Finite Group Databasebeta

← #10698319

Subgroups — #10698319

Order 768.

416 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 93 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
2#70792C211yesmaximal, characteristic
3#70515C211yesmaximal, characteristic
4#69471C211yesmaximal, characteristic
5C3×#1303C211yesmaximal, characteristic
6#67737C211yesmaximal, characteristic
7#67715C211yesmaximal, characteristic
8#10371—31nomaximal, Sylow
9#5470C2211yescharacteristic
10C2×C4×Dic6—21no—
11C3×#431C2211yescharacteristic
12C3×#423C2211yescharacteristic
13C3×C4×C4⋊C4C2211yescharacteristic
14#4982—21no—
15#4979—21no—
16C4×C4⋊Dic3C2211yescharacteristic
17#4932C2211yescharacteristic
18C4×Dic3⋊C4—21no—
19#4925C2211yescharacteristic
20#4657—21no—
21#4656—21no—
22#4555—21no—
23#4555—21no—
24#4467—21no—
25#4467—21no—
26#4452—21no—
27#4452—21no—
28#2399—31no—
29#1995—31no—
30#1503—31no—
31#1503—31no—
32#1303—31no—
33#1303D311yescharacteristic
34#1274—31no—
35C22×Dic6—21no—
36C2×C3×M16—21no—
37C2×C3×C4⋊C4—21no—
38C2×C3×C42—21no—
39C2×C3×C42C2311yescharacteristic
40#926—21no—
41C2×C4⋊Dic3—21no—
42C2×C4⋊Dic3—21no—
43#924—21no—
44#924—21no—
45C2×Dic3⋊C4—21no—
46C2×Dic3⋊C4—21no—
47C2×Dic3⋊C4—21no—
48C2×Dic3⋊C4—21no—
49C2×C4×Dic3—21no—
50C2×C4×Dic3D411yescharacteristic
51C2×C4×Dic3D411yescharacteristic
52#870—21no—
53#869Q811yescharacteristic, complemented
54#869Q811yescharacteristic, complemented
55C4×Dic6—21no—
56C4×Dic6—21no—
57C4×Dic6—42no—
58C4×Dic6—42no—
59C3×C4⋊C8—21no—
60C3×C4⋊C8—21no—
61C3×C4⋊C8—21no—
62C3×C22⋊C8—21no—
63C3×C4×C8—21no—
64C3×#95D411yescharacteristic
65C3×#95D411yescharacteristic
66C3⋊#95—21no—
67C3⋊#95—21no—
68C3⋊#95—21no—
69C3⋊#95—21no—
70C3⋊#95D411yescharacteristic
71C3⋊#95D411yescharacteristic
72#830—21no—
73C4⋊(C3⋊C8)—21no—
74C4×C3⋊C8—21no—
75#533—31no—
76C2×C4×Q8—61no—
77#431—31no—
78#431C2×D311yescharacteristic
79#423C2×D311yescharacteristic
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—61no—
91#328—61no—
92#328—61no—
93#328—61no—
94C22×C3×C4—21no—
95C22×C3×C4—21no—
96C22×C3×C4C2×D411yescharacteristic
97C22×C3×C4C2×D411yescharacteristic
98C22×C3×C4C2×D411yescharacteristic
99C22×Dic3—21no—
100C22×Dic3—21no—
101C22×Dic3—21no—
102C22×Dic3—21no—
103C2×Dic6—21no—
104C2×Dic6—21no—
105C2×Dic6—42no—
106C2×Dic6—42no—
107C2×Dic6—42no—
108C3×M16—41no—
109C3×M16—41no—
110C2×C3×C8—41no—
111C2×C3×C8—41no—
112C3×C4⋊C4—42no—
113C3×C42—41no—
114C3×C42—41no—
115C3×C42C2×Q811yescharacteristic
116C3×C42#4111yescharacteristic
117C3×C42#4111yescharacteristic
118C3×C42#4111yescharacteristic
119#217—41no—
120C4⋊Dic3—21no—
121C4⋊Dic3—21no—
122C4⋊Dic3—21no—
123C4⋊Dic3—21no—
124C4⋊Dic3—42no—
125C4⋊Dic3—42no—
126Dic3⋊C4—42no—
127Dic3⋊C4—42no—
128Dic3⋊C4—42no—
129Dic3⋊C4—42no—
130Dic3⋊C4—41no—
131Dic3⋊C4—41no—
132C4×Dic3—21no—
133C4×Dic3—21no—
134C4×Dic3—41no—
135C4×Dic3—41no—
136C4×Dic3—41no—
137C4×Dic3SD1612yes—
138C4×Dic3Q1612yes—
139C2×C3⋊C8—41no—
140C22×Q8—61no—
141C2×M16—21no—
142C4⋊Q8—31no—
143C4⋊Q8—31no—
144#125—61no—
145C22⋊Q8—61no—
146C22⋊Q8—61no—
147C4×Q8—61no—
148C4×Q8—61no—
149C4×Q8—122no—
150C4×Q8—122no—
151#117—61no—
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—21no—
166C4⋊C8—21no—
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#95D1211yescharacteristic
179#95D1211yescharacteristic
180C23×C3#12011yescharacteristic
181C2×C3×C4—22no—
182C2×C3×C4—41no—
183C2×C3×C4—41no—
184C2×C3×C4—41no—
185C2×C3×C4—41no—
186C2×C3×C4—41no—
187C2×C3×C4—41no—
188C2×C3×C4—41no—
189C2×C3×C4C22⋊Q811yescharacteristic
190C2×C3×C4C22⋊Q811yescharacteristic
191C2×C3×C4C22⋊Q811yescharacteristic
192C2×C3×C4#12411yescharacteristic
193C2×C3×C4#12411yescharacteristic
194C2×C3×C4#12411yescharacteristic
195C2×C3×C4C2×SD1611yescharacteristic
196C2×C3×C4C2×Q1611yescharacteristic
197C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
198C2×C3×C4#13711yescharacteristic
199C2×Dic3—22no—
200C2×Dic3—22no—
201C2×Dic3—22no—
202C2×Dic3—41no—
203C2×Dic3—41no—
204C2×Dic3—41no—
205C2×Dic3—41no—
206C2×Dic3—41no—
207C2×Dic3—41no—
208C2×Dic3—41no—
209C2×Dic3—41no—
210C2×Dic3—41no—
211C2×Dic3—41no—
212C2×Dic3—41no—
213Dic6—48no—
214Dic6—81no—
215Dic6—81no—
216C3×C8—81no—
217C3×C8—81no—
218C3⋊C8—81no—
219C2×Q8—61no—
220C2×Q8—61no—
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×C4C2×D1211yescharacteristic
231C22×C4D3×D411yescharacteristic
232C22×C4D3×D411yescharacteristic
233M16—41no—
234M16—41no—
235C2×C8—41no—
236C2×C8—41no—
237C2×C8—121no—
238C4⋊C4—42no—
239C4⋊C4—61no—
240C4⋊C4—61no—
241C4⋊C4—61no—
242C4⋊C4—61no—
243C4⋊C4—122no—
244C4⋊C4—122no—
245C4⋊C4—122no—
246C4⋊C4—122no—
247C4⋊C4—122no—
248C4⋊C4—122no—
249C4⋊C4—121no—
250C4⋊C4—121no—
251C22⋊C4—121no—
252C42—32no—
253C42—32no—
254C42—41no—
255C42—41no—
256C42—61no—
257C42—61no—
258C42—121no—
259C42—121no—
260C42—121no—
261C42#23511yescharacteristic
262C42#23511yescharacteristic
263C42#23711yescharacteristic
264C42D3×Q811yescharacteristic
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#47511yescharacteristic
277C3×C4#47711yescharacteristic
278C3×C4#48711yescharacteristic
279C3×C4#48911yescharacteristic
280Dic3—42no—
281Dic3—42no—
282Dic3—42no—
283Dic3—81no—
284Dic3—81no—
285Dic3—81no—
286C23#88211yescharacteristic
287Q8—128no—
288Q8—241no—
289Q8—241no—
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—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×C4—121no—
308C2×C4—121no—
309C2×C4—121no—
310C2×C4—121no—
311C2×C4—121no—
312C2×C4#87511yescharacteristic
313C2×C4#88511yescharacteristic
314C2×C4#88511yescharacteristic
315C2×C4#89611yescharacteristic
316C2×C4#89611yescharacteristic
317C2×C4#89711yescharacteristic
318C2×C4#91111yescharacteristic
319C2×C4D3×SD1611yescharacteristic
320C2×C4#91511yescharacteristic
321C2×C4D3×Q1611yescharacteristic
322C8—81no—
323C8—81no—
324C8—241no—
325C2×C3—21no—
326C2×C3—21no—
327C2×C3#202411yescharacteristic
328C2×C3#203111yescharacteristic
329C2×C3#203111yescharacteristic
330C22—21no—
331C22—21no—
332C22#467311yescharacteristic
333C22#477611yescharacteristic
334C22#480911yescharacteristic
335C4—21no—
336C4—21no—
337C4—21no—
338C4—21no—
339C4—42no—
340C4—81no—
341C4—122no—
342C4—122no—
343C4—122no—
344C4—241no—
345C4—241no—
346C4—241no—
347C4#476511yescharacteristic
348C4#479211yescharacteristic
349C4#485411yescharacteristic
350C4#487511yescharacteristic
351C3#1037111yescharacteristic, complemented, Sylow
352C2—21no—
353C2—21no—
354C2#6877911yescharacteristic
355C2#6891011yescharacteristic
356C2#6904211yescharacteristic
subgroupG/H
centerC2×C4#885
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1303C2
socleC22×C3#393
prime radicalC23×C3#120
hypercenter#1303D3
hypocenterC3#10371
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1303

Complements

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

NH ≅ G/Ntype
53. #869Q8semidirect
54. #869Q8semidirect
351. C3#10371semidirect

Maximal subgroups

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

subgroupindex
#736692
#707922
#705152
#694712
C3×#13032
#677372
#677152
#103713