Finite Group Databasebeta

← #10698318

Subgroups — #10698318

Order 768.

416 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 95 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#73669C211yesmaximal, characteristic
2C3×#1503C211yesmaximal, characteristic
3#71840C211yesmaximal, characteristic
4#71835C211yesmaximal, characteristic
5#67746C211yesmaximal, characteristic
6#67737C211yesmaximal, characteristic
7#67620C211yesmaximal, characteristic
8#10371—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C4×Dic6—21no—
11C3×#431C2211yescharacteristic
12C3×C4×C4⋊C4C2211yescharacteristic
13#4982—21no—
14#4979—21no—
15C4×C4⋊Dic3C2211yescharacteristic
16#4932C2211yescharacteristic
17C4×Dic3⋊C4—21no—
18#4925C2211yescharacteristic
19#4924C2211yescharacteristic
20#4657—21no—
21#4656—21no—
22C3×#328—21no—
23C3×#328—21no—
24#4538—21no—
25#4538—21no—
26#4532—21no—
27#4532—21no—
28#2399—31no—
29#1995—31no—
30#1503—31no—
31#1503D311yescharacteristic
32#1303—31no—
33#1303—31no—
34#1274—31no—
35C22×Dic6—21no—
36C3×C4⋊Q8Q811yescharacteristic, complemented
37C3×C4⋊Q8Q811yescharacteristic, complemented
38C3×#125—21no—
39C3×C22⋊Q8—21no—
40C3×C22⋊Q8—21no—
41C3×C4×Q8—21no—
42C3×C4×Q8—21no—
43C3×#117—21no—
44C2×C3×C4⋊C4—21no—
45C2×C3×C42—21no—
46C2×C3×C42C2311yescharacteristic
47C2×C4⋊Dic3—21no—
48C2×C4⋊Dic3—21no—
49C2×Dic3⋊C4—21no—
50C2×Dic3⋊C4—21no—
51C2×Dic3⋊C4—21no—
52C2×Dic3⋊C4—21no—
53C2×C4×Dic3—21no—
54C2×C4×Dic3D411yescharacteristic
55C2×C4×Dic3D411yescharacteristic
56C2×#208—21no—
57C4×Dic6—42no—
58C4×Dic6—42no—
59C3×C4⋊C8—21no—
60C3×C22⋊C8—21no—
61C3×C4×C8—21no—
62C3×#95D411yescharacteristic
63C3×#95D411yescharacteristic
64C3⋊#95—21no—
65C3⋊#95—21no—
66C3⋊#95—21no—
67C3⋊#95—21no—
68C3⋊#95D411yescharacteristic
69C3⋊#95D411yescharacteristic
70#830—21no—
71C4⋊(C3⋊C8)—21no—
72C4⋊(C3⋊C8)—21no—
73C4⋊(C3⋊C8)—21no—
74C4×C3⋊C8—21no—
75#533C2×D311yescharacteristic
76C2×C4×Q8—61no—
77#431—31no—
78#431C2×D311yescharacteristic
79#423—31no—
80#398—61no—
81#398—61no—
82#395—61no—
83#395—61no—
84C4×C4⋊C4—31no—
85C4×C4⋊C4—61no—
86C4×C4⋊C4C2×D311yescharacteristic
87#376—31no—
88#339—61no—
89#339—61no—
90#328—21no—
91#328—21no—
92#328—61no—
93#328—61no—
94C2×C3×Q8—21no—
95C2×C3×Q8—21no—
96C22×C3×C4—21no—
97C22×C3×C4—21no—
98C22×C3×C4C2×D411yescharacteristic
99C22×C3×C4C2×D411yescharacteristic
100C22×C3×C4C2×D411yescharacteristic
101C22×Dic3—21no—
102C22×Dic3—21no—
103C22×Dic3—21no—
104C22×Dic3—21no—
105C2×Dic6—42no—
106C2×Dic6—42no—
107C2×Dic6—42no—
108C2×C3×C8—41no—
109C3×C4⋊C4—21no—
110C3×C4⋊C4—21no—
111C3×C4⋊C4—21no—
112C3×C4⋊C4—21no—
113C3×C4⋊C4—42no—
114C3×C4⋊C4—41no—
115C3×C4⋊C4—41no—
116C3×C22⋊C4—41no—
117C3×C42—41no—
118C3×C42—41no—
119C3×C42—41no—
120C3×C42C2×Q811yescharacteristic
121C3×C42#4111yescharacteristic
122C3×C42#4111yescharacteristic
123C3×C42#4111yescharacteristic
124C4⋊Dic3—42no—
125C4⋊Dic3—42no—
126Dic3⋊C4—42no—
127Dic3⋊C4—42no—
128Dic3⋊C4—42no—
129Dic3⋊C4—42no—
130C4×Dic3—21no—
131C4×Dic3—21no—
132C4×Dic3—41no—
133C4×Dic3—41no—
134C4×Dic3SD1612yes—
135C4×Dic3Q1612yes—
136#208—41no—
137#208—41no—
138C2×C3⋊C8—41no—
139C2×C3⋊C8—41no—
140C22×Q8—61no—
141C2×M16—61no—
142C4⋊Q8Dic611yescharacteristic, complemented
143C4⋊Q8Dic611yescharacteristic, complemented
144#125—21no—
145C22⋊Q8—21no—
146C22⋊Q8—21no—
147C4×Q8—21no—
148C4×Q8—21no—
149C4×Q8—122no—
150C4×Q8—122no—
151#117—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—61no—
154C2×C4⋊C4—61no—
155C2×C4⋊C4—61no—
156C2×C4⋊C4—61no—
157C2×C4⋊C4—61no—
158C2×C4⋊C4—61no—
159C2×C42—21no—
160C2×C42—31no—
161C2×C42—31no—
162C2×C42—61no—
163C2×C42C22×D311yescharacteristic
164C4⋊C8—21no—
165C4⋊C8—61no—
166C4⋊C8—61no—
167C4⋊C8—61no—
168C22⋊C8—21no—
169C22⋊C8—61no—
170C4×C8—21no—
171C4×C8—61no—
172#95—31no—
173#95—31no—
174#95—61no—
175#95—61no—
176#95—61no—
177#95—61no—
178#95#6711yescharacteristic
179#95#6711yescharacteristic
180C23×C3#12011yescharacteristic
181C3×Q8—81no—
182C3×Q8—81no—
183C2×C3×C4—22no—
184C2×C3×C4—41no—
185C2×C3×C4—41no—
186C2×C3×C4—41no—
187C2×C3×C4—41no—
188C2×C3×C4—41no—
189C2×C3×C4—41no—
190C2×C3×C4—41no—
191C2×C3×C4—41no—
192C2×C3×C4—41no—
193C2×C3×C4C22⋊Q811yescharacteristic
194C2×C3×C4C22⋊Q811yescharacteristic
195C2×C3×C4C22⋊Q811yescharacteristic
196C2×C3×C4#12411yescharacteristic
197C2×C3×C4#12411yescharacteristic
198C2×C3×C4#12411yescharacteristic
199C2×C3×C4C2×SD1611yescharacteristic
200C2×C3×C4C2×Q1611yescharacteristic
201C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
202C2×C3×C4#13711yescharacteristic
203C2×Dic3—22no—
204C2×Dic3—22no—
205C2×Dic3—22no—
206C2×Dic3—41no—
207C2×Dic3—41no—
208C2×Dic3—41no—
209C2×Dic3—41no—
210C2×Dic3—41no—
211C2×Dic3—41no—
212C2×Dic3—41no—
213C2×Dic3—41no—
214C2×Dic3—41no—
215Dic6—48no—
216C3×C8—81no—
217C3⋊C8—81no—
218C3⋊C8—81no—
219C2×Q8—21no—
220C2×Q8—21no—
221C2×Q8—122no—
222C2×Q8—122no—
223C2×Q8—122no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—61no—
227C22×C4—61no—
228C22×C4—61no—
229C22×C4—61no—
230C22×C4D3×D411yescharacteristic
231C22×C4D3×D411yescharacteristic
232C22×C4C2×#6711yescharacteristic
233M16—121no—
234M16—121no—
235C2×C8—41no—
236C2×C8—121no—
237C2×C8—121no—
238C4⋊C4—21no—
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—42no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4—122no—
246C4⋊C4—122no—
247C4⋊C4—122no—
248C4⋊C4—122no—
249C4⋊C4—122no—
250C4⋊C4—122no—
251C22⋊C4—41no—
252C42—32no—
253C42—32no—
254C42—41no—
255C42—41no—
256C42—41no—
257C42—61no—
258C42—61no—
259C42—121no—
260C42—121no—
261C42C2×Dic611yescharacteristic
262C42#23511yescharacteristic
263C42#23711yescharacteristic
264C42#23711yescharacteristic
265C22×C3—21no—
266C22×C3—21no—
267C22×C3#39311yescharacteristic
268C22×C3#45011yescharacteristic
269C22×C3#45111yescharacteristic
270C3×C4—21no—
271C3×C4—21no—
272C3×C4—21no—
273C3×C4—21no—
274C3×C4—42no—
275C3×C4—81no—
276C3×C4—81no—
277C3×C4—81no—
278C3×C4#47511yescharacteristic
279C3×C4#47711yescharacteristic
280C3×C4#48711yescharacteristic
281C3×C4#48911yescharacteristic
282Dic3—42no—
283Dic3—42no—
284Dic3—42no—
285Dic3—81no—
286C23#93711yescharacteristic
287Q8—81no—
288Q8—81no—
289Q8—128no—
290C2×C4—22no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—62no—
301C2×C4—62no—
302C2×C4—62no—
303C2×C4—121no—
304C2×C4—121no—
305C2×C4—121no—
306C2×C4—121no—
307C2×C4—121no—
308C2×C4—121no—
309C2×C4—121no—
310C2×C4—121no—
311C2×C4—121no—
312C2×C4#87811yescharacteristic
313C2×C4#87811yescharacteristic
314C2×C4#88511yescharacteristic
315C2×C4#88511yescharacteristic
316C2×C4D3×SD1611yescharacteristic
317C2×C4#91411yescharacteristic
318C2×C4D3×Q1611yescharacteristic
319C2×C4#91811yescharacteristic
320C2×C4#92411yescharacteristic
321C2×C4#93611yescharacteristic
322C8—81no—
323C8—241no—
324C8—241no—
325C2×C3—21no—
326C2×C3—21no—
327C2×C3#202411yescharacteristic
328C2×C3#203111yescharacteristic
329C2×C3#203111yescharacteristic
330C22—21no—
331C22—21no—
332C22#494711yescharacteristic
333C22#516911yescharacteristic
334C22#518611yescharacteristic
335C4—21no—
336C4—21no—
337C4—21no—
338C4—21no—
339C4—42no—
340C4—81no—
341C4—81no—
342C4—81no—
343C4—122no—
344C4—122no—
345C4—122no—
346C4—241no—
347C4#479111yescharacteristic
348C4#479211yescharacteristic
349C4#479311yescharacteristic
350C4#479411yescharacteristic
351C3#1037111yescharacteristic, complemented, Sylow
352C2—21no—
353C2—21no—
354C2#7003811yescharacteristic
355C2#7023111yescharacteristic
356C2#7036811yescharacteristic
subgroupG/H
centerC2×C4#885
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1503C2
socleC22×C3#393
prime radicalC23×C3#120
hypercenter#1503D3
hypocenterC3#10371
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1503

Complements

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

NH ≅ G/Ntype
36. C3×C4⋊Q8Q8semidirect
37. C3×C4⋊Q8Q8semidirect
142. C4⋊Q8Dic6semidirect
143. C4⋊Q8Dic6semidirect
351. C3#10371semidirect

Maximal subgroups

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

subgroupindex
#736692
C3×#15032
#718402
#718352
#677462
#677372
#676202
#103713