Finite Group Databasebeta

← #10692803

Subgroups — #10692803

Order 768.

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

normal subgroups only

This group has 490 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#83338C211yesmaximal, characteristic
2#73395C211yesmaximal, characteristic
3C3×#1506C211yesmaximal, characteristic, complemented
4#71836C211yesmaximal, characteristic, complemented
5#70808C211yesmaximal, characteristic
6#67748C211yesmaximal, characteristic, complemented
7#67615C211yesmaximal, characteristic, complemented
8#9439—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×#875C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C3×#376C2211yescharacteristic
13#5089C412yescomplemented
14#5089C412yescomplemented
15#5068—22no—
16#5004—21no—
17#5001—21no—
18#4940—21no—
19C4×(C2×D3)⋊C4—21no—
20#4932C2211yescharacteristic
21#4926C2211yescharacteristic
22C4×#208C2211yescharacteristic
23#4669—21no—
24#4666—21no—
25#4665—21no—
26#4646—21no—
27C3×#328—21no—
28C3×#328—21no—
29#4538—21no—
30#4538—21no—
31#4464—22no—
32#3140—31no—
33#2372—31no—
34#1533—31no—
35#1506—31no—
36#1506D311yescharacteristic, complemented
37#1300—31no—
38#1300—31no—
39C22×D12D411yescharacteristic
40C22×C4×D3—21no—
41C22×Dic6D411yescharacteristic
42C3×C4⋊Q8C2×C411yescharacteristic, complemented
43C3×C4⋊Q8C2×C411yescharacteristic, complemented
44C3×#125—21no—
45C3×C22⋊Q8—21no—
46C3×C22⋊Q8—21no—
47C3×C4×Q8—21no—
48C3×C4×Q8—21no—
49C3×#117—21no—
50C2×C3×C42—21no—
51C2×C3×C42C2311yescharacteristic
52C2×(C2×D3)⋊C4—21no—
53C2×(C2×D3)⋊C4—21no—
54C2×(C2×D3)⋊C4—21no—
55C2×(C2×D3)⋊C4—21no—
56C2×(C2×D3)⋊C4—21no—
57C2×Dic3⋊C4—21no—
58C2×C4×Dic3—21no—
59C2×C4×Dic3—21no—
60C2×#208—21no—
61#875—21no—
62#875—21no—
63#875C2×C412yes—
64#875D412yes—
65C3×C4⋊C8—21no—
66C3×C22⋊C8—21no—
67C3×#97—21no—
68C3×#95—21no—
69C3×#95D411yescharacteristic
70C3×#95D411yescharacteristic
71C3⋊#95—21no—
72C3⋊#95—21no—
73C3⋊#95—21no—
74C3⋊#95D411yescharacteristic
75C3⋊#95D411yescharacteristic
76#830—21no—
77#810—22no—
78#810—22no—
79#809—22no—
80#809—22no—
81C4⋊(C3⋊C8)—21no—
82#803—21no—
83#803C2×C412yescomplemented
84C4×C3⋊C8—21no—
85#533C2×D311yescharacteristic
86C2×#124—31no—
87#489—32no—
88#489—32no—
89#488—62no—
90#432—31no—
91#432C2×D311yescharacteristic
92#419—61no—
93#418—61no—
94C4×M16—31no—
95#391—61no—
96#388—61no—
97#388—61no—
98#386—61no—
99C4×C22⋊C4—61no—
100C4×C22⋊C4—61no—
101#376—31no—
102#376C2×D311yescharacteristic
103#329—62no—
104#328—21no—
105#328—21no—
106#328—61no—
107#328—61no—
108C23×D3—21no—
109C2×C3×Q8—21no—
110C2×C3×Q8—21no—
111C22×C3×C4—21no—
112C22×C3×C4—21no—
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C22×Dic3—21no—
117C22×Dic3—21no—
118C22×Dic3—21no—
119C2×D12—21no—
120C2×D12—41no—
121C2×D12—41no—
122C2×D12C22⋊C412yes—
123C2×C4×D3—42no—
124C2×C4×D3—42no—
125C2×C4×D3—81no—
126C2×Dic6—21no—
127C2×Dic6—41no—
128C2×Dic6—41no—
129C2×Dic6C22⋊C412yes—
130C2×C3×C8—41no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—21no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—41no—
136C3×C4⋊C4—41no—
137C3×C22⋊C4—41no—
138C3×C42—42no—
139C3×C42—41no—
140C3×C42C22×C411yescharacteristic
141C3×C42C2×D411yescharacteristic
142C3×C42#4111yescharacteristic
143C3×C42#4111yescharacteristic
144(C2×D3)⋊C4—22no—
145(C2×D3)⋊C4—22no—
146(C2×D3)⋊C4—22no—
147(C2×D3)⋊C4—22no—
148(C2×D3)⋊C4—42no—
149(C2×D3)⋊C4—42no—
150(C2×D3)⋊C4—81no—
151Dic3⋊C4—81no—
152C4×Dic3—42no—
153C4×Dic3—81no—
154#208—41no—
155#208—41no—
156C2×C3⋊C8—22no—
157C2×C3⋊C8—41no—
158C22×Q8—31no—
159C22×D4—31no—
160C23×C4—61no—
161C2×M16—61no—
162C4⋊Q8C4×D311yescharacteristic, complemented
163C4⋊Q8C4×D311yescharacteristic, complemented
164#125—21no—
165#124—32no—
166#124—32no—
167#124—61no—
168#124—61no—
169C22⋊Q8—21no—
170C22⋊Q8—21no—
171C4×Q8—21no—
172C4×Q8—21no—
173#117—21no—
174C2×C4⋊C4—61no—
175C2×C22⋊C4—61no—
176C2×C22⋊C4—61no—
177C2×C22⋊C4—61no—
178C2×C22⋊C4—61no—
179C2×C22⋊C4—61no—
180C2×C42—21no—
181C2×C42—61no—
182C2×C42—61no—
183C2×C42C22×D311yescharacteristic
184C4⋊C8—21no—
185C4⋊C8—61no—
186#103—62no—
187#103—62no—
188#102—62no—
189#102—62no—
190C22⋊C8—21no—
191C22⋊C8—61no—
192#97—21no—
193#97—32no—
194#97—61no—
195C4×C8—61no—
196#95—21no—
197#95—31no—
198#95—31no—
199#95—61no—
200#95—61no—
201#95—61no—
202#95#6711yescharacteristic
203#95#6711yescharacteristic
204C23×C3#12011yescharacteristic
205C22×D3—22no—
206C22×D3—41no—
207C22×D3—41no—
208C22×D3—41no—
209C22×D3—41no—
210C22×D3—41no—
211C22×D3—41no—
212C3×Q8—81no—
213C3×Q8—81no—
214C2×C3×C4—21no—
215C2×C3×C4—21no—
216C2×C3×C4—22no—
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×C4C2×C22⋊C411yescharacteristic
227C2×C3×C4C4×D411yescharacteristic
228C2×C3×C4C4×D411yescharacteristic
229C2×C3×C4#12111yescharacteristic
230C2×C3×C4#12111yescharacteristic
231C2×C3×C4#12311yescharacteristic
232C2×Dic3—22no—
233C2×Dic3—22no—
234C2×Dic3—41no—
235C2×Dic3—41no—
236C2×Dic3—41no—
237C2×Dic3—41no—
238C2×Dic3—41no—
239C2×Dic3—41no—
240C2×Dic3—41no—
241D12—42no—
242D12—81no—
243C4×D3—84no—
244Dic6—42no—
245Dic6—81no—
246C3×C8—81no—
247C3⋊C8—42no—
248C3⋊C8—81no—
249C24—61no—
250C2×Q8—21no—
251C2×Q8—21no—
252C2×Q8—32no—
253C2×Q8—61no—
254C2×Q8—121no—
255C2×Q8—121no—
256C2×D4—32no—
257C2×D4—61no—
258C2×D4—121no—
259C2×D4—121no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—61no—
263C22×C4—61no—
264C22×C4—61no—
265C22×C4—122no—
266C22×C4—122no—
267C22×C4—241no—
268C22×C4D3×D411yescharacteristic
269C22×C4D3×D411yescharacteristic
270C22×C4C2×#6711yescharacteristic
271M16—121no—
272M16—121no—
273C2×C8—41no—
274C2×C8—62no—
275C2×C8—121no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—241no—
283C22⋊C4—41no—
284C22⋊C4—62no—
285C22⋊C4—62no—
286C22⋊C4—62no—
287C22⋊C4—62no—
288C22⋊C4—122no—
289C22⋊C4—122no—
290C22⋊C4—241no—
291C42—42no—
292C42—41no—
293C42—122no—
294C42—241no—
295C42C2×C4×D311yescharacteristic
296C42#23511yescharacteristic
297C42D3×D411yescharacteristic
298C42#23711yescharacteristic
299C22×C3—21no—
300C22×C3—21no—
301C22×C3#38611yescharacteristic
302C22×C3#45511yescharacteristic
303C22×C3#45511yescharacteristic
304C2×D3—42no—
305C2×D3—42no—
306C2×D3—42no—
307C2×D3—81no—
308C2×D3—81no—
309C2×D3—81no—
310C2×D3—81no—
311C3×C4—21no—
312C3×C4—21no—
313C3×C4—21no—
314C3×C4—21no—
315C3×C4—21no—
316C3×C4—21no—
317C3×C4—42no—
318C3×C4—81no—
319C3×C4—81no—
320C3×C4—81no—
321Dic3—42no—
322Dic3—42no—
323Dic3—81no—
324C23—62no—
325C23—121no—
326C23—121no—
327C23—121no—
328C23—121no—
329C23—121no—
330C23—121no—
331C23#93711yescharacteristic
332Q8—81no—
333Q8—81no—
334Q8—122no—
335Q8—241no—
336D4—122no—
337D4—241no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—22no—
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—41no—
350C2×C4—62no—
351C2×C4—62no—
352C2×C4—121no—
353C2×C4—121no—
354C2×C4—121no—
355C2×C4—121no—
356C2×C4—121no—
357C2×C4—121no—
358C2×C4—121no—
359C2×C4—244no—
360C2×C4D3×C22⋊C411yescharacteristic
361C2×C4#67⋊C411yescharacteristic
362C2×C4#88311yescharacteristic
363C2×C4#88411yescharacteristic
364C2×C4C4×#6711yescharacteristic
365C2×C4#93811yescharacteristic
366C8—81no—
367C8—122no—
368C8—241no—
369C2×C3—21no—
370C2×C3—21no—
371C2×C3#188511yescharacteristic
372C2×C3#190611yescharacteristic
373C2×C3#190611yescharacteristic
374D3—82no—
375C22—21no—
376C22—21no—
377C22—122no—
378C22—122no—
379C22—122no—
380C22—241no—
381C22—241no—
382C22—241no—
383C22—241no—
384C22#495611yescharacteristic
385C22#520211yescharacteristic
386C22#520211yescharacteristic
387C4—21no—
388C4—21no—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—21no—
393C4—42no—
394C4—81no—
395C4—81no—
396C4—81no—
397C4—122no—
398C4—122no—
399C4—241no—
400C3#943911yescharacteristic, complemented, Sylow
401C2—21no—
402C2—21no—
403C2—242no—
404C2#7005511yescharacteristic
405C2#7034311yescharacteristic
406C2#7039111yescharacteristic
subgroupG/H
centerC22#4956
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1506C2
socleC22×C3#386
prime radicalC22×D12D4
cosocleC2×C3×C42C23
hypercenter#1506D3
hypocenterC3#9439
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1506

Complements

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

NH ≅ G/Ntype
3. C3×#1506C2semidirect
4. #71836C2semidirect
6. #67748C2semidirect
7. #67615C2semidirect
13. #5089C4semidirect
14. #5089C4semidirect
36. #1506D3semidirect
42. C3×C4⋊Q8C2×C4semidirect
43. C3×C4⋊Q8C2×C4semidirect
83. #803C2×C4semidirect
162. C4⋊Q8C4×D3semidirect
163. C4⋊Q8C4×D3semidirect
400. C3#9439semidirect

Maximal subgroups

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

subgroupindex
#833382
#733952
C3×#15062
#718362
#708082
#677482
#676152
#94393