Finite Group Databasebeta

← #10696110

Subgroups — #10696110

Order 768.

470 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 81 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
1C3×#3166C211yesmaximal, characteristic
2#83369C211yesmaximal, characteristic
3#83042C211yesmaximal, characteristic
4#71837C211yesmaximal, characteristic
5#71736C211yesmaximal, characteristic
6#69857C211yesmaximal, characteristic
7#67666C211yesmaximal, characteristic
8#10025—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×#125C2211yescharacteristic
11C2×C4×Dic6C2211yescharacteristic
12C3×#501—21no—
13C3×#501—21no—
14C3×#499—22no—
15C3×#428—21no—
16C3×#428—21no—
17C3×C4×M16C2211yescharacteristic
18#5094—21no—
19#5090—21no—
20#5050—21no—
21#5043—21no—
22#5030—22no—
23#4986—21no—
24#4985—21no—
25#4983—21no—
26#4979—21no—
27#4978—22no—
28#4937C2211yescharacteristic
29#4926C2211yescharacteristic
30#4926C2211yescharacteristic
31#4543—22no—
32#4538—22no—
33#4456—22no—
34#3166D311yescharacteristic
35#3113—31no—
36#2634—31no—
37#1534—31no—
38#1515—31no—
39#1491—31no—
40#1288—31no—
41C22×Dic6D411yescharacteristic
42C2×C3×M16—21no—
43C3×C4⋊Q8D411yescharacteristic
44C3×C4⋊Q8D411yescharacteristic
45C3×#125—21no—
46C3×#125—21no—
47C3×#125—21no—
48C3×#125—21no—
49C3×#125—21no—
50C3×C22⋊Q8—21no—
51C3×C22⋊Q8—21no—
52C3×C4×Q8—21no—
53C3×C4×Q8—21no—
54C3×#117—21no—
55C2×C3×C4⋊C4—21no—
56C2×C3×C4⋊C4—21no—
57C2×C3×C4⋊C4D411yescharacteristic
58C2×C3×C4⋊C4D411yescharacteristic
59C2×C3×C42C2311yescharacteristic
60C2×#216—22no—
61C2×C4⋊Dic3D411yescharacteristic
62C2×Dic3⋊C4—22no—
63C2×Dic3⋊C4—21no—
64C2×C4×Dic3—21no—
65C4×Dic6—21no—
66C4×Dic6—21no—
67C4×Dic6—21no—
68C4×Dic6—21no—
69C3×#107—22no—
70C3×#107—22no—
71C3×#106—22no—
72C3×#106—22no—
73C3×#97—21no—
74C3×C4×C8D412yes—
75#835—22no—
76C3⋊#95—22no—
77C3⋊#95—22no—
78C3⋊#95—22no—
79C3⋊#95—21no—
80C3⋊#95—21no—
81C3⋊#95D412yes—
82#830—21no—
83#830—21no—
84#810—22no—
85#808—22no—
86C4⋊(C3⋊C8)—21no—
87C4⋊(C3⋊C8)—21no—
88#803—21no—
89#803D412yes—
90#533C2×D311yescharacteristic
91C2×#125C2×D311yescharacteristic
92C2×C4×Q8—31no—
93#501—21no—
94#501—21no—
95#499—22no—
96#483—61no—
97#475—61no—
98#462—61no—
99#451—61no—
100#441—62no—
101#432—31no—
102#432—31no—
103#428—21no—
104#428—21no—
105C4×M16C2×D311yescharacteristic
106#400—61no—
107#400—61no—
108#398—61no—
109#395—61no—
110#387—62no—
111#383—31no—
112#330—62no—
113#328—62no—
114Q8⋊C8—62no—
115C2×C3×Q8—21no—
116C2×C3×Q8—21no—
117C22×C3×C4—21no—
118C22×C3×C4—21no—
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×Dic3—21no—
123C22×Dic3—22no—
124C22×Dic3—21no—
125C2×Dic6—21no—
126C2×Dic6—21no—
127C2×Dic6—42no—
128C3×M16—24no—
129C2×C3×C8—22no—
130C3×C4⋊C4—21no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—21no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—41no—
139C3×C4⋊C4—41no—
140C3×C4⋊C4—41no—
141C3×C4⋊C4—41no—
142C3×C4⋊C4—41no—
143C3×C4⋊C4—41no—
144C3×C22⋊C4—41no—
145C3×C42—41no—
146C3×C42C2×D411yescharacteristic
147C3×C42C2×D411yescharacteristic
148C3×C42C2×D411yescharacteristic
149C3×C42#4111yescharacteristic
150#216—44no—
151C4⋊Dic3—21no—
152C4⋊Dic3—21no—
153Dic3⋊C4—44no—
154Dic3⋊C4—41no—
155Dic3⋊C4—41no—
156C4×Dic3—41no—
157C4×Dic3—41no—
158C2×C3⋊C8—22no—
159C2×C3⋊C8—41no—
160C22×Q8—31no—
161C2×Q16—62no—
162C2×M16—21no—
163C4⋊Q8#6711yescharacteristic
164C4⋊Q8#6711yescharacteristic
165#125—21no—
166#125—21no—
167#125—21no—
168#125—21no—
169#125—21no—
170C22⋊Q8—21no—
171C22⋊Q8—21no—
172C4×Q8—21no—
173C4×Q8—21no—
174C4×Q8—61no—
175C4×Q8—61no—
176C4×Q8—61no—
177C4×Q8—61no—
178#117—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—31no—
182C2×C4⋊C4—62no—
183C2×C4⋊C4—61no—
184C2×C4⋊C4#6711yescharacteristic
185C2×C4⋊C4#6711yescharacteristic
186C2×C42—61no—
187C2×C42C22×D311yescharacteristic
188#107—22no—
189#107—22no—
190#106—22no—
191#106—22no—
192#106—62no—
193C4⋊C8—61no—
194C4⋊C8—61no—
195#103—62no—
196#103—62no—
197C22⋊C8—61no—
198C22⋊C8—61no—
199#97—21no—
200#97—32no—
201#97—61no—
202C4×C8#6712yes—
203#95—32no—
204#95—62no—
205#95—62no—
206#95—62no—
207#95—61no—
208#95—61no—
209C23×C3#12011yescharacteristic
210C3×Q8—42no—
211C3×Q8—81no—
212C2×C3×C4—22no—
213C2×C3×C4—41no—
214C2×C3×C4—41no—
215C2×C3×C4—42no—
216C2×C3×C4—42no—
217C2×C3×C4—41no—
218C2×C3×C4—41no—
219C2×C3×C4—41no—
220C2×C3×C4—41no—
221C2×C3×C4#12011yescharacteristic
222C2×C3×C4#12011yescharacteristic
223C2×C3×C4#12111yescharacteristic
224C2×C3×C4#12111yescharacteristic
225C2×C3×C4#12111yescharacteristic
226C2×C3×C4#12711yescharacteristic
227C2×Dic3—22no—
228C2×Dic3—41no—
229C2×Dic3—41no—
230C2×Dic3—42no—
231C2×Dic3—42no—
232C2×Dic3—41no—
233C2×Dic3—42no—
234C2×Dic3—42no—
235Dic6—22no—
236Dic6—22no—
237Dic6—41no—
238Dic6—41no—
239C3×C8—24no—
240C3⋊C8—42no—
241C3⋊C8—81no—
242C2×Q8—21no—
243C2×Q8—21no—
244C2×Q8—61no—
245C2×Q8—61no—
246C2×Q8—122no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—61no—
250C22×C4—62no—
251C22×C4—61no—
252C22×C4D3×D411yescharacteristic
253C22×C4D3×D411yescharacteristic
254C22×C4C2×#6711yescharacteristic
255Q16—124no—
256M16—24no—
257C2×C8—22no—
258C2×C8—62no—
259C2×C8—121no—
260C4⋊C4—21no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—61no—
275C4⋊C4—61no—
276C4⋊C4—124no—
277C4⋊C4—121no—
278C4⋊C4—121no—
279C22⋊C4—41no—
280C42—41no—
281C42—121no—
282C42—121no—
283C42D3×D411yescharacteristic
284C42#23711yescharacteristic
285C42C2×#6711yescharacteristic
286C42C2×#6711yescharacteristic
287C22×C3—21no—
288C22×C3—21no—
289C22×C3#39211yescharacteristic
290C22×C3#45612yes—
291C3×C4—21no—
292C3×C4—21no—
293C3×C4—21no—
294C3×C4—21no—
295C3×C4—21no—
296C3×C4—21no—
297C3×C4—81no—
298C3×C4—81no—
299C3×C4—81no—
300C3×C4—81no—
301Dic3—42no—
302Dic3—82no—
303Dic3—81no—
304C23#93711yescharacteristic
305Q8—42no—
306Q8—62no—
307Q8—62no—
308Q8—81no—
309Q8—121no—
310Q8—121no—
311C2×C4—22no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—62no—
321C2×C4—121no—
322C2×C4—121no—
323C2×C4—122no—
324C2×C4—122no—
325C2×C4—121no—
326C2×C4—122no—
327C2×C4—122no—
328C2×C4#93711yescharacteristic
329C2×C4#93811yescharacteristic
330C2×C4#93911yescharacteristic
331C2×C4#93911yescharacteristic
332C2×C4#94011yescharacteristic
333C2×C4#95311yescharacteristic
334C8—24no—
335C8—122no—
336C8—241no—
337C2×C3—22no—
338C2×C3#200511yescharacteristic
339C2×C3#200911yescharacteristic
340C2×C3#201011yescharacteristic
341C22—21no—
342C22—21no—
343C22#509512yes—
344C22#522311yescharacteristic
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—21no—
350C4—21no—
351C4—81no—
352C4—81no—
353C4—81no—
354C4—81no—
355C4—122no—
356C4—242no—
357C4—241no—
358C3#1002511yescharacteristic, complemented, Sylow
359C2—22no—
360C2#7179511yescharacteristic
361C2#7211811yescharacteristic
362C2#7212011yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3166C2
socleC22×C3#392
prime radicalC23×C3#120
hypercenter#3166D3
hypocenterC3#10025
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3166

Complements

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

NH ≅ G/Ntype
358. C3#10025semidirect

Maximal subgroups

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

subgroupindex
C3×#31662
#833692
#830422
#718372
#717362
#698572
#676662
#100253