Finite Group Databasebeta

← #10691850

Subgroups — #10691850

Order 768.

422 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 109 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#79313C211yesmaximal, characteristic
2#78480C211yesmaximal, characteristic
3#70793C211yesmaximal, characteristic
4#70780C211yesmaximal, characteristic
5#69828C211yesmaximal, characteristic
6C3×#1275C211yesmaximal, characteristic
7#67738C211yesmaximal, characteristic
8#9290—31nomaximal, Sylow
9#5470C2211yescharacteristic
10C2×#869C2211yescharacteristic
11C3×#431C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C2×C3×#97C2211yescharacteristic
14#5102—21no—
15#5101—21no—
16#4936C2211yescharacteristic
17#4924C2211yescharacteristic
18#4849—22no—
19#4739—22no—
20#4701—22no—
21#4701—22no—
22#4699C412yescomplemented
23#4681C412yescomplemented
24#4551—21no—
25#4551—21no—
26#4482—22no—
27#4467—22no—
28#3209—31no—
29#2961—31no—
30#1993—31no—
31#1549—31no—
32#1518—31no—
33#1286—31no—
34#1275D311yescharacteristic
35C22×Dic6—21no—
36C22×C3×C8D411yescharacteristic
37C22×C3×C8D411yescharacteristic
38C2×C3×C42C2311yescharacteristic
39#926—21no—
40C2×C4⋊Dic3—21no—
41C2×C4⋊Dic3—21no—
42C2×C4⋊Dic3—21no—
43#924—21no—
44#924D412yes—
45C2×#208—21no—
46C2×Dic12—24no—
47#870—21no—
48#869—21no—
49#869—21no—
50#869—21no—
51#869C2×C412yes—
52#869C2×C411yescharacteristic
53#869C2×C411yescharacteristic
54C4×Dic6—21no—
55C4×Dic6D412yes—
56C3×C4⋊C8—21no—
57C3×C4⋊C8—21no—
58C3×C22⋊C8—21no—
59C3×C22⋊C8—21no—
60C3×#97—21no—
61C3×#97—21no—
62C3×C4×C8—21no—
63C3×C4×C8C2×C412yes—
64C3⋊#95—21no—
65C3⋊#95—21no—
66#816—24no—
67#815—22no—
68Dic3⋊C8—24no—
69C8×Dic3—22no—
70C4⋊(C3⋊C8)—21no—
71C4⋊(C3⋊C8)D412yes—
72#533—31no—
73C2×C4⋊Q8—31no—
74#494—32no—
75#487—32no—
76#462—62no—
77#451—62no—
78#446—62no—
79#446—62no—
80#431C2×D311yescharacteristic
81#431C2×D311yescharacteristic
82#424—61no—
83#423—31no—
84#405—61no—
85C2×#97C2×D311yescharacteristic
86#382—31no—
87#340—61no—
88#340—61no—
89#332—62no—
90#328—62no—
91C22×C3×C4C2×D411yescharacteristic
92C22×C3×C4C2×D411yescharacteristic
93C22×C3×C4#4111yescharacteristic
94C22×Dic3—41no—
95C22×Dic3—41no—
96C2×Dic6—22no—
97C2×Dic6—21no—
98C2×Dic6—41no—
99C2×Dic6—82no—
100C2×Dic6C22⋊C412yes—
101C2×C3×C8—21no—
102C2×C3×C8—21no—
103C2×C3×C8—22no—
104C2×C3×C8—21no—
105C2×C3×C8—21no—
106C2×C3×C8—21no—
107C2×C3×C8—41no—
108C2×C3×C8Q1612yescomplemented
109C3×C42C22×C411yescharacteristic
110C3×C42C2×D411yescharacteristic
111C3×C42C2×D411yescharacteristic
112C3×C42#4111yescharacteristic
113#217—22no—
114C4⋊Dic3—21no—
115C4⋊Dic3—21no—
116C4⋊Dic3—21no—
117C4⋊Dic3—22no—
118C4⋊Dic3—22no—
119C4⋊Dic3—41no—
120C4⋊Dic3—41no—
121C4⋊Dic3—41no—
122C4⋊Dic3—41no—
123C4⋊Dic3C22⋊C412yes—
124Dic3⋊C4—22no—
125Dic3⋊C4—22no—
126C4×Dic3—22no—
127#208—41no—
128#208—41no—
129C2×C3⋊C8—22no—
130Dic12—84no—
131C22×Q8—61no—
132C2×Q16—64no—
133C2×M16—61no—
134C22×C8D1211yescharacteristic
135C22×C8D1211yescharacteristic
136C4⋊Q8—31no—
137C4⋊Q8—31no—
138C4⋊Q8—32no—
139C4⋊Q8—61no—
140C4⋊Q8—61no—
141C4⋊Q8—61no—
142#125—61no—
143C22⋊Q8—32no—
144C22⋊Q8—61no—
145C4×Q8—32no—
146C4×Q8—61no—
147#117—61no—
148C2×C4⋊C4—61no—
149C2×C4⋊C4—61no—
150C2×C4⋊C4—61no—
151C2×C42C22×D311yescharacteristic
152C4⋊C8—21no—
153C4⋊C8—21no—
154C4⋊C8—32no—
155C4⋊C8—64no—
156C4⋊C8—61no—
157#103—64no—
158C22⋊C8—21no—
159C22⋊C8—21no—
160#97—21no—
161#97—21no—
162#97—62no—
163C4×C8—21no—
164C4×C8—62no—
165C4×C8C4×D312yes—
166#95—61no—
167#95—61no—
168C23×C3#12111yescharacteristic
169C2×C3×C4—41no—
170C2×C3×C4—41no—
171C2×C3×C4C2×C22⋊C411yescharacteristic
172C2×C3×C4C4×D411yescharacteristic
173C2×C3×C4C4×D411yescharacteristic
174C2×C3×C4#12011yescharacteristic
175C2×C3×C4#12111yescharacteristic
176C2×C3×C4#12311yescharacteristic
177C2×C3×C4C2×Q1611yescharacteristic
178C2×C3×C4#13511yescharacteristic
179C2×C3×C4#13711yescharacteristic
180C2×C3×C4#13711yescharacteristic
181C2×Dic3—22no—
182C2×Dic3—22no—
183C2×Dic3—42no—
184C2×Dic3—82no—
185C2×Dic3—81no—
186C2×Dic3—81no—
187C2×Dic3—81no—
188Dic6—42no—
189Dic6—44no—
190Dic6—81no—
191Dic6—82no—
192C3×C8—22no—
193C3×C8—42no—
194C3×C8—41no—
195C3×C8—81no—
196C3⋊C8—42no—
197C2×Q8—32no—
198C2×Q8—61no—
199C2×Q8—62no—
200C2×Q8—121no—
201C2×Q8—242no—
202C22×C4—121no—
203C22×C4—121no—
204C22×C4C2×D1211yescharacteristic
205C22×C4D3×D411yescharacteristic
206C22×C4#23911yescharacteristic
207Q16—244no—
208M16—121no—
209M16—121no—
210C2×C8—21no—
211C2×C8—21no—
212C2×C8—22no—
213C2×C8—21no—
214C2×C8—21no—
215C2×C8—21no—
216C2×C8—41no—
217C2×C8—62no—
218C2×C8Dic1212yescomplemented
219C4⋊C4—32no—
220C4⋊C4—61no—
221C4⋊C4—61no—
222C4⋊C4—62no—
223C4⋊C4—61no—
224C4⋊C4—62no—
225C4⋊C4—62no—
226C4⋊C4—62no—
227C4⋊C4—121no—
228C4⋊C4—121no—
229C4⋊C4—121no—
230C4⋊C4—121no—
231C22⋊C4—62no—
232C42—62no—
233C42C2×C4×D311yescharacteristic
234C42#23511yescharacteristic
235C42D3×D411yescharacteristic
236C42D3×D411yescharacteristic
237C22×C3—21no—
238C22×C3—21no—
239C22×C3#38611yescharacteristic
240C22×C3#46211yescharacteristic
241C22×C3#46411yescharacteristic
242C3×C4—21no—
243C3×C4—21no—
244C3×C4—21no—
245C3×C4—21no—
246C3×C4C4×Q1611yescharacteristic
247C3×C4#44111yescharacteristic
248C3×C4#45111yescharacteristic
249C3×C4#45211yescharacteristic
250Dic3—42no—
251Dic3—42no—
252Dic3—82no—
253Dic3—161no—
254C23#89511yescharacteristic
255Q8—122no—
256Q8—124no—
257Q8—241no—
258Q8—242no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—62no—
262C2×C4—62no—
263C2×C4—122no—
264C2×C4—242no—
265C2×C4—241no—
266C2×C4—241no—
267C2×C4—241no—
268C2×C4C4×D1211yescharacteristic
269C2×C4D3×C22⋊C411yescharacteristic
270C2×C4#88211yescharacteristic
271C2×C4#88411yescharacteristic
272C2×C4#89311yescharacteristic
273C2×C4#89411yescharacteristic
274C2×C4#90411yescharacteristic
275C2×C4C2×Dic1211yescharacteristic
276C2×C4#90911yescharacteristic
277C2×C4#90911yescharacteristic
278C8—22no—
279C8—42no—
280C8—41no—
281C8—81no—
282C8—122no—
283C2×C3—21no—
284C2×C3—21no—
285C2×C3#187811yescharacteristic
286C2×C3#190211yescharacteristic
287C2×C3#191011yescharacteristic
288C22—21no—
289C22—21no—
290C22#466511yescharacteristic
291C22#484511yescharacteristic
292C22#484911yescharacteristic
293C4—21no—
294C4—21no—
295C4—21no—
296C4—21no—
297C4—122no—
298C4—122no—
299C4—242no—
300C4—481no—
301C4C4×Dic1211yescharacteristic
302C4#471611yescharacteristic
303C4#473311yescharacteristic
304C4#473911yescharacteristic
305C3#929011yescharacteristic, complemented, Sylow
306C2—21no—
307C2—21no—
308C2#6868511yescharacteristic
309C2#6871311yescharacteristic
310C2#6872511yescharacteristic
subgroupG/H
centerC22#4665
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1275C2
socleC22×C3#386
prime radicalC23×C3#121
cosocleC2×C3×C42C23
hypercenter#1275D3
hypocenterC3#9290
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1275

Complements

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

NH ≅ G/Ntype
22. #4699C4semidirect
23. #4681C4semidirect
108. C2×C3×C8Q16semidirect
218. C2×C8Dic12semidirect
305. C3#9290semidirect

Maximal subgroups

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

subgroupindex
#793132
#784802
#707932
#707802
#698282
C3×#12752
#677382
#92903