Finite Group Databasebeta

← #10696547

Subgroups — #10696547

Order 768.

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

normal subgroups only

This group has 560 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×#3152C211yesmaximal, characteristic
2#83273C211yesmaximal, characteristic
3#83246C211yesmaximal, characteristic, complemented
4#71812C211yesmaximal, characteristic
5#71777C211yesmaximal, characteristic, complemented
6#69863C211yesmaximal, characteristic, complemented
7#67665C211yesmaximal, characteristic, complemented
8#10091—31nomaximal, Sylow
9C2×C3×#137—22no—
10C3×#532C2211yescharacteristic
11C2×C3×C4⋊Q8C2211yescharacteristic
12C2×#870C2211yescharacteristic
13C3×#497—22no—
14C3×#494—21no—
15C3×#492—21no—
16C3×C4×M16C2211yescharacteristic
17#5230—22no—
18#5081—21no—
19#5071—21no—
20#5034—21no—
21#5033—21no—
22#5009—22no—
23#4983—22no—
24#4977—21no—
25#4975—21no—
26#4932C2211yescharacteristic
27#4926C2211yescharacteristic
28#4926C2211yescharacteristic
29#4546—22no—
30#4537—22no—
31#4465—22no—
32#3234—31no—
33#3152D311yescharacteristic
34#2660—31no—
35#1558—31no—
36#1528—31no—
37#1505—31no—
38#1300—31no—
39C2×C3×#41—21no—
40C22×C3×Q8D412yes—
41C3×#137—48no—
42C2×C3×Q16—24no—
43C2×C3×SD16—24no—
44C2×C3×M16—21no—
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8—21no—
47C3×C4⋊Q8—21no—
48C3×C4⋊Q8—21no—
49C3×C4⋊Q8D411yescharacteristic
50C3×#127D411yescharacteristic
51C3×#124—21no—
52C3×#121—21no—
53C3×#121—21no—
54C3×C4×D4—21no—
55C3×C4×D4—21no—
56C2×C3×C4⋊C4—21no—
57C2×C3×C4⋊C4—21no—
58C2×C3×C42C2311yescharacteristic
59C2×C4⋊Dic3D411yescharacteristic
60C2×C4⋊Dic3D411yescharacteristic
61C2×Dic3⋊C4—21no—
62C2×Dic3⋊C4—21no—
63C2×Dic3⋊C4—24no—
64C2×C4×Dic3—21no—
65#870—21no—
66#870—21no—
67#870—21no—
68#870—21no—
69C3×#97D412yes—
70C3×C4×C8—21no—
71#832—22no—
72#832—22no—
73C3⋊#95—24no—
74C3⋊#95—21no—
75C3⋊#95D412yes—
76#830—21no—
77#830—21no—
78#808—22no—
79#807—22no—
80C4⋊(C3⋊C8)—21no—
81C4⋊(C3⋊C8)D412yes—
82#803—21no—
83#803—21no—
84C2×#137—22no—
85#532C2×D311yescharacteristic
86C2×C4⋊Q8C2×D311yescharacteristic
87C2×#125—31no—
88#501—61no—
89#497—22no—
90#494—21no—
91#492—21no—
92#490—61no—
93#481—61no—
94#480—61no—
95#478—62no—
96#432—31no—
97#432—31no—
98C4×M16C2×D311yescharacteristic
99#398—62no—
100#395—62no—
101#384—61no—
102#382—61no—
103#376—31no—
104#333—62no—
105#330—62no—
106#327—62no—
107C3×#41—44no—
108C2×C3×Q8—21no—
109C2×C3×Q8—22no—
110C2×C3×Q8—22no—
111C2×C3×Q8—44no—
112C2×C3×D4—21no—
113C2×C3×D4—21no—
114C2×C3×D4—21no—
115C2×C3×D4—41no—
116C22×C3×C4—22no—
117C22×C3×C4—41no—
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C22×Dic3—22no—
122C22×Dic3—21no—
123C22×Dic3—21no—
124C3×Q16—48no—
125C3×SD16—48no—
126C3×M16—24no—
127C2×C3×C8—22no—
128C3×C4⋊C4—21no—
129C3×C4⋊C4—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×C42C2×D411yescharacteristic
137C3×C42C2×D411yescharacteristic
138C3×C42C2×D411yescharacteristic
139C3×C42#4111yescharacteristic
140C4⋊Dic3—21no—
141C4⋊Dic3—21no—
142C4⋊Dic3—21no—
143C4⋊Dic3—21no—
144Dic3⋊C4—41no—
145Dic3⋊C4—41no—
146Dic3⋊C4—41no—
147Dic3⋊C4—41no—
148Dic3⋊C4—84no—
149C4×Dic3—24no—
150C2×C3⋊C8—22no—
151C2×C3⋊C8—41no—
152C2×#41—21no—
153C22×Q8#6712yes—
154#137—48no—
155C2×Q16—24no—
156C2×SD16—24no—
157C2×M16—21no—
158C4⋊Q8—21no—
159C4⋊Q8—21no—
160C4⋊Q8—21no—
161C4⋊Q8—21no—
162C4⋊Q8#6711yescharacteristic
163#127#6711yescharacteristic
164#125—61no—
165#125—61no—
166#125—61no—
167#125—61no—
168#124—21no—
169#121—21no—
170#121—21no—
171C4×D4—21no—
172C4×D4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—31no—
176C2×C4⋊C4—31no—
177C2×C4⋊C4—61no—
178C2×C4⋊C4—61no—
179C2×C4⋊C4—64no—
180C2×C42—61no—
181C2×C42C22×D311yescharacteristic
182#107—62no—
183#106—62no—
184C4⋊C8—32no—
185C4⋊C8—61no—
186#102—62no—
187#102—62no—
188C22⋊C8—61no—
189C22⋊C8—61no—
190#97—61no—
191#97—61no—
192#97#6712yes—
193C4×C8—21no—
194#95—32no—
195#95—64no—
196#95—61no—
197C23×C3—41no—
198C23×C3#12011yescharacteristic
199C3×Q8—44no—
200C3×Q8—42no—
201C3×Q8—82no—
202C3×D4—42no—
203C3×D4—42no—
204C3×D4—81no—
205C3×D4—82no—
206C2×C3×C4—22no—
207C2×C3×C4—22no—
208C2×C3×C4—41no—
209C2×C3×C4—41no—
210C2×C3×C4—42no—
211C2×C3×C4—44no—
212C2×C3×C4—82no—
213C2×C3×C4#12011yescharacteristic
214C2×C3×C4#12011yescharacteristic
215C2×C3×C4#12111yescharacteristic
216C2×C3×C4#12111yescharacteristic
217C2×C3×C4#12111yescharacteristic
218C2×C3×C4#12711yescharacteristic
219C2×Dic3—24no—
220C2×Dic3—42no—
221C2×Dic3—42no—
222C2×Dic3—42no—
223C2×Dic3—41no—
224C2×Dic3—41no—
225C2×Dic3—42no—
226C3×C8—24no—
227C3⋊C8—42no—
228C3⋊C8—81no—
229#41—44no—
230C2×Q8—22no—
231C2×Q8—21no—
232C2×Q8—22no—
233C2×Q8—44no—
234C2×D4—21no—
235C2×D4—21no—
236C2×D4—21no—
237C2×D4—41no—
238C22×C4—22no—
239C22×C4—41no—
240C22×C4—62no—
241C22×C4—61no—
242C22×C4—61no—
243C22×C4D3×D411yescharacteristic
244C22×C4D3×D411yescharacteristic
245C22×C4C2×#6711yescharacteristic
246Q16—48no—
247SD16—48no—
248M16—24no—
249C2×C8—22no—
250C2×C8—62no—
251C2×C8—121no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—61no—
259C4⋊C4—61no—
260C4⋊C4—61no—
261C4⋊C4—61no—
262C4⋊C4—121no—
263C4⋊C4—121no—
264C4⋊C4—121no—
265C4⋊C4—121no—
266C4⋊C4—244no—
267C22⋊C4—41no—
268C22⋊C4—41no—
269C42—64no—
270C42D3×D411yescharacteristic
271C42#23711yescharacteristic
272C42C2×#6711yescharacteristic
273C42C2×#6711yescharacteristic
274C22×C3—21no—
275C22×C3—21no—
276C22×C3—81no—
277C22×C3—81no—
278C22×C3—81no—
279C22×C3#39211yescharacteristic
280C22×C3#45312yes—
281C3×C4—21no—
282C3×C4—21no—
283C3×C4—21no—
284C3×C4—21no—
285C3×C4—21no—
286C3×C4—82no—
287C3×C4—81no—
288C3×C4#45612yes—
289Dic3—44no—
290Dic3—81no—
291Dic3—81no—
292C23—41no—
293C23#93711yescharacteristic
294Q8—44no—
295Q8—42no—
296Q8—82no—
297D4—42no—
298D4—42no—
299D4—81no—
300D4—82no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—42no—
306C2×C4—44no—
307C2×C4—64no—
308C2×C4—82no—
309C2×C4—122no—
310C2×C4—122no—
311C2×C4—122no—
312C2×C4—121no—
313C2×C4—121no—
314C2×C4—122no—
315C2×C4#93711yescharacteristic
316C2×C4#93811yescharacteristic
317C2×C4#93911yescharacteristic
318C2×C4#93911yescharacteristic
319C2×C4#94011yescharacteristic
320C2×C4#95311yescharacteristic
321C8—24no—
322C8—122no—
323C8—241no—
324C2×C3—22no—
325C2×C3—161no—
326C2×C3#200211yescharacteristic
327C2×C3#200511yescharacteristic
328C2×C3#200911yescharacteristic
329C22—21no—
330C22—21no—
331C22—81no—
332C22—81no—
333C22—81no—
334C22#507712yes—
335C22#522311yescharacteristic
336C4—21no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—21no—
341C4—82no—
342C4—81no—
343C4—124no—
344C4—241no—
345C4—241no—
346C4#520512yes—
347C3#1009111yescharacteristic, complemented, Sylow
348C2—22no—
349C2—161no—
350C2#7176911yescharacteristic
351C2#7210811yescharacteristic
352C2#7211111yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3152C2
socleC22×C3#392
prime radicalC3×#532C22
hypercenter#3152D3
hypocenterC3#10091
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3152

Complements

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

NH ≅ G/Ntype
3. #83246C2semidirect
5. #71777C2semidirect
6. #69863C2semidirect
7. #67665C2semidirect
347. C3#10091semidirect

Maximal subgroups

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

subgroupindex
C3×#31522
#832732
#832462
#718122
#717772
#698632
#676652
#100913