Finite Group Databasebeta

← #10696084

Subgroups — #10696084

Order 768.

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

normal subgroups only

This group has 492 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×#2634C211yesmaximal, characteristic, complemented
2#83012C211yesmaximal, characteristic
3#82444C211yesmaximal, characteristic
4#71736C211yesmaximal, characteristic, complemented
5#71623C211yesmaximal, characteristic, complemented
6#69564C211yesmaximal, characteristic
7#67633C211yesmaximal, characteristic, complemented
8#10021—31nomaximal, Sylow
9C2×C3×#125C2211yescharacteristic
10C2×C3×C4×Q8C2211yescharacteristic
11#5477C2211yescharacteristic
12C3×#400—21no—
13C3×#400—21no—
14C3×#398—21no—
15C3×#395—21no—
16C3×#387—22no—
17C3×#383C2211yescharacteristic
18#5227—21no—
19#5069—21no—
20#5068—22no—
21#5067—21no—
22#5027—21no—
23#5026—22no—
24#5021—21no—
25#4967—21no—
26#4966—21no—
27#4966—21no—
28#4926C2211yescharacteristic
29#4926C2211yescharacteristic
30C4×#208C2211yescharacteristic
31#4543—22no—
32#4535—22no—
33#4451—22no—
34#3142—31no—
35#3112—31no—
36#2634D311yescharacteristic, complemented
37#1534—31no—
38#1514—31no—
39#1491—31no—
40#1288—31no—
41C22×C3×Q8D411yescharacteristic
42C2×#235—21no—
43C3×#125—21no—
44C3×#125—21no—
45C3×#125—21no—
46C3×#125—21no—
47C3×C4×Q8—21no—
48C3×C4×Q8—21no—
49C3×C4×Q8—21no—
50C3×C4×Q8—21no—
51C2×C3×C4⋊C4—22no—
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4—21no—
55C2×C3×C4⋊C4D411yescharacteristic
56C2×C3×C4⋊C4D411yescharacteristic
57C2×C3×C4⋊C4D411yescharacteristic
58C2×C3×C42—21no—
59C2×C3×C42C2311yescharacteristic
60C2×#215—22no—
61#930—21no—
62#930—21no—
63C2×#208—21no—
64#875—21no—
65#874D411yescharacteristic
66C4×D12—21no—
67C4×D12—21no—
68#869D411yescharacteristic
69C3×#95—22no—
70C3×#95—22no—
71C3×#95—22no—
72C3×#95—21no—
73C3×#95—21no—
74C3×#95D412yes—
75#835—22no—
76#830—21no—
77#830—21no—
78#810—22no—
79#810—22no—
80#809—22no—
81#809—22no—
82#809—22no—
83#807—22no—
84C4⋊(C3⋊C8)—21no—
85C4⋊(C3⋊C8)—21no—
86#803—21no—
87#803—21no—
88#803D412yes—
89C4×C3⋊C8D412yes—
90#532—31no—
91C2×#125C2×D311yescharacteristic
92C2×C4×Q8C2×D311yescharacteristic
93#491—61no—
94#490—61no—
95#488—62no—
96#480—61no—
97#477—61no—
98#460—61no—
99#448—61no—
100#440—62no—
101#432—31no—
102#432—31no—
103#417—61no—
104#417—61no—
105C4×M16—31no—
106#400—21no—
107#400—21no—
108#398—21no—
109#395—21no—
110#387—22no—
111#383C2×D311yescharacteristic
112#330—62no—
113#327—62no—
114Q8⋊C8—62no—
115C2×C3×Q8—21no—
116C2×C3×Q8—21no—
117C2×C3×Q8—42no—
118C22×C3×C4—21no—
119C22×C3×C4—22no—
120C22×C3×C4—21no—
121C22×C3×C4—21no—
122C22×C3×C4—21no—
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C2×#67—41no—
127#235—44no—
128C2×D12—21no—
129C2×D12—21no—
130C2×D12—21no—
131C2×C4×D3—41no—
132C2×Dic6—21no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—44no—
140C3×C4⋊C4—41no—
141C3×C4⋊C4—41no—
142C3×C4⋊C4—41no—
143C3×C4⋊C4—41no—
144C3×C4⋊C4—41no—
145C3×C4⋊C4—41no—
146C3×C42—41no—
147C3×C42—41no—
148C3×C42C2×D411yescharacteristic
149C3×C42C2×D411yescharacteristic
150C3×C42C2×D411yescharacteristic
151C3×C42#4111yescharacteristic
152#215—44no—
153(C2×D3)⋊C4—41no—
154(C2×D3)⋊C4—41no—
155C4⋊Dic3—21no—
156C4⋊Dic3—21no—
157#208—42no—
158C2×C3⋊C8—22no—
159C2×C3⋊C8—22no—
160C2×C3⋊C8—41no—
161C2×#41—61no—
162C22×Q8#6711yescharacteristic
163C2×SD16—62no—
164C2×M16—61no—
165C4⋊Q8—31no—
166#127—31no—
167#125—21no—
168#125—21no—
169#125—21no—
170#125—21no—
171#124—61no—
172#121—61no—
173#121—61no—
174C4×Q8—21no—
175C4×Q8—21no—
176C4×Q8—21no—
177C4×Q8—21no—
178C4×D4—61no—
179C4×D4—61no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4#6711yescharacteristic
185C2×C4⋊C4#6711yescharacteristic
186C2×C4⋊C4#6711yescharacteristic
187C2×C42—21no—
188C2×C42C22×D311yescharacteristic
189#107—62no—
190C4⋊C8—61no—
191C4⋊C8—61no—
192#103—62no—
193#103—62no—
194#103—62no—
195#102—62no—
196#102—62no—
197#102—62no—
198C22⋊C8—61no—
199C22⋊C8—61no—
200#97—32no—
201#97—61no—
202#97—61no—
203C4×C8—32no—
204#95—22no—
205#95—22no—
206#95—22no—
207#95—21no—
208#95—21no—
209#95#6712yes—
210C23×C3#12011yescharacteristic
211C22×D3—41no—
212C3×Q8—22no—
213C3×Q8—22no—
214C3×Q8—41no—
215C3×Q8—41no—
216C2×C3×C4—22no—
217C2×C3×C4—22no—
218C2×C3×C4—41no—
219C2×C3×C4—41no—
220C2×C3×C4—41no—
221C2×C3×C4—41no—
222C2×C3×C4—42no—
223C2×C3×C4—42no—
224C2×C3×C4—42no—
225C2×C3×C4—42no—
226C2×C3×C4—41no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4—42no—
230C2×C3×C4—42no—
231C2×C3×C4#12011yescharacteristic
232C2×C3×C4#12011yescharacteristic
233C2×C3×C4#12111yescharacteristic
234C2×C3×C4#12111yescharacteristic
235C2×C3×C4#12111yescharacteristic
236C2×C3×C4#12711yescharacteristic
237#67—82no—
238C2×Dic3—41no—
239D12—42no—
240D12—42no—
241D12—81no—
242C4×D3—82no—
243Dic6—42no—
244C3⋊C8—42no—
245C3⋊C8—42no—
246C3⋊C8—81no—
247#41—124no—
248C2×Q8—21no—
249C2×Q8—21no—
250C2×Q8—42no—
251C2×Q8—61no—
252C2×D4—61no—
253C2×D4—61no—
254C2×D4—61no—
255C2×D4—121no—
256C22×C4—21no—
257C22×C4—22no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—121no—
262C22×C4C2×#6711yescharacteristic
263C22×C4C2×#6711yescharacteristic
264C22×C4C2×#6711yescharacteristic
265SD16—124no—
266M16—122no—
267C2×C8—62no—
268C2×C8—62no—
269C2×C8—121no—
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—44no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—61no—
284C4⋊C4—61no—
285C22⋊C4—121no—
286C22⋊C4—121no—
287C42—41no—
288C42—41no—
289C42D3×D411yescharacteristic
290C42D3×D411yescharacteristic
291C42D3×D411yescharacteristic
292C42#23711yescharacteristic
293C22×C3—21no—
294C22×C3—21no—
295C22×C3#39211yescharacteristic
296C22×C3#45412yes—
297C2×D3—81no—
298C2×D3—81no—
299C2×D3—81no—
300C3×C4—21no—
301C3×C4—21no—
302C3×C4—21no—
303C3×C4—21no—
304C3×C4—21no—
305C3×C4—21no—
306C3×C4—42no—
307C3×C4—82no—
308C3×C4—81no—
309C3×C4—81no—
310C3×C4—81no—
311Dic3—81no—
312C23—121no—
313C23#95311yescharacteristic
314Q8—22no—
315Q8—22no—
316Q8—41no—
317Q8—41no—
318Q8—122no—
319D4—122no—
320D4—122no—
321D4—241no—
322D4—242no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—121no—
339C2×C4—242no—
340C2×C4#93711yescharacteristic
341C2×C4#93711yescharacteristic
342C2×C4#93811yescharacteristic
343C2×C4#93911yescharacteristic
344C2×C4#93911yescharacteristic
345C2×C4#94011yescharacteristic
346C8—122no—
347C8—122no—
348C8—241no—
349C2×C3—22no—
350C2×C3#200311yescharacteristic
351C2×C3#200911yescharacteristic
352C2×C3#201011yescharacteristic
353D3—161no—
354C22—21no—
355C22—21no—
356C22—241no—
357C22—241no—
358C22—241no—
359C22#522311yescharacteristic
360C22#524512yes—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—42no—
368C4—82no—
369C4—81no—
370C4—81no—
371C4—81no—
372C4—241no—
373C3#1002111yescharacteristic, complemented, Sylow
374C2—22no—
375C2—481no—
376C2#7171511yescharacteristic
377C2#7186711yescharacteristic
378C2#7186911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#2634C2
socleC22×C3#392
prime radical#5477C22
hypercenter#2634D3
hypocenterC3#10021
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2634

Complements

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

NH ≅ G/Ntype
1. C3×#2634C2semidirect
4. #71736C2semidirect
5. #71623C2semidirect
7. #67633C2semidirect
36. #2634D3semidirect
373. C3#10021semidirect

Maximal subgroups

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

subgroupindex
C3×#26342
#830122
#824442
#717362
#716232
#695642
#676332
#100213