Finite Group Databasebeta

← #10848810

Subgroups — #10848810

Order 768.

1052 conjugacy classes of subgroups; only the 191 normal subgroups are shown.

normal subgroups only

This group has 1052 conjugacy classes of subgroups — too many to draw. Only its 191 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×#5698C211yesmaximal, characteristic
2C2×C3×#491C211yesmaximal, characteristic, complemented
3#83261C211yesmaximal, characteristic
4C2×#5092C211yesmaximal, characteristic, complemented
5#71740C211yesmaximal, characteristic, complemented
6#70281C214yesmaximal, complemented
7#70273C214yesmaximal
8#69830C211yesmaximal, characteristic
9#69822C211yesmaximal, characteristic, complemented
10#25609—31nomaximal, Sylow
11C2×C3×C4⋊Q8C2211yescharacteristic
12C2×C3×#125C2211yescharacteristic
13C2×#946C2212yes—
14C22×#216C2214yes—
15#5698—24no—
16#5698—24no—
17C2×#898—22no—
18C2×#897C2212yes—
19C2×#890—22no—
20C2×#887C2212yes—
21C2×#872C2211yescharacteristic
22C2×#869C2211yescharacteristic
23C3×#491—24no—
24C2×C3×#103C2214yes—
25C2×C3×#97C2211yescharacteristic
26#5229C2212yes—
27C2×(C2×D3)⋊C8C2212yes—
28C2×#816C2214yes—
29#5092—24no—
30#4992—22no—
31#4983—22no—
32#4936C2211yescharacteristic
33C2×C4×C3⋊C8C2211yescharacteristic
34#4550C2212yes—
35#4529C2214yes—
36C2×#541—31no—
37C2×#494—31no—
38C2×#491D311yescharacteristic, complemented
39#2621—31no—
40#2001—34no—
41#1991—31no—
42#1960—31no—
43#1878—34no—
44#1750—31no—
45C22×C3×Q8C2312yes—
46C22×C4×D3C2311yescharacteristic
47C22×Dic6C2312yes—
48C22×C3×C8C2312yes—
49C3×C4⋊Q8—22no—
50C3×C4⋊Q8—22no—
51C3×#125—22no—
52C3×#125D414yes—
53C2×C3×C4⋊C4—22no—
54C2×C3×C4⋊C4—22no—
55C2×C3×C4⋊C4C2312yes—
56C2×C3×C42C2311yescharacteristic
57#946—44no—
58C2×#216—28no—
59C2×#216—28no—
60C2×#216—44no—
61C2×(C2×D3)⋊C4—22no—
62C2×(C2×D3)⋊C4—22no—
63C2×(C2×D3)⋊C4D412yes—
64C2×C4⋊Dic3—22no—
65C2×C4⋊Dic3—22no—
66C2×C4⋊Dic3C2312yes—
67C2×Dic3⋊C4—22no—
68C2×Dic3⋊C4—22no—
69C2×Dic3⋊C4D412yes—
70C2×C4×Dic3—22no—
71C2×C4×Dic3C2311yescharacteristic
72C22×C3⋊C8C2312yes—
73#898—216no—
74#897—24no—
75#897—24no—
76#890—216no—
77#887—24no—
78#887—24no—
79#872—22no—
80#872—22no—
81#869—22no—
82#869D414yes—
83C3×#103—28no—
84C3×#97—22no—
85C3⋊#95—22no—
86C3⋊#95—24no—
87(C2×D3)⋊C8—24no—
88#816—44no—
89C4×C3⋊C8—22no—
90C22×Q16—34no—
91#541—64no—
92#541—64no—
93C2×C4⋊Q8—31no—
94C2×C4⋊Q8C2×D311yescharacteristic
95C2×#126—62no—
96C2×#125—62no—
97C2×#125C2×D311yescharacteristic
98C2×C22⋊Q8—32no—
99C2×C22⋊Q8—32no—
100C2×C4×Q8—32no—
101C2×#117—31no—
102#494—64no—
103#491—24no—
104C2×#103—34no—
105C2×#103C2×D314yes—
106C2×C22⋊C8—32no—
107C2×#97C2×D311yescharacteristic
108C2×C4×C8—31no—
109#398—62no—
110#397—62no—
111#391—32no—
112#382—31no—
113#340—34no—
114#336—32no—
115C23×D3C2×D411yescharacteristic
116C2×C3×Q8—22no—
117C2×C3×Q8—22no—
118C2×C3×Q8—24no—
119C2×C3×Q8—24no—
120C22×C3×C4—22no—
121C22×C3×C4—22no—
122C22×C3×C4C2×D412yes—
123C22×C3×C4C2411yescharacteristic
124C22×Dic3—22no—
125C22×Dic3—22no—
126C22×Dic3C2×D412yes—
127C22×Dic3C2×D411yescharacteristic
128C2×C4×D3—21no—
129C2×C4×D3—21no—
130C2×C4×D3—22no—
131C2×C4×D3—22no—
132C2×Dic6—24no—
133C2×Dic6—24no—
134C2×Dic6—22no—
135C2×Dic6C2×D414yes—
136C2×C3×C8—22no—
137C2×C3×C8—22no—
138C2×C3×C8—22no—
139C3×C4⋊C4—28no—
140C3×C4⋊C4—22no—
141C3×C4⋊C4—44no—
142C3×C4⋊C4C2×D414yes—
143C3×C42C2×D412yes—
144C3×C42C2×D412yes—
145#216—416no—
146(C2×D3)⋊C4—28no—
147(C2×D3)⋊C4—24no—
148(C2×D3)⋊C4—44no—
149C4⋊Dic3—28no—
150C4⋊Dic3—28no—
151C4⋊Dic3—24no—
152Dic3⋊C4—28no—
153Dic3⋊C4—24no—
154Dic3⋊C4—44no—
155C4×Dic3—28no—
156C4×Dic3—21no—
157C4×Dic3—21no—
158C2×C3⋊C8—22no—
159C2×C3⋊C8—22no—
160C2×C3⋊C8#4114yes—
161C22×Q8—32no—
162C22×Q8C22×D312yes—
163C23×C4—31no—
164C2×Q16—68no—
165C2×Q16—68no—
166C2×Q16—124no—
167C22×C8—32no—
168C22×C8C22×D312yes—
169C4⋊Q8—22no—
170C4⋊Q8—22no—
171C4⋊Q8—34no—
172C4⋊Q8—62no—
173#126—616no—
174#125—22no—
175#125—616no—
176#125#6714yes—
177C22⋊Q8—64no—
178C22⋊Q8—64no—
179C22⋊Q8—124no—
180C4×Q8—64no—
181C4×Q8—64no—
182#117—62no—
183#117—62no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—32no—
187C2×C4⋊C4—32no—
188C2×C4⋊C4—62no—
189C2×C4⋊C4—62no—
190C2×C4⋊C4—62no—
191C2×C4⋊C4—62no—
192C2×C4⋊C4C22×D312yes—
193C2×C22⋊C4—32no—
194C2×C22⋊C4—62no—
195C2×C22⋊C4—62no—
196C2×C42—31no—
197C2×C42—62no—
198C2×C42C22×D311yescharacteristic
199#103—28no—
200#103—124no—
201C22⋊C8—64no—
202#97—22no—
203C4×C8—62no—
204#95—64no—
205#95—62no—
206C23×C3C22×D411yescharacteristic
207C22×D3—21no—
208C22×D3—21no—
209C22×D3—21no—
210C22×D3—24no—
211C3×Q8—44no—
212C3×Q8—44no—
213C2×C3×C4—24no—
214C2×C3×C4—24no—
215C2×C3×C4—44no—
216C2×C3×C4—42no—
217C2×C3×C4—42no—
218C2×C3×C4—42no—
219C2×C3×C4#12014yes—
220C2×C3×C4C22×D411yescharacteristic
221C2×C3×C4C22×D411yescharacteristic
222C2×C3×C4C2×#4112yes—
223C2×C3×C4H(5,𝔽2)12yes—
224C2×Dic3—22no—
225C2×Dic3—24no—
226C2×Dic3—21no—
227C2×Dic3—21no—
228C2×Dic3—21no—
229C2×Dic3—24no—
230C2×Dic3—42no—
231C2×Dic3—42no—
232C2×Dic3—42no—
233C2×Dic3—44no—
234C4×D3—42no—
235C4×D3—42no—
236Dic6—28no—
237Dic6—44no—
238C3×C8—42no—
239C3⋊C8—24no—
240C24—31no—
241C2×Q8—22no—
242C2×Q8—24no—
243C2×Q8—24no—
244C2×Q8—22no—
245C2×Q8—34no—
246C2×Q8—62no—
247C2×Q8—64no—
248C2×Q8—64no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—32no—
252C22×C4—31no—
253C22×C4—62no—
254C22×C4—61no—
255C22×C4—62no—
256C22×C4—62no—
257C22×C4—61no—
258C22×C4—62no—
259C22×C4D3×D412yes—
260C22×C4C23×D311yescharacteristic
261Q16—1216no—
262C2×C8—22no—
263C2×C8—22no—
264C2×C8—22no—
265C2×C8—34no—
266C2×C8—62no—
267C2×C8—62no—
268C4⋊C4—28no—
269C4⋊C4—22no—
270C4⋊C4—44no—
271C4⋊C4—68no—
272C4⋊C4—68no—
273C4⋊C4—64no—
274C4⋊C4—68no—
275C4⋊C4—64no—
276C4⋊C4—124no—
277C4⋊C4C2×#6714yes—
278C22⋊C4—68no—
279C22⋊C4—64no—
280C22⋊C4—124no—
281C42—68no—
282C42—61no—
283C42—61no—
284C42D3×D412yes—
285C42C2×#6712yes—
286C22×C3C2×#12011yescharacteristic
287C22×C3#53211yescharacteristic
288C22×C3#53511yescharacteristic
289C22×C3#57814yes—
290C2×D3—42no—
291C2×D3—41no—
292C2×D3—42no—
293C2×D3—42no—
294C3×C4—24no—
295C3×C4—44no—
296C3×C4—82no—
297C3×C4#54612yes—
298C3×C4#55012yes—
299Dic3—44no—
300Dic3—41no—
301Dic3—42no—
302Dic3—82no—
303C23—61no—
304C23—61no—
305C23—64no—
306C23—61no—
307C23C2×D3×D411yescharacteristic
308Q8—44no—
309Q8—44no—
310Q8—68no—
311Q8—124no—
312C2×C4—24no—
313C2×C4—24no—
314C2×C4—44no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—61no—
319C2×C4—61no—
320C2×C4—64no—
321C2×C4—62no—
322C2×C4—64no—
323C2×C4—61no—
324C2×C4—122no—
325C2×C4—122no—
326C2×C4—122no—
327C2×C4—122no—
328C2×C4—122no—
329C2×C4—124no—
330C2×C4#93714yes—
331C2×C4C2×D3×D411yescharacteristic
332C2×C4D3×#4112yes—
333C2×C4#100912yes—
334C2×C4C22×#6711yescharacteristic
335C8—42no—
336C8—64no—
337C2×C3#243111yescharacteristic
338C2×C3#301112yes—
339C2×C3#317712yes—
340C2×C3#322312yes—
341D3—81no—
342C22—121no—
343C22—122no—
344C22—122no—
345C22—122no—
346C22#566811yescharacteristic
347C22#567411yescharacteristic
348C22#577914yes—
349C22C2×#93711yescharacteristic
350C4—24no—
351C4—44no—
352C4—82no—
353C4—121no—
354C4—124no—
355C4—122no—
356C4—242no—
357C4#567612yes—
358C4#583412yes—
359C3#2560911yescharacteristic, complemented, Sylow
360C2—241no—
361C2#8173012yes—
362C2#8174412yes—
363C2#8295811yescharacteristic
364C2#8398312yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#491C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenterC2×#491D3
hypocenterC3#25609
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#491

Complements

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

NH ≅ G/Ntype
2. C2×C3×#491C2semidirect
4. C2×#5092C2semidirect
5. #71740C2semidirect
6. #70281C2semidirect
9. #69822C2semidirect
38. C2×#491D3semidirect
359. C3#25609semidirect

Maximal subgroups

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

subgroupindex
C2×#56982
C2×C3×#4912
#832612
C2×#50922
#717402
#702812
#702732
#698302
#698222
#256093