Finite Group Databasebeta

← #10697005

Subgroups — #10697005

Order 768.

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

normal subgroups only

This group has 498 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
1C3×#3227C211yesmaximal, characteristic
2#83283C211yesmaximal, characteristic
3C2×#5093C211yesmaximal, characteristic, complemented
4#71802C211yesmaximal, characteristic
5#71790C211yesmaximal, characteristic, complemented
6#70780C211yesmaximal, characteristic, complemented
7#67739C211yesmaximal, characteristic, complemented
8#10156—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×C3×C4⋊Q8C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#498—21no—
13C3×#486—21no—
14C3×#481—22no—
15C3×#476—22no—
16C3×#431C2211yescharacteristic
17#5232—22no—
18#5093—22no—
19#5093—22no—
20#5079—21no—
21#5070—21no—
22#5055—22no—
23C2×#810—22no—
24#4924C2211yescharacteristic
25C4×#208C2211yescharacteristic
26C2×C4×C3⋊C8C2211yescharacteristic
27D4⋊(C3⋊C8)—22no—
28#4530—22no—
29#4482—22no—
30#3227D311yescharacteristic
31#3141—31no—
32C2×#487—31no—
33#1997—31no—
34#1549—31no—
35#1492—31no—
36#1278—31no—
37C2×C3×#41—21no—
38C22×C3×Q8—21no—
39C22×Dic6—21no—
40C3×C4⋊Q8—21no—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8D411yescharacteristic
45C3×#127D411yescharacteristic
46C3×#124—21no—
47C3×#121—21no—
48C3×#121—21no—
49C3×C4×D4—21no—
50C3×C4×D4—21no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C4⋊C4D412yes—
53C2×C3×C42C2311yescharacteristic
54C2×C4⋊Dic3—21no—
55C2×C4⋊Dic3—21no—
56C2×#208—21no—
57C2×#208D412yes—
58C22×C3⋊C8—21no—
59#869—21no—
60#869—21no—
61#869—21no—
62#869D412yes—
63C3×#106—24no—
64C3×C4⋊C8D412yes—
65C3×#102—24no—
66C3×C22⋊C8—21no—
67C3×C4×C8—21no—
68#835—24no—
69#832—24no—
70#810—28no—
71C4⋊(C3⋊C8)—21no—
72C4⋊(C3⋊C8)—21no—
73#803—21no—
74C4×C3⋊C8—21no—
75C4×C3⋊C8—21no—
76C4×C3⋊C8D412yes—
77#532C2×D311yescharacteristic
78C2×C4⋊Q8—31no—
79C2×C4⋊Q8C2×D311yescharacteristic
80#498—21no—
81#489—62no—
82#487—62no—
83#487—62no—
84#487—61no—
85#486—21no—
86#486—61no—
87#481—22no—
88#476—22no—
89#431C2×D311yescharacteristic
90#423—31no—
91#417—62no—
92C2×#103—62no—
93C4×M16—31no—
94C2×C4×C8—31no—
95#343—62no—
96#332—62no—
97D4⋊C8—62no—
98C3×#41—42no—
99C3×#41—42no—
100C2×C3×Q8—21no—
101C2×C3×Q8—41no—
102C2×C3×Q8—41no—
103C2×C3×Q8—82no—
104C2×C3×D4—21no—
105C2×C3×D4—21no—
106C2×C3×D4—21no—
107C2×C3×D4—41no—
108C22×C3×C4—22no—
109C22×C3×C4—41no—
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4#4111yescharacteristic
113C22×Dic3—41no—
114C2×Dic6—22no—
115C2×Dic6—44no—
116C2×Dic6—41no—
117C2×C3×C8—22no—
118C3×C4⋊C4—22no—
119C3×C4⋊C4—22no—
120C3×C4⋊C4—21no—
121C3×C4⋊C4—21no—
122C3×C4⋊C4—41no—
123C3×C4⋊C4—41no—
124C3×C22⋊C4—41no—
125C3×C22⋊C4—41no—
126C3×C42C2×D411yescharacteristic
127C3×C42C2×D411yescharacteristic
128C3×C42C2×D411yescharacteristic
129C3×C42C2×D411yescharacteristic
130C4⋊Dic3—22no—
131C4⋊Dic3—22no—
132C4⋊Dic3—41no—
133C4⋊Dic3—41no—
134#208—22no—
135#208—22no—
136#208—42no—
137C2×C3⋊C8—22no—
138C2×C3⋊C8—22no—
139C2×C3⋊C8—42no—
140C2×C3⋊C8—41no—
141C2×#41—21no—
142C22×Q8—21no—
143C22×Q8—61no—
144C2×M16—32no—
145C2×M16—61no—
146C22×C8—61no—
147C4⋊Q8—21no—
148C4⋊Q8—21no—
149C4⋊Q8—21no—
150C4⋊Q8—21no—
151C4⋊Q8—32no—
152C4⋊Q8—61no—
153C4⋊Q8—61no—
154C4⋊Q8—61no—
155C4⋊Q8#6711yescharacteristic
156#127#6711yescharacteristic
157#124—21no—
158#121—21no—
159#121—21no—
160C4×D4—21no—
161C4×D4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—61no—
164C2×C4⋊C4—61no—
165C2×C4⋊C4#6712yes—
166C2×C42C22×D311yescharacteristic
167#106—24no—
168C4⋊C8—61no—
169C4⋊C8—61no—
170C4⋊C8#6712yes—
171#103—68no—
172#103—64no—
173#102—24no—
174#102—64no—
175C22⋊C8—21no—
176#97—61no—
177C4×C8—21no—
178C4×C8—32no—
179C4×C8—61no—
180C4×C8—61no—
181C23×C3—41no—
182C23×C3#12111yescharacteristic
183C3×Q8—42no—
184C3×Q8—82no—
185C3×Q8—82no—
186C3×D4—42no—
187C3×D4—42no—
188C3×D4—42no—
189C3×D4—82no—
190C2×C3×C4—22no—
191C2×C3×C4—22no—
192C2×C3×C4—41no—
193C2×C3×C4—42no—
194C2×C3×C4—81no—
195C2×C3×C4—81no—
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—
205C2×Dic3—42no—
206C2×Dic3—82no—
207Dic6—44no—
208Dic6—82no—
209C3×C8—42no—
210C3⋊C8—42no—
211C3⋊C8—42no—
212C3⋊C8—81no—
213#41—42no—
214#41—42no—
215C2×Q8—21no—
216C2×Q8—41no—
217C2×Q8—41no—
218C2×Q8—62no—
219C2×Q8—82no—
220C2×Q8—124no—
221C2×Q8—121no—
222C2×D4—21no—
223C2×D4—21no—
224C2×D4—21no—
225C2×D4—41no—
226C22×C4—22no—
227C22×C4—41no—
228C22×C4—121no—
229C22×C4D3×D411yescharacteristic
230C22×C4#23711yescharacteristic
231C22×C4C2×#6711yescharacteristic
232M16—62no—
233M16—62no—
234M16—122no—
235C2×C8—22no—
236C2×C8—62no—
237C2×C8—62no—
238C2×C8—122no—
239C2×C8—121no—
240C4⋊C4—22no—
241C4⋊C4—21no—
242C4⋊C4—22no—
243C4⋊C4—21no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—62no—
247C4⋊C4—62no—
248C4⋊C4—121no—
249C4⋊C4—121no—
250C22⋊C4—41no—
251C22⋊C4—41no—
252C42D3×D411yescharacteristic
253C42D3×D411yescharacteristic
254C42C2×#6711yescharacteristic
255C42C2×#6711yescharacteristic
256C22×C3—21no—
257C22×C3—21no—
258C22×C3—81no—
259C22×C3—81no—
260C22×C3—81no—
261C22×C3#39211yescharacteristic
262C22×C3#46412yes—
263C3×C4—21no—
264C3×C4—21no—
265C3×C4—81no—
266C3×C4—82no—
267C3×C4#45212yes—
268C3×C4#45412yes—
269C3×C4#47112yes—
270C3×C4#49511yescharacteristic
271C3×C4#49711yescharacteristic
272Dic3—82no—
273C23—41no—
274C23#93911yescharacteristic
275Q8—42no—
276Q8—82no—
277Q8—82no—
278Q8—124no—
279Q8—242no—
280D4—42no—
281D4—42no—
282D4—42no—
283D4—82no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—82no—
291C2×C4—122no—
292C2×C4—242no—
293C2×C4#91512yes—
294C2×C4#91612yes—
295C2×C4#93711yescharacteristic
296C2×C4#93711yescharacteristic
297C2×C4#93811yescharacteristic
298C2×C4#93911yescharacteristic
299C2×C4#94011yescharacteristic
300C2×C4#95311yescharacteristic
301C8—42no—
302C8—122no—
303C8—122no—
304C8—241no—
305C2×C3—22no—
306C2×C3—161no—
307C2×C3#200311yescharacteristic
308C2×C3#201511yescharacteristic
309C2×C3#201911yescharacteristic
310C22—21no—
311C22—21no—
312C22—81no—
313C22—81no—
314C22—81no—
315C22#516612yes—
316C22#522311yescharacteristic
317C4—21no—
318C4—21no—
319C4—81no—
320C4—82no—
321C4—242no—
322C4#516711yescharacteristic
323C4#516811yescharacteristic
324C4#517312yes—
325C4#520012yes—
326C4#524512yes—
327C3#1015611yescharacteristic, complemented, Sylow
328C2—22no—
329C2—161no—
330C2#7186711yescharacteristic
331C2#7205811yescharacteristic
332C2#7209711yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3227C2
socleC22×C3#392
prime radicalC3×#532C22
hypercenter#3227D3
hypocenterC3#10156
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3227

Complements

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

NH ≅ G/Ntype
3. C2×#5093C2semidirect
5. #71790C2semidirect
6. #70780C2semidirect
7. #67739C2semidirect
327. C3#10156semidirect

Maximal subgroups

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

subgroupindex
C3×#32272
#832832
C2×#50932
#718022
#717902
#707802
#677392
#101563