Finite Group Databasebeta

← #10697326

Subgroups — #10697326

Order 768.

512 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 99 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
1#83224C211yesmaximal, characteristic, complemented
2#83199C211yesmaximal, characteristic
3#79027C211yesmaximal, characteristic
4C3×#1981C211yesmaximal, characteristic, complemented
5#69493C211yesmaximal, characteristic, complemented
6#69469C211yesmaximal, characteristic
7#67706C211yesmaximal, characteristic, complemented
8#10202—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10#5477C2211yescharacteristic
11#5470C2211yescharacteristic
12C2×C3×C4⋊C8C2211yescharacteristic
13C2×C3×#103—22no—
14C3×C4×C4⋊C4C2211yescharacteristic
15#5114—22no—
16#5093—21no—
17#5093—21no—
18#5091—22no—
19#5089—22no—
20#5088—21no—
21#5078—21no—
22#4970—22no—
23#4966—22no—
24#4925C2211yescharacteristic
25C4×#208C2211yescharacteristic
26#4851—21no—
27#4848—21no—
28#4842—21no—
29#4841—21no—
30#4523—22no—
31#4451—22no—
32#3246—31no—
33#3230—31no—
34#3141—31no—
35#1986—31no—
36#1981D311yescharacteristic, complemented
37#1502—31no—
38#1283—31no—
39C22×C3×Q8—21no—
40C2×#235—21no—
41C22×C3×C8D412yes—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8D412yes—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C42—21no—
50C2×C3×C42C2311yescharacteristic
51#930—21no—
52#930—21no—
53#926—21no—
54#924—21no—
55#924—21no—
56C2×#208D412yes—
57C2×#204—22no—
58#875—21no—
59#874D411yescharacteristic
60C4×D12—21no—
61C4×D12—21no—
62#870—21no—
63#869D411yescharacteristic
64#869D411yescharacteristic
65#869D411yescharacteristic
66C4×Dic6—21no—
67C4×Dic6—21no—
68C3×C4⋊C8—21no—
69C3×C4⋊C8—21no—
70C3×#103—44no—
71C3×#95—21no—
72#830D412yes—
73#821—22no—
74#817—22no—
75#816—22no—
76#810—24no—
77#810—24no—
78#809—24no—
79#807—24no—
80C4⋊(C3⋊C8)—21no—
81#803—21no—
82C4×C3⋊C8—21no—
83C4×C3⋊C8—21no—
84#533—31no—
85#532—31no—
86C2×C4⋊Q8C2×D311yescharacteristic
87#500—61no—
88#489—62no—
89#487—61no—
90#487—61no—
91#486—61no—
92#484—62no—
93#477—62no—
94#476—61no—
95#475—61no—
96#465—62no—
97#461—61no—
98#460—61no—
99#431—31no—
100C2×C4⋊C8C2×D311yescharacteristic
101#417—62no—
102C2×#103—22no—
103C4×M16—31no—
104C4×C4⋊C4C2×D311yescharacteristic
105#339—62no—
106#327—62no—
107C2×C3×Q8—22no—
108C2×C3×Q8—42no—
109C2×C3×Q8—42no—
110C2×C3×Q8—41no—
111C22×C3×C4—41no—
112C22×C3×C4—41no—
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C2×#67—41no—
117#235—42no—
118#235—42no—
119C2×D12—21no—
120C2×D12—21no—
121C2×D12—21no—
122C2×C4×D3—41no—
123C2×Dic6—21no—
124C2×Dic6—21no—
125C2×Dic6—21no—
126C2×C3×C8—22no—
127C2×C3×C8—42no—
128C3×C4⋊C4—22no—
129C3×C4⋊C4—22no—
130C3×C4⋊C4—22no—
131C3×C4⋊C4—41no—
132C3×C4⋊C4—41no—
133C3×C4⋊C4—41no—
134C3×C42—42no—
135C3×C42C2×D411yescharacteristic
136C3×C42C2×D411yescharacteristic
137C3×C42C2×D411yescharacteristic
138C3×C42#4111yescharacteristic
139#217—41no—
140(C2×D3)⋊C4—41no—
141(C2×D3)⋊C4—41no—
142C4⋊Dic3—21no—
143C4⋊Dic3—21no—
144C4⋊Dic3—21no—
145C4⋊Dic3—21no—
146C4⋊Dic3—21no—
147C4⋊Dic3—21no—
148Dic3⋊C4—41no—
149Dic3⋊C4—41no—
150C4×Dic3—41no—
151#208—22no—
152#208—22no—
153C2×C3⋊C8—22no—
154C2×C3⋊C8—22no—
155#204—82no—
156C2×#41—61no—
157C22×Q8—21no—
158C2×SD16—62no—
159C2×M16—32no—
160C22×C8#6712yes—
161C4⋊Q8—21no—
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164C4⋊Q8—31no—
165C4⋊Q8—31no—
166C4⋊Q8—31no—
167C4⋊Q8D1212yes—
168#127—31no—
169#125—61no—
170#124—61no—
171C22⋊Q8—61no—
172C22⋊Q8—61no—
173#121—61no—
174#121—61no—
175C4×Q8—61no—
176C4×Q8—61no—
177C4×D4—61no—
178C4×D4—61no—
179#117—61no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C42—21no—
184C2×C42C22×D311yescharacteristic
185#107—64no—
186#106—62no—
187C4⋊C8—21no—
188C4⋊C8—21no—
189C4⋊C8—61no—
190#103—44no—
191#103—64no—
192#103—64no—
193#103—62no—
194#102—64no—
195#102—62no—
196C22⋊C8—32no—
197#97—61no—
198C4×C8—61no—
199C4×C8—61no—
200#95—21no—
201C23×C3#12711yescharacteristic
202C22×D3—41no—
203C3×Q8—44no—
204C3×Q8—81no—
205C3×Q8—81no—
206C2×C3×C4—21no—
207C2×C3×C4—21no—
208C2×C3×C4—41no—
209C2×C3×C4—42no—
210C2×C3×C4—42no—
211C2×C3×C4—81no—
212C2×C3×C4—82no—
213C2×C3×C4—81no—
214C2×C3×C4#12011yescharacteristic
215C2×C3×C4#12011yescharacteristic
216C2×C3×C4#12011yescharacteristic
217C2×C3×C4#12111yescharacteristic
218C2×C3×C4#12111yescharacteristic
219C2×C3×C4#12111yescharacteristic
220C2×C3×C4#13512yes—
221C2×C3×C4#13712yes—
222#67—81no—
223#67—81no—
224C2×Dic3—41no—
225C2×Dic3—41no—
226C2×Dic3—41no—
227D12—42no—
228D12—41no—
229D12—41no—
230D12—81no—
231C4×D3—82no—
232Dic6—42no—
233Dic6—42no—
234Dic6—41no—
235Dic6—41no—
236C3×C8—42no—
237C3⋊C8—42no—
238C3⋊C8—42no—
239#41—122no—
240#41—122no—
241C2×Q8—22no—
242C2×Q8—42no—
243C2×Q8—42no—
244C2×Q8—41no—
245C2×Q8—61no—
246C2×Q8—61no—
247C2×Q8—61no—
248C2×D4—61no—
249C2×D4—61no—
250C2×D4—61no—
251C2×D4—121no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—121no—
255C22×C4D3×D411yescharacteristic
256C22×C4D3×D411yescharacteristic
257C22×C4C2×#6711yescharacteristic
258SD16—242no—
259M16—62no—
260M16—62no—
261C2×C8—22no—
262C2×C8—42no—
263C2×C8—62no—
264C2×C8—62no—
265C4⋊C4—22no—
266C4⋊C4—22no—
267C4⋊C4—22no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—61no—
272C4⋊C4—61no—
273C4⋊C4—61no—
274C4⋊C4—61no—
275C4⋊C4—61no—
276C4⋊C4—61no—
277C4⋊C4—121no—
278C4⋊C4—121no—
279C22⋊C4—121no—
280C22⋊C4—121no—
281C22⋊C4—121no—
282C42—42no—
283C42—121no—
284C42C2×D1211yescharacteristic
285C42#23511yescharacteristic
286C42D3×D411yescharacteristic
287C42D3×D411yescharacteristic
288C22×C3—21no—
289C22×C3—21no—
290C22×C3#39211yescharacteristic
291C22×C3#49511yescharacteristic
292C22×C3#49711yescharacteristic
293C2×D3—81no—
294C2×D3—81no—
295C2×D3—81no—
296C3×C4—21no—
297C3×C4—21no—
298C3×C4—21no—
299C3×C4—82no—
300C3×C4—82no—
301C3×C4#45212yes—
302C3×C4#45311yescharacteristic
303C3×C4#45411yescharacteristic
304C3×C4#46412yes—
305Dic3—81no—
306Dic3—81no—
307Dic3—81no—
308C23—121no—
309C23#94011yescharacteristic
310Q8—44no—
311Q8—81no—
312Q8—81no—
313Q8—122no—
314Q8—121no—
315Q8—122no—
316Q8—121no—
317D4—121no—
318D4—122no—
319D4—121no—
320D4—241no—
321D4—241no—
322D4—241no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—82no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4—121no—
332C2×C4—121no—
333C2×C4—121no—
334C2×C4—242no—
335C2×C4#88211yescharacteristic
336C2×C4#88211yescharacteristic
337C2×C4#88411yescharacteristic
338C2×C4#88411yescharacteristic
339C2×C4#91612yes—
340C2×C4#91812yes—
341C2×C4#93011yescharacteristic
342C2×C4#93711yescharacteristic
343C8—42no—
344C8—122no—
345C8—122no—
346C2×C3—21no—
347C2×C3—21no—
348C2×C3#200611yescharacteristic
349C2×C3#200911yescharacteristic
350C2×C3#201511yescharacteristic
351D3—161no—
352C22—21no—
353C22—21no—
354C22—241no—
355C22—241no—
356C22—241no—
357C22#496011yescharacteristic
358C22#516811yescharacteristic
359C22#519011yescharacteristic
360C4—21no—
361C4—21no—
362C4—21no—
363C4—82no—
364C4—82no—
365C4—241no—
366C4—241no—
367C4—241no—
368C4#479812yes—
369C4#480812yes—
370C4#506211yescharacteristic
371C4#507711yescharacteristic
372C3#1020211yescharacteristic, complemented, Sylow
373C2—21no—
374C2—21no—
375C2—481no—
376C2#6966411yescharacteristic
377C2#7027811yescharacteristic
378C2#7035111yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1981C2
socleC22×C3#392
prime radical#5477C22
hypercenter#1981D3
hypocenterC3#10202
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1981

Complements

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

NH ≅ G/Ntype
1. #83224C2semidirect
4. C3×#1981C2semidirect
5. #69493C2semidirect
7. #67706C2semidirect
36. #1981D3semidirect
372. C3#10202semidirect

Maximal subgroups

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

subgroupindex
#832242
#831992
#790272
C3×#19812
#694932
#694692
#677062
#102023