Finite Group Databasebeta

← #10696627

Subgroups — #10696627

Order 768.

568 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 85 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×#2658C211yesmaximal, characteristic
2#83207C211yesmaximal, characteristic, complemented
3#82949C211yesmaximal, characteristic
4#71776C211yesmaximal, characteristic, complemented
5#71718C211yesmaximal, characteristic
6#69433C211yesmaximal, characteristic, complemented
7#67615C211yesmaximal, characteristic, complemented
8#10102—31nomaximal, Sylow
9C2×C3×C4⋊Q8—21no—
10C2×C3×C4⋊Q8C2211yescharacteristic
11C2×C3×#124—21no—
12C2×C3×#124C2211yescharacteristic
13#5470C2211yescharacteristic
14C3×#399—22no—
15C3×#399—22no—
16C3×#376C2211yescharacteristic
17#5228—21no—
18#5218—21no—
19#5081—22no—
20#5080—21no—
21#5078—21no—
22#5060—21no—
23#5055—21no—
24#5054—22no—
25#4964—22no—
26#4926C2211yescharacteristic
27#4926C2211yescharacteristic
28C4×#208C2211yescharacteristic
29#4547—22no—
30#4540—22no—
31#4452—22no—
32#3215—31no—
33#3165—31no—
34#2658D311yescharacteristic
35#1556—31no—
36#1527—31no—
37#1506—31no—
38#1300—31no—
39C22×C3×Q8—21no—
40C22×C3×Q8D411yescharacteristic
41C22×C3×D4D411yescharacteristic
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8—44no—
47C3×#124—21no—
48C3×#124—21no—
49C3×#124—21no—
50C3×#124—21no—
51C3×#124—44no—
52C2×C3×C4⋊C4—24no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4D412yes—
55C2×C3×C22⋊C4—24no—
56C2×C3×C22⋊C4—21no—
57C2×C3×C22⋊C4—21no—
58C2×C3×C42—21no—
59C2×C3×C42C2311yescharacteristic
60C2×#216—22no—
61C2×#214—22no—
62#926—21no—
63#924—21no—
64#924—21no—
65C2×#208—21no—
66#870—21no—
67#869D411yescharacteristic
68#869D411yescharacteristic
69C4×Dic6—21no—
70C4×Dic6—21no—
71C3×#95—21no—
72C3×#95D412yes—
73#835—22no—
74#832—22no—
75#830—21no—
76#830—21no—
77#808—24no—
78#807—24no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)D412yes—
81#803—21no—
82#803—21no—
83#803D412yes—
84C4×C3⋊C8—21no—
85#533—31no—
86C2×C4⋊Q8—21no—
87C2×C4⋊Q8C2×D311yescharacteristic
88C2×#124—21no—
89C2×#124C2×D311yescharacteristic
90#501—62no—
91#500—61no—
92#498—61no—
93#497—61no—
94#489—61no—
95#464—62no—
96#451—61no—
97#450—61no—
98#432—31no—
99#432—31no—
100#428—62no—
101C4×M16—31no—
102#399—22no—
103#399—22no—
104#376C2×D311yescharacteristic
105#332—62no—
106#329—62no—
107#328—62no—
108C24×C3—21no—
109C2×C3×Q8—21no—
110C2×C3×Q8—21no—
111C2×C3×Q8—42no—
112C2×C3×Q8—42no—
113C2×C3×Q8—42no—
114C2×C3×Q8—41no—
115C2×C3×Q8—41no—
116C2×C3×D4—21no—
117C2×C3×D4—21no—
118C2×C3×D4—42no—
119C22×C3×C4—22no—
120C22×C3×C4—21no—
121C22×C3×C4—22no—
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C2×Dic6—21no—
126C2×Dic6—21no—
127C3×C4⋊C4—22no—
128C3×C4⋊C4—22no—
129C3×C4⋊C4—48no—
130C3×C4⋊C4—41no—
131C3×C4⋊C4—41no—
132C3×C22⋊C4—48no—
133C3×C22⋊C4—41no—
134C3×C22⋊C4—41no—
135C3×C22⋊C4—41no—
136C3×C22⋊C4—41no—
137C3×C42—24no—
138C3×C42C2×D411yescharacteristic
139C3×C42C2×D411yescharacteristic
140C3×C42C2×D411yescharacteristic
141C3×C42#4111yescharacteristic
142#217—41no—
143#216—82no—
144#214—82no—
145C4⋊Dic3—21no—
146C4⋊Dic3—21no—
147C4⋊Dic3—21no—
148C4⋊Dic3—21no—
149Dic3⋊C4—41no—
150Dic3⋊C4—41no—
151C4×Dic3—41no—
152#208—24no—
153C2×C3⋊C8—22no—
154C2×C3⋊C8—22no—
155C2×C3⋊C8—41no—
156C22×Q8—21no—
157C22×Q8#6711yescharacteristic
158C22×D4#6711yescharacteristic
159C2×Q16—62no—
160C2×SD16—62no—
161C2×M16—61no—
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164C4⋊Q8—21no—
165C4⋊Q8—21no—
166C4⋊Q8—31no—
167C4⋊Q8—31no—
168C4⋊Q8—44no—
169#125—61no—
170#124—21no—
171#124—21no—
172#124—21no—
173#124—21no—
174#124—44no—
175C22⋊Q8—61no—
176C22⋊Q8—61no—
177C4×Q8—61no—
178C4×Q8—61no—
179#117—61no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4#6712yes—
183C2×C22⋊C4—24no—
184C2×C22⋊C4—21no—
185C2×C22⋊C4—21no—
186C2×C42—21no—
187C2×C42C22×D311yescharacteristic
188#107—64no—
189#106—64no—
190C4⋊C8—32no—
191C4⋊C8—61no—
192#103—62no—
193#102—62no—
194C22⋊C8—61no—
195C22⋊C8—61no—
196#97—32no—
197#97—61no—
198#97—61no—
199C4×C8—61no—
200#95—21no—
201#95#6712yes—
202C23×C3—42no—
203C23×C3—42no—
204C23×C3—41no—
205C23×C3—42no—
206C23×C3#12011yescharacteristic
207C3×Q8—42no—
208C3×Q8—81no—
209C3×Q8—82no—
210C3×Q8—82no—
211C3×D4—42no—
212C3×D4—81no—
213C2×C3×C4—22no—
214C2×C3×C4—22no—
215C2×C3×C4—24no—
216C2×C3×C4—41no—
217C2×C3×C4—42no—
218C2×C3×C4—42no—
219C2×C3×C4—42no—
220C2×C3×C4—42no—
221C2×C3×C4—42no—
222C2×C3×C4—41no—
223C2×C3×C4—42no—
224C2×C3×C4#12011yescharacteristic
225C2×C3×C4#12011yescharacteristic
226C2×C3×C4#12111yescharacteristic
227C2×C3×C4#12111yescharacteristic
228C2×C3×C4#12111yescharacteristic
229C2×C3×C4#12711yescharacteristic
230C2×Dic3—41no—
231C2×Dic3—41no—
232Dic6—42no—
233Dic6—81no—
234C3⋊C8—24no—
235C3⋊C8—42no—
236C3⋊C8—81no—
237C24—21no—
238C2×Q8—21no—
239C2×Q8—21no—
240C2×Q8—42no—
241C2×Q8—42no—
242C2×Q8—42no—
243C2×Q8—41no—
244C2×Q8—41no—
245C2×Q8—61no—
246C2×Q8—61no—
247C2×D4—21no—
248C2×D4—21no—
249C2×D4—42no—
250C22×C4—22no—
251C22×C4—21no—
252C22×C4—22no—
253C22×C4C2×#6711yescharacteristic
254C22×C4C2×#6711yescharacteristic
255C22×C4C2×#6711yescharacteristic
256Q16—242no—
257SD16—242no—
258M16—64no—
259C2×C8—62no—
260C2×C8—62no—
261C2×C8—121no—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—48no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—61no—
268C4⋊C4—61no—
269C4⋊C4—61no—
270C4⋊C4—61no—
271C4⋊C4—121no—
272C4⋊C4—121no—
273C22⋊C4—48no—
274C22⋊C4—41no—
275C22⋊C4—41no—
276C22⋊C4—41no—
277C22⋊C4—41no—
278C22⋊C4—121no—
279C42—24no—
280C42—121no—
281C42D3×D411yescharacteristic
282C42D3×D411yescharacteristic
283C42D3×D411yescharacteristic
284C42#23711yescharacteristic
285C22×C3—21no—
286C22×C3—21no—
287C22×C3—84no—
288C22×C3—81no—
289C22×C3—81no—
290C22×C3—81no—
291C22×C3#39211yescharacteristic
292C22×C3#45612yes—
293C3×C4—21no—
294C3×C4—21no—
295C3×C4—21no—
296C3×C4—21no—
297C3×C4—21no—
298C3×C4—44no—
299C3×C4—81no—
300C3×C4—82no—
301C3×C4#45312yes—
302Dic3—81no—
303Dic3—81no—
304C23—42no—
305C23—42no—
306C23—41no—
307C23—42no—
308C23#95311yescharacteristic
309Q8—42no—
310Q8—81no—
311Q8—82no—
312Q8—82no—
313Q8—122no—
314Q8—241no—
315D4—42no—
316D4—81no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—24no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—41no—
327C2×C4—42no—
328C2×C4—121no—
329C2×C4—121no—
330C2×C4#93711yescharacteristic
331C2×C4#93711yescharacteristic
332C2×C4#93811yescharacteristic
333C2×C4#93911yescharacteristic
334C2×C4#93911yescharacteristic
335C2×C4#94011yescharacteristic
336C8—64no—
337C8—122no—
338C8—241no—
339C2×C3—22no—
340C2×C3—161no—
341C2×C3#200211yescharacteristic
342C2×C3#200511yescharacteristic
343C2×C3#200911yescharacteristic
344C22—21no—
345C22—21no—
346C22—84no—
347C22—81no—
348C22—81no—
349C22—81no—
350C22#522311yescharacteristic
351C22#524612yes—
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—44no—
358C4—81no—
359C4—82no—
360C4—241no—
361C4—241no—
362C4#507712yes—
363C3#1010211yescharacteristic, complemented, Sylow
364C2—22no—
365C2—161no—
366C2#7176911yescharacteristic
367C2#7186811yescharacteristic
368C2#7186911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#2658C2
socleC22×C3#392
prime radicalC22×C3×D4D4
hypercenter#2658D3
hypocenterC3#10102
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2658

Complements

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

NH ≅ G/Ntype
2. #83207C2semidirect
4. #71776C2semidirect
6. #69433C2semidirect
7. #67615C2semidirect
363. C3#10102semidirect

Maximal subgroups

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

subgroupindex
C3×#26582
#832072
#829492
#717762
#717182
#694332
#676152
#101023