Finite Group Databasebeta

← #10694391

Subgroups — #10694391

Order 768.

392 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 99 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#83344C211yesmaximal, characteristic
2C3×#1485C211yesmaximal, characteristic
3#71821C211yesmaximal, characteristic
4#70769C211yesmaximal, characteristic
5#70741C211yesmaximal, characteristic
6C4×#822C211yesmaximal, characteristic
7#69862C211yesmaximal, characteristic
8#9727—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10#5470C2211yescharacteristic
11C3×#431C2211yescharacteristic
12C3×C4×M16C2211yescharacteristic
13C2×#822—21no—
14#5093C411yescharacteristic
15#5092C411yescharacteristic
16#5079C411yescharacteristic
17#5078C411yescharacteristic
18#5051—21no—
19#5051—21no—
20C4×#216—21no—
21C4×#216—21no—
22C4×#208C2211yescharacteristic
23C4×#208C2211yescharacteristic
24C2×C4×C3⋊C8C2211yescharacteristic
25C3×Q8⋊C8—22no—
26#4558—21no—
27#4543—22no—
28#4484—22no—
29#4465—22no—
30#4455—21no—
31#4454—21no—
32#3170—31no—
33C4×#108—31no—
34#1526—31no—
35#1526—31no—
36#1485—31no—
37#1485D311yescharacteristic
38#1451—31no—
39C2×C3×M16—21no—
40C3×C4⋊Q8C2×C411yescharacteristic
41C3×C4⋊Q8C2×C411yescharacteristic
42C3×#125—21no—
43C3×C22⋊Q8—21no—
44C3×C22⋊Q8—21no—
45C3×C4×Q8—21no—
46C3×C4×Q8—21no—
47C3×#117—21no—
48C2×C3×C42C2311yescharacteristic
49C2×#216—21no—
50C2×#216—21no—
51#926—21no—
52#924—21no—
53#924—21no—
54C2×#208—21no—
55C2×#208—21no—
56C22×C3⋊C8D411yescharacteristic
57C22×C3⋊C8D411yescharacteristic
58#870—21no—
59#869C2×C411yescharacteristic
60#869C2×C411yescharacteristic
61C4×Dic6—21no—
62C4×Dic6—21no—
63C3×C4⋊C8D412yes—
64C3×C22⋊C8—21no—
65C3×#97—21no—
66C3×C4×C8—21no—
67C3×C4×C8—21no—
68#835—21no—
69#835—21no—
70#822—24no—
71#810—21no—
72#810—21no—
73#807—21no—
74#807—21no—
75#803—21no—
76#803D412yes—
77C4×C3⋊C8—21no—
78C4×C3⋊C8—21no—
79C4×C3⋊C8C2×C411yescharacteristic
80C4×C3⋊C8C2×C411yescharacteristic
81C4×C3⋊C8D411yescharacteristic
82C4×C3⋊C8D411yescharacteristic
83#533—31no—
84#533C2×D311yescharacteristic
85#500—31no—
86#494—31no—
87#487—31no—
88#487—31no—
89#467—61no—
90#467—61no—
91C4×Q16—61no—
92C4×Q16—61no—
93#431C2×D311yescharacteristic
94C2×#108—61no—
95C4×M16—31no—
96C4×M16—31no—
97C4×M16C2×D311yescharacteristic
98C2×C4×C8—31no—
99#341—61no—
100#330—62no—
101#330—62no—
102Q8⋊C8—22no—
103Q8⋊C8—62no—
104#322—61no—
105#322—61no—
106C2×C3×Q8—21no—
107C2×C3×Q8—21no—
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4#4111yescharacteristic
110C22×C3×C4#4111yescharacteristic
111C2×Dic6—21no—
112C2×Dic6—21no—
113C3×M16—22no—
114C3×M16—41no—
115C2×C3×C8—22no—
116C2×C3×C8—41no—
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—41no—
125C3×C42C22×C411yescharacteristic
126C3×C42C2×D411yescharacteristic
127C3×C42C2×D411yescharacteristic
128C3×C42C2×D411yescharacteristic
129#217—41no—
130#216—42no—
131#216—42no—
132C4⋊Dic3—21no—
133C4⋊Dic3—21no—
134C4⋊Dic3—21no—
135C4⋊Dic3—21no—
136Dic3⋊C4—41no—
137Dic3⋊C4—41no—
138C4×Dic3—41no—
139#208—22no—
140#208—42no—
141#208—41no—
142C2×C3⋊C8—22no—
143C2×C3⋊C8—22no—
144C2×C3⋊C8—41no—
145C2×C3⋊C8—41no—
146C2×C3⋊C8C22⋊C411yescharacteristic
147C2×C3⋊C8C22⋊C411yescharacteristic
148C2×C3⋊C8C22⋊C411yescharacteristic
149C2×C3⋊C8C22⋊C411yescharacteristic
150C2×Q16—61no—
151C2×Q16—61no—
152C2×M16—21no—
153C2×M16—61no—
154C2×M16—61no—
155C22×C8—31no—
156C22×C8—31no—
157C4⋊Q8—31no—
158C4⋊Q8—31no—
159C4⋊Q8C4×D311yescharacteristic
160C4⋊Q8C4×D311yescharacteristic
161#125—21no—
162#125—61no—
163C22⋊Q8—21no—
164C22⋊Q8—21no—
165C22⋊Q8—61no—
166C22⋊Q8—61no—
167C4×Q8—21no—
168C4×Q8—21no—
169C4×Q8—61no—
170C4×Q8—61no—
171#117—21no—
172#117—61no—
173C2×C42C22×D311yescharacteristic
174#108—64no—
175#107—61no—
176#107—61no—
177C4⋊C8#6712yes—
178#103—61no—
179#103—61no—
180#103—61no—
181#103—61no—
182C22⋊C8—21no—
183#97—21no—
184#97—32no—
185#97—61no—
186C4×C8—21no—
187C4×C8—21no—
188C4×C8—31no—
189C4×C8—31no—
190C4×C8—31no—
191C4×C8—31no—
192C4×C8—61no—
193C4×C8—61no—
194C23×C3#12411yescharacteristic
195C3×Q8—41no—
196C3×Q8—41no—
197C3×Q8—42no—
198C2×C3×C4—21no—
199C2×C3×C4—21no—
200C2×C3×C4—22no—
201C2×C3×C4—22no—
202C2×C3×C4—41no—
203C2×C3×C4—41no—
204C2×C3×C4C2×C22⋊C411yescharacteristic
205C2×C3×C4C4×D411yescharacteristic
206C2×C3×C4C4×D411yescharacteristic
207C2×C3×C4#12111yescharacteristic
208C2×C3×C4#12111yescharacteristic
209C2×C3×C4#12711yescharacteristic
210C2×Dic3—41no—
211C2×Dic3—41no—
212Dic6—41no—
213Dic6—41no—
214Dic6—42no—
215C3×C8—42no—
216C3×C8—42no—
217C3⋊C8—21no—
218C3⋊C8—21no—
219C3⋊C8—21no—
220C3⋊C8—42no—
221C3⋊C8—42no—
222C3⋊C8#10412yes—
223C2×Q8—21no—
224C2×Q8—21no—
225C2×Q8—61no—
226C2×Q8—61no—
227C22×C4#23511yescharacteristic
228C22×C4D3×D411yescharacteristic
229C22×C4#23711yescharacteristic
230Q16—122no—
231Q16—122no—
232M16—22no—
233M16—41no—
234M16—62no—
235M16—122no—
236M16—121no—
237C2×C8—22no—
238C2×C8—31no—
239C2×C8—31no—
240C2×C8—31no—
241C2×C8—31no—
242C2×C8—41no—
243C2×C8—62no—
244C2×C8—62no—
245C2×C8—121no—
246C2×C8—121no—
247C4⋊C4—21no—
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—61no—
254C4⋊C4—61no—
255C4⋊C4—61no—
256C4⋊C4—61no—
257C4⋊C4—121no—
258C4⋊C4—121no—
259C22⋊C4—41no—
260C22⋊C4—121no—
261C42—41no—
262C42—121no—
263C42C2×C4×D311yescharacteristic
264C42D3×D411yescharacteristic
265C42D3×D411yescharacteristic
266C42C2×#6711yescharacteristic
267C22×C3—21no—
268C22×C3—21no—
269C22×C3—21no—
270C22×C3#39011yescharacteristic
271C3×C4—21no—
272C3×C4—21no—
273C3×C4—21no—
274C3×C4—81no—
275C3×C4—81no—
276C3×C4C2×#10411yescharacteristic
277C3×C4#42111yescharacteristic
278C3×C4#44412yes—
279C3×C4#47312yes—
280Dic3—81no—
281Dic3—81no—
282C23#88511yescharacteristic
283Q8—42no—
284Q8—41no—
285Q8—41no—
286Q8—122no—
287Q8—121no—
288Q8—121no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—121no—
296C2×C4—121no—
297C2×C4D3×C22⋊C411yescharacteristic
298C2×C4#67⋊C411yescharacteristic
299C2×C4#88411yescharacteristic
300C2×C4C4×#6711yescharacteristic
301C2×C4#93911yescharacteristic
302C2×C4#94011yescharacteristic
303C8—32no—
304C8—42no—
305C8—42no—
306C8—61no—
307C8—61no—
308C8—61no—
309C8—122no—
310C8—122no—
311C2×C3—41no—
312C2×C3#197511yescharacteristic
313C2×C3#197911yescharacteristic
314C2×C3#198011yescharacteristic
315C22—21no—
316C22—21no—
317C22—21no—
318C22#495711yescharacteristic
319C4—21no—
320C4—21no—
321C4—21no—
322C4—81no—
323C4—81no—
324C4—241no—
325C4—241no—
326C4D3×#10411yescharacteristic
327C4#482111yescharacteristic
328C4#514512yes—
329C4#520412yes—
330C3#972711yescharacteristic, complemented, Sylow
331C2—41no—
332C2#7000011yescharacteristic
333C2#7033611yescharacteristic
334C2#7035811yescharacteristic
subgroupG/H
centerC2×C4#939
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1485C2
socleC22×C3#390
prime radicalC23×C3#124
cosocleC2×C3×C42C23
hypercenter#1485D3
hypocenterC3#9727
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1485

Complements

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

NH ≅ G/Ntype
330. C3#9727semidirect

Maximal subgroups

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

subgroupindex
#833442
C3×#14852
#718212
#707692
#707412
C4×#8222
#698622
#97273