Finite Group Databasebeta

← #10696827

Subgroups — #10696827

Order 768.

498 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 85 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×#490C211yesmaximal, characteristic
2#83169C211yesmaximal, characteristic
3#83021C211yesmaximal, characteristic, complemented
4#71760C211yesmaximal, characteristic
5#71737C211yesmaximal, characteristic, complemented
6#69801C211yesmaximal, characteristic, complemented
7#67662C211yesmaximal, characteristic, complemented
8#10131—31nomaximal, Sylow
9C2×C3×#127C2211yescharacteristic
10C2×C3×#125C2211yescharacteristic
11#5470C2211yescharacteristic
12C3×#490—22no—
13C3×#490—22no—
14C2×C3×#102—21no—
15C2×C3×#102—21no—
16C2×C3×#97C2211yescharacteristic
17#5218—22no—
18#5076—21no—
19#5075—22no—
20#5070—21no—
21#5069—21no—
22#5064—21no—
23#5063—22no—
24#4971—21no—
25#4970—21no—
26#4926C2211yescharacteristic
27#4926C2211yescharacteristic
28#4925C2211yescharacteristic
29#4544—22no—
30#4528—22no—
31#4465—22no—
32#3236—31no—
33#3208—31no—
34C2×#490D311yescharacteristic
35#1555—31no—
36#1531—31no—
37#1470—31no—
38#1302—31no—
39C22×C3×D4—21no—
40C22×C3×D4—21no—
41C22×C3×D4—21no—
42C22×C3×C8—21no—
43C3×#127—21no—
44C3×#127—21no—
45C3×#127—21no—
46C3×#127D412yes—
47C3×#125—21no—
48C3×#125—21no—
49C3×#125—21no—
50C3×#125—21no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4D411yescharacteristic
54C2×C3×C4⋊C4D411yescharacteristic
55C2×C3×C42C2311yescharacteristic
56C2×#214—24no—
57#926—21no—
58#924—21no—
59#924—21no—
60#870—21no—
61#869D411yescharacteristic
62#869D411yescharacteristic
63C4×Dic6—21no—
64C4×Dic6—21no—
65C3×#102—24no—
66C3×#102—24no—
67C3×#97—21no—
68C3×#97D412yes—
69#832—24no—
70#830—21no—
71#830—21no—
72#830D412yes—
73#810—22no—
74#810—22no—
75#808—22no—
76#807—22no—
77C4⋊(C3⋊C8)—21no—
78C4⋊(C3⋊C8)—21no—
79C4⋊(C3⋊C8)D412yes—
80#803—21no—
81#803—21no—
82C4×C3⋊C8—21no—
83#533—31no—
84C2×#127C2×D311yescharacteristic
85C2×#125C2×D311yescharacteristic
86#501—61no—
87#492—61no—
88#491—61no—
89#490—22no—
90#490—22no—
91#486—61no—
92#484—61no—
93#483—61no—
94#479—62no—
95#460—62no—
96#450—62no—
97#432—31no—
98#432—31no—
99#431—31no—
100C2×#102—21no—
101C2×#102—21no—
102C2×#97C2×D311yescharacteristic
103#331—62no—
104#330—62no—
105#324—62no—
106C24×C3—41no—
107C2×C3×D4—22no—
108C2×C3×D4—22no—
109C2×C3×D4—22no—
110C2×C3×D4—44no—
111C2×C3×D4—41no—
112C2×C3×D4—41no—
113C2×C3×D4—41no—
114C2×C3×D4—82no—
115C2×C3×D4—82no—
116C22×C3×C4—21no—
117C22×C3×C4—21no—
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4#4111yescharacteristic
121C2×Dic6—21no—
122C2×Dic6—21no—
123C2×C3×C8—22no—
124C2×C3×C8—42no—
125C3×C4⋊C4—21no—
126C3×C4⋊C4—21no—
127C3×C4⋊C4—21no—
128C3×C4⋊C4—21no—
129C3×C4⋊C4—41no—
130C3×C4⋊C4—41no—
131C3×C4⋊C4—41no—
132C3×C4⋊C4—41no—
133C3×C42C2×D411yescharacteristic
134C3×C42C2×D411yescharacteristic
135C3×C42C2×D411yescharacteristic
136C3×C42C2×D411yescharacteristic
137#217—41no—
138#214—84no—
139C4⋊Dic3—21no—
140C4⋊Dic3—21no—
141C4⋊Dic3—21no—
142C4⋊Dic3—21no—
143Dic3⋊C4—41no—
144Dic3⋊C4—41no—
145C4×Dic3—41no—
146C2×C3⋊C8—22no—
147C2×C3⋊C8—22no—
148C2×C3⋊C8—41no—
149C22×D4—21no—
150C22×D4—21no—
151C22×D4—21no—
152C2×SD16—64no—
153C22×C8—21no—
154C4⋊Q8—31no—
155C4⋊Q8—31no—
156#127—21no—
157#127—21no—
158#127—21no—
159#127#6712yes—
160#125—21no—
161#125—21no—
162#125—21no—
163#125—21no—
164#125—61no—
165C22⋊Q8—61no—
166C22⋊Q8—61no—
167C4×Q8—61no—
168C4×Q8—61no—
169#117—61no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4#6711yescharacteristic
173C2×C4⋊C4#6711yescharacteristic
174C2×C42C22×D311yescharacteristic
175#107—62no—
176#106—62no—
177C4⋊C8—32no—
178C4⋊C8—61no—
179C4⋊C8—61no—
180#103—62no—
181#103—62no—
182#102—24no—
183#102—24no—
184#102—64no—
185C22⋊C8—32no—
186C22⋊C8—61no—
187C22⋊C8—61no—
188#97—21no—
189#97—61no—
190#97—61no—
191#97#6712yes—
192C4×C8—61no—
193C23×C3—42no—
194C23×C3—82no—
195C23×C3—82no—
196C23×C3—82no—
197C23×C3#12111yescharacteristic
198C3×D4—44no—
199C3×D4—44no—
200C3×D4—82no—
201C3×D4—82no—
202C3×D4—82no—
203C3×D4—82no—
204C2×C3×C4—22no—
205C2×C3×C4—22no—
206C2×C3×C4—41no—
207C2×C3×C4—41no—
208C2×C3×C4—41no—
209C2×C3×C4—42no—
210C2×C3×C4—81no—
211C2×C3×C4#12011yescharacteristic
212C2×C3×C4#12011yescharacteristic
213C2×C3×C4#12011yescharacteristic
214C2×C3×C4#12111yescharacteristic
215C2×C3×C4#12111yescharacteristic
216C2×C3×C4#12711yescharacteristic
217C2×Dic3—41no—
218C2×Dic3—41no—
219Dic6—42no—
220Dic6—42no—
221C3×C8—42no—
222C3⋊C8—42no—
223C3⋊C8—42no—
224C3⋊C8—81no—
225C24—41no—
226C2×Q8—61no—
227C2×Q8—61no—
228C2×D4—22no—
229C2×D4—22no—
230C2×D4—22no—
231C2×D4—44no—
232C2×D4—41no—
233C2×D4—41no—
234C2×D4—41no—
235C2×D4—82no—
236C2×D4—82no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4D3×D411yescharacteristic
240C22×C4#23711yescharacteristic
241C22×C4C2×#6711yescharacteristic
242SD16—244no—
243C2×C8—22no—
244C2×C8—42no—
245C2×C8—62no—
246C2×C8—62no—
247C2×C8—121no—
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—61no—
257C4⋊C4—61no—
258C4⋊C4—61no—
259C4⋊C4—61no—
260C4⋊C4—121no—
261C4⋊C4—121no—
262C22⋊C4—121no—
263C42—121no—
264C42D3×D411yescharacteristic
265C42D3×D411yescharacteristic
266C42C2×#6711yescharacteristic
267C42C2×#6711yescharacteristic
268C22×C3—21no—
269C22×C3—21no—
270C22×C3—82no—
271C22×C3—82no—
272C22×C3—82no—
273C22×C3—164no—
274C22×C3#39211yescharacteristic
275C22×C3#47112yes—
276C3×C4—21no—
277C3×C4—21no—
278C3×C4—21no—
279C3×C4—21no—
280C3×C4—21no—
281C3×C4—81no—
282C3×C4—81no—
283C3×C4#45412yes—
284Dic3—81no—
285Dic3—81no—
286C23—42no—
287C23—82no—
288C23—82no—
289C23—82no—
290C23#93911yescharacteristic
291Q8—122no—
292Q8—122no—
293D4—44no—
294D4—44no—
295D4—82no—
296D4—82no—
297D4—82no—
298D4—82no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—41no—
302C2×C4—42no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—81no—
306C2×C4—121no—
307C2×C4—121no—
308C2×C4#93711yescharacteristic
309C2×C4#93711yescharacteristic
310C2×C4#93811yescharacteristic
311C2×C4#93911yescharacteristic
312C2×C4#94011yescharacteristic
313C2×C4#95311yescharacteristic
314C8—42no—
315C8—122no—
316C8—122no—
317C8—241no—
318C2×C3—22no—
319C2×C3—162no—
320C2×C3#200311yescharacteristic
321C2×C3#200811yescharacteristic
322C2×C3#201911yescharacteristic
323C22—21no—
324C22—21no—
325C22—82no—
326C22—82no—
327C22—82no—
328C22—164no—
329C22#520012yes—
330C22#522311yescharacteristic
331C4—21no—
332C4—21no—
333C4—21no—
334C4—21no—
335C4—21no—
336C4—81no—
337C4—81no—
338C4—241no—
339C4—241no—
340C4#506212yes—
341C3#1013111yescharacteristic, complemented, Sylow
342C2—22no—
343C2—162no—
344C2#7173011yescharacteristic
345C2#7209711yescharacteristic
346C2#7209911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×#490C2
socleC22×C3#392
prime radicalC2×C3×#127C22
hypercenterC2×#490D3
hypocenterC3#10131
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#490

Complements

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

NH ≅ G/Ntype
3. #83021C2semidirect
5. #71737C2semidirect
6. #69801C2semidirect
7. #67662C2semidirect
341. C3#10131semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4902
#831692
#830212
#717602
#717372
#698012
#676622
#101313