Finite Group Databasebeta

← #10697248

Subgroups — #10697248

Order 768.

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

normal subgroups only

This group has 464 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×#475C211yesmaximal, characteristic
2#83224C211yesmaximal, characteristic
3#82431C211yesmaximal, characteristic
4#71774C211yesmaximal, characteristic
5#71624C211yesmaximal, characteristic
6#70429C211yesmaximal, characteristic
7#67707C211yesmaximal, characteristic
8#10191—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×C3×C4×Q8C2211yescharacteristic
11#5470C2211yescharacteristic
12C3×#475—22no—
13C3×#475—22no—
14C2×C3×#106—21no—
15C2×C3×C4⋊C8C2211yescharacteristic
16C2×C3×#103—21no—
17#5225—21no—
18#5093—21no—
19#5091—22no—
20#5078—21no—
21#5022—22no—
22#5021—21no—
23#5020—21no—
24#4970—22no—
25#4964—21no—
26#4925C2211yescharacteristic
27#4925C2211yescharacteristic
28#4924C2211yescharacteristic
29#4546—22no—
30#4528—22no—
31#4484—22no—
32#3230—31no—
33#3078—31no—
34C2×#475D311yescharacteristic
35#1548—31no—
36#1481—31no—
37#1461—31no—
38#1301—31no—
39C22×C3×Q8—21no—
40C22×C3×Q8D411yescharacteristic
41C22×C3×C8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8D412yes—
46C3×C4×Q8—21no—
47C3×C4×Q8—21no—
48C3×C4×Q8—21no—
49C3×C4×Q8—21no—
50C2×C3×C4⋊C4—21no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4D411yescharacteristic
54C2×C3×C42—21no—
55C2×C3×C42C2311yescharacteristic
56#926—21no—
57#924—21no—
58#924—21no—
59C2×#208—21no—
60#870—21no—
61#869D411yescharacteristic
62#869D411yescharacteristic
63C4×Dic6—21no—
64C4×Dic6—21no—
65C3×#106—24no—
66C3×C4⋊C8—21no—
67C3×C4⋊C8D412yes—
68C3×#103—24no—
69#835—22no—
70#835—22no—
71#830—21no—
72#830D412yes—
73#810—24no—
74#808—22no—
75#807—22no—
76#807—24no—
77C4⋊(C3⋊C8)—21no—
78C4⋊(C3⋊C8)—21no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)D412yes—
81C4×C3⋊C8—21no—
82C4×C3⋊C8—21no—
83#533—31no—
84C2×C4⋊Q8C2×D311yescharacteristic
85C2×C4×Q8C2×D311yescharacteristic
86#500—61no—
87#487—61no—
88#484—62no—
89#479—62no—
90#477—61no—
91#477—62no—
92#475—22no—
93#475—22no—
94#475—61no—
95#431—31no—
96#431—31no—
97#428—61no—
98C2×#106—21no—
99#423—31no—
100C2×C4⋊C8C2×D311yescharacteristic
101#419—61no—
102C2×#103—21no—
103#333—62no—
104Q8⋊C8—62no—
105#324—62no—
106C2×C3×Q8—21no—
107C2×C3×Q8—21no—
108C2×C3×Q8—21no—
109C2×C3×Q8—22no—
110C2×C3×Q8—21no—
111C2×C3×Q8—21no—
112C2×C3×Q8—41no—
113C2×C3×Q8—82no—
114C2×C3×Q8Q1612yes—
115C22×C3×C4—21no—
116C22×C3×C4—21no—
117C22×C3×C4—41no—
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—41no—
125C2×C3×C8—41no—
126C3×C4⋊C4—21no—
127C3×C4⋊C4—22no—
128C3×C4⋊C4—22no—
129C3×C4⋊C4—21no—
130C3×C4⋊C4—41no—
131C3×C4⋊C4—41no—
132C3×C4⋊C4—41no—
133C3×C4⋊C4—41no—
134C3×C42—41no—
135C3×C42—41no—
136C3×C42C2×D411yescharacteristic
137C3×C42C2×D411yescharacteristic
138C3×C42C2×D411yescharacteristic
139C3×C42C2×D411yescharacteristic
140#217—41no—
141C4⋊Dic3—21no—
142C4⋊Dic3—21no—
143C4⋊Dic3—21no—
144C4⋊Dic3—21no—
145Dic3⋊C4—41no—
146Dic3⋊C4—41no—
147C4×Dic3—41no—
148#208—41no—
149#208—41no—
150C2×C3⋊C8—22no—
151C2×C3⋊C8—22no—
152C2×C3⋊C8—41no—
153C22×Q8—21no—
154C22×Q8#6711yescharacteristic
155C2×M16—61no—
156C22×C8—21no—
157C4⋊Q8—21no—
158C4⋊Q8—21no—
159C4⋊Q8—21no—
160C4⋊Q8—31no—
161C4⋊Q8—31no—
162C4⋊Q8#6712yes—
163#125—61no—
164C22⋊Q8—61no—
165C22⋊Q8—61no—
166C4×Q8—21no—
167C4×Q8—21no—
168C4×Q8—21no—
169C4×Q8—21no—
170C4×Q8—61no—
171C4×Q8—61no—
172#117—61no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4#6711yescharacteristic
177C2×C42—21no—
178C2×C42C22×D311yescharacteristic
179#107—62no—
180#107—64no—
181#106—24no—
182#106—62no—
183C4⋊C8—21no—
184C4⋊C8—32no—
185C4⋊C8—61no—
186C4⋊C8—61no—
187C4⋊C8—61no—
188C4⋊C8#6712yes—
189#103—24no—
190#103—64no—
191#103—62no—
192#103—62no—
193C22⋊C8—32no—
194C22⋊C8—61no—
195C4×C8—61no—
196C4×C8—61no—
197C23×C3#12111yescharacteristic
198C3×Q8—22no—
199C3×Q8—22no—
200C3×Q8—41no—
201C3×Q8—41no—
202C3×Q8—82no—
203C3×Q8—82no—
204C2×C3×C4—22no—
205C2×C3×C4—22no—
206C2×C3×C4—41no—
207C2×C3×C4—41no—
208C2×C3×C4—41no—
209C2×C3×C4—41no—
210C2×C3×C4—42no—
211C2×C3×C4—82no—
212C2×C3×C4—81no—
213C2×C3×C4#12011yescharacteristic
214C2×C3×C4#12011yescharacteristic
215C2×C3×C4#12011yescharacteristic
216C2×C3×C4#12111yescharacteristic
217C2×C3×C4#12111yescharacteristic
218C2×C3×C4#12711yescharacteristic
219C2×C3×C4C2×Q1611yescharacteristic
220C2×C3×C4#13511yescharacteristic
221C2×C3×C4#13711yescharacteristic
222C2×C3×C4#13711yescharacteristic
223C2×Dic3—41no—
224C2×Dic3—41no—
225Dic6—42no—
226Dic6—42no—
227C3×C8—42no—
228C3⋊C8—42no—
229C3⋊C8—42no—
230C3⋊C8—81no—
231C2×Q8—21no—
232C2×Q8—21no—
233C2×Q8—22no—
234C2×Q8—21no—
235C2×Q8—21no—
236C2×Q8—21no—
237C2×Q8—41no—
238C2×Q8—61no—
239C2×Q8—61no—
240C2×Q8—82no—
241C2×Q8#21612yes—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—41no—
245C22×C4D3×D411yescharacteristic
246C22×C4#23711yescharacteristic
247C22×C4C2×#6711yescharacteristic
248M16—121no—
249M16—121no—
250C2×C8—22no—
251C2×C8—41no—
252C2×C8—41no—
253C2×C8—62no—
254C2×C8—62no—
255C2×C8—121no—
256C4⋊C4—22no—
257C4⋊C4—21no—
258C4⋊C4—22no—
259C4⋊C4—21no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—61no—
265C4⋊C4—61no—
266C4⋊C4—61no—
267C4⋊C4—61no—
268C4⋊C4—121no—
269C4⋊C4—121no—
270C22⋊C4—121no—
271C42—41no—
272C42—41no—
273C42—121no—
274C42D3×D411yescharacteristic
275C42D3×D411yescharacteristic
276C42C2×#6711yescharacteristic
277C42C2×#6711yescharacteristic
278C22×C3—21no—
279C22×C3—21no—
280C22×C3#39211yescharacteristic
281C22×C3#46711yescharacteristic
282C22×C3#47011yescharacteristic
283C3×C4—21no—
284C3×C4—21no—
285C3×C4—21no—
286C3×C4—42no—
287C3×C4—81no—
288C3×C4—82no—
289C3×C4#45111yescharacteristic
290C3×C4#45111yescharacteristic
291C3×C4#45211yescharacteristic
292C3×C4#45211yescharacteristic
293C3×C4#47112yes—
294Dic3—81no—
295Dic3—81no—
296C23#93911yescharacteristic
297Q8—22no—
298Q8—22no—
299Q8—41no—
300Q8—41no—
301Q8—82no—
302Q8—82no—
303Q8—122no—
304Q8—122no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—82no—
313C2×C4—81no—
314C2×C4—121no—
315C2×C4—121no—
316C2×C4#93711yescharacteristic
317C2×C4#93711yescharacteristic
318C2×C4#93811yescharacteristic
319C2×C4#93911yescharacteristic
320C2×C4#94011yescharacteristic
321C2×C4#94211yescharacteristic
322C2×C4C2×#21611yescharacteristic
323C2×C4#95011yescharacteristic
324C2×C4#95111yescharacteristic
325C2×C4#95311yescharacteristic
326C8—42no—
327C8—122no—
328C8—122no—
329C8—241no—
330C2×C3—21no—
331C2×C3—21no—
332C2×C3#200111yescharacteristic
333C2×C3#200911yescharacteristic
334C2×C3#201911yescharacteristic
335C22—21no—
336C22—21no—
337C22#505111yescharacteristic
338C22#505211yescharacteristic
339C22#522311yescharacteristic
340C4—21no—
341C4—21no—
342C4—21no—
343C4—42no—
344C4—82no—
345C4—81no—
346C4—241no—
347C4—241no—
348C4#504711yescharacteristic
349C4#505011yescharacteristic
350C4#517712yes—
351C4#522811yescharacteristic
352C4#523911yescharacteristic
353C3#1019111yescharacteristic, complemented, Sylow
354C2—21no—
355C2—21no—
356C2#7170011yescharacteristic
357C2#7205911yescharacteristic
358C2#7211111yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×#475C2
socleC22×C3#392
prime radicalC23×C3#121
hypercenterC2×#475D3
hypocenterC3#10191
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#475

Complements

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

NH ≅ G/Ntype
353. C3#10191semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4752
#832242
#824312
#717742
#716242
#704292
#677072
#101913