Finite Group Databasebeta

← #10692092

Subgroups — #10692092

Order 768.

536 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 97 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
1#83369C211yesmaximal, characteristic
2#73616C211yesmaximal, characteristic
3C3×#1986C211yesmaximal, characteristic
4#71836C211yesmaximal, characteristic
5#69857C211yesmaximal, characteristic
6#67665C211yesmaximal, characteristic
7#67621C211yesmaximal, characteristic
8#9324—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×#887—21no—
11C2×C4×Dic6C2211yescharacteristic
12C3×C4×M16C2211yescharacteristic
13C3×C4×C4⋊C4C2211yescharacteristic
14#5094C412yescomplemented
15#5090C412yescomplemented
16#5050—21no—
17#5043—21no—
18#5030—22no—
19Dic3×C4⋊C4—21no—
20#4932C2211yescharacteristic
21C4×Dic3⋊C4—22no—
22#4926C2211yescharacteristic
23#4926C2211yescharacteristic
24#4649—21no—
25#4648—21no—
26C3×#339—22no—
27#4538—21no—
28#4538—21no—
29#4456—22no—
30#3113—31no—
31#2276—31no—
32#1986D311yescharacteristic
33#1506—31no—
34#1491—31no—
35#1300—31no—
36#1285—31no—
37C22×Dic6D411yescharacteristic
38C2×C3×M16—21no—
39C3×C4⋊Q8C2×C411yescharacteristic
40C3×C4⋊Q8C2×C411yescharacteristic
41C3×#125—21no—
42C3×C22⋊Q8—21no—
43C3×C22⋊Q8—21no—
44C3×C4×Q8—21no—
45C3×C4×Q8—21no—
46C3×#117—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C42D412yes—
49C2×C3×C42C2311yescharacteristic
50C2×#216—22no—
51C2×C4⋊Dic3D411yescharacteristic
52C2×Dic3⋊C4—22no—
53C2×Dic3⋊C4—21no—
54C2×C4×Dic3—22no—
55C2×C4×Dic3—21no—
56C2×C4×Dic3—21no—
57C2×C4×Dic3—22no—
58#887—28no—
59C4×Dic6—21no—
60C4×Dic6—21no—
61C4×Dic6C2×C412yes—
62C4×Dic6D412yes—
63C3×#97—21no—
64C3×C4×C8—21no—
65C3×#95—21no—
66#835—22no—
67C3⋊#95—22no—
68C3⋊#95—21no—
69C3⋊#95D411yescharacteristic
70C3⋊#95D411yescharacteristic
71#830—21no—
72#830—21no—
73#810—22no—
74#808—22no—
75C4⋊(C3⋊C8)—21no—
76C4⋊(C3⋊C8)C2×C412yes—
77#803—21no—
78#803—21no—
79#533C2×D311yescharacteristic
80C2×C4×Q8—31no—
81C2×C4×Q8—61no—
82#483—61no—
83#475—32no—
84#462—32no—
85#451—61no—
86#441—62no—
87#432—31no—
88#432—31no—
89C4×M16C2×D311yescharacteristic
90C4×C4⋊C4—61no—
91C4×C4⋊C4—61no—
92C4×C4⋊C4—62no—
93C4×C4⋊C4C2×D311yescharacteristic
94#376—31no—
95#376—61no—
96#339—22no—
97#328—61no—
98#328—61no—
99Q8⋊C8—62no—
100C2×C3×Q8—21no—
101C2×C3×Q8—21no—
102C22×C3×C4—22no—
103C22×C3×C4C2×D411yescharacteristic
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C22×Dic3—21no—
107C22×Dic3—21no—
108C22×Dic3—21no—
109C22×Dic3—21no—
110C2×Dic6—21no—
111C2×Dic6—24no—
112C2×Dic6C22⋊C412yes—
113C3×M16—22no—
114C3×M16—41no—
115C2×C3×C8—41no—
116C3×C4⋊C4—24no—
117C3×C4⋊C4—21no—
118C3×C4⋊C4—21no—
119C3×C4⋊C4—21no—
120C3×C4⋊C4—21no—
121C3×C4⋊C4—41no—
122C3×C4⋊C4—41no—
123C3×C22⋊C4—41no—
124C3×C42—42no—
125C3×C42—41no—
126C3×C42C22×C411yescharacteristic
127C3×C42C2×D411yescharacteristic
128C3×C42#4111yescharacteristic
129C3×C42#4111yescharacteristic
130#216—44no—
131C4⋊Dic3—21no—
132C4⋊Dic3C22⋊C412yes—
133Dic3⋊C4—28no—
134Dic3⋊C4—22no—
135Dic3⋊C4—22no—
136C4×Dic3—28no—
137C4×Dic3—24no—
138C4×Dic3—22no—
139C4×Dic3—22no—
140C4×Dic3—82no—
141C2×C3⋊C8—22no—
142C2×C3⋊C8—41no—
143C22×Q8—31no—
144C2×Q16—62no—
145C2×M16—21no—
146C4⋊Q8C4×D311yescharacteristic
147C4⋊Q8C4×D311yescharacteristic
148#125—21no—
149C22⋊Q8—21no—
150C22⋊Q8—21no—
151C4×Q8—21no—
152C4×Q8—21no—
153C4×Q8—32no—
154C4×Q8—32no—
155C4×Q8—68no—
156C4×Q8—61no—
157C4×Q8—61no—
158#117—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—31no—
161C2×C4⋊C4—62no—
162C2×C4⋊C4—61no—
163C2×C42—62no—
164C2×C42—61no—
165C2×C42—61no—
166C2×C42—62no—
167C2×C42#6712yes—
168C2×C42C22×D311yescharacteristic
169#106—62no—
170C4⋊C8—32no—
171C4⋊C8—61no—
172#103—62no—
173#103—62no—
174C22⋊C8—61no—
175C22⋊C8—61no—
176#97—21no—
177#97—61no—
178#97—61no—
179C4×C8—21no—
180#95—21no—
181#95—31no—
182#95—31no—
183#95—62no—
184#95—61no—
185C23×C3#12011yescharacteristic
186C3×Q8—42no—
187C3×Q8—81no—
188C2×C3×C4—21no—
189C2×C3×C4—21no—
190C2×C3×C4—22no—
191C2×C3×C4—24no—
192C2×C3×C4—41no—
193C2×C3×C4—42no—
194C2×C3×C4—42no—
195C2×C3×C4—41no—
196C2×C3×C4—41no—
197C2×C3×C4C2×C22⋊C411yescharacteristic
198C2×C3×C4C4×D411yescharacteristic
199C2×C3×C4C4×D411yescharacteristic
200C2×C3×C4#12111yescharacteristic
201C2×C3×C4#12111yescharacteristic
202C2×C3×C4#12311yescharacteristic
203C2×Dic3—24no—
204C2×Dic3—22no—
205C2×Dic3—22no—
206C2×Dic3—22no—
207C2×Dic3—22no—
208C2×Dic3—41no—
209C2×Dic3—41no—
210C2×Dic3—41no—
211C2×Dic3—41no—
212C2×Dic3—41no—
213C2×Dic3—41no—
214Dic6—24no—
215Dic6—22no—
216Dic6#10414yes—
217C3×C8—42no—
218C3⋊C8—42no—
219C3⋊C8—81no—
220C2×Q8—21no—
221C2×Q8—21no—
222C2×Q8—32no—
223C2×Q8—64no—
224C2×Q8—61no—
225C22×C4—22no—
226C22×C4—61no—
227C22×C4—61no—
228C22×C4—61no—
229C22×C4—61no—
230C22×C4D3×D411yescharacteristic
231C22×C4D3×D411yescharacteristic
232C22×C4C2×#6711yescharacteristic
233Q16—124no—
234M16—22no—
235M16—41no—
236C2×C8—41no—
237C2×C8—62no—
238C2×C8—121no—
239C4⋊C4—24no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4—21no—
244C4⋊C4—32no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—68no—
248C4⋊C4—61no—
249C4⋊C4—62no—
250C4⋊C4—62no—
251C22⋊C4—41no—
252C42—42no—
253C42—41no—
254C42—64no—
255C42—68no—
256C42—62no—
257C42—62no—
258C42—242no—
259C42C2×C4×D311yescharacteristic
260C42#23511yescharacteristic
261C42D3×D411yescharacteristic
262C42#23711yescharacteristic
263C22×C3—21no—
264C22×C3—21no—
265C22×C3#38611yescharacteristic
266C22×C3#45511yescharacteristic
267C22×C3#45611yescharacteristic
268C3×C4—21no—
269C3×C4—21no—
270C3×C4—21no—
271C3×C4—21no—
272C3×C4—21no—
273C3×C4—44no—
274C3×C4—81no—
275C3×C4—81no—
276C3×C4C2×#10412yes—
277Dic3—24no—
278Dic3—42no—
279Dic3—42no—
280Dic3—42no—
281C23#93711yescharacteristic
282Q8—34no—
283Q8—42no—
284Q8—64no—
285Q8—62no—
286Q8—81no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—22no—
290C2×C4—24no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—64no—
297C2×C4—62no—
298C2×C4—62no—
299C2×C4—62no—
300C2×C4—62no—
301C2×C4—121no—
302C2×C4—121no—
303C2×C4—121no—
304C2×C4—121no—
305C2×C4—121no—
306C2×C4—121no—
307C2×C4D3×C22⋊C411yescharacteristic
308C2×C4#67⋊C411yescharacteristic
309C2×C4#88311yescharacteristic
310C2×C4#88411yescharacteristic
311C2×C4C4×#6711yescharacteristic
312C2×C4#93811yescharacteristic
313C8—42no—
314C8—122no—
315C8—241no—
316C2×C3—21no—
317C2×C3—21no—
318C2×C3#188211yescharacteristic
319C2×C3#189611yescharacteristic
320C2×C3#190511yescharacteristic
321C22—21no—
322C22—21no—
323C22#495611yescharacteristic
324C22#506111yescharacteristic
325C22#509511yescharacteristic
326C4—21no—
327C4—21no—
328C4—21no—
329C4—21no—
330C4—21no—
331C4—44no—
332C4—64no—
333C4—81no—
334C4—81no—
335C4—122no—
336C4—122no—
337C4—122no—
338C4D3×#10412yes—
339C3#932411yescharacteristic, complemented, Sylow
340C2—21no—
341C2—21no—
342C2#6969311yescharacteristic
343C2#7038211yescharacteristic
344C2#7041011yescharacteristic
subgroupG/H
centerC22#4956
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1986C2
socleC22×C3#386
prime radicalC23×C3#120
cosocleC2×C3×C42C23
hypercenter#1986D3
hypocenterC3#9324
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1986

Complements

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

NH ≅ G/Ntype
14. #5094C4semidirect
15. #5090C4semidirect
339. C3#9324semidirect

Maximal subgroups

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

subgroupindex
#833692
#736162
C3×#19862
#718362
#698572
#676652
#676212
#93243