Finite Group Databasebeta

← #10692495

Subgroups — #10692495

Order 768.

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

normal subgroups only

This group has 462 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
1C3×#2336C211yesmaximal, characteristic, complemented
2#83019C211yesmaximal, characteristic
3#71735C211yesmaximal, characteristic, complemented
4#69564C211yesmaximal, characteristic
5#69492C211yesmaximal, characteristic
6#67633C211yesmaximal, characteristic, complemented
7#67621C211yesmaximal, characteristic, complemented
8#9382—31nomaximal, Sylow
9C2×C3×#125C2211yescharacteristic
10#5477C2211yescharacteristic
11C3×#389—21no—
12C3×#387—21no—
13C3×#387—21no—
14C3×#387—21no—
15C3×#387—21no—
16C3×#383C2211yescharacteristic
17C3×C4×C4⋊C4—21no—
18C3×C4×C4⋊C4C2211yescharacteristic
19#5067C412yescomplemented
20#5066—22no—
21#5064C412yescomplemented
22#4968—21no—
23#4967—21no—
24C2×#805—21no—
25C2×#805—21no—
26#4926C2211yescharacteristic
27#4926C2211yescharacteristic
28C4×#208C2211yescharacteristic
29#4543—22no—
30#4451—21no—
31#4451—21no—
32#3234—31no—
33#2336D311yescharacteristic, complemented
34#1980—31no—
35#1528—31no—
36#1514—31no—
37#1288—31no—
38#1285—31no—
39C2×#235—21no—
40C3×#125—21no—
41C3×#125—21no—
42C3×#125C2×C412yes—
43C3×#125D412yes—
44C2×C3×C4⋊C4—21no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4D411yescharacteristic
50C2×C3×C4⋊C4D411yescharacteristic
51C2×C3×C42—21no—
52C2×C3×C42—21no—
53C2×C3×C42D411yescharacteristic
54C2×C3×C42D411yescharacteristic
55C2×C3×C42C2311yescharacteristic
56#930—21no—
57#930—21no—
58C2×#208—21no—
59#875—21no—
60#874C2×C411yescharacteristic
61C4×D12—21no—
62C4×D12—21no—
63#869C2×C411yescharacteristic
64C3×#95—21no—
65C3×#95—21no—
66C3×#95—21no—
67C3×#95—21no—
68C3×#95—21no—
69C3×#95—21no—
70C3×#95—21no—
71C3×#95D411yescharacteristic
72C3×#95D411yescharacteristic
73#830—21no—
74#830—21no—
75#809—22no—
76#809—22no—
77#808—22no—
78#807—22no—
79#805—42no—
80#805—42no—
81C4⋊(C3⋊C8)—21no—
82C4⋊(C3⋊C8)—21no—
83#803—21no—
84#803—21no—
85#803C2×C412yes—
86C4×C3⋊C8—21no—
87#532—31no—
88C2×#125C2×D311yescharacteristic
89#501—32no—
90#490—32no—
91#481—61no—
92#480—61no—
93#478—62no—
94#432—31no—
95#432—31no—
96C2×#104—61no—
97C2×#104—61no—
98C4×M16—31no—
99#389—21no—
100#387—21no—
101#387—21no—
102#387—21no—
103#387—21no—
104#383C2×D311yescharacteristic
105C4×C4⋊C4—21no—
106C4×C4⋊C4C2×D311yescharacteristic
107#330—62no—
108#327—61no—
109#327—61no—
110C22×C3×C4—21no—
111C22×C3×C4—21no—
112C22×C3×C4—21no—
113C22×C3×C4—21no—
114C22×C3×C4—21no—
115C22×C3×C4—21no—
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C2×#67—41no—
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—
129C3×C4⋊C4—22no—
130C3×C4⋊C4—22no—
131C3×C4⋊C4—22no—
132C3×C4⋊C4—22no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—42no—
136C3×C4⋊C4—42no—
137C3×C4⋊C4—81no—
138C3×C4⋊C4C22⋊C412yes—
139C3×C4⋊C4C22⋊C412yes—
140C3×C42—41no—
141C3×C42—41no—
142C3×C42—42no—
143C3×C42—81no—
144C3×C42C22×C411yescharacteristic
145C3×C42C2×D411yescharacteristic
146C3×C42#4111yescharacteristic
147C3×C42#4111yescharacteristic
148(C2×D3)⋊C4—41no—
149(C2×D3)⋊C4—41no—
150C4⋊Dic3—21no—
151C4⋊Dic3—21no—
152#208—22no—
153#208—41no—
154C2×C3⋊C8—22no—
155C2×C3⋊C8—41no—
156C2×C3⋊C8—41no—
157C2×#41—61no—
158C2×M16—61no—
159C4⋊Q8—31no—
160#127—31no—
161#125—21no—
162#125—21no—
163#125C4×D312yes—
164#125D1212yes—
165#124—61no—
166#121—61no—
167#121—61no—
168C4×D4—61no—
169C4×D4—61no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4#6711yescharacteristic
176C2×C4⋊C4#6711yescharacteristic
177C2×C42—21no—
178C2×C42—21no—
179C2×C42#6711yescharacteristic
180C2×C42#6711yescharacteristic
181C2×C42C22×D311yescharacteristic
182#107—62no—
183#106—62no—
184C4⋊C8—61no—
185C4⋊C8—61no—
186#104—122no—
187#104—122no—
188#102—62no—
189#102—62no—
190C22⋊C8—61no—
191C22⋊C8—61no—
192#97—32no—
193#97—61no—
194#97—61no—
195C4×C8—61no—
196#95—21no—
197#95—21no—
198#95—21no—
199#95—21no—
200#95—21no—
201#95—21no—
202#95—21no—
203#95#6711yescharacteristic
204#95#6711yescharacteristic
205C23×C3#12011yescharacteristic
206C22×D3—41no—
207C2×C3×C4—21no—
208C2×C3×C4—21no—
209C2×C3×C4—22no—
210C2×C3×C4—22no—
211C2×C3×C4—22no—
212C2×C3×C4—41no—
213C2×C3×C4—41no—
214C2×C3×C4—41no—
215C2×C3×C4—41no—
216C2×C3×C4—41no—
217C2×C3×C4—41no—
218C2×C3×C4—41no—
219C2×C3×C4—41no—
220C2×C3×C4—41no—
221C2×C3×C4—41no—
222C2×C3×C4—41no—
223C2×C3×C4—41no—
224C2×C3×C4—41no—
225C2×C3×C4—41no—
226C2×C3×C4—41no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4C2×C22⋊C411yescharacteristic
230C2×C3×C4C4×D411yescharacteristic
231C2×C3×C4C4×D411yescharacteristic
232C2×C3×C4#12111yescharacteristic
233C2×C3×C4#12111yescharacteristic
234C2×C3×C4#12311yescharacteristic
235#67—81no—
236#67—81no—
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—42no—
248C3⋊C8—81no—
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—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—121no—
265C22×C4C2×#6711yescharacteristic
266C22×C4C2×#6711yescharacteristic
267C22×C4C2×#6711yescharacteristic
268M16—62no—
269M16—121no—
270C2×C8—62no—
271C2×C8—121no—
272C2×C8—121no—
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—61no—
282C4⋊C4—61no—
283C4⋊C4—81no—
284C4⋊C4(C2×D3)⋊C412yes—
285C4⋊C4(C2×D3)⋊C412yes—
286C22⋊C4—121no—
287C22⋊C4—121no—
288C42—41no—
289C42—41no—
290C42—42no—
291C42—81no—
292C42C2×C4×D311yescharacteristic
293C42C2×D1211yescharacteristic
294C42#23511yescharacteristic
295C42#23511yescharacteristic
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—42no—
312C3×C4—42no—
313C3×C4—81no—
314C3×C4—81no—
315C3×C4—81no—
316Dic3—81no—
317C23—121no—
318C23#95311yescharacteristic
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—22no—
331C2×C4—22no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—121no—
350C2×C4—241no—
351C2×C4—241no—
352C2×C4C2×(C2×D3)⋊C411yescharacteristic
353C2×C4C4×#6711yescharacteristic
354C2×C4C4×#6711yescharacteristic
355C2×C4#92911yescharacteristic
356C2×C4#93011yescharacteristic
357C2×C4#93011yescharacteristic
358C8—122no—
359C8—122no—
360C8—241no—
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#521311yescharacteristic
373C22#524111yescharacteristic
374C22#524511yescharacteristic
375C4—21no—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—21no—
380C4—21no—
381C4—42no—
382C4—42no—
383C4—42no—
384C4—81no—
385C4—81no—
386C4—81no—
387C4—241no—
388C3#938211yescharacteristic, complemented, Sylow
389C2—21no—
390C2—21no—
391C2—481no—
392C2#7164511yescharacteristic
393C2#7166311yescharacteristic
394C2#7166711yescharacteristic
subgroupG/H
centerC22#5213
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2336C2
socleC22×C3#386
prime radical#5477C22
cosocleC2×C3×C42C23
hypercenter#2336D3
hypocenterC3#9382
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2336

Complements

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

NH ≅ G/Ntype
1. C3×#2336C2semidirect
3. #71735C2semidirect
6. #67633C2semidirect
7. #67621C2semidirect
19. #5067C4semidirect
21. #5064C4semidirect
33. #2336D3semidirect
388. C3#9382semidirect

Maximal subgroups

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

subgroupindex
C3×#23362
#830192
#717352
#695642
#694922
#676332
#676212
#93823