Finite Group Databasebeta

← #10696543

Subgroups — #10696543

Order 768.

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

normal subgroups only

This group has 598 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×#2640C211yesmaximal, characteristic, complemented
2#83203C211yesmaximal, characteristic, complemented
3#82947C211yesmaximal, characteristic
4#71777C211yesmaximal, characteristic, complemented
5#71718C211yesmaximal, characteristic, complemented
6#69432C211yesmaximal, characteristic, complemented
7#67615C211yesmaximal, characteristic, complemented
8#10090—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×C3×#124C2211yescharacteristic
11C2×#942—22no—
12#5477C2211yescharacteristic
13C3×#397—22no—
14C3×#393—22no—
15C3×#391—21no—
16C3×#390—21no—
17C3×#376C2211yescharacteristic
18#5227—21no—
19#5217—21no—
20#5092—21no—
21#5086—22no—
22#5085—21no—
23#5058—21no—
24#5057—22no—
25#5055—21no—
26#4926C2211yescharacteristic
27#4926C2211yescharacteristic
28C4×#208C2211yescharacteristic
29#4546—22no—
30#4540—22no—
31#4451—22no—
32#3213—31no—
33#3152—31no—
34#2640D311yescharacteristic, complemented
35#1558—31no—
36#1527—31no—
37#1505—31no—
38#1300—31no—
39C22×C3×Q8D411yescharacteristic
40C22×C3×Q8D412yes—
41C22×C3×D4D411yescharacteristic
42C2×#235—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8—21no—
47C3×#124—21no—
48C3×#124—21no—
49C3×#124—21no—
50C3×#124—21no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C4⋊C4—21no—
53C2×C3×C22⋊C4—21no—
54C2×C3×C22⋊C4—21no—
55C2×C3×C22⋊C4—24no—
56C2×C3×C42—21no—
57C2×C3×C42C2311yescharacteristic
58C2×#216—24no—
59#942—48no—
60C2×#215—24no—
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—24no—
72C3×#95—21no—
73C3×#95D412yes—
74#835—22no—
75#832—22no—
76#830—21no—
77#830—21no—
78C4⋊(C3⋊C8)—21no—
79C4⋊(C3⋊C8)D412yes—
80#803—21no—
81#803—21no—
82#803D412yes—
83C4×C3⋊C8—21no—
84C2×#137—62no—
85#532—31no—
86C2×C4⋊Q8C2×D311yescharacteristic
87C2×#124C2×D311yescharacteristic
88#497—62no—
89#496—61no—
90#494—61no—
91#492—61no—
92#489—61no—
93#463—62no—
94#448—61no—
95#447—61no—
96#432—31no—
97#432—31no—
98C4×M16—31no—
99#397—22no—
100#393—22no—
101#391—21no—
102#390—21no—
103#376C2×D311yescharacteristic
104#333—62no—
105#329—62no—
106#327—62no—
107C24×C3—21no—
108C2×C3×Q8—21no—
109C2×C3×Q8—22no—
110C2×C3×Q8—21no—
111C2×C3×Q8—22no—
112C2×C3×Q8—42no—
113C2×C3×Q8—44no—
114C2×C3×D4—21no—
115C2×C3×D4—21no—
116C2×C3×D4—42no—
117C22×C3×C4—21no—
118C22×C3×C4—22no—
119C22×C3×C4—22no—
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C2×#67—41no—
124#235—44no—
125C2×D12—21no—
126C2×D12—21no—
127C2×D12—21no—
128C2×C4×D3—41no—
129C2×Dic6—21no—
130C3×C4⋊C4—41no—
131C3×C4⋊C4—41no—
132C3×C4⋊C4—41no—
133C3×C4⋊C4—41no—
134C3×C22⋊C4—41no—
135C3×C22⋊C4—41no—
136C3×C22⋊C4—41no—
137C3×C22⋊C4—41no—
138C3×C22⋊C4—84no—
139C3×C42—42no—
140C3×C42C2×D411yescharacteristic
141C3×C42C2×D411yescharacteristic
142C3×C42C2×D411yescharacteristic
143C3×C42#4111yescharacteristic
144#216—48no—
145#215—48no—
146#215—82no—
147#213—82no—
148(C2×D3)⋊C4—41no—
149(C2×D3)⋊C4—41no—
150C4⋊Dic3—21no—
151C4⋊Dic3—21no—
152#208—24no—
153C2×C3⋊C8—22no—
154C2×C3⋊C8—22no—
155C2×C3⋊C8—41no—
156C2×#41—61no—
157C22×Q8#6711yescharacteristic
158C22×Q8#6712yes—
159C22×D4#6711yescharacteristic
160#137—128no—
161C2×Q16—64no—
162C2×SD16—64no—
163C2×SD16—62no—
164C2×D8—62no—
165C2×M16—61no—
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170C4⋊Q8—31no—
171#127—31no—
172#124—21no—
173#124—21no—
174#124—21no—
175#124—21no—
176#124—61no—
177#121—61no—
178#121—61no—
179C4×D4—61no—
180C4×D4—61no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C22⋊C4—21no—
184C2×C22⋊C4—21no—
185C2×C22⋊C4—24no—
186C2×C42—21no—
187C2×C42C22×D311yescharacteristic
188C4⋊C8—32no—
189C4⋊C8—61no—
190#103—62no—
191#102—62no—
192C22⋊C8—61no—
193C22⋊C8—61no—
194#97—32no—
195#97—61no—
196#97—61no—
197C4×C8—61no—
198#95—24no—
199#95—21no—
200#95#6712yes—
201C23×C3—42no—
202C23×C3—42no—
203C23×C3—41no—
204C23×C3—42no—
205C23×C3#12011yescharacteristic
206C22×D3—41no—
207C3×Q8—44no—
208C3×Q8—42no—
209C3×Q8—82no—
210C3×Q8—81no—
211C3×D4—42no—
212C3×D4—81no—
213C2×C3×C4—22no—
214C2×C3×C4—22no—
215C2×C3×C4—41no—
216C2×C3×C4—42no—
217C2×C3×C4—42no—
218C2×C3×C4—42no—
219C2×C3×C4—41no—
220C2×C3×C4—42no—
221C2×C3×C4—42no—
222C2×C3×C4—44no—
223C2×C3×C4#12011yescharacteristic
224C2×C3×C4#12011yescharacteristic
225C2×C3×C4#12111yescharacteristic
226C2×C3×C4#12111yescharacteristic
227C2×C3×C4#12111yescharacteristic
228C2×C3×C4#12711yescharacteristic
229#67—82no—
230C2×Dic3—41no—
231D12—42no—
232D12—42no—
233D12—81no—
234C4×D3—82no—
235Dic6—42no—
236C3⋊C8—24no—
237C3⋊C8—42no—
238C3⋊C8—81no—
239C24—21no—
240#41—124no—
241C2×Q8—21no—
242C2×Q8—22no—
243C2×Q8—21no—
244C2×Q8—22no—
245C2×Q8—42no—
246C2×Q8—44no—
247C2×Q8—61no—
248C2×D4—21no—
249C2×D4—21no—
250C2×D4—42no—
251C2×D4—61no—
252C2×D4—61no—
253C2×D4—61no—
254C2×D4—121no—
255C22×C4—21no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—121no—
259C22×C4C2×#6711yescharacteristic
260C22×C4C2×#6711yescharacteristic
261C22×C4C2×#6711yescharacteristic
262Q16—128no—
263SD16—128no—
264SD16—242no—
265D8—242no—
266M16—64no—
267C2×C8—62no—
268C2×C8—62no—
269C2×C8—121no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—61no—
275C4⋊C4—61no—
276C22⋊C4—41no—
277C22⋊C4—41no—
278C22⋊C4—41no—
279C22⋊C4—41no—
280C22⋊C4—84no—
281C22⋊C4—121no—
282C22⋊C4—121no—
283C42—42no—
284C42D3×D411yescharacteristic
285C42D3×D411yescharacteristic
286C42D3×D411yescharacteristic
287C42#23711yescharacteristic
288C22×C3—21no—
289C22×C3—21no—
290C22×C3—84no—
291C22×C3—81no—
292C22×C3—81no—
293C22×C3—81no—
294C22×C3#39211yescharacteristic
295C22×C3#45312yes—
296C2×D3—81no—
297C2×D3—81no—
298C2×D3—81no—
299C3×C4—21no—
300C3×C4—21no—
301C3×C4—21no—
302C3×C4—21no—
303C3×C4—21no—
304C3×C4—82no—
305C3×C4—82no—
306C3×C4—81no—
307C3×C4#45512yes—
308Dic3—81no—
309C23—42no—
310C23—42no—
311C23—41no—
312C23—42no—
313C23—121no—
314C23#95311yescharacteristic
315Q8—44no—
316Q8—42no—
317Q8—81no—
318Q8—82no—
319Q8—122no—
320D4—42no—
321D4—81no—
322D4—122no—
323D4—122no—
324D4—241no—
325D4—242no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—41no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—44no—
336C2×C4—121no—
337C2×C4—242no—
338C2×C4#93711yescharacteristic
339C2×C4#93711yescharacteristic
340C2×C4#93811yescharacteristic
341C2×C4#93911yescharacteristic
342C2×C4#93911yescharacteristic
343C2×C4#94011yescharacteristic
344C8—64no—
345C8—122no—
346C8—241no—
347C2×C3—22no—
348C2×C3—161no—
349C2×C3#200211yescharacteristic
350C2×C3#200411yescharacteristic
351C2×C3#200911yescharacteristic
352D3—161no—
353C22—21no—
354C22—21no—
355C22—84no—
356C22—81no—
357C22—81no—
358C22—81no—
359C22—241no—
360C22—241no—
361C22—241no—
362C22#522311yescharacteristic
363C22#524112yes—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—21no—
369C4—82no—
370C4—81no—
371C4—82no—
372C4—241no—
373C4#506112yes—
374C3#1009011yescharacteristic, complemented, Sylow
375C2—22no—
376C2—161no—
377C2—481no—
378C2#7173111yescharacteristic
379C2#7185611yescharacteristic
380C2#7186911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#2640C2
socleC22×C3#392
prime radical#82947C2
hypercenter#2640D3
hypocenterC3#10090
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2640

Complements

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

NH ≅ G/Ntype
1. C3×#2640C2semidirect
2. #83203C2semidirect
4. #71777C2semidirect
5. #71718C2semidirect
6. #69432C2semidirect
7. #67615C2semidirect
34. #2640D3semidirect
374. C3#10090semidirect

Maximal subgroups

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

subgroupindex
C3×#26402
#832032
#829472
#717772
#717182
#694322
#676152
#100903