Finite Group Databasebeta

← #10696571

Subgroups — #10696571

Order 768.

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

normal subgroups only

This group has 590 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, complemented
2#83199C211yesmaximal, characteristic, complemented
3#82947C211yesmaximal, characteristic
4#71776C211yesmaximal, characteristic, complemented
5#71718C211yesmaximal, characteristic, complemented
6#69432C211yesmaximal, characteristic, complemented
7#67615C211yesmaximal, characteristic, complemented
8#10094—31nomaximal, Sylow
9C2×C3×C4⋊Q8—21no—
10C2×C3×C4⋊Q8C2211yescharacteristic
11C2×C3×#124—21no—
12C2×C3×#124C2211yescharacteristic
13#5477C2211yescharacteristic
14C3×#399—22no—
15C3×#399—22no—
16C3×#376C2211yescharacteristic
17#5227—21no—
18#5217—21no—
19#5093—21no—
20#5089—22no—
21#5088—21no—
22#5058—21no—
23#5057—22no—
24#5055—21no—
25#4966—22no—
26#4926C2211yescharacteristic
27#4926C2211yescharacteristic
28C4×#208C2211yescharacteristic
29#4547—22no—
30#4540—22no—
31#4451—22no—
32#3213—31no—
33#3141—31no—
34#2658D311yescharacteristic, complemented
35#1556—31no—
36#1527—31no—
37#1505—31no—
38#1300—31no—
39C22×C3×Q8—21no—
40C22×C3×Q8D411yescharacteristic
41C22×C3×D4D411yescharacteristic
42C2×#235—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8—21no—
47C3×C4⋊Q8—44no—
48C3×#124—21no—
49C3×#124—21no—
50C3×#124—21no—
51C3×#124—21no—
52C3×#124—44no—
53C2×C3×C4⋊C4—24no—
54C2×C3×C4⋊C4—21no—
55C2×C3×C4⋊C4D412yes—
56C2×C3×C22⋊C4—24no—
57C2×C3×C22⋊C4—21no—
58C2×C3×C22⋊C4—21no—
59C2×C3×C42—21no—
60C2×C3×C42C2311yescharacteristic
61C2×#215—22no—
62C2×#213—22no—
63#930—21no—
64#930—21no—
65C2×#208—21no—
66#875—21no—
67#874D411yescharacteristic
68C4×D12—21no—
69C4×D12—21no—
70#869D411yescharacteristic
71C3×#95—21no—
72C3×#95D412yes—
73#835—22no—
74#832—22no—
75#830—21no—
76#830—21no—
77#810—24no—
78#809—24no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)D412yes—
81#803—21no—
82#803—21no—
83#803D412yes—
84C4×C3⋊C8—21no—
85#532—31no—
86C2×C4⋊Q8—21no—
87C2×C4⋊Q8C2×D311yescharacteristic
88C2×#124—21no—
89C2×#124C2×D311yescharacteristic
90#496—61no—
91#489—62no—
92#489—61no—
93#487—61no—
94#486—61no—
95#463—62no—
96#448—61no—
97#447—61no—
98#432—31no—
99#432—31no—
100#417—62no—
101C4×M16—31no—
102#399—22no—
103#399—22no—
104#376C2×D311yescharacteristic
105#332—62no—
106#329—62no—
107#327—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×#67—41no—
126#235—44no—
127C2×D12—21no—
128C2×D12—21no—
129C2×D12—21no—
130C2×C4×D3—41no—
131C2×Dic6—21no—
132C3×C4⋊C4—22no—
133C3×C4⋊C4—22no—
134C3×C4⋊C4—48no—
135C3×C4⋊C4—41no—
136C3×C4⋊C4—41no—
137C3×C22⋊C4—48no—
138C3×C22⋊C4—41no—
139C3×C22⋊C4—41no—
140C3×C22⋊C4—41no—
141C3×C22⋊C4—41no—
142C3×C42—24no—
143C3×C42C2×D411yescharacteristic
144C3×C42C2×D411yescharacteristic
145C3×C42C2×D411yescharacteristic
146C3×C42#4111yescharacteristic
147#215—82no—
148#213—82no—
149(C2×D3)⋊C4—41no—
150(C2×D3)⋊C4—41no—
151C4⋊Dic3—21no—
152C4⋊Dic3—21no—
153#208—42no—
154C2×C3⋊C8—22no—
155C2×C3⋊C8—22no—
156C2×C3⋊C8—41no—
157C2×#41—61no—
158C22×Q8—21no—
159C22×Q8#6711yescharacteristic
160C22×D4#6711yescharacteristic
161C2×SD16—62no—
162C2×D8—62no—
163C2×M16—61no—
164C4⋊Q8—21no—
165C4⋊Q8—21no—
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C4⋊Q8—31no—
169C4⋊Q8—44no—
170#127—31no—
171#124—21no—
172#124—21no—
173#124—21no—
174#124—21no—
175#124—44no—
176#124—61no—
177#121—61no—
178#121—61no—
179C4×D4—61no—
180C4×D4—61no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4#6712yes—
184C2×C22⋊C4—24no—
185C2×C22⋊C4—21no—
186C2×C22⋊C4—21no—
187C2×C42—21no—
188C2×C42C22×D311yescharacteristic
189C4⋊C8—32no—
190C4⋊C8—61no—
191#103—64no—
192#103—62no—
193#102—64no—
194#102—62no—
195C22⋊C8—61no—
196C22⋊C8—61no—
197#97—32no—
198#97—61no—
199#97—61no—
200C4×C8—61no—
201#95—21no—
202#95#6712yes—
203C23×C3—42no—
204C23×C3—42no—
205C23×C3—41no—
206C23×C3—42no—
207C23×C3#12011yescharacteristic
208C22×D3—41no—
209C3×Q8—42no—
210C3×Q8—81no—
211C3×Q8—82no—
212C3×Q8—82no—
213C3×D4—42no—
214C3×D4—81no—
215C2×C3×C4—22no—
216C2×C3×C4—22no—
217C2×C3×C4—24no—
218C2×C3×C4—41no—
219C2×C3×C4—42no—
220C2×C3×C4—42no—
221C2×C3×C4—42no—
222C2×C3×C4—42no—
223C2×C3×C4—42no—
224C2×C3×C4—41no—
225C2×C3×C4—42no—
226C2×C3×C4#12011yescharacteristic
227C2×C3×C4#12011yescharacteristic
228C2×C3×C4#12111yescharacteristic
229C2×C3×C4#12111yescharacteristic
230C2×C3×C4#12111yescharacteristic
231C2×C3×C4#12711yescharacteristic
232#67—82no—
233C2×Dic3—41no—
234D12—42no—
235D12—42no—
236D12—81no—
237C4×D3—82no—
238Dic6—42no—
239C3⋊C8—42no—
240C3⋊C8—42no—
241C3⋊C8—81no—
242C24—21no—
243#41—124no—
244C2×Q8—21no—
245C2×Q8—21no—
246C2×Q8—42no—
247C2×Q8—42no—
248C2×Q8—42no—
249C2×Q8—41no—
250C2×Q8—41no—
251C2×Q8—61no—
252C2×D4—21no—
253C2×D4—21no—
254C2×D4—42no—
255C2×D4—61no—
256C2×D4—61no—
257C2×D4—61no—
258C2×D4—121no—
259C22×C4—22no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—121no—
263C22×C4C2×#6711yescharacteristic
264C22×C4C2×#6711yescharacteristic
265C22×C4C2×#6711yescharacteristic
266SD16—242no—
267D8—242no—
268M16—122no—
269C2×C8—62no—
270C2×C8—62no—
271C2×C8—121no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—48no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—61no—
278C4⋊C4—61no—
279C22⋊C4—48no—
280C22⋊C4—41no—
281C22⋊C4—41no—
282C22⋊C4—41no—
283C22⋊C4—41no—
284C22⋊C4—121no—
285C22⋊C4—121no—
286C42—24no—
287C42D3×D411yescharacteristic
288C42D3×D411yescharacteristic
289C42D3×D411yescharacteristic
290C42#23711yescharacteristic
291C22×C3—21no—
292C22×C3—21no—
293C22×C3—84no—
294C22×C3—81no—
295C22×C3—81no—
296C22×C3—81no—
297C22×C3#39211yescharacteristic
298C22×C3#45412yes—
299C2×D3—81no—
300C2×D3—81no—
301C2×D3—81no—
302C3×C4—21no—
303C3×C4—21no—
304C3×C4—21no—
305C3×C4—21no—
306C3×C4—21no—
307C3×C4—44no—
308C3×C4—81no—
309C3×C4—82no—
310C3×C4#45312yes—
311Dic3—81no—
312C23—42no—
313C23—42no—
314C23—41no—
315C23—42no—
316C23—121no—
317C23#95311yescharacteristic
318Q8—42no—
319Q8—81no—
320Q8—82no—
321Q8—82no—
322Q8—122no—
323D4—42no—
324D4—81no—
325D4—122no—
326D4—122no—
327D4—241no—
328D4—242no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—24no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—121no—
341C2×C4—242no—
342C2×C4#93711yescharacteristic
343C2×C4#93711yescharacteristic
344C2×C4#93811yescharacteristic
345C2×C4#93911yescharacteristic
346C2×C4#93911yescharacteristic
347C2×C4#94011yescharacteristic
348C8—122no—
349C8—122no—
350C8—241no—
351C2×C3—22no—
352C2×C3—161no—
353C2×C3#200211yescharacteristic
354C2×C3#200311yescharacteristic
355C2×C3#200911yescharacteristic
356D3—161no—
357C22—21no—
358C22—21no—
359C22—84no—
360C22—81no—
361C22—81no—
362C22—81no—
363C22—241no—
364C22—241no—
365C22—241no—
366C22#522311yescharacteristic
367C22#524512yes—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—44no—
374C4—81no—
375C4—82no—
376C4—241no—
377C4#507712yes—
378C3#1009411yescharacteristic, complemented, Sylow
379C2—22no—
380C2—161no—
381C2—481no—
382C2#7176911yescharacteristic
383C2#7186711yescharacteristic
384C2#7186911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#2658C2
socleC22×C3#392
prime radical#82947C2
hypercenter#2658D3
hypocenterC3#10094
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
1. C3×#2658C2semidirect
2. #83199C2semidirect
4. #71776C2semidirect
5. #71718C2semidirect
6. #69432C2semidirect
7. #67615C2semidirect
34. #2658D3semidirect
378. C3#10094semidirect

Maximal subgroups

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

subgroupindex
C3×#26582
#831992
#829472
#717762
#717182
#694322
#676152
#100943