Finite Group Databasebeta

← #10698764

Subgroups — #10698764

Order 768.

460 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 101 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×#460C211yesmaximal, characteristic
2#71759C211yesmaximal, characteristic
3#71759C211yesmaximal, characteristic
4#71637C211yesmaximal, characteristic, complemented
5#71621C211yesmaximal, characteristic, complemented
6#70422C211yesmaximal, characteristic, complemented
7#67707C211yesmaximal, characteristic, complemented
8#10442—31nomaximal, Sylow
9C22×C3×SD16—21no—
10C2×C3×#127C2211yescharacteristic
11C2×C3×C4×Q8C2211yescharacteristic
12C3×#460—22no—
13C3×#460—22no—
14C2×C3×C4⋊C8C2211yescharacteristic
15C2×C3×#102—21no—
16C2×#836—21no—
17#4925C2211yescharacteristic
18#4925C2211yescharacteristic
19#4924C2211yescharacteristic
20#4924C2211yescharacteristic
21#4544—22no—
22#4544—22no—
23#4535—22no—
24#4530—21no—
25#4479—22no—
26C2×#460D311yescharacteristic
27#1801—31no—
28#1571—31no—
29#1545—31no—
30#1545—31no—
31#1479—31no—
32#1301—31no—
33C22×C3×Q8D411yescharacteristic
34C22×C3×D4—21no—
35C22×C3×D4—21no—
36C22×C3×D4—21no—
37C2×C3×SD16—24no—
38C2×C3×SD16—42no—
39C2×C3×SD16—42no—
40C22×C3×C8—21no—
41C3×#127—21no—
42C3×#127—21no—
43C3×#127—21no—
44C3×#127D412yes—
45C3×C4×Q8—21no—
46C3×C4×Q8—21no—
47C3×C4×Q8—21no—
48C3×C4×Q8—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C4⋊C4D411yescharacteristic
51C2×C3×C42—21no—
52C2×C3×C42C2311yescharacteristic
53C2×#208—21no—
54C2×#208—21no—
55C3×C4⋊C8—21no—
56C3×C4⋊C8D412yes—
57C3×#102—24no—
58#836—42no—
59#830—21no—
60#830—21no—
61C4⋊(C3⋊C8)—21no—
62C4⋊(C3⋊C8)—21no—
63C4⋊(C3⋊C8)—21no—
64C4⋊(C3⋊C8)—21no—
65C4⋊(C3⋊C8)—21no—
66C4⋊(C3⋊C8)D412yes—
67C4×C3⋊C8—21no—
68C4×C3⋊C8—21no—
69C22×SD16—21no—
70C2×#127C2×D311yescharacteristic
71C2×C4×Q8C2×D311yescharacteristic
72#460—22no—
73#460—22no—
74#431—31no—
75#431—31no—
76#423—31no—
77#423—31no—
78C2×C4⋊C8C2×D311yescharacteristic
79C2×#102—21no—
80C2×#101—61no—
81#343—61no—
82#335—62no—
83#331—62no—
84#331—62no—
85Q8⋊C8—62no—
86C24×C3—41no—
87C2×C3×Q8—21no—
88C2×C3×Q8—21no—
89C2×C3×Q8—21no—
90C2×C3×Q8—21no—
91C2×C3×Q8—21no—
92C2×C3×Q8D812yes—
93C2×C3×D4—22no—
94C2×C3×D4—22no—
95C2×C3×D4—22no—
96C2×C3×D4—41no—
97C2×C3×D4—41no—
98C2×C3×D4—41no—
99C2×C3×D4—41no—
100C2×C3×D4—41no—
101C2×C3×D4—41no—
102C2×C3×D4—41no—
103C2×C3×D4—82no—
104C2×C3×D4—82no—
105C22×C3×C4—21no—
106C22×C3×C4—21no—
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4#4111yescharacteristic
109C22×C3×C4#4111yescharacteristic
110C3×SD16—48no—
111C2×C3×C8—22no—
112C2×C3×C8—41no—
113C2×C3×C8—41no—
114C3×C4⋊C4—21no—
115C3×C4⋊C4—21no—
116C3×C4⋊C4—41no—
117C3×C4⋊C4—41no—
118C3×C42—41no—
119C3×C42—41no—
120C3×C42C2×D411yescharacteristic
121C3×C42C2×D411yescharacteristic
122C3×C42C2×D411yescharacteristic
123C3×C42#4111yescharacteristic
124#208—41no—
125#208—41no—
126#208—41no—
127#208—41no—
128C2×C3⋊C8—22no—
129C2×C3⋊C8—41no—
130C2×C3⋊C8—41no—
131C2×C3⋊C8—41no—
132C22×Q8#6711yescharacteristic
133C22×D4—21no—
134C22×D4—21no—
135C22×D4—21no—
136C2×SD16—24no—
137C2×SD16—42no—
138C2×SD16—42no—
139C2×M16—61no—
140C2×M16—61no—
141C22×C8—21no—
142#127—21no—
143#127—21no—
144#127—21no—
145#127#6712yes—
146C4×Q8—21no—
147C4×Q8—21no—
148C4×Q8—21no—
149C4×Q8—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4#6711yescharacteristic
152C2×C42—21no—
153C2×C42C22×D311yescharacteristic
154C4⋊C8—21no—
155C4⋊C8—32no—
156C4⋊C8—61no—
157C4⋊C8—61no—
158C4⋊C8—61no—
159C4⋊C8—61no—
160C4⋊C8—61no—
161C4⋊C8#6712yes—
162#102—24no—
163#101—122no—
164C22⋊C8—61no—
165C22⋊C8—61no—
166C4×C8—61no—
167C4×C8—61no—
168C23×C3—42no—
169C23×C3—82no—
170C23×C3—82no—
171C23×C3—81no—
172C23×C3—81no—
173C23×C3#12311yescharacteristic
174C3×Q8—22no—
175C3×Q8—22no—
176C3×Q8—41no—
177C3×Q8—41no—
178C3×D4—42no—
179C3×D4—42no—
180C3×D4—82no—
181C3×D4—82no—
182C3×D4—82no—
183C3×D4—81no—
184C3×D4—81no—
185C3×D4—82no—
186C2×C3×C4—22no—
187C2×C3×C4—22no—
188C2×C3×C4—41no—
189C2×C3×C4—41no—
190C2×C3×C4—41no—
191C2×C3×C4—41no—
192C2×C3×C4—81no—
193C2×C3×C4#12011yescharacteristic
194C2×C3×C4#12111yescharacteristic
195C2×C3×C4#12111yescharacteristic
196C2×C3×C4#12111yescharacteristic
197C2×C3×C4#12311yescharacteristic
198C2×C3×C4#12411yescharacteristic
199C2×C3×C4C2×D811yescharacteristic
200C2×C3×C4#13511yescharacteristic
201C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
202C2×C3×C4#13711yescharacteristic
203C3×C8—42no—
204C3⋊C8—42no—
205C3⋊C8—81no—
206C3⋊C8—81no—
207C3⋊C8—81no—
208C24—41no—
209C2×Q8—21no—
210C2×Q8—21no—
211C2×Q8—21no—
212C2×Q8—21no—
213C2×Q8—21no—
214C2×Q8#21312yes—
215C2×D4—22no—
216C2×D4—22no—
217C2×D4—22no—
218C2×D4—41no—
219C2×D4—41no—
220C2×D4—41no—
221C2×D4—41no—
222C2×D4—41no—
223C2×D4—41no—
224C2×D4—41no—
225C2×D4—82no—
226C2×D4—82no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4#23711yescharacteristic
230C22×C4#23711yescharacteristic
231C22×C4C2×#6711yescharacteristic
232SD16—48no—
233M16—121no—
234M16—121no—
235M16—121no—
236M16—121no—
237C2×C8—22no—
238C2×C8—41no—
239C2×C8—41no—
240C2×C8—62no—
241C2×C8—121no—
242C2×C8—121no—
243C2×C8—121no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C42—41no—
249C42—41no—
250C42D3×D411yescharacteristic
251C42#23711yescharacteristic
252C42C2×#6711yescharacteristic
253C42C2×#6711yescharacteristic
254C22×C3—21no—
255C22×C3—21no—
256C22×C3—82no—
257C22×C3—82no—
258C22×C3—82no—
259C22×C3—162no—
260C22×C3—162no—
261C22×C3#39411yescharacteristic
262C22×C3#48011yescharacteristic
263C22×C3#48211yescharacteristic
264C3×C4—21no—
265C3×C4—21no—
266C3×C4—21no—
267C3×C4—42no—
268C3×C4—81no—
269C3×C4#44711yescharacteristic
270C3×C4#45211yescharacteristic
271C3×C4#46611yescharacteristic
272C3×C4#46811yescharacteristic
273C3×C4#47112yes—
274C23—42no—
275C23—82no—
276C23—82no—
277C23—81no—
278C23—81no—
279C23#93511yescharacteristic
280Q8—22no—
281Q8—22no—
282Q8—41no—
283Q8—41no—
284D4—42no—
285D4—42no—
286D4—82no—
287D4—82no—
288D4—82no—
289D4—82no—
290D4—81no—
291D4—81no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—81no—
299C2×C4C2×#21311yescharacteristic
300C2×C4#93211yescharacteristic
301C2×C4#93511yescharacteristic
302C2×C4#93611yescharacteristic
303C2×C4#93811yescharacteristic
304C2×C4#93911yescharacteristic
305C2×C4#93911yescharacteristic
306C2×C4#95011yescharacteristic
307C2×C4#95111yescharacteristic
308C2×C4#95311yescharacteristic
309C8—42no—
310C8—122no—
311C8—241no—
312C8—241no—
313C8—241no—
314C2×C3—21no—
315C2×C3—21no—
316C2×C3—162no—
317C2×C3#203311yescharacteristic
318C2×C3#204611yescharacteristic
319C2×C3#204811yescharacteristic
320C22—21no—
321C22—21no—
322C22—82no—
323C22—82no—
324C22—82no—
325C22—162no—
326C22—162no—
327C22#503311yescharacteristic
328C22#503511yescharacteristic
329C22#522211yescharacteristic
330C4—21no—
331C4—21no—
332C4—21no—
333C4—42no—
334C4—81no—
335C4#503811yescharacteristic
336C4#504211yescharacteristic
337C4#517712yes—
338C4#521711yescharacteristic
339C4#523911yescharacteristic
340C3#1044211yescharacteristic, complemented, Sylow
341C2—21no—
342C2—21no—
343C2—162no—
344C2#7167911yescharacteristic
345C2#7205311yescharacteristic
346C2#7209211yescharacteristic
subgroupG/H
centerC22#5222
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×#460C2
socleC22×C3#394
prime radicalC2×C3×#127C22
hypercenterC2×#460D3
hypocenterC3#10442
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#460

Complements

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

NH ≅ G/Ntype
4. #71637C2semidirect
5. #71621C2semidirect
6. #70422C2semidirect
7. #67707C2semidirect
340. C3#10442semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4602
#717592
#716372
#716212
#704222
#677072
#104423