Finite Group Databasebeta

← #10692494

Subgroups — #10692494

Order 768.

462 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 89 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#79318C211yesmaximal, characteristic
2#73630C211yesmaximal, characteristic, complemented
3#70835C211yesmaximal, characteristic
4#70812C211yesmaximal, characteristic, complemented
5#69492C211yesmaximal, characteristic
6C3×#1285C211yesmaximal, characteristic, complemented
7#67752C211yesmaximal, characteristic, complemented
8#9382—31nomaximal, Sylow
9#5477C2211yescharacteristic
10C2×#870C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C3×#432C2211yescharacteristic
13C3×C4×C4⋊C4C2211yescharacteristic
14#4978—21no—
15#4976—21no—
16#4937C2211yescharacteristic
17#4933—21no—
18C4×Dic3⋊C4—21no—
19C2×#805—21no—
20C2×#805—21no—
21C4×#208C2211yescharacteristic
22#4847—22no—
23#4736—21no—
24#4735—21no—
25#4713C412yescomplemented
26#4702C412yescomplemented
27#4654—21no—
28#4653—21no—
29#4470—21no—
30#4470—21no—
31#4465—22no—
32#3234—31no—
33#2336—31no—
34#1980—31no—
35#1528—31no—
36#1514—31no—
37#1288—31no—
38#1285D311yescharacteristic, complemented
39C2×#235—21no—
40C2×C3×C4⋊C4—21no—
41C2×C3×C42D411yescharacteristic
42C2×C3×C42D411yescharacteristic
43C2×C3×C42C2311yescharacteristic
44#930—21no—
45#930—21no—
46C2×C4⋊Dic3D411yescharacteristic
47C2×C4⋊Dic3D411yescharacteristic
48C2×Dic3⋊C4—21no—
49C2×Dic3⋊C4—21no—
50C2×Dic3⋊C4—21no—
51C2×Dic3⋊C4—21no—
52C2×C4×Dic3—21no—
53C2×C4×Dic3—21no—
54C2×#208—21no—
55#875—21no—
56#874C2×C411yescharacteristic
57C4×D12—21no—
58C4×D12—21no—
59#870—21no—
60#870—21no—
61#870C2×C412yes—
62#870D412yes—
63#869C2×C411yescharacteristic
64C3×C4⋊C8—21no—
65C3×C4⋊C8—21no—
66C3×C22⋊C8—21no—
67C3×C22⋊C8—21no—
68C3×#97—21no—
69C3×#97C2×C412yes—
70C3×#95—21no—
71C3⋊#95—21no—
72C3⋊#95—21no—
73C3⋊#95—21no—
74C3⋊#95—21no—
75C3⋊#95—21no—
76C3⋊#95—21no—
77C3⋊#95D411yescharacteristic
78C3⋊#95D411yescharacteristic
79#821—22no—
80#821—22no—
81#818—22no—
82#817—22no—
83#805—42no—
84#805—42no—
85#803—21no—
86C4×C3⋊C8—21no—
87#532—31no—
88C2×#125—31no—
89#501—32no—
90#490—32no—
91#481—61no—
92#480—61no—
93#478—62no—
94#432C2×D311yescharacteristic
95#432C2×D311yescharacteristic
96C2×#104—61no—
97C2×#104—61no—
98C4×M16—31no—
99#389—61no—
100#387—61no—
101#387—61no—
102#387—61no—
103#387—61no—
104#383—31no—
105C4×C4⋊C4—61no—
106C4×C4⋊C4C2×D311yescharacteristic
107#330—62no—
108#327—61no—
109#327—61no—
110C22×C3×C4—21no—
111C22×C3×C4—21no—
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C2×#67—41no—
116C22×Dic3—21no—
117C22×Dic3—21no—
118C22×Dic3—21no—
119C22×Dic3—21no—
120#235—41no—
121#235—41no—
122#235—41no—
123#235—41no—
124C2×D12—21no—
125C2×D12—21no—
126C2×D12—21no—
127C2×C4×D3—41no—
128C2×Dic6—21no—
129C2×C3×C8—22no—
130C2×C3×C8—41no—
131C3×C4⋊C4—42no—
132C3×C42—41no—
133C3×C42—41no—
134C3×C42C22×C411yescharacteristic
135C3×C42C2×D411yescharacteristic
136C3×C42#4111yescharacteristic
137C3×C42#4111yescharacteristic
138(C2×D3)⋊C4—41no—
139(C2×D3)⋊C4—41no—
140C4⋊Dic3—21no—
141C4⋊Dic3—21no—
142C4⋊Dic3—21no—
143C4⋊Dic3—21no—
144C4⋊Dic3C22⋊C412yes—
145C4⋊Dic3C22⋊C412yes—
146Dic3⋊C4—22no—
147Dic3⋊C4—22no—
148Dic3⋊C4—22no—
149Dic3⋊C4—22no—
150Dic3⋊C4—42no—
151Dic3⋊C4—81no—
152C4×Dic3—42no—
153C4×Dic3—81no—
154#208—22no—
155#208—41no—
156C2×C3⋊C8—41no—
157C2×#41—61no—
158C2×M16—61no—
159C4⋊Q8—31no—
160#127—31no—
161#125—32no—
162#125—32no—
163#125—61no—
164#125—61no—
165#124—61no—
166#121—61no—
167#121—61no—
168C4×D4—61no—
169C4×D4—61no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—31no—
172C2×C4⋊C4—31no—
173C2×C4⋊C4—61no—
174C2×C4⋊C4—61no—
175C2×C4⋊C4—61no—
176C2×C4⋊C4—61no—
177C2×C42—61no—
178C2×C42—61no—
179C2×C42D1211yescharacteristic
180C2×C42D1211yescharacteristic
181C2×C42C22×D311yescharacteristic
182#107—62no—
183#106—62no—
184C4⋊C8—21no—
185C4⋊C8—21no—
186#104—122no—
187#104—122no—
188#102—62no—
189#102—62no—
190C22⋊C8—21no—
191C22⋊C8—21no—
192#97—21no—
193#97—61no—
194#97C4×D312yes—
195C4×C8—61no—
196#95—21no—
197#95—31no—
198#95—31no—
199#95—61no—
200#95—61no—
201#95—61no—
202#95—61no—
203#95—61no—
204#95—61no—
205C23×C3#12011yescharacteristic
206C22×D3—41no—
207C2×C3×C4—21no—
208C2×C3×C4—21no—
209C2×C3×C4—22no—
210C2×C3×C4—41no—
211C2×C3×C4—41no—
212C2×C3×C4—41no—
213C2×C3×C4—41no—
214C2×C3×C4—41no—
215C2×C3×C4—41no—
216C2×C3×C4—41no—
217C2×C3×C4C2×C22⋊C411yescharacteristic
218C2×C3×C4C4×D411yescharacteristic
219C2×C3×C4C4×D411yescharacteristic
220C2×C3×C4#12111yescharacteristic
221C2×C3×C4#12111yescharacteristic
222C2×C3×C4#12311yescharacteristic
223#67—81no—
224#67—81no—
225C2×Dic3—22no—
226C2×Dic3—22no—
227C2×Dic3—41no—
228C2×Dic3—41no—
229C2×Dic3—41no—
230C2×Dic3—41no—
231C2×Dic3—41no—
232C2×Dic3—41no—
233C2×Dic3—41no—
234C2×Dic3—41no—
235C2×Dic3—41no—
236C2×Dic3—41no—
237C2×Dic3—41no—
238D12—42no—
239D12—41no—
240D12—41no—
241D12—81no—
242C4×D3—81no—
243C4×D3—81no—
244Dic6—41no—
245Dic6—41no—
246C3×C8—42no—
247C3×C8—81no—
248C3⋊C8—42no—
249#41—121no—
250#41—121no—
251#41—121no—
252#41—121no—
253C2×Q8—61no—
254C2×D4—61no—
255C2×D4—61no—
256C2×D4—61no—
257C2×D4—121no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—61no—
261C22×C4—61no—
262C22×C4—61no—
263C22×C4—61no—
264C22×C4—121no—
265C22×C4C2×D1211yescharacteristic
266C22×C4D3×D411yescharacteristic
267C22×C4D3×D411yescharacteristic
268M16—62no—
269M16—121no—
270C2×C8—22no—
271C2×C8—41no—
272C2×C8—121no—
273C4⋊C4—32no—
274C4⋊C4—32no—
275C4⋊C4—42no—
276C4⋊C4—62no—
277C4⋊C4—62no—
278C4⋊C4—62no—
279C4⋊C4—61no—
280C4⋊C4—61no—
281C4⋊C4—62no—
282C4⋊C4—61no—
283C4⋊C4—61no—
284C4⋊C4—122no—
285C4⋊C4—241no—
286C22⋊C4—121no—
287C22⋊C4—121no—
288C42—41no—
289C42—41no—
290C42—122no—
291C42—241no—
292C42C2×C4×D311yescharacteristic
293C42#23511yescharacteristic
294C42D3×D411yescharacteristic
295C42#23911yescharacteristic
296C22×C3—21no—
297C22×C3—21no—
298C22×C3#38611yescharacteristic
299C22×C3#45311yescharacteristic
300C22×C3#45411yescharacteristic
301C2×D3—81no—
302C2×D3—81no—
303C2×D3—81no—
304C3×C4—21no—
305C3×C4—21no—
306C3×C4—21no—
307C3×C4—21no—
308C3×C4—21no—
309C3×C4—21no—
310C3×C4—42no—
311C3×C4—81no—
312Dic3—42no—
313Dic3—42no—
314Dic3—81no—
315Dic3—81no—
316Dic3—81no—
317C23—121no—
318C23#88211yescharacteristic
319Q8—121no—
320Q8—121no—
321D4—121no—
322D4—121no—
323D4—122no—
324D4—241no—
325D4—241no—
326D4—241no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—22no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—62no—
338C2×C4—62no—
339C2×C4—121no—
340C2×C4—121no—
341C2×C4—121no—
342C2×C4—121no—
343C2×C4—121no—
344C2×C4—121no—
345C2×C4—121no—
346C2×C4—121no—
347C2×C4—121no—
348C2×C4—121no—
349C2×C4—121no—
350C2×C4—241no—
351C2×C4—241no—
352C2×C4C4×D1211yescharacteristic
353C2×C4D3×C22⋊C411yescharacteristic
354C2×C4#88411yescharacteristic
355C2×C4#89311yescharacteristic
356C2×C4#89411yescharacteristic
357C2×C4#89511yescharacteristic
358C8—42no—
359C8—81no—
360C8—122no—
361C2×C3—21no—
362C2×C3—21no—
363C2×C3#188311yescharacteristic
364C2×C3#190611yescharacteristic
365C2×C3#191211yescharacteristic
366D3—161no—
367C22—21no—
368C22—21no—
369C22—241no—
370C22—241no—
371C22—241no—
372C22#466511yescharacteristic
373C22#474711yescharacteristic
374C22#475311yescharacteristic
375C4—21no—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—21no—
380C4—21no—
381C4—42no—
382C4—81no—
383C4—122no—
384C4—122no—
385C4—241no—
386C4—241no—
387C4—241no—
388C3#938211yescharacteristic, complemented, Sylow
389C2—21no—
390C2—21no—
391C2—481no—
392C2#6870811yescharacteristic
393C2#6877811yescharacteristic
394C2#6900511yescharacteristic
subgroupG/H
centerC22#4665
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1285C2
socleC22×C3#386
prime radical#5477C22
cosocleC2×C3×C42C23
hypercenter#1285D3
hypocenterC3#9382
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1285

Complements

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

NH ≅ G/Ntype
2. #73630C2semidirect
4. #70812C2semidirect
6. C3×#1285C2semidirect
7. #67752C2semidirect
25. #4713C4semidirect
26. #4702C4semidirect
38. #1285D3semidirect
388. C3#9382semidirect

Maximal subgroups

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

subgroupindex
#793182
#736302
#708352
#708122
#694922
C3×#12852
#677522
#93823