Finite Group Databasebeta

← #10696583

Subgroups — #10696583

Order 768.

684 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 85 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×#3215C211yesmaximal, characteristic, complemented
2#83336C211yesmaximal, characteristic, complemented
3#82979C211yesmaximal, characteristic
4#71836C211yesmaximal, characteristic, complemented
5#71718C211yesmaximal, characteristic, complemented
6#70818C211yesmaximal, characteristic, complemented
7#67748C211yesmaximal, characteristic, complemented
8#10096—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×#124C2211yescharacteristic
11C2×#947—21no—
12C2×#940—21no—
13C2×#874C2211yescharacteristic
14C3×#497—21no—
15C3×#489—21no—
16C3×#464—22no—
17C3×#451—21no—
18C3×#450—21no—
19C3×#432C2211yescharacteristic
20#5088—21no—
21#5088—21no—
22#5067—22no—
23#5001—22no—
24#4960—22no—
25#4960—22no—
26#4932C2211yescharacteristic
27#4926C2211yescharacteristic
28C4×#208C2211yescharacteristic
29#4540—22no—
30#4538—22no—
31#4485—22no—
32#3215D311yescharacteristic, complemented
33#3138—31no—
34#2657—31no—
35#1555—31no—
36#1527—31no—
37#1506—31no—
38#1300—31no—
39C22×C3×Q8D411yescharacteristic
40C22×C3×D4D411yescharacteristic
41C22×#67—24no—
42C22×D12—21no—
43C22×D12—21no—
44C22×D12D412yes—
45C2×C3×Q16—22no—
46C2×C3×SD16—22no—
47C3×C4⋊Q8D411yescharacteristic, complemented
48C3×C4⋊Q8D411yescharacteristic, complemented
49C3×#125—21no—
50C3×#124—21no—
51C3×#124—21no—
52C3×#124—21no—
53C3×#124—21no—
54C3×C22⋊Q8—21no—
55C3×C22⋊Q8—21no—
56C3×C4×Q8—21no—
57C3×C4×Q8—21no—
58C3×#117—21no—
59C2×C3×C22⋊C4—21no—
60C2×C3×C22⋊C4—21no—
61C2×C3×C42C2311yescharacteristic
62#947—44no—
63#940—44no—
64C2×(C2×D3)⋊C4—24no—
65C2×C4×Dic3—21no—
66C2×#208—21no—
67#874—21no—
68#874—21no—
69#874—21no—
70#874—21no—
71C3×C4⋊C8D412yescomplemented
72C3×#103—22no—
73C3×#102—22no—
74C3×C22⋊C8—21no—
75C3×#97—21no—
76C3⋊#95—21no—
77C3⋊#95D412yes—
78#830—21no—
79#809—24no—
80#809—24no—
81C4⋊(C3⋊C8)—21no—
82#803—21no—
83#803D412yescomplemented
84C4×C3⋊C8—21no—
85#533C2×D311yescharacteristic
86C2×#127—31no—
87C2×#127—61no—
88C2×#124—61no—
89C2×#124C2×D311yescharacteristic
90#497—21no—
91#490—62no—
92#489—21no—
93#486—61no—
94#486—61no—
95#464—22no—
96#451—21no—
97#450—21no—
98#432—31no—
99#432C2×D311yescharacteristic
100#418—62no—
101C4×M16—31no—
102#392—62no—
103#392—62no—
104#376—31no—
105#331—62no—
106#329—62no—
107#328—62no—
108C24×C3—21no—
109C23×D3—22no—
110C2×C3×Q8—21no—
111C2×C3×Q8—21no—
112C2×C3×Q8—21no—
113C2×C3×Q8—21no—
114C2×C3×Q8—42no—
115C2×C3×D4—21no—
116C2×C3×D4—21no—
117C2×C3×D4—42no—
118C22×C3×C4—21no—
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C2×#67—48no—
123C2×#67—84no—
124C2×#67—84no—
125C22×Dic3—22no—
126C2×D12—22no—
127C2×D12—22no—
128C2×D12—44no—
129C2×D12—42no—
130C2×D12—42no—
131C2×D12—41no—
132C2×D12—41no—
133C2×D12—41no—
134C2×D12—41no—
135C2×D12—82no—
136C3×Q16—82no—
137C3×SD16—82no—
138C2×C3×C8—22no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—21no—
142C3×C4⋊C4—21no—
143C3×C4⋊C4—41no—
144C3×C4⋊C4—41no—
145C3×C22⋊C4—41no—
146C3×C22⋊C4—41no—
147C3×C22⋊C4—41no—
148C3×C22⋊C4—41no—
149C3×C22⋊C4—41no—
150C3×C42—41no—
151C3×C42C2×D411yescharacteristic
152C3×C42C2×D411yescharacteristic
153C3×C42C2×D411yescharacteristic
154C3×C42#4111yescharacteristic
155(C2×D3)⋊C4—48no—
156C4×Dic3—24no—
157#208—42no—
158C2×C3⋊C8—22no—
159C2×C3⋊C8—41no—
160C22×Q8#6711yescharacteristic
161C22×D4—32no—
162C22×D4—64no—
163C22×D4—61no—
164C22×D4—61no—
165C22×D4#6711yescharacteristic
166C2×Q16—22no—
167C2×SD16—22no—
168C2×M16—61no—
169C4⋊Q8#6711yescharacteristic, complemented
170C4⋊Q8#6711yescharacteristic, complemented
171#127—61no—
172#127—61no—
173#127—61no—
174#127—61no—
175#127—124no—
176#125—21no—
177#124—21no—
178#124—21no—
179#124—21no—
180#124—21no—
181#124—124no—
182C22⋊Q8—21no—
183C22⋊Q8—21no—
184C4×Q8—21no—
185C4×Q8—21no—
186#117—21no—
187C2×C22⋊C4—21no—
188C2×C22⋊C4—21no—
189C2×C22⋊C4—64no—
190C2×C42—61no—
191C2×C42C22×D311yescharacteristic
192C4⋊C8—61no—
193C4⋊C8#6712yescomplemented
194#103—22no—
195#102—22no—
196#102—64no—
197#102—64no—
198C22⋊C8—21no—
199C22⋊C8—61no—
200#97—21no—
201#97—32no—
202#97—61no—
203C4×C8—61no—
204#95—32no—
205#95—61no—
206C23×C3—42no—
207C23×C3—42no—
208C23×C3—41no—
209C23×C3—42no—
210C23×C3#12011yescharacteristic
211C22×D3—42no—
212C22×D3—42no—
213C22×D3—44no—
214C22×D3—42no—
215C22×D3—44no—
216C3×Q8—42no—
217C3×Q8—42no—
218C3×Q8—81no—
219C3×Q8—81no—
220C3×D4—42no—
221C3×D4—81no—
222C2×C3×C4—22no—
223C2×C3×C4—22no—
224C2×C3×C4—41no—
225C2×C3×C4—42no—
226C2×C3×C4—41no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4#12011yescharacteristic
230C2×C3×C4#12011yescharacteristic
231C2×C3×C4#12111yescharacteristic
232C2×C3×C4#12111yescharacteristic
233C2×C3×C4#12111yescharacteristic
234C2×C3×C4#12711yescharacteristic
235#67—816no—
236C2×Dic3—24no—
237C2×Dic3—42no—
238C2×Dic3—42no—
239D12—44no—
240D12—82no—
241D12—82no—
242D12—82no—
243D12—82no—
244D12—82no—
245C3×C8—42no—
246C3⋊C8—42no—
247C3⋊C8—81no—
248C24—21no—
249C24—62no—
250C2×Q8—21no—
251C2×Q8—21no—
252C2×Q8—21no—
253C2×Q8—21no—
254C2×Q8—42no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—42no—
258C2×D4—62no—
259C2×D4—62no—
260C2×D4—124no—
261C2×D4—122no—
262C2×D4—122no—
263C2×D4—128no—
264C2×D4—121no—
265C2×D4—121no—
266C2×D4—121no—
267C2×D4—121no—
268C2×D4—242no—
269C2×D4—244no—
270C2×D4—244no—
271C22×C4—21no—
272C22×C4—62no—
273C22×C4D3×D411yescharacteristic
274C22×C4D3×D411yescharacteristic
275C22×C4C2×#6711yescharacteristic
276Q16—82no—
277SD16—82no—
278M16—122no—
279C2×C8—22no—
280C2×C8—62no—
281C2×C8—121no—
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C22⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—41no—
291C22⋊C4—41no—
292C22⋊C4—41no—
293C22⋊C4—128no—
294C42—41no—
295C42—64no—
296C42D3×D411yescharacteristic
297C42#23711yescharacteristic
298C42C2×#6711yescharacteristic
299C42C2×#6711yescharacteristic
300C22×C3—21no—
301C22×C3—21no—
302C22×C3—84no—
303C22×C3—81no—
304C22×C3—81no—
305C22×C3—81no—
306C22×C3#39211yescharacteristic
307C22×C3#45512yes—
308C2×D3—84no—
309C2×D3—84no—
310C2×D3—82no—
311C2×D3—82no—
312C2×D3—82no—
313C3×C4—21no—
314C3×C4—21no—
315C3×C4—21no—
316C3×C4—21no—
317C3×C4—21no—
318C3×C4—81no—
319C3×C4—81no—
320C3×C4—81no—
321C3×C4#45312yes—
322Dic3—44no—
323C23—42no—
324C23—42no—
325C23—42no—
326C23—41no—
327C23—122no—
328C23—122no—
329C23—124no—
330C23—124no—
331C23—122no—
332C23#93711yescharacteristic
333Q8—42no—
334Q8—42no—
335Q8—81no—
336Q8—81no—
337D4—42no—
338D4—81no—
339D4—124no—
340D4—242no—
341D4—242no—
342D4—242no—
343D4—242no—
344D4—2416no—
345D4—242no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—64no—
354C2×C4—122no—
355C2×C4—122no—
356C2×C4#93711yescharacteristic
357C2×C4#93811yescharacteristic
358C2×C4#93911yescharacteristic
359C2×C4#93911yescharacteristic
360C2×C4#94011yescharacteristic
361C2×C4#95311yescharacteristic
362C8—42no—
363C8—122no—
364C8—241no—
365C2×C3—22no—
366C2×C3—161no—
367C2×C3#200211yescharacteristic
368C2×C3#200411yescharacteristic
369C2×C3#200811yescharacteristic
370D3—162no—
371C22—21no—
372C22—21no—
373C22—84no—
374C22—81no—
375C22—81no—
376C22—81no—
377C22—244no—
378C22—244no—
379C22—242no—
380C22—242no—
381C22—242no—
382C22#520212yes—
383C22#522311yescharacteristic
384C4—21no—
385C4—21no—
386C4—21no—
387C4—21no—
388C4—21no—
389C4—81no—
390C4—81no—
391C4—81no—
392C4—124no—
393C4#524112yes—
394C3#1009611yescharacteristic, complemented, Sylow
395C2—22no—
396C2—161no—
397C2—482no—
398C2#7185611yescharacteristic
399C2#7210611yescharacteristic
400C2#7211911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3215C2
socleC22×C3#392
prime radical#82979C2
hypercenter#3215D3
hypocenterC3#10096
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3215

Complements

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

NH ≅ G/Ntype
1. C3×#3215C2semidirect
2. #83336C2semidirect
4. #71836C2semidirect
5. #71718C2semidirect
6. #70818C2semidirect
7. #67748C2semidirect
32. #3215D3semidirect
47. C3×C4⋊Q8D4semidirect
48. C3×C4⋊Q8D4semidirect
71. C3×C4⋊C8D4semidirect
83. #803D4semidirect
169. C4⋊Q8#67semidirect
170. C4⋊Q8#67semidirect
193. C4⋊C8#67semidirect
394. C3#10096semidirect

Maximal subgroups

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

subgroupindex
C3×#32152
#833362
#829792
#718362
#717182
#708182
#677482
#100963