Finite Group Databasebeta

← #10691102

Subgroups — #10691102

Order 768.

490 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 113 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#79068C211yesmaximal, characteristic
2#74157C211yesmaximal, characteristic
3#70792C211yesmaximal, characteristic
4#70766C211yesmaximal, characteristic
5#69546C211yesmaximal, characteristic
6C3×#1286C211yesmaximal, characteristic
7#67715C211yesmaximal, characteristic
8#9176—31nomaximal, Sylow
9C2×#887—21no—
10#5470C2211yescharacteristic
11C2×C4×Dic6C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#423C2211yescharacteristic
14C3×#382C2211yescharacteristic
15#5121—21no—
16#5120—21no—
17#4972—21no—
18C4×C4⋊Dic3C2211yescharacteristic
19#4925C2211yescharacteristic
20#4851C412yescomplemented
21#4850C412yescomplemented
22#4829—22no—
23#4650—21no—
24#4649—21no—
25#4647—22no—
26#4484—22no—
27#4467—21no—
28#4467—21no—
29#4452—22no—
30#3078—31no—
31#2351—31no—
32#1518—31no—
33#1503—31no—
34#1479—31no—
35#1303—31no—
36#1286D311yescharacteristic
37C22×Dic6D411yescharacteristic
38C2×C3×M16—21no—
39C2×C3×C4⋊C4—21no—
40C2×C3×C42C2311yescharacteristic
41#926—21no—
42C2×C4⋊Dic3—21no—
43C2×C4⋊Dic3D411yescharacteristic
44#924—21no—
45#924—21no—
46C2×Dic3⋊C4—21no—
47C2×Dic3⋊C4—22no—
48C2×Dic3⋊C4—22no—
49C2×C4×Dic3—21no—
50C2×C4×Dic3—22no—
51C2×C4×Dic3—21no—
52#887—42no—
53#887—42no—
54#870—21no—
55#869C2×C411yescharacteristic
56#869C2×C411yescharacteristic
57C4×Dic6—21no—
58C4×Dic6—21no—
59C4×Dic6—21no—
60C4×Dic6—21no—
61C4×Dic6C2×C412yes—
62C4×Dic6D412yes—
63C3×C4⋊C8—21no—
64C3×C4⋊C8—21no—
65C3×C4⋊C8C2×C412yes—
66C3×C22⋊C8—21no—
67C3×C4×C8—21no—
68C3×#95—21no—
69C3×#95D412yes—
70C3⋊#95—22no—
71C3⋊#95D411yescharacteristic
72C3⋊#95D411yescharacteristic
73#830—21no—
74#818—22no—
75#817—22no—
76#816—22no—
77#816—22no—
78C4⋊(C3⋊C8)—21no—
79C4×C3⋊C8—21no—
80#533—31no—
81C2×C4×Q8—31no—
82C2×C4×Q8—61no—
83#479—62no—
84#477—32no—
85#475—32no—
86#431—31no—
87#431C2×D311yescharacteristic
88#428—61no—
89#423C2×D311yescharacteristic
90#419—61no—
91#389—61no—
92#389—62no—
93#382—61no—
94#382C2×D311yescharacteristic
95C4×C4⋊C4—31no—
96C4×C4⋊C4—61no—
97#328—61no—
98#328—61no—
99#328—62no—
100Q8⋊C8—62no—
101C22×C3×C4—22no—
102C22×C3×C4C2×D411yescharacteristic
103C22×C3×C4C2×D411yescharacteristic
104C22×C3×C4C2×D411yescharacteristic
105C22×Dic3—21no—
106C22×Dic3—21no—
107C22×Dic3—21no—
108C22×Dic3—21no—
109C2×Dic6—21no—
110C2×Dic6—21no—
111C2×Dic6—21no—
112C2×Dic6—21no—
113C2×Dic6—21no—
114C2×Dic6C22⋊C412yes—
115C2×Dic6SD1612yes—
116C2×Dic6Q1612yes—
117C3×M16—41no—
118C3×M16—41no—
119C2×C3×C8—22no—
120C2×C3×C8—41no—
121C3×C4⋊C4—41no—
122C3×C4⋊C4—41no—
123C3×C42C22×C411yescharacteristic
124C3×C42C2×D411yescharacteristic
125C3×C42#4111yescharacteristic
126C3×C42#4111yescharacteristic
127#217—41no—
128C4⋊Dic3—21no—
129C4⋊Dic3—21no—
130C4⋊Dic3—21no—
131C4⋊Dic3—21no—
132C4⋊Dic3—21no—
133C4⋊Dic3—42no—
134C4⋊Dic3C22⋊C412yes—
135Dic3⋊C4—22no—
136Dic3⋊C4—22no—
137Dic3⋊C4—44no—
138Dic3⋊C4—44no—
139Dic3⋊C4—41no—
140Dic3⋊C4—41no—
141C4×Dic3—22no—
142C4×Dic3—22no—
143C4×Dic3—44no—
144C4×Dic3—41no—
145C4×Dic3—41no—
146C4×Dic3—41no—
147C2×C3⋊C8—41no—
148C22×Q8—31no—
149C2×M16—21no—
150C4⋊Q8—31no—
151C4⋊Q8—31no—
152#125—61no—
153C22⋊Q8—61no—
154C22⋊Q8—61no—
155C4×Q8—32no—
156C4×Q8—32no—
157C4×Q8—61no—
158C4×Q8—61no—
159C4×Q8—61no—
160C4×Q8—61no—
161C4×Q8—122no—
162C4×Q8—122no—
163#117—61no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—31no—
166C2×C4⋊C4—61no—
167C2×C4⋊C4—62no—
168C2×C4⋊C4—62no—
169C2×C4⋊C4—61no—
170C2×C42—61no—
171C2×C42—62no—
172C2×C42—61no—
173C2×C42C22×D311yescharacteristic
174#107—62no—
175#106—62no—
176C4⋊C8—21no—
177C4⋊C8—21no—
178C4⋊C8—61no—
179C4⋊C8C4×D312yes—
180#103—62no—
181#103—62no—
182C22⋊C8—21no—
183C22⋊C8—61no—
184C4×C8—21no—
185C4×C8—61no—
186#95—21no—
187#95—31no—
188#95—31no—
189#95—62no—
190#95D1212yes—
191C23×C3#12011yescharacteristic
192C2×C3×C4—41no—
193C2×C3×C4—41no—
194C2×C3×C4—42no—
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×SD1611yescharacteristic
204C2×C3×C4C2×Q1611yescharacteristic
205C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
206C2×C3×C4#13711yescharacteristic
207C2×Dic3—22no—
208C2×Dic3—22no—
209C2×Dic3—22no—
210C2×Dic3—22no—
211C2×Dic3—22no—
212C2×Dic3—42no—
213C2×Dic3—41no—
214C2×Dic3—41no—
215C2×Dic3—41no—
216C2×Dic3—41no—
217C2×Dic3—41no—
218C2×Dic3—41no—
219C2×Dic3—41no—
220Dic6—22no—
221Dic6—22no—
222Dic6—22no—
223Dic6—81no—
224Dic6—81no—
225Dic6#10314yes—
226C3×C8—42no—
227C3×C8—81no—
228C3⋊C8—81no—
229C2×Q8—32no—
230C2×Q8—32no—
231C2×Q8—32no—
232C2×Q8—61no—
233C2×Q8—61no—
234C2×Q8—61no—
235C2×Q8—61no—
236C2×Q8—61no—
237C22×C4—22no—
238C22×C4—61no—
239C22×C4—61no—
240C22×C4—61no—
241C22×C4—61no—
242C22×C4C2×D1211yescharacteristic
243C22×C4D3×D411yescharacteristic
244C22×C4D3×D411yescharacteristic
245M16—41no—
246M16—41no—
247C2×C8—22no—
248C2×C8—41no—
249C2×C8—121no—
250C4⋊C4—32no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—62no—
254C4⋊C4—61no—
255C4⋊C4—62no—
256C4⋊C4—61no—
257C4⋊C4—61no—
258C4⋊C4—61no—
259C4⋊C4—61no—
260C4⋊C4—124no—
261C4⋊C4—124no—
262C4⋊C4—122no—
263C4⋊C4—121no—
264C4⋊C4—121no—
265C22⋊C4—121no—
266C42—62no—
267C42—62no—
268C42—124no—
269C42—121no—
270C42—121no—
271C42—121no—
272C42C2×C4×D311yescharacteristic
273C42#23511yescharacteristic
274C42D3×D411yescharacteristic
275C42#23911yescharacteristic
276C22×C3—21no—
277C22×C3—21no—
278C22×C3#38611yescharacteristic
279C22×C3#45011yescharacteristic
280C22×C3#45111yescharacteristic
281C3×C4—21no—
282C3×C4—21no—
283C3×C4—21no—
284C3×C4—21no—
285C3×C4—82no—
286C3×C4C2×#10311yescharacteristic
287C3×C4#41711yescharacteristic
288C3×C4#48311yescharacteristic
289C3×C4#48411yescharacteristic
290Dic3—24no—
291Dic3—42no—
292Dic3—42no—
293Dic3—81no—
294Dic3—81no—
295Dic3—81no—
296C23#88211yescharacteristic
297Q8—34no—
298Q8—62no—
299Q8—62no—
300Q8—62no—
301Q8—241no—
302Q8—241no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—62no—
309C2×C4—62no—
310C2×C4—62no—
311C2×C4—62no—
312C2×C4—62no—
313C2×C4—122no—
314C2×C4—121no—
315C2×C4—121no—
316C2×C4—121no—
317C2×C4—121no—
318C2×C4—121no—
319C2×C4—121no—
320C2×C4—121no—
321C2×C4C4×D1211yescharacteristic
322C2×C4D3×C22⋊C411yescharacteristic
323C2×C4#88411yescharacteristic
324C2×C4#89311yescharacteristic
325C2×C4#89411yescharacteristic
326C2×C4#89511yescharacteristic
327C2×C4#91111yescharacteristic
328C2×C4D3×SD1611yescharacteristic
329C2×C4#91511yescharacteristic
330C2×C4D3×Q1611yescharacteristic
331C8—42no—
332C8—81no—
333C8—241no—
334C2×C3—21no—
335C2×C3—21no—
336C2×C3#187211yescharacteristic
337C2×C3#190511yescharacteristic
338C2×C3#190611yescharacteristic
339C22—21no—
340C22—21no—
341C22#466511yescharacteristic
342C22#477611yescharacteristic
343C22#480911yescharacteristic
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—64no—
349C4—82no—
350C4—122no—
351C4—122no—
352C4—241no—
353C4—241no—
354C4—241no—
355C4#477111yescharacteristic
356C4D3×#10311yescharacteristic
357C4#486211yescharacteristic
358C4#488411yescharacteristic
359C3#917611yescharacteristic, complemented, Sylow
360C2—21no—
361C2—21no—
362C2#6870711yescharacteristic
363C2#6887811yescharacteristic
364C2#6903911yescharacteristic
subgroupG/H
centerC22#4665
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1286C2
socleC22×C3#386
prime radicalC23×C3#120
cosocleC2×C3×C42C23
hypercenter#1286D3
hypocenterC3#9176
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1286

Complements

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

NH ≅ G/Ntype
20. #4851C4semidirect
21. #4850C4semidirect
359. C3#9176semidirect

Maximal subgroups

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

subgroupindex
#790682
#741572
#707922
#707662
#695462
C3×#12862
#677152
#91763