Finite Group Databasebeta

← #10691116

Subgroups — #10691116

Order 768.

446 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 105 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#74159C211yesmaximal, characteristic
3#70792C211yesmaximal, characteristic
4#70766C211yesmaximal, characteristic
5#69546C211yesmaximal, characteristic
6C3×#1286C211yesmaximal, characteristic
7#67715C211yesmaximal, characteristic
8#9178—31nomaximal, Sylow
9#5470C2211yescharacteristic
10C2×C4×Dic6C2211yescharacteristic
11C3×#431C2211yescharacteristic
12C3×#423C2211yescharacteristic
13C3×#382C2211yescharacteristic
14#5121—21no—
15#5120—21no—
16Dic3×C4⋊C4—21no—
17#4973—21no—
18C4×C4⋊Dic3C2211yescharacteristic
19#4925C2211yescharacteristic
20#4851C412yescomplemented
21#4850C412yescomplemented
22#4829—22no—
23#4655—22no—
24#4651—21no—
25#4648—21no—
26#4484—22no—
27#4467—21no—
28#4467—21no—
29#4452—22no—
30#3078—31no—
31#2356—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×C4×Dic3—21no—
48C2×C4×Dic3—22no—
49C2×C4×Dic3—21no—
50#870—21no—
51#869C2×C411yescharacteristic
52#869C2×C411yescharacteristic
53C4×Dic6—21no—
54C4×Dic6—21no—
55C4×Dic6—21no—
56C4×Dic6—21no—
57C4×Dic6C2×C412yes—
58C4×Dic6D412yes—
59C3×C4⋊C8—21no—
60C3×C4⋊C8—21no—
61C3×C4⋊C8C2×C412yes—
62C3×C22⋊C8—21no—
63C3×C4×C8—21no—
64C3×#95—21no—
65C3×#95D412yes—
66C3⋊#95—22no—
67C3⋊#95—22no—
68C3⋊#95—22no—
69C3⋊#95D411yescharacteristic
70C3⋊#95D411yescharacteristic
71#830—21no—
72#818—22no—
73#817—22no—
74#816—22no—
75#816—22no—
76C4⋊(C3⋊C8)—21no—
77C4×C3⋊C8—21no—
78#533—31no—
79C2×C4×Q8—31no—
80#479—62no—
81#477—32no—
82#475—32no—
83#431—31no—
84#431C2×D311yescharacteristic
85#428—61no—
86#423C2×D311yescharacteristic
87#419—61no—
88#391—61no—
89#387—62no—
90#383—61no—
91#382C2×D311yescharacteristic
92C4×C4⋊C4—31no—
93C4×C4⋊C4—61no—
94#376—61no—
95#328—61no—
96#328—61no—
97#328—62no—
98Q8⋊C8—62no—
99C22×C3×C4—22no—
100C22×C3×C4C2×D411yescharacteristic
101C22×C3×C4C2×D411yescharacteristic
102C22×C3×C4C2×D411yescharacteristic
103C22×Dic3—21no—
104C22×Dic3—21no—
105C22×Dic3—21no—
106C22×Dic3—21no—
107C2×Dic6—21no—
108C2×Dic6—21no—
109C2×Dic6—21no—
110C2×Dic6—21no—
111C2×Dic6—21no—
112C2×Dic6—21no—
113C2×Dic6—21no—
114C2×Dic6C22⋊C412yes—
115C3×M16—41no—
116C3×M16—41no—
117C2×C3×C8—22no—
118C2×C3×C8—41no—
119C3×C4⋊C4—41no—
120C3×C4⋊C4—41no—
121C3×C42C22×C411yescharacteristic
122C3×C42C2×D411yescharacteristic
123C3×C42#4111yescharacteristic
124C3×C42#4111yescharacteristic
125#217—41no—
126C4⋊Dic3—21no—
127C4⋊Dic3—21no—
128C4⋊Dic3—21no—
129C4⋊Dic3—21no—
130C4⋊Dic3—21no—
131C4⋊Dic3—42no—
132C4⋊Dic3C22⋊C412yes—
133Dic3⋊C4—22no—
134Dic3⋊C4—22no—
135Dic3⋊C4—41no—
136Dic3⋊C4—41no—
137C4×Dic3—22no—
138C4×Dic3—22no—
139C4×Dic3—44no—
140C4×Dic3—41no—
141C4×Dic3—41no—
142C4×Dic3—41no—
143C2×C3⋊C8—41no—
144C22×Q8—31no—
145C2×M16—21no—
146C4⋊Q8—31no—
147C4⋊Q8—31no—
148#125—61no—
149C22⋊Q8—61no—
150C22⋊Q8—61no—
151C4×Q8—32no—
152C4×Q8—32no—
153C4×Q8—61no—
154C4×Q8—61no—
155C4×Q8—61no—
156C4×Q8—61no—
157#117—61no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—31no—
160C2×C4⋊C4—61no—
161C2×C4⋊C4—61no—
162C2×C42—61no—
163C2×C42—62no—
164C2×C42—61no—
165C2×C42C22×D311yescharacteristic
166#107—62no—
167#106—62no—
168C4⋊C8—21no—
169C4⋊C8—21no—
170C4⋊C8—61no—
171C4⋊C8C4×D312yes—
172#103—62no—
173#103—62no—
174C22⋊C8—21no—
175C22⋊C8—61no—
176C4×C8—21no—
177C4×C8—61no—
178#95—21no—
179#95—31no—
180#95—31no—
181#95—62no—
182#95—62no—
183#95—62no—
184#95D1212yes—
185C23×C3#12011yescharacteristic
186C2×C3×C4—41no—
187C2×C3×C4—41no—
188C2×C3×C4—42no—
189C2×C3×C4—42no—
190C2×C3×C4—42no—
191C2×C3×C4C2×C22⋊C411yescharacteristic
192C2×C3×C4C4×D411yescharacteristic
193C2×C3×C4C4×D411yescharacteristic
194C2×C3×C4#12111yescharacteristic
195C2×C3×C4#12111yescharacteristic
196C2×C3×C4#12311yescharacteristic
197C2×C3×C4#13511yescharacteristic
198C2×C3×C4#13511yescharacteristic
199C2×C3×C4#13711yescharacteristic
200C2×C3×C4#13711yescharacteristic
201C2×Dic3—22no—
202C2×Dic3—22no—
203C2×Dic3—22no—
204C2×Dic3—42no—
205C2×Dic3—41no—
206C2×Dic3—41no—
207C2×Dic3—41no—
208C2×Dic3—41no—
209C2×Dic3—41no—
210C2×Dic3—41no—
211C2×Dic3—41no—
212C2×Dic3—41no—
213C2×Dic3—41no—
214Dic6—22no—
215Dic6—22no—
216Dic6—41no—
217Dic6—41no—
218Dic6—81no—
219Dic6—81no—
220C3×C8—42no—
221C3×C8—81no—
222C3⋊C8—81no—
223C2×Q8—32no—
224C2×Q8—61no—
225C2×Q8—61no—
226C2×Q8—61no—
227C2×Q8—61no—
228C2×Q8—61no—
229C2×Q8—61no—
230C2×Q8—61no—
231C22×C4—22no—
232C22×C4—61no—
233C22×C4—61no—
234C22×C4—61no—
235C22×C4—61no—
236C22×C4C2×D1211yescharacteristic
237C22×C4D3×D411yescharacteristic
238C22×C4D3×D411yescharacteristic
239M16—41no—
240M16—41no—
241C2×C8—22no—
242C2×C8—41no—
243C2×C8—121no—
244C4⋊C4—32no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—62no—
248C4⋊C4—61no—
249C4⋊C4—62no—
250C4⋊C4—61no—
251C4⋊C4—61no—
252C4⋊C4—61no—
253C4⋊C4—61no—
254C4⋊C4—122no—
255C4⋊C4—121no—
256C4⋊C4—121no—
257C22⋊C4—121no—
258C42—62no—
259C42—62no—
260C42—124no—
261C42—121no—
262C42—121no—
263C42—121no—
264C42C2×C4×D311yescharacteristic
265C42#23511yescharacteristic
266C42D3×D411yescharacteristic
267C42#23911yescharacteristic
268C22×C3—21no—
269C22×C3—21no—
270C22×C3#38611yescharacteristic
271C22×C3#45211yescharacteristic
272C22×C3#45211yescharacteristic
273C3×C4—21no—
274C3×C4—21no—
275C3×C4—21no—
276C3×C4—21no—
277C3×C4—82no—
278C3×C4#41611yescharacteristic
279C3×C4#41911yescharacteristic
280C3×C4#48511yescharacteristic
281C3×C4#48511yescharacteristic
282Dic3—42no—
283Dic3—42no—
284Dic3—42no—
285Dic3—81no—
286Dic3—81no—
287Dic3—81no—
288C23#88211yescharacteristic
289Q8—62no—
290Q8—62no—
291Q8—121no—
292Q8—121no—
293Q8—241no—
294Q8—241no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—62no—
301C2×C4—62no—
302C2×C4—62no—
303C2×C4—122no—
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—121no—
313C2×C4C4×D1211yescharacteristic
314C2×C4D3×C22⋊C411yescharacteristic
315C2×C4#88411yescharacteristic
316C2×C4#89311yescharacteristic
317C2×C4#89411yescharacteristic
318C2×C4#89511yescharacteristic
319C2×C4#91211yescharacteristic
320C2×C4#91511yescharacteristic
321C2×C4#91611yescharacteristic
322C2×C4#91811yescharacteristic
323C8—42no—
324C8—81no—
325C8—241no—
326C2×C3—21no—
327C2×C3—21no—
328C2×C3#187311yescharacteristic
329C2×C3#190511yescharacteristic
330C2×C3#190611yescharacteristic
331C22—21no—
332C22—21no—
333C22#466511yescharacteristic
334C22#478311yescharacteristic
335C22#480811yescharacteristic
336C4—21no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—82no—
341C4—122no—
342C4—122no—
343C4—122no—
344C4—241no—
345C4—241no—
346C4—241no—
347C4#477211yescharacteristic
348C4#480211yescharacteristic
349C4#486811yescharacteristic
350C4#488811yescharacteristic
351C3#917811yescharacteristic, complemented, Sylow
352C2—21no—
353C2—21no—
354C2#6870711yescharacteristic
355C2#6887911yescharacteristic
356C2#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#9178
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
351. C3#9178semidirect

Maximal subgroups

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

subgroupindex
#790682
#741592
#707922
#707662
#695462
C3×#12862
#677152
#91783