Finite Group Databasebeta

← #10691446

Subgroups — #10691446

Order 768.

566 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 115 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#79310C211yesmaximal, characteristic
2C4×#887C211yesmaximal, characteristic
3#70514C211yesmaximal, characteristic
4#70500C211yesmaximal, characteristic
5#69471C211yesmaximal, characteristic
6C3×#1303C211yesmaximal, characteristic
7#67736C211yesmaximal, characteristic
8#9232—31nomaximal, Sylow
9C2×#887—21no—
10#5470C2211yescharacteristic
11C2×C4×Dic6C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#423C2211yescharacteristic
14C3×C4×C4⋊C4C2211yescharacteristic
15Dic3×C4⋊C4—21no—
16C4×Dic3⋊C4—21no—
17C42×Dic3—21no—
18C42×Dic3C2211yescharacteristic
19#4925C2211yescharacteristic
20#4850C412yescomplemented
21#4849C412yescomplemented
22#4739—21no—
23#4722—21no—
24C4×Dic12—22no—
25#4649—22no—
26#4555—21no—
27#4555—21no—
28#4484—22no—
29#4452—22no—
30#3104—31no—
31C42×Q8—31no—
32#1994—31no—
33#1503—31no—
34#1487—31no—
35#1303D311yescharacteristic
36#1276—31no—
37C22×Dic6D411yescharacteristic
38C2×C3×M16—21no—
39C2×C3×C4⋊C4—21no—
40C2×C3×C42—21no—
41C2×C3×C42C2311yescharacteristic
42#926—21no—
43C2×C4⋊Dic3D411yescharacteristic
44#924—21no—
45#924—21no—
46C2×Dic3⋊C4—22no—
47C2×Dic3⋊C4—21no—
48C2×C4×Dic3—22no—
49C2×C4×Dic3—21no—
50C2×C4×Dic3—21no—
51C2×C4×Dic3—22no—
52C2×C4×Dic3—21no—
53C2×C4×Dic3D411yescharacteristic
54C2×C4×Dic3D411yescharacteristic
55C2×Dic12—22no—
56#887—28no—
57#870—21no—
58#869C2×C411yescharacteristic
59#869C2×C411yescharacteristic
60C4×Dic6—21no—
61C4×Dic6—21no—
62C4×Dic6—21no—
63C4×Dic6—21no—
64C4×Dic6C2×C412yes—
65C4×Dic6D412yes—
66C3×C4⋊C8—21no—
67C3×C4⋊C8—21no—
68C3×C4⋊C8C2×C412yes—
69C3×C22⋊C8—21no—
70C3×C4×C8—21no—
71C3×#95D412yes—
72C3⋊#95—22no—
73#830—21no—
74#818—22no—
75#816—22no—
76#816—22no—
77C4⋊(C3⋊C8)—21no—
78C4×C3⋊C8—21no—
79#533—31no—
80C2×C4×Q8—31no—
81C2×C4×Q8—61no—
82#484—61no—
83#477—32no—
84#462—32no—
85#451—61no—
86C4×Q16—62no—
87#431—31no—
88#431C2×D311yescharacteristic
89#423C2×D311yescharacteristic
90C4×C4⋊C4—61no—
91C4×C4⋊C4—61no—
92C4×C4⋊C4—62no—
93C4×C4⋊C4C2×D311yescharacteristic
94C43—31no—
95C43—61no—
96#339—61no—
97#339—61no—
98#328—62no—
99Q8⋊C8—62no—
100C22×C3×C4—22no—
101C22×C3×C4C2×D411yescharacteristic
102C22×C3×C4C2×D411yescharacteristic
103C22×C3×C4C2×D411yescharacteristic
104C22×Dic3—21no—
105C22×Dic3—21no—
106C22×Dic3—21no—
107C22×Dic3—21no—
108C2×Dic6—21no—
109C2×Dic6—24no—
110C2×Dic6—21no—
111C2×Dic6—21no—
112C2×Dic6C22⋊C412yes—
113C3×M16—41no—
114C3×M16—41no—
115C2×C3×C8—22no—
116C2×C3×C8—41no—
117C3×C4⋊C4—24no—
118C3×C42—41no—
119C3×C42—41no—
120C3×C42C22×C411yescharacteristic
121C3×C42C2×D411yescharacteristic
122C3×C42#4111yescharacteristic
123C3×C42#4111yescharacteristic
124#217—41no—
125C4⋊Dic3—21no—
126C4⋊Dic3—21no—
127C4⋊Dic3—21no—
128C4⋊Dic3—21no—
129C4⋊Dic3—21no—
130C4⋊Dic3C22⋊C412yes—
131Dic3⋊C4—28no—
132Dic3⋊C4—22no—
133Dic3⋊C4—22no—
134Dic3⋊C4—41no—
135Dic3⋊C4—41no—
136C4×Dic3—28no—
137C4×Dic3—24no—
138C4×Dic3—22no—
139C4×Dic3—22no—
140C4×Dic3—21no—
141C4×Dic3—21no—
142C4×Dic3—21no—
143C4×Dic3—42no—
144C4×Dic3—44no—
145C4×Dic3—41no—
146C4×Dic3Q1612yescomplemented
147C2×C3⋊C8—41no—
148Dic12—44no—
149C22×Q8—31no—
150C2×Q16—62no—
151C2×M16—21no—
152C4⋊Q8—31no—
153C4⋊Q8—31no—
154#125—61no—
155C22⋊Q8—61no—
156C22⋊Q8—61no—
157C4×Q8—32no—
158C4×Q8—32no—
159C4×Q8—68no—
160C4×Q8—61no—
161C4×Q8—61no—
162C4×Q8—61no—
163C4×Q8—61no—
164#117—61no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—31no—
167C2×C4⋊C4—62no—
168C2×C4⋊C4—61no—
169C2×C42—21no—
170C2×C42—31no—
171C2×C42—31no—
172C2×C42—62no—
173C2×C42—61no—
174C2×C42—61no—
175C2×C42—62no—
176C2×C42—61no—
177C2×C42C22×D311yescharacteristic
178#107—62no—
179C4⋊C8—21no—
180C4⋊C8—21no—
181C4⋊C8—61no—
182C4⋊C8C4×D312yes—
183#103—62no—
184#103—62no—
185C22⋊C8—21no—
186C22⋊C8—61no—
187C4×C8—21no—
188C4×C8—61no—
189#95—62no—
190#95D1212yes—
191C23×C3#12011yescharacteristic
192C2×C3×C4—22no—
193C2×C3×C4—24no—
194C2×C3×C4—41no—
195C2×C3×C4—42no—
196C2×C3×C4—42no—
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×C3×C4C2×Q1611yescharacteristic
204C2×C3×C4#13511yescharacteristic
205C2×C3×C4#13711yescharacteristic
206C2×C3×C4#13711yescharacteristic
207C2×Dic3—22no—
208C2×Dic3—24no—
209C2×Dic3—22no—
210C2×Dic3—22no—
211C2×Dic3—22no—
212C2×Dic3—22no—
213C2×Dic3—22no—
214C2×Dic3—41no—
215C2×Dic3—42no—
216C2×Dic3—41no—
217C2×Dic3—41no—
218C2×Dic3—41no—
219Dic6—24no—
220Dic6—22no—
221Dic6—42no—
222Dic6—81no—
223Dic6#10414yes—
224C3×C8—42no—
225C3×C8—81no—
226C3⋊C8—81no—
227C2×Q8—32no—
228C2×Q8—64no—
229C2×Q8—61no—
230C2×Q8—61no—
231C2×Q8—61no—
232C22×C4—22no—
233C22×C4—61no—
234C22×C4—61no—
235C22×C4—61no—
236C22×C4—61no—
237C22×C4C2×D1211yescharacteristic
238C22×C4D3×D411yescharacteristic
239C22×C4D3×D411yescharacteristic
240Q16—124no—
241M16—41no—
242M16—41no—
243C2×C8—22no—
244C2×C8—41no—
245C2×C8—121no—
246C4⋊C4—24no—
247C4⋊C4—32no—
248C4⋊C4—68no—
249C4⋊C4—62no—
250C4⋊C4—61no—
251C4⋊C4—62no—
252C4⋊C4—61no—
253C4⋊C4—61no—
254C4⋊C4—61no—
255C4⋊C4—61no—
256C4⋊C4—121no—
257C4⋊C4—121no—
258C22⋊C4—121no—
259C42—32no—
260C42—41no—
261C42—41no—
262C42—64no—
263C42—68no—
264C42—62no—
265C42—62no—
266C42—61no—
267C42—61no—
268C42—61no—
269C42—122no—
270C42—124no—
271C42—121no—
272C42C2×C4×D311yescharacteristic
273C42#23511yescharacteristic
274C42D3×D411yescharacteristic
275C42#23911yescharacteristic
276C22×C3—21no—
277C22×C3—21no—
278C22×C3#38611yescharacteristic
279C22×C3#45111yescharacteristic
280C22×C3#45211yescharacteristic
281C3×C4—21no—
282C3×C4—21no—
283C3×C4—21no—
284C3×C4—44no—
285C3×C4C2×#10412yes—
286C3×C4C4×Q1611yescharacteristic
287C3×C4#44111yescharacteristic
288C3×C4#46211yescharacteristic
289C3×C4#46411yescharacteristic
290Dic3—24no—
291Dic3—24no—
292Dic3—42no—
293Dic3—42no—
294Dic3—81no—
295Dic3—81no—
296C23#88211yescharacteristic
297Q8—34no—
298Q8—64no—
299Q8—62no—
300Q8—122no—
301Q8—241no—
302C2×C4—22no—
303C2×C4—24no—
304C2×C4—41no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—62no—
308C2×C4—64no—
309C2×C4—62no—
310C2×C4—62no—
311C2×C4—62no—
312C2×C4—62no—
313C2×C4—62no—
314C2×C4—121no—
315C2×C4—122no—
316C2×C4—121no—
317C2×C4—121no—
318C2×C4—121no—
319C2×C4C4×D1211yescharacteristic
320C2×C4D3×C22⋊C411yescharacteristic
321C2×C4#88411yescharacteristic
322C2×C4#89311yescharacteristic
323C2×C4#89411yescharacteristic
324C2×C4#89511yescharacteristic
325C2×C4#91211yescharacteristic
326C2×C4#91511yescharacteristic
327C2×C4#91511yescharacteristic
328C2×C4D3×Q1611yescharacteristic
329C8—42no—
330C8—81no—
331C8—241no—
332C2×C3—21no—
333C2×C3—21no—
334C2×C3#187811yescharacteristic
335C2×C3#189611yescharacteristic
336C2×C3#190611yescharacteristic
337C22—21no—
338C22—21no—
339C22#466511yescharacteristic
340C22#478311yescharacteristic
341C22#480911yescharacteristic
342C4—21no—
343C4—21no—
344C4—21no—
345C4—44no—
346C4—64no—
347C4—64no—
348C4—122no—
349C4—122no—
350C4—241no—
351C4—241no—
352C4#476711yescharacteristic
353C4#479511yescharacteristic
354C4D3×#10412yes—
355C4#485311yescharacteristic
356C4#487311yescharacteristic
357C3#923211yescharacteristic, complemented, Sylow
358C2—21no—
359C2—21no—
360C2#6875611yescharacteristic
361C2#6890811yescharacteristic
362C2#6903911yescharacteristic
subgroupG/H
centerC2×C4#884
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1303C2
socleC22×C3#386
prime radicalC23×C3#120
cosocleC2×C3×C42C23
hypercenter#1303D3
hypocenterC3#9232
LCSG ▷ 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
20. #4850C4semidirect
21. #4849C4semidirect
146. C4×Dic3Q16semidirect
357. C3#9232semidirect

Maximal subgroups

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

subgroupindex
#793102
C4×#8872
#705142
#705002
#694712
C3×#13032
#677362
#92323