Finite Group Databasebeta

← #10697298

Subgroups — #10697298

Order 768.

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

normal subgroups only

This group has 508 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#83222C211yesmaximal, characteristic, complemented
2#83199C211yesmaximal, characteristic
3#79027C211yesmaximal, characteristic
4C3×#1987C211yesmaximal, characteristic, complemented
5#69493C211yesmaximal, characteristic, complemented
6#69469C211yesmaximal, characteristic
7#67706C211yesmaximal, characteristic, complemented
8#10198—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10#5477C2211yescharacteristic
11#5470C2211yescharacteristic
12C2×C3×C4⋊C8C2211yescharacteristic
13C3×C4×C4⋊C4C2211yescharacteristic
14#5228—22no—
15#5114—22no—
16#5093—21no—
17#5092—21no—
18#5090—22no—
19#5089—22no—
20#5088—21no—
21#5079—21no—
22#4966—22no—
23#4925C2211yescharacteristic
24C4×#208C2211yescharacteristic
25#4851—21no—
26#4848—21no—
27#4842—21no—
28#4841—21no—
29C3×#340—22no—
30#4523—22no—
31#4451—22no—
32#3246—31no—
33#3209—31no—
34#3141—31no—
35#1987D311yescharacteristic, complemented
36#1986—31no—
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
51C2×#216—24no—
52#930—21no—
53#930—21no—
54#926—21no—
55#924—21no—
56#924—21no—
57C2×#208D412yes—
58C2×#204—22no—
59#875—21no—
60#874D411yescharacteristic
61C4×D12—21no—
62C4×D12—21no—
63#870—21no—
64#869D411yescharacteristic
65#869D411yescharacteristic
66#869D411yescharacteristic
67C4×Dic6—21no—
68C4×Dic6—21no—
69C3×C4⋊C8—21no—
70C3×C4⋊C8—21no—
71C3×#95—21no—
72#835—24no—
73#830D412yes—
74#821—22no—
75#817—22no—
76#816—22no—
77#810—24no—
78#809—24no—
79C4⋊(C3⋊C8)—21no—
80#803—21no—
81C4×C3⋊C8—21no—
82C4×C3⋊C8—21no—
83#533—31no—
84#532—31no—
85C2×C4⋊Q8C2×D311yescharacteristic
86#494—61no—
87#489—62no—
88#487—61no—
89#487—61no—
90#486—61no—
91#476—61no—
92#475—61no—
93#465—62no—
94#462—62no—
95#461—61no—
96#460—61no—
97#451—62no—
98#431—31no—
99C2×C4⋊C8C2×D311yescharacteristic
100#417—62no—
101C4×M16—31no—
102C4×C4⋊C4C2×D311yescharacteristic
103#340—22no—
104#339—62no—
105#327—62no—
106C2×C3×Q8—22no—
107C2×C3×Q8—41no—
108C2×C3×Q8—82no—
109C22×C3×C4—41no—
110C22×C3×C4—41no—
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4C2×D411yescharacteristic
114C2×#67—41no—
115#235—42no—
116#235—42no—
117C2×D12—21no—
118C2×D12—21no—
119C2×D12—21no—
120C2×C4×D3—41no—
121C2×Dic6—21no—
122C2×Dic6—21no—
123C2×Dic6—21no—
124C2×C3×C8—22no—
125C2×C3×C8—42no—
126C3×C4⋊C4—22no—
127C3×C4⋊C4—22no—
128C3×C4⋊C4—22no—
129C3×C4⋊C4—41no—
130C3×C4⋊C4—41no—
131C3×C4⋊C4—41no—
132C3×C42—42no—
133C3×C42C2×D411yescharacteristic
134C3×C42C2×D411yescharacteristic
135C3×C42C2×D411yescharacteristic
136C3×C42#4111yescharacteristic
137#217—41no—
138#216—84no—
139(C2×D3)⋊C4—41no—
140(C2×D3)⋊C4—41no—
141C4⋊Dic3—21no—
142C4⋊Dic3—21no—
143C4⋊Dic3—21no—
144C4⋊Dic3—21no—
145C4⋊Dic3—21no—
146C4⋊Dic3—21no—
147Dic3⋊C4—41no—
148Dic3⋊C4—41no—
149C4×Dic3—41no—
150#208—22no—
151#208—22no—
152C2×C3⋊C8—22no—
153C2×C3⋊C8—22no—
154#204—82no—
155C2×#41—61no—
156C22×Q8—21no—
157C2×Q16—64no—
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#106—62no—
186C4⋊C8—21no—
187C4⋊C8—21no—
188C4⋊C8—61no—
189#103—64no—
190#103—62no—
191#103—64no—
192#102—64no—
193#102—62no—
194C22⋊C8—32no—
195#97—61no—
196C4×C8—61no—
197C4×C8—61no—
198#95—21no—
199C23×C3#12711yescharacteristic
200C22×D3—41no—
201C3×Q8—44no—
202C3×Q8—82no—
203C2×C3×C4—21no—
204C2×C3×C4—21no—
205C2×C3×C4—41no—
206C2×C3×C4—42no—
207C2×C3×C4—42no—
208C2×C3×C4—81no—
209C2×C3×C4—82no—
210C2×C3×C4—81no—
211C2×C3×C4#12011yescharacteristic
212C2×C3×C4#12011yescharacteristic
213C2×C3×C4#12011yescharacteristic
214C2×C3×C4#12111yescharacteristic
215C2×C3×C4#12111yescharacteristic
216C2×C3×C4#12111yescharacteristic
217C2×C3×C4#13512yes—
218C2×C3×C4#13712yes—
219#67—81no—
220#67—81no—
221C2×Dic3—41no—
222C2×Dic3—41no—
223C2×Dic3—41no—
224D12—42no—
225D12—41no—
226D12—41no—
227D12—81no—
228C4×D3—82no—
229Dic6—42no—
230Dic6—42no—
231Dic6—41no—
232Dic6—41no—
233C3×C8—42no—
234C3⋊C8—42no—
235C3⋊C8—42no—
236#41—122no—
237#41—122no—
238C2×Q8—22no—
239C2×Q8—41no—
240C2×Q8—61no—
241C2×Q8—61no—
242C2×Q8—61no—
243C2×Q8—82no—
244C2×D4—61no—
245C2×D4—61no—
246C2×D4—61no—
247C2×D4—121no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—121no—
251C22×C4D3×D411yescharacteristic
252C22×C4D3×D411yescharacteristic
253C22×C4C2×#6711yescharacteristic
254Q16—244no—
255SD16—242no—
256M16—62no—
257M16—62no—
258C2×C8—22no—
259C2×C8—42no—
260C2×C8—62no—
261C2×C8—62no—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—22no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—61no—
269C4⋊C4—61no—
270C4⋊C4—61no—
271C4⋊C4—61no—
272C4⋊C4—61no—
273C4⋊C4—61no—
274C4⋊C4—121no—
275C4⋊C4—121no—
276C22⋊C4—121no—
277C22⋊C4—121no—
278C22⋊C4—121no—
279C42—42no—
280C42—121no—
281C42C2×D1211yescharacteristic
282C42#23511yescharacteristic
283C42D3×D411yescharacteristic
284C42D3×D411yescharacteristic
285C22×C3—21no—
286C22×C3—21no—
287C22×C3#39211yescharacteristic
288C22×C3#49511yescharacteristic
289C22×C3#49711yescharacteristic
290C2×D3—81no—
291C2×D3—81no—
292C2×D3—81no—
293C3×C4—21no—
294C3×C4—21no—
295C3×C4—21no—
296C3×C4—82no—
297C3×C4—82no—
298C3×C4#45212yes—
299C3×C4#45511yescharacteristic
300C3×C4#45611yescharacteristic
301C3×C4#46412yes—
302Dic3—81no—
303Dic3—81no—
304Dic3—81no—
305C23—121no—
306C23#94011yescharacteristic
307Q8—44no—
308Q8—82no—
309Q8—122no—
310Q8—121no—
311Q8—122no—
312Q8—121no—
313D4—121no—
314D4—122no—
315D4—121no—
316D4—241no—
317D4—241no—
318D4—241no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—41no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—82no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—121no—
328C2×C4—121no—
329C2×C4—121no—
330C2×C4—242no—
331C2×C4#88211yescharacteristic
332C2×C4#88211yescharacteristic
333C2×C4#88411yescharacteristic
334C2×C4#88411yescharacteristic
335C2×C4#91612yes—
336C2×C4#91812yes—
337C2×C4#93011yescharacteristic
338C2×C4#93711yescharacteristic
339C8—42no—
340C8—122no—
341C8—122no—
342C2×C3—21no—
343C2×C3—21no—
344C2×C3#200711yescharacteristic
345C2×C3#200811yescharacteristic
346C2×C3#201511yescharacteristic
347D3—161no—
348C22—21no—
349C22—21no—
350C22—241no—
351C22—241no—
352C22—241no—
353C22#496011yescharacteristic
354C22#516811yescharacteristic
355C22#519011yescharacteristic
356C4—21no—
357C4—21no—
358C4—21no—
359C4—82no—
360C4—82no—
361C4—241no—
362C4—241no—
363C4—241no—
364C4#479812yes—
365C4#480812yes—
366C4#506111yescharacteristic
367C4#509511yescharacteristic
368C3#1019811yescharacteristic, complemented, Sylow
369C2—21no—
370C2—21no—
371C2—481no—
372C2#6969411yescharacteristic
373C2#7027811yescharacteristic
374C2#7033111yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1987C2
socleC22×C3#392
prime radical#5477C22
hypercenter#1987D3
hypocenterC3#10198
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1987

Complements

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

NH ≅ G/Ntype
1. #83222C2semidirect
4. C3×#1987C2semidirect
5. #69493C2semidirect
7. #67706C2semidirect
35. #1987D3semidirect
368. C3#10198semidirect

Maximal subgroups

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

subgroupindex
#832222
#831992
#790272
C3×#19872
#694932
#694692
#677062
#101983