Finite Group Databasebeta

← #10696793

Subgroups — #10696793

Order 768.

454 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 81 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
1C3×#3216C211yesmaximal, characteristic
2#83342C211yesmaximal, characteristic, complemented
3#83273C211yesmaximal, characteristic
4#71827C211yesmaximal, characteristic, complemented
5#71801C211yesmaximal, characteristic
6#70831C211yesmaximal, characteristic, complemented
7#67754C211yesmaximal, characteristic, complemented
8#10126—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C3×#532C2211yescharacteristic
11C2×#870C2211yescharacteristic
12C3×#497—21no—
13C3×#489—21no—
14C3×#462—21no—
15C3×#460—21no—
16C3×#452—22no—
17C3×#432C2211yescharacteristic
18#5081—21no—
19#5081—21no—
20#5081—21no—
21#5071—21no—
22#5065—22no—
23#5034—21no—
24#5033—21no—
25#5009—22no—
26#4995—21no—
27#4995—21no—
28#4926C2211yescharacteristic
29#4925C2211yescharacteristic
30C4×#208C2211yescharacteristic
31#4535—22no—
32D4⋊(C3⋊C8)—22no—
33#4469—22no—
34#3234—31no—
35#3216D311yescharacteristic
36#3166—31no—
37#1497—31no—
38#1484—31no—
39#1470—31no—
40#1291—31no—
41C2×C3×#41—21no—
42C2×C3×Q16—22no—
43C2×C3×SD16—22no—
44C3×C4⋊Q8D411yescharacteristic
45C3×C4⋊Q8D411yescharacteristic
46C3×C4⋊Q8D411yescharacteristic
47C3×#127D411yescharacteristic
48C3×#125—21no—
49C3×#124—21no—
50C3×C22⋊Q8—21no—
51C3×C22⋊Q8—21no—
52C3×#121—21no—
53C3×#121—21no—
54C3×C4×Q8—21no—
55C3×C4×Q8—21no—
56C3×C4×D4—21no—
57C3×C4×D4—21no—
58C3×#117—21no—
59C2×C3×C42C2311yescharacteristic
60C2×C4⋊Dic3D411yescharacteristic
61C2×C4⋊Dic3D411yescharacteristic
62C2×Dic3⋊C4—21no—
63C2×Dic3⋊C4—21no—
64C2×#208—21no—
65#870—21no—
66#870—21no—
67#870—21no—
68#870—21no—
69C3×C4⋊C8—21no—
70C3×#103—22no—
71C3×#102—22no—
72C3×C22⋊C8D412yes—
73C3×#97—21no—
74#832—22no—
75#832—22no—
76#830—21no—
77#830—21no—
78#808—22no—
79#808—22no—
80#808—22no—
81#807—22no—
82#807—22no—
83#807—22no—
84C4⋊(C3⋊C8)—21no—
85C4⋊(C3⋊C8)D412yes—
86#803—21no—
87#803—21no—
88C4×C3⋊C8—21no—
89C4×C3⋊C8D412yes—
90#533C2×D311yescharacteristic
91#532C2×D311yescharacteristic
92C2×#125—31no—
93#501—61no—
94#501—61no—
95#501—61no—
96#499—62no—
97#497—21no—
98#490—61no—
99#489—21no—
100#481—61no—
101#480—61no—
102#478—62no—
103#462—21no—
104#460—21no—
105#452—22no—
106#432—31no—
107#432C2×D311yescharacteristic
108#431—31no—
109#428—61no—
110#428—61no—
111C4×M16—31no—
112Q8⋊C8—62no—
113D4⋊C8—62no—
114#324—62no—
115C3×#41—82no—
116C2×C3×Q8—21no—
117C2×C3×Q8—21no—
118C2×C3×Q8—21no—
119C2×C3×D4—21no—
120C2×C3×D4—21no—
121C2×C3×D4—21no—
122C2×C3×D4—41no—
123C22×C3×C4—41no—
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4#4111yescharacteristic
127C22×Dic3—21no—
128C22×Dic3—21no—
129C3×Q16—82no—
130C3×SD16—82no—
131C2×C3×C8—22no—
132C3×C4⋊C4—21no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—41no—
139C3×C4⋊C4—41no—
140C3×C22⋊C4—41no—
141C3×C22⋊C4—41no—
142C3×C22⋊C4—41no—
143C3×C42—41no—
144C3×C42C2×D411yescharacteristic
145C3×C42C2×D411yescharacteristic
146C3×C42C2×D411yescharacteristic
147C3×C42C2×D411yescharacteristic
148C4⋊Dic3—21no—
149C4⋊Dic3—21no—
150C4⋊Dic3—21no—
151C4⋊Dic3—21no—
152Dic3⋊C4—41no—
153Dic3⋊C4—41no—
154Dic3⋊C4—41no—
155Dic3⋊C4—41no—
156#208—24no—
157C2×C3⋊C8—22no—
158C2×C3⋊C8—22no—
159C2×C3⋊C8—41no—
160C2×#41—21no—
161C2×Q16—22no—
162C2×SD16—22no—
163C2×M16—61no—
164C4⋊Q8#6711yescharacteristic
165C4⋊Q8#6711yescharacteristic
166C4⋊Q8#6711yescharacteristic
167#127#6711yescharacteristic
168#125—21no—
169#125—61no—
170#125—61no—
171#125—61no—
172#125—61no—
173#124—21no—
174C22⋊Q8—21no—
175C22⋊Q8—21no—
176#121—21no—
177#121—21no—
178C4×Q8—21no—
179C4×Q8—21no—
180C4×D4—21no—
181C4×D4—21no—
182#117—21no—
183C2×C4⋊C4—31no—
184C2×C4⋊C4—31no—
185C2×C4⋊C4—61no—
186C2×C4⋊C4—61no—
187C2×C42C22×D311yescharacteristic
188#107—62no—
189#107—62no—
190#107—62no—
191#106—62no—
192#106—62no—
193#106—62no—
194C4⋊C8—21no—
195C4⋊C8—32no—
196C4⋊C8—61no—
197#103—22no—
198#102—22no—
199#102—62no—
200#102—62no—
201C22⋊C8—61no—
202C22⋊C8—61no—
203C22⋊C8#6712yes—
204#97—21no—
205#97—61no—
206#97—61no—
207C4×C8—32no—
208C4×C8—61no—
209C23×C3—41no—
210C23×C3#12111yescharacteristic
211C3×Q8—41no—
212C3×Q8—41no—
213C3×Q8—42no—
214C3×Q8—42no—
215C3×D4—42no—
216C3×D4—42no—
217C3×D4—42no—
218C3×D4—82no—
219C2×C3×C4—22no—
220C2×C3×C4—41no—
221C2×C3×C4—41no—
222C2×C3×C4—41no—
223C2×C3×C4—41no—
224C2×C3×C4—41no—
225C2×C3×C4—82no—
226C2×C3×C4#12011yescharacteristic
227C2×C3×C4#12011yescharacteristic
228C2×C3×C4#12011yescharacteristic
229C2×C3×C4#12111yescharacteristic
230C2×C3×C4#12111yescharacteristic
231C2×C3×C4#12711yescharacteristic
232C2×Dic3—42no—
233C2×Dic3—41no—
234C2×Dic3—41no—
235C2×Dic3—81no—
236C3×C8—42no—
237C3⋊C8—24no—
238C3⋊C8—42no—
239C3⋊C8—81no—
240#41—82no—
241C2×Q8—21no—
242C2×Q8—21no—
243C2×Q8—21no—
244C2×D4—21no—
245C2×D4—21no—
246C2×D4—21no—
247C2×D4—41no—
248C22×C4—41no—
249C22×C4—61no—
250C22×C4—61no—
251C22×C4D3×D411yescharacteristic
252C22×C4#23711yescharacteristic
253C22×C4C2×#6711yescharacteristic
254Q16—82no—
255SD16—82no—
256M16—64no—
257C2×C8—22no—
258C2×C8—62no—
259C2×C8—62no—
260C2×C8—121no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—61no—
270C4⋊C4—61no—
271C4⋊C4—61no—
272C4⋊C4—61no—
273C4⋊C4—121no—
274C4⋊C4—121no—
275C4⋊C4—121no—
276C4⋊C4—121no—
277C22⋊C4—41no—
278C22⋊C4—41no—
279C22⋊C4—41no—
280C42—41no—
281C42D3×D411yescharacteristic
282C42D3×D411yescharacteristic
283C42C2×#6711yescharacteristic
284C42C2×#6711yescharacteristic
285C22×C3—21no—
286C22×C3—21no—
287C22×C3—21no—
288C22×C3—81no—
289C22×C3—81no—
290C22×C3—81no—
291C22×C3#39211yescharacteristic
292C3×C4—21no—
293C3×C4—21no—
294C3×C4—21no—
295C3×C4—21no—
296C3×C4—21no—
297C3×C4—81no—
298C3×C4—81no—
299C3×C4—81no—
300C3×C4#45612yes—
301Dic3—81no—
302Dic3—81no—
303C23—41no—
304C23#93811yescharacteristic
305Q8—41no—
306Q8—41no—
307Q8—42no—
308Q8—42no—
309D4—42no—
310D4—42no—
311D4—42no—
312D4—82no—
313C2×C4—22no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—82no—
320C2×C4—122no—
321C2×C4—121no—
322C2×C4—121no—
323C2×C4—241no—
324C2×C4#93711yescharacteristic
325C2×C4#93711yescharacteristic
326C2×C4#93911yescharacteristic
327C2×C4#93911yescharacteristic
328C2×C4#94011yescharacteristic
329C2×C4#95311yescharacteristic
330C8—42no—
331C8—64no—
332C8—122no—
333C8—241no—
334C2×C3—41no—
335C2×C3—161no—
336C2×C3#200511yescharacteristic
337C2×C3#200811yescharacteristic
338C2×C3#201711yescharacteristic
339C22—21no—
340C22—21no—
341C22—21no—
342C22—81no—
343C22—81no—
344C22—81no—
345C22#522311yescharacteristic
346C4—21no—
347C4—21no—
348C4—21no—
349C4—21no—
350C4—21no—
351C4—81no—
352C4—81no—
353C4—81no—
354C4—241no—
355C4—241no—
356C4#520512yes—
357C3#1012611yescharacteristic, complemented, Sylow
358C2—41no—
359C2—161no—
360C2#7185811yescharacteristic
361C2#7210811yescharacteristic
362C2#7212111yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3216C2
socleC22×C3#392
prime radicalC3×#532C22
hypercenter#3216D3
hypocenterC3#10126
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3216

Complements

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

NH ≅ G/Ntype
2. #83342C2semidirect
4. #71827C2semidirect
6. #70831C2semidirect
7. #67754C2semidirect
357. C3#10126semidirect

Maximal subgroups

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

subgroupindex
C3×#32162
#833422
#832732
#718272
#718012
#708312
#677542
#101263