Finite Group Databasebeta

← #10690502

Subgroups — #10690502

Order 768.

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

normal subgroups only

This group has 404 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#79267C211yesmaximal, characteristic
2#78442C211yesmaximal, characteristic
3#70769C211yesmaximal, characteristic
4#69862C211yesmaximal, characteristic
5#69856C211yesmaximal, characteristic
6C3×#1278C211yesmaximal, characteristic
7#67656C211yesmaximal, characteristic
8#9068—31nomaximal, Sylow
9#5470C2211yescharacteristic
10#5470C2211yescharacteristic
11C3×#431C2211yescharacteristic
12C3×C4×M16C2211yescharacteristic
13C2×C3×C4×C8C2211yescharacteristic
14#5106—22no—
15#4925C2211yescharacteristic
16C4×#208C2211yescharacteristic
17#4831—21no—
18#4830—21no—
19#4720—22no—
20#4683—22no—
21#4682C411yescharacteristic
22#4682C411yescharacteristic
23#4680C411yescharacteristic
24#4680C411yescharacteristic
25#4679—21no—
26C8×Dic6—21no—
27#4484—21no—
28#4484—21no—
29#4465—22no—
30#4456—21no—
31#4456—21no—
32#3160—31no—
33#2986—31no—
34#1526—31no—
35#1485—31no—
36#1485—31no—
37#1278—31no—
38#1278D311yescharacteristic
39C2×C3×M16—21no—
40C22×C3×C8—21no—
41C2×C3×C42C2311yescharacteristic
42#926—21no—
43#926D412yes—
44#924—21no—
45#924—21no—
46#924—21no—
47#924—21no—
48C2×#208—21no—
49#870—21no—
50#870D412yes—
51#869C2×C411yescharacteristic
52#869C2×C411yescharacteristic
53#869C2×C411yescharacteristic
54#869C2×C411yescharacteristic
55C4×Dic6—21no—
56C4×Dic6—21no—
57C4×Dic6—21no—
58C4×Dic6—21no—
59C3×C4⋊C8D411yescharacteristic
60C3×C4⋊C8D411yescharacteristic
61C3×C22⋊C8—21no—
62C3×#97—21no—
63C3×C4×C8—21no—
64C3×C4×C8—21no—
65C3×C4×C8—21no—
66C3×C4×C8C2×C411yescharacteristic
67C3×C4×C8C2×C411yescharacteristic
68#830—21no—
69#818—22no—
70#818—22no—
71#817—22no—
72#817—22no—
73#815—22no—
74Dic3⋊C8—22no—
75Dic3⋊C8—22no—
76C8×Dic3—22no—
77C4⋊(C3⋊C8)D411yescharacteristic
78C4⋊(C3⋊C8)D411yescharacteristic
79#803—21no—
80C4×C3⋊C8—21no—
81C4×C3⋊C8—21no—
82#533—31no—
83#533—31no—
84#500—31no—
85#500—31no—
86#499—62no—
87#498—31no—
88#498—31no—
89#446—61no—
90#446—61no—
91C8×Q8—61no—
92C8×Q8—61no—
93#433—62no—
94#431—31no—
95#431C2×D311yescharacteristic
96#427—62no—
97C4×M16—31no—
98C4×M16C2×D311yescharacteristic
99C2×C4×C8C2×D311yescharacteristic
100#330—62no—
101Q8⋊C8—61no—
102Q8⋊C8—61no—
103Q8⋊C8—61no—
104Q8⋊C8—61no—
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4#4111yescharacteristic
107C22×C3×C4#4111yescharacteristic
108C2×Dic6—21no—
109C2×Dic6—21no—
110C2×Dic6—21no—
111C2×Dic6—21no—
112C3×M16—22no—
113C3×M16—41no—
114C2×C3×C8—22no—
115C2×C3×C8—22no—
116C2×C3×C8—21no—
117C2×C3×C8—21no—
118C2×C3×C8—21no—
119C2×C3×C8—21no—
120C2×C3×C8—41no—
121C3×C42C22×C411yescharacteristic
122C3×C42C2×D411yescharacteristic
123C3×C42C2×D411yescharacteristic
124C3×C42C2×D411yescharacteristic
125#217—22no—
126#217—41no—
127C4⋊Dic3—21no—
128C4⋊Dic3—21no—
129C4⋊Dic3—21no—
130C4⋊Dic3—21no—
131C4⋊Dic3—21no—
132C4⋊Dic3—21no—
133C4⋊Dic3C22⋊C412yes—
134C4⋊Dic3C22⋊C412yes—
135Dic3⋊C4—22no—
136Dic3⋊C4—22no—
137Dic3⋊C4—41no—
138Dic3⋊C4—41no—
139C4×Dic3—22no—
140C4×Dic3—41no—
141#208—22no—
142#208—41no—
143C2×C3⋊C8—21no—
144C2×C3⋊C8—21no—
145C2×C3⋊C8—41no—
146C2×M16—21no—
147C2×M16—61no—
148C22×C8—21no—
149C4⋊Q8—31no—
150C4⋊Q8—31no—
151C4⋊Q8—31no—
152C4⋊Q8—31no—
153#125—32no—
154#125—61no—
155C22⋊Q8—61no—
156C22⋊Q8—61no—
157C22⋊Q8—61no—
158C22⋊Q8—61no—
159C4×Q8—61no—
160C4×Q8—61no—
161C4×Q8—61no—
162C4×Q8—61no—
163#117—32no—
164#117—61no—
165C2×C42C22×D311yescharacteristic
166#107—62no—
167#107—62no—
168#106—62no—
169#106—62no—
170C4⋊C8—31no—
171C4⋊C8—31no—
172C4⋊C8—62no—
173C4⋊C8—62no—
174C4⋊C8D1211yescharacteristic
175C4⋊C8D1211yescharacteristic
176C22⋊C8—21no—
177C22⋊C8—61no—
178#97—21no—
179#97—62no—
180#97—61no—
181C4×C8—21no—
182C4×C8—21no—
183C4×C8—21no—
184C4×C8—62no—
185C4×C8—61no—
186C4×C8—61no—
187C4×C8C4×D311yescharacteristic
188C4×C8C4×D311yescharacteristic
189C23×C3#12311yescharacteristic
190C2×C3×C4—21no—
191C2×C3×C4—21no—
192C2×C3×C4—21no—
193C2×C3×C4—21no—
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4C2×C22⋊C411yescharacteristic
197C2×C3×C4C4×D411yescharacteristic
198C2×C3×C4C4×D411yescharacteristic
199C2×C3×C4#12011yescharacteristic
200C2×C3×C4#12111yescharacteristic
201C2×C3×C4#12111yescharacteristic
202C2×Dic3—22no—
203C2×Dic3—22no—
204C2×Dic3—41no—
205C2×Dic3—41no—
206Dic6—41no—
207Dic6—41no—
208Dic6—41no—
209Dic6—41no—
210Dic6—41no—
211Dic6—41no—
212Dic6—41no—
213Dic6—41no—
214C3×C8—22no—
215C3×C8—22no—
216C3×C8—42no—
217C3×C8—41no—
218C3×C8—41no—
219C3⋊C8—42no—
220C3⋊C8—41no—
221C3⋊C8—41no—
222C2×Q8—61no—
223C2×Q8—61no—
224C2×Q8—61no—
225C2×Q8—61no—
226C22×C4#23511yescharacteristic
227C22×C4D3×D411yescharacteristic
228C22×C4#23911yescharacteristic
229M16—22no—
230M16—41no—
231M16—62no—
232M16—121no—
233C2×C8—22no—
234C2×C8—22no—
235C2×C8—21no—
236C2×C8—21no—
237C2×C8—21no—
238C2×C8—21no—
239C2×C8—41no—
240C2×C8—61no—
241C2×C8—61no—
242C2×C8—121no—
243C4⋊C4—32no—
244C4⋊C4—32no—
245C4⋊C4—61no—
246C4⋊C4—62no—
247C4⋊C4—61no—
248C4⋊C4—62no—
249C4⋊C4—61no—
250C4⋊C4—61no—
251C4⋊C4—61no—
252C4⋊C4—61no—
253C4⋊C4—121no—
254C4⋊C4—121no—
255C22⋊C4—62no—
256C22⋊C4—121no—
257C42—62no—
258C42—121no—
259C42C2×C4×D311yescharacteristic
260C42C2×D1211yescharacteristic
261C42D3×D411yescharacteristic
262C42D3×D411yescharacteristic
263C22×C3—21no—
264C22×C3—21no—
265C22×C3—21no—
266C22×C3#38611yescharacteristic
267C3×C4—21no—
268C3×C4—21no—
269C3×C4—21no—
270C3×C4#44311yescharacteristic
271C3×C4#44311yescharacteristic
272C3×C4#44411yescharacteristic
273C3×C4#44411yescharacteristic
274C3×C4#45611yescharacteristic
275C3×C4#45611yescharacteristic
276Dic3—42no—
277Dic3—42no—
278Dic3—81no—
279Dic3—81no—
280C23#89411yescharacteristic
281Q8—121no—
282Q8—121no—
283Q8—121no—
284Q8—121no—
285Q8—121no—
286Q8—121no—
287Q8—121no—
288Q8—121no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—62no—
296C2×C4—62no—
297C2×C4—121no—
298C2×C4—121no—
299C2×C4C4×D1211yescharacteristic
300C2×C4D3×C22⋊C411yescharacteristic
301C2×C4#88211yescharacteristic
302C2×C4#88411yescharacteristic
303C2×C4#89311yescharacteristic
304C2×C4#89511yescharacteristic
305C8—22no—
306C8—22no—
307C8—42no—
308C8—41no—
309C8—41no—
310C8—122no—
311C8—121no—
312C8—121no—
313C2×C3—41no—
314C2×C3#187611yescharacteristic
315C2×C3#187611yescharacteristic
316C2×C3#188611yescharacteristic
317C22—21no—
318C22—21no—
319C22—21no—
320C22#466511yescharacteristic
321C4—21no—
322C4—21no—
323C4—21no—
324C4—122no—
325C4—122no—
326C4—241no—
327C4—241no—
328C4#470011yescharacteristic
329C4#471711yescharacteristic
330C4#482611yescharacteristic
331C4#482611yescharacteristic
332C4#489411yescharacteristic
333C4#489511yescharacteristic
334C3#906811yescharacteristic, complemented, Sylow
335C2—41no—
336C2#6869511yescharacteristic
337C2#6876511yescharacteristic
338C2#6894611yescharacteristic
subgroupG/H
centerC2×C4#882
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1278C2
socleC22×C3#386
prime radicalC23×C3#123
cosocleC2×C3×C42C23
hypercenter#1278D3
hypocenterC3#9068
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1278

Complements

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

NH ≅ G/Ntype
334. C3#9068semidirect

Maximal subgroups

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

subgroupindex
#792672
#784422
#707692
#698622
#698562
C3×#12782
#676562
#90683