121no—
329C2×C4—121no—
330C2×C4—121no—
331C2×C4—121no—
332C2×C4—121no—
333C2×C4—121no—
334C2×C4—121no—
335C2×C4—121no—
336C2×C4—121no—
337C2×C4—121no—
338C2×C4—121no—
339C2×C4—241no—
340C2×C4—241no—
341C2×C4#67⋊C411yescharacteristic
342C2×C4#67⋊C411yescharacteristic
343C2×C4#88611yescharacteristic
344C2×C4C2×(C2×D3)⋊C411yescharacteristic
345C2×C4#93911yescharacteristic
346C2×C4#93911yescharacteristic
347C8—42no—
348C8—42no—
349C8—64no—
350C2×C3#166011yescharacteristic
351C2×C3#166111yescharacteristic
352C2×C3#166311yescharacteristic
353C2×C3#166411yescharacteristic
354C2×C3#188211yescharacteristic
355C2×C3#188311yescharacteristic
356C2×C3#188611yescharacteristic
357C22#475211yescharacteristic
358C22#475211yescharacteristic
359C22#482211yescharacteristic
360C22#482511yescharacteristic
361C22#482611yescharacteristic
362C22#482711yescharacteristic
363C22#495811yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—21no—
369C4—21no—
370C4—42no—
371C4—42no—
372C4—241no—
373C4—241no—
374C4—241no—
375C4—241no—
376C4—242no—
377C3#900011yescharacteristic, complemented, Sylow
378C2#6837211yescharacteristic
379C2#6837311yescharacteristic
380C2#6837611yescharacteristic
381C2#6837811yescharacteristic
382C2#7010311yescharacteristic
383C2#7036511yescharacteristic
384C2#7036611yescharacteristic
subgroupG/H
centerC23#882
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC2×C3×#330C2
socle, prime radicalC23×C3#120
cosocleC2×C3×C42C23
hypercenterC2×#330D3
hypocenterC3#9000
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#330

Complements

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

NH ≅ G/Ntype
16. #5064C4semidirect
377. C3#9000semidirect

Maximal subgroups

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

subgroupindex
#830272
C2×#50642
C2×C3×#3302
#698282
#698222
#676592
#90003