Finite Group Databasebeta

← #10848792

Subgroups — #10848792

Order 768.

1152 conjugacy classes of subgroups; only the 191 normal subgroups are shown.

normal subgroups only

This group has 1152 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, complemented
2C2×C3×#490C211yesmaximal, characteristic, complemented
3#83189C211yesmaximal, characteristic
4C2×#5076C211yesmaximal, characteristic, complemented
5#71740C211yesmaximal, characteristic, complemented
6#70213C214yesmaximal, complemented
7#70205C214yesmaximal, complemented
8#69830C211yesmaximal, characteristic, complemented
9#69822C211yesmaximal, characteristic, complemented
10#25607—31nomaximal, Sylow
11C2×C3×#127C2211yescharacteristic
12C2×C3×#125C2211yescharacteristic
13C2×#938C2212yes—
14C22×#214C2214yes—
15#5698—24no—
16#5698—24no—
17C2×#898—22no—
18C2×#897C2212yes—
19C2×#890—22no—
20C2×#887C2212yes—
21C2×#872C2211yescharacteristic
22C2×#869C2211yescharacteristic
23C3×#490—24no—
24C2×C3×#102C2214yes—
25C2×C3×#97C2211yescharacteristic
26#5223—22no—
27#5222—22no—
28#5221C2212yes—
29C2×(C2×D3)⋊C8C2212yes—
30C2×#816C2214yes—
31#5076—24no—
32#4936C2211yescharacteristic
33C2×C4×C3⋊C8C2211yescharacteristic
34#4550C2212yes—
35#4529C2214yes—
36C2×#541—31no—
37C2×#492—31no—
38C2×#490D311yescharacteristic, complemented
39#2619—31no—
40#2000—34no—
41#1991—31no—
42#1960—31no—
43#1880—34no—
44#1750—31no—
45C22×C3×D4—22no—
46C22×C3×D4C2312yes—
47C22×#67—22no—
48C22×C4×D3C2311yescharacteristic
49C22×Dic6C2312yes—
50C22×C3×C8C2312yes—
51C3×#127—22no—
52C3×#127—22no—
53C3×#125—22no—
54C3×#125D414yescomplemented
55C2×C3×C4⋊C4—22no—
56C2×C3×C4⋊C4C2312yes—
57C2×C3×C42C2311yescharacteristic
58C2×#217—22no—
59C2×#217—24no—
60#938—44no—
61C2×#214—28no—
62C2×#214—28no—
63C2×#214—44no—
64C2×(C2×D3)⋊C4—22no—
65C2×(C2×D3)⋊C4D412yes—
66C2×C4⋊Dic3—22no—
67C2×C4⋊Dic3—22no—
68C2×C4⋊Dic3C2312yes—
69C2×Dic3⋊C4—22no—
70C2×Dic3⋊C4D412yes—
71C2×C4×Dic3—22no—
72C2×C4×Dic3C2311yescharacteristic
73C22×C3⋊C8C2312yes—
74#898—216no—
75#897—24no—
76#897—24no—
77#890—216no—
78#887—24no—
79#887—24no—
80#872—22no—
81#872—22no—
82#869—22no—
83#869D414yescomplemented
84C3×#102—28no—
85C3×#97—22no—
86C3⋊#95—22no—
87(C2×D3)⋊C8—24no—
88#816—44no—
89C4×C3⋊C8—22no—
90C22×SD16—34no—
91#541—64no—
92#541—64no—
93C2×C4⋊Q8—31no—
94C2×#127C2×D311yescharacteristic
95C2×#126—62no—
96C2×#125—62no—
97C2×#125C2×D311yescharacteristic
98C2×C22⋊Q8—32no—
99C2×#121—32no—
100C2×C4×Q8—32no—
101C2×#117—31no—
102#492—64no—
103#490—24no—
104C2×#103—34no—
105C2×#102C2×D314yes—
106C2×C22⋊C8—32no—
107C2×#97C2×D311yescharacteristic
108C2×C4×C8—31no—
109#394—62no—
110#392—62no—
111#390—32no—
112#382—31no—
113#340—34no—
114#336—32no—
115C24×C3—22no—
116C23×D3C2×D411yescharacteristic
117C2×C3×D4—22no—
118C2×C3×D4—22no—
119C2×C3×D4—24no—
120C2×C3×D4—24no—
121C2×C3×D4—44no—
122C2×C3×D4—84no—
123C22×C3×C4—22no—
124C22×C3×C4C2×D412yes—
125C22×C3×C4C2411yescharacteristic
126C2×#67—44no—
127C2×#67—84no—
128C22×Dic3—22no—
129C22×Dic3—22no—
130C22×Dic3C2×D412yes—
131C22×Dic3C2×D411yescharacteristic
132C2×C4×D3—21no—
133C2×C4×D3—21no—
134C2×C4×D3—22no—
135C2×C4×D3—22no—
136C2×Dic6—24no—
137C2×Dic6—24no—
138C2×Dic6—22no—
139C2×Dic6C2×D414yes—
140C2×C3×C8—22no—
141C2×C3×C8—22no—
142C2×C3×C8—22no—
143C3×C4⋊C4—28no—
144C3×C4⋊C4—22no—
145C3×C4⋊C4C2×D414yes—
146C3×C42C2×D412yes—
147C3×C42C2×D412yes—
148#217—44no—
149#217—84no—
150#214—416no—
151(C2×D3)⋊C4—28no—
152(C2×D3)⋊C4—24no—
153C4⋊Dic3—28no—
154C4⋊Dic3—28no—
155C4⋊Dic3—24no—
156Dic3⋊C4—28no—
157Dic3⋊C4—24no—
158C4×Dic3—28no—
159C4×Dic3—21no—
160C4×Dic3—21no—
161C2×C3⋊C8—22no—
162C2×C3⋊C8—22no—
163C2×C3⋊C8#4114yes—
164C22×Q8—32no—
165C22×D4—22no—
166C22×D4—62no—
167C22×D4C22×D312yes—
168C23×C4—31no—
169C2×SD16—68no—
170C2×SD16—68no—
171C2×SD16—124no—
172C22×C8—32no—
173C22×C8C22×D312yes—
174C4⋊Q8—34no—
175C4⋊Q8—62no—
176#127—22no—
177#127—22no—
178#126—616no—
179#125—22no—
180#125—616no—
181#125#6714yescomplemented
182C22⋊Q8—64no—
183C22⋊Q8—64no—
184#121—124no—
185C4×Q8—64no—
186C4×Q8—64no—
187#117—62no—
188#117—62no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—32no—
191C2×C4⋊C4—32no—
192C2×C4⋊C4—62no—
193C2×C4⋊C4—62no—
194C2×C4⋊C4—62no—
195C2×C4⋊C4C22×D312yes—
196C2×C22⋊C4—32no—
197C2×C22⋊C4—62no—
198C2×C22⋊C4—62no—
199C2×C22⋊C4—64no—
200C2×C42—31no—
201C2×C42—62no—
202C2×C42C22×D311yescharacteristic
203#103—124no—
204#102—28no—
205C22⋊C8—64no—
206#97—22no—
207C4×C8—62no—
208#95—62no—
209C23×C3—42no—
210C23×C3—42no—
211C23×C3—42no—
212C23×C3—44no—
213C23×C3—44no—
214C23×C3C22×D411yescharacteristic
215C22×D3—21no—
216C22×D3—21no—
217C22×D3—21no—
218C22×D3—24no—
219C3×D4—44no—
220C3×D4—44no—
221C3×D4—88no—
222C2×C3×C4—24no—
223C2×C3×C4—24no—
224C2×C3×C4—44no—
225C2×C3×C4#12014yes—
226C2×C3×C4C22×D411yescharacteristic
227C2×C3×C4C22×D411yescharacteristic
228C2×C3×C4C2×#4112yes—
229C2×C3×C4H(5,𝔽2)12yes—
230#67—88no—
231C2×Dic3—22no—
232C2×Dic3—24no—
233C2×Dic3—21no—
234C2×Dic3—21no—
235C2×Dic3—21no—
236C2×Dic3—24no—
237C2×Dic3—42no—
238C2×Dic3—42no—
239C2×Dic3—42no—
240C2×Dic3—44no—
241C4×D3—42no—
242C4×D3—42no—
243Dic6—28no—
244Dic6—44no—
245C3×C8—42no—
246C3⋊C8—24no—
247C24—22no—
248C24—31no—
249C2×Q8—34no—
250C2×Q8—62no—
251C2×Q8—64no—
252C2×Q8—64no—
253C2×D4—22no—
254C2×D4—24no—
255C2×D4—24no—
256C2×D4—22no—
257C2×D4—44no—
258C2×D4—84no—
259C2×D4—124no—
260C2×D4—244no—
261C22×C4—22no—
262C22×C4—32no—
263C22×C4—31no—
264C22×C4—62no—
265C22×C4—61no—
266C22×C4—62no—
267C22×C4—62no—
268C22×C4—61no—
269C22×C4—62no—
270C22×C4D3×D412yes—
271C22×C4C23×D311yescharacteristic
272SD16—1216no—
273C2×C8—22no—
274C2×C8—22no—
275C2×C8—22no—
276C2×C8—34no—
277C2×C8—62no—
278C2×C8—62no—
279C4⋊C4—28no—
280C4⋊C4—22no—
281C4⋊C4—68no—
282C4⋊C4—68no—
283C4⋊C4—64no—
284C4⋊C4—68no—
285C4⋊C4—64no—
286C4⋊C4C2×#6714yes—
287C22⋊C4—68no—
288C22⋊C4—64no—
289C22⋊C4—124no—
290C22⋊C4—244no—
291C42—68no—
292C42—61no—
293C42—61no—
294C42D3×D412yes—
295C42C2×#6712yes—
296C22×C3—84no—
297C22×C3—82no—
298C22×C3—82no—
299C22×C3—82no—
300C22×C3—84no—
301C22×C3C2×#12011yescharacteristic
302C22×C3#53211yescharacteristic
303C22×C3#53511yescharacteristic
304C22×C3#57714yes—
305C2×D3—42no—
306C2×D3—41no—
307C2×D3—42no—
308C2×D3—42no—
309C3×C4—24no—
310C3×C4—44no—
311C3×C4#54612yes—
312C3×C4#55012yes—
313Dic3—44no—
314Dic3—41no—
315Dic3—42no—
316Dic3—82no—
317C23—42no—
318C23—42no—
319C23—44no—
320C23—44no—
321C23—42no—
322C23—61no—
323C23—61no—
324C23—64no—
325C23—61no—
326C23C2×D3×D411yescharacteristic
327Q8—68no—
328Q8—124no—
329D4—44no—
330D4—44no—
331D4—88no—
332D4—248no—
333C2×C4—24no—
334C2×C4—24no—
335C2×C4—44no—
336C2×C4—61no—
337C2×C4—61no—
338C2×C4—64no—
339C2×C4—62no—
340C2×C4—64no—
341C2×C4—61no—
342C2×C4—122no—
343C2×C4—122no—
344C2×C4—122no—
345C2×C4—122no—
346C2×C4—122no—
347C2×C4—124no—
348C2×C4#93714yes—
349C2×C4C2×D3×D411yescharacteristic
350C2×C4D3×#4112yes—
351C2×C4#100912yes—
352C2×C4C22×#6711yescharacteristic
353C8—42no—
354C8—64no—
355C2×C3—162no—
356C2×C3#243111yescharacteristic
357C2×C3#301012yes—
358C2×C3#317412yes—
359C2×C3#322212yes—
360D3—81no—
361C22—82no—
362C22—84no—
363C22—82no—
364C22—82no—
365C22—84no—
366C22—121no—
367C22—122no—
368C22—122no—
369C22—122no—
370C22#566811yescharacteristic
371C22#567411yescharacteristic
372C22#577514yes—
373C22C2×#93711yescharacteristic
374C4—24no—
375C4—44no—
376C4—121no—
377C4—124no—
378C4—122no—
379C4—242no—
380C4#567612yes—
381C4#583412yes—
382C3#2560711yescharacteristic, complemented, Sylow
383C2—162no—
384C2—241no—
385C2#8170512yes—
386C2#8172012yes—
387C2#8295811yescharacteristic
388C2#8394812yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#490C2
socleC23×C3C22×D4
prime radical#83189C2
hypercenterC2×#490D3
hypocenterC3#25607
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#490

Complements

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

NH ≅ G/Ntype
1. C2×#5698C2semidirect
2. C2×C3×#490C2semidirect
4. C2×#5076C2semidirect
5. #71740C2semidirect
6. #70213C2semidirect
7. #70205C2semidirect
8. #69830C2semidirect
9. #69822C2semidirect
38. C2×#490D3semidirect
54. C3×#125D4semidirect
83. #869D4semidirect
181. #125#67semidirect
382. C3#25607semidirect

Maximal subgroups

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

subgroupindex
C2×#56982
C2×C3×#4902
#831892
C2×#50762
#717402
#702132
#702052
#698302
#698222
#256073