Finite Group Databasebeta

← #10696988

Subgroups — #10696988

Order 768.

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

normal subgroups only

This group has 574 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
1C2×C3×#494C211yesmaximal, characteristic, complemented
2#83213C211yesmaximal, characteristic
3#83199C211yesmaximal, characteristic
4#71790C211yesmaximal, characteristic, complemented
5#71775C211yesmaximal, characteristic, complemented
6#69735C211yesmaximal, characteristic
7#67656C211yesmaximal, characteristic, complemented
8#10154—31nomaximal, Sylow
9C22×C3×Q16—22no—
10C2×C3×C4⋊Q8C2211yescharacteristic
11C2×C3×C4⋊Q8C2211yescharacteristic
12#5477C2211yescharacteristic
13C3×#494—22no—
14C3×#494—22no—
15C2×C3×C4×C8C2211yescharacteristic
16#5227—22no—
17#5093—21no—
18#5089—22no—
19#5088—21no—
20#5085—21no—
21#5083—22no—
22#5079—21no—
23#4966—22no—
24#4925C2211yescharacteristic
25#4924C2211yescharacteristic
26C4×#208C2211yescharacteristic
27#4547—22no—
28#4530—22no—
29#4459—22no—
30#3206—31no—
31C2×#494D311yescharacteristic, complemented
32#3141—31no—
33#1997—31no—
34#1549—31no—
35#1492—31no—
36#1278—31no—
37C22×C3×Q8—21no—
38C22×C3×Q8D412yes—
39C2×#235—21no—
40C2×C3×Q16—28no—
41C2×C3×Q16—48no—
42C22×C3×C8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8—21no—
47C3×C4⋊Q8—21no—
48C3×C4⋊Q8—21no—
49C3×C4⋊Q8—21no—
50C3×C4⋊Q8D412yes—
51C2×C3×C4⋊C4—21no—
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4—21no—
55C2×C3×C42C2311yescharacteristic
56C2×#215—24no—
57#930—21no—
58#930—21no—
59C2×#208—21no—
60C2×#208D412yes—
61#875—21no—
62#874D411yescharacteristic
63C4×D12—21no—
64C4×D12—21no—
65#869D411yescharacteristic
66C3×C4×C8—21no—
67C3×C4×C8D412yes—
68#835—24no—
69#830—21no—
70#810—24no—
71#809—24no—
72C4⋊(C3⋊C8)—21no—
73C4⋊(C3⋊C8)—21no—
74C4⋊(C3⋊C8)D412yes—
75#803—21no—
76C4×C3⋊C8—21no—
77C4×C3⋊C8—21no—
78C22×Q16—22no—
79#532—31no—
80C2×C4⋊Q8C2×D311yescharacteristic
81C2×C4⋊Q8C2×D311yescharacteristic
82#494—22no—
83#494—22no—
84#492—61no—
85#489—62no—
86#487—61no—
87#487—61no—
88#486—61no—
89#461—62no—
90#448—62no—
91#431—31no—
92#423—31no—
93#417—62no—
94C4×M16—31no—
95C2×C4×C8C2×D311yescharacteristic
96#343—62no—
97#332—62no—
98D4⋊C8—62no—
99C2×C3×Q8—22no—
100C2×C3×Q8—22no—
101C2×C3×Q8—22no—
102C2×C3×Q8—44no—
103C2×C3×Q8—44no—
104C2×C3×Q8—41no—
105C22×C3×C4—22no—
106C22×C3×C4—41no—
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4#4111yescharacteristic
110C2×#67—41no—
111#235—42no—
112#235—42no—
113C2×D12—21no—
114C2×D12—21no—
115C2×D12—21no—
116C2×C4×D3—41no—
117C2×Dic6—21no—
118C3×Q16—416no—
119C2×C3×C8—24no—
120C2×C3×C8—22no—
121C3×C4⋊C4—22no—
122C3×C4⋊C4—22no—
123C3×C4⋊C4—41no—
124C3×C4⋊C4—41no—
125C3×C4⋊C4—41no—
126C3×C4⋊C4—41no—
127C3×C4⋊C4—41no—
128C3×C4⋊C4—41no—
129C3×C42C2×D411yescharacteristic
130C3×C42C2×D411yescharacteristic
131C3×C42C2×D411yescharacteristic
132C3×C42C2×D411yescharacteristic
133#215—84no—
134(C2×D3)⋊C4—41no—
135(C2×D3)⋊C4—41no—
136C4⋊Dic3—21no—
137C4⋊Dic3—21no—
138#208—22no—
139#208—22no—
140#208—42no—
141C2×C3⋊C8—22no—
142C2×C3⋊C8—22no—
143C2×C3⋊C8—41no—
144C2×#41—61no—
145C22×Q8—21no—
146C22×Q8#6712yes—
147C2×Q16—28no—
148C2×Q16—48no—
149C2×SD16—64no—
150C2×M16—32no—
151C2×M16—61no—
152C22×C8—21no—
153C4⋊Q8—21no—
154C4⋊Q8—21no—
155C4⋊Q8—21no—
156C4⋊Q8—21no—
157C4⋊Q8—21no—
158C4⋊Q8—21no—
159C4⋊Q8—21no—
160C4⋊Q8—31no—
161C4⋊Q8#6712yes—
162#127—31no—
163#124—61no—
164#121—61no—
165#121—61no—
166C4×D4—61no—
167C4×D4—61no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C42C22×D311yescharacteristic
173C4⋊C8—32no—
174C4⋊C8—61no—
175C4⋊C8—61no—
176#103—64no—
177#103—64no—
178#102—64no—
179C22⋊C8—61no—
180#97—61no—
181C4×C8—21no—
182C4×C8—61no—
183C4×C8—61no—
184C4×C8#6712yes—
185C23×C3#12111yescharacteristic
186C22×D3—41no—
187C3×Q8—44no—
188C3×Q8—44no—
189C3×Q8—44no—
190C3×Q8—82no—
191C2×C3×C4—22no—
192C2×C3×C4—22no—
193C2×C3×C4—42no—
194C2×C3×C4—42no—
195C2×C3×C4—82no—
196C2×C3×C4—82no—
197C2×C3×C4#12011yescharacteristic
198C2×C3×C4#12011yescharacteristic
199C2×C3×C4#12011yescharacteristic
200C2×C3×C4#12111yescharacteristic
201C2×C3×C4#12111yescharacteristic
202C2×C3×C4#12711yescharacteristic
203C2×C3×C4#13512yes—
204C2×C3×C4#13712yes—
205#67—82no—
206C2×Dic3—41no—
207D12—42no—
208D12—42no—
209D12—42no—
210C4×D3—81no—
211C4×D3—81no—
212Dic6—42no—
213C3×C8—24no—
214C3⋊C8—42no—
215C3⋊C8—42no—
216C3⋊C8—81no—
217#41—122no—
218#41—122no—
219C2×Q8—22no—
220C2×Q8—22no—
221C2×Q8—22no—
222C2×Q8—44no—
223C2×Q8—44no—
224C2×Q8—41no—
225C2×Q8—61no—
226C2×D4—61no—
227C2×D4—61no—
228C2×D4—61no—
229C2×D4—121no—
230C22×C4—22no—
231C22×C4—41no—
232C22×C4—121no—
233C22×C4D3×D411yescharacteristic
234C22×C4#23711yescharacteristic
235C22×C4C2×#6711yescharacteristic
236Q16—416no—
237SD16—244no—
238M16—62no—
239M16—62no—
240M16—122no—
241C2×C8—24no—
242C2×C8—22no—
243C2×C8—62no—
244C2×C8—62no—
245C2×C8—121no—
246C4⋊C4—22no—
247C4⋊C4—22no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—61no—
255C4⋊C4—61no—
256C22⋊C4—121no—
257C22⋊C4—121no—
258C42D3×D411yescharacteristic
259C42D3×D411yescharacteristic
260C42C2×#6711yescharacteristic
261C42C2×#6711yescharacteristic
262C22×C3—21no—
263C22×C3—21no—
264C22×C3#39211yescharacteristic
265C22×C3#46412yes—
266C2×D3—81no—
267C2×D3—81no—
268C2×D3—81no—
269C3×C4—21no—
270C3×C4—21no—
271C3×C4—82no—
272C3×C4—82no—
273C3×C4#45212yes—
274C3×C4#45312yes—
275C3×C4#47312yes—
276C3×C4#49511yescharacteristic
277C3×C4#49711yescharacteristic
278Dic3—81no—
279C23—121no—
280C23#93911yescharacteristic
281Q8—44no—
282Q8—44no—
283Q8—44no—
284Q8—82no—
285Q8—122no—
286D4—122no—
287D4—122no—
288D4—122no—
289D4—242no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4—121no—
297C2×C4—241no—
298C2×C4—241no—
299C2×C4#91812yes—
300C2×C4#91912yes—
301C2×C4#93711yescharacteristic
302C2×C4#93711yescharacteristic
303C2×C4#93811yescharacteristic
304C2×C4#93911yescharacteristic
305C2×C4#94011yescharacteristic
306C2×C4#95311yescharacteristic
307C8—24no—
308C8—122no—
309C8—122no—
310C8—241no—
311C2×C3—22no—
312C2×C3#200211yescharacteristic
313C2×C3#201511yescharacteristic
314C2×C3#202011yescharacteristic
315D3—161no—
316C22—21no—
317C22—21no—
318C22—241no—
319C22—241no—
320C22—241no—
321C22#518512yes—
322C22#522311yescharacteristic
323C4—21no—
324C4—21no—
325C4—82no—
326C4—82no—
327C4—241no—
328C4#507712yes—
329C4#518712yes—
330C4#519011yescharacteristic
331C4#519111yescharacteristic
332C4#519212yes—
333C3#1015411yescharacteristic, complemented, Sylow
334C2—22no—
335C2—481no—
336C2#7176911yescharacteristic
337C2#7207211yescharacteristic
338C2#7207411yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×#494C2
socleC22×C3#392
prime radical#5477C22
hypercenterC2×#494D3
hypocenterC3#10154
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#494

Complements

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

NH ≅ G/Ntype
1. C2×C3×#494C2semidirect
4. #71790C2semidirect
5. #71775C2semidirect
7. #67656C2semidirect
31. C2×#494D3semidirect
333. C3#10154semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4942
#832132
#831992
#717902
#717752
#697352
#676562
#101543