Finite Group Databasebeta

← #10698667

Subgroups — #10698667

Order 768.

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

normal subgroups only

This group has 444 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#83071C211yesmaximal, characteristic
2C3×#1992C211yesmaximal, characteristic, complemented
3C2×#4543C211yesmaximal, characteristic, complemented
4C2×#4464C211yesmaximal, characteristic
5#69581C211yesmaximal, characteristic
6#69575C211yesmaximal, characteristic, complemented
7#67659C211yesmaximal, characteristic, complemented
8#10426—31nomaximal, Sylow
9C2×C3×#125C2211yescharacteristic
10C2×#875C2211yescharacteristic
11C2×C3×#97C2211yescharacteristic
12C3×#384C2211yescharacteristic
13#4994—21no—
14#4994—21no—
15#4993—21no—
16#4992—21no—
17#4991—21no—
18#4979—21no—
19C2×#810—21no—
20C2×#809—21no—
21#4936C2211yescharacteristic
22C2×#803C2211yescharacteristic
23C2×#803C2211yescharacteristic
24C3×#340—21no—
25C3×#340—21no—
26#4543—22no—
27#4529—21no—
28#4529—21no—
29#4464—22no—
30#2820—31no—
31#1992D311yescharacteristic, complemented
32#1959—31no—
33#1836—31no—
34C2×#330—31no—
35C2×#329—31no—
36#1295—31no—
37C22×D12D411yescharacteristic
38C22×Dic6D411yescharacteristic
39C22×C3×C8—21no—
40C3×#125—21no—
41C3×#125—21no—
42C3×#125—21no—
43C3×#125—21no—
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×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×(C2×D3)⋊C4—21no—
58C2×C4⋊Dic3D411yescharacteristic
59C2×C4⋊Dic3D411yescharacteristic
60C2×Dic3⋊C4—21no—
61C2×Dic3⋊C4—21no—
62C22×C3⋊C8—21no—
63C22×C3⋊C8—21no—
64#875—21no—
65#875—21no—
66#875—21no—
67#875—21no—
68C3×#97—21no—
69C3×#97—21no—
70C3⋊#95—21no—
71C3⋊#95—21no—
72C3⋊#95—21no—
73C3⋊#95—21no—
74#810—42no—
75#809—42no—
76#803—21no—
77#803—21no—
78#803—21no—
79#803D412yes—
80C2×#125C2×D311yescharacteristic
81C2×#124—31no—
82C2×#103—61no—
83C2×#102—61no—
84C2×#97—31no—
85C2×#97—31no—
86C2×#97C2×D311yescharacteristic
87#399—61no—
88#397—61no—
89#396—61no—
90#396—61no—
91#395—61no—
92#394—61no—
93#384C2×D311yescharacteristic
94#382—31no—
95#340—21no—
96#340—21no—
97#340—61no—
98#340—61no—
99#330—62no—
100#329—62no—
101C23×D3—21no—
102C22×C3×C4—21no—
103C22×C3×C4—21no—
104C22×C3×C4—41no—
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4C2×D411yescharacteristic
108C22×Dic3—21no—
109C22×Dic3—21no—
110C22×Dic3—21no—
111C2×D12—21no—
112C2×D12—21no—
113C2×D12—41no—
114C2×D12—41no—
115C2×Dic6—21no—
116C2×Dic6—21no—
117C2×Dic6—41no—
118C2×Dic6—41no—
119C2×C3×C8—22no—
120C2×C3×C8—41no—
121C2×C3×C8—41no—
122C3×C4⋊C4—21no—
123C3×C4⋊C4—21no—
124C3×C4⋊C4—21no—
125C3×C4⋊C4—21no—
126C3×C4⋊C4—41no—
127C3×C4⋊C4—41no—
128C3×C4⋊C4—42no—
129C3×C4⋊C4—42no—
130C3×C4⋊C4—41no—
131C3×C4⋊C4—41no—
132C3×C4⋊C4—41no—
133C3×C4⋊C4—41no—
134C3×C42C2×D411yescharacteristic
135C3×C42#4111yescharacteristic
136C3×C42#4111yescharacteristic
137C3×C42#4111yescharacteristic
138(C2×D3)⋊C4—41no—
139(C2×D3)⋊C4—41no—
140(C2×D3)⋊C4—41no—
141(C2×D3)⋊C4—41no—
142(C2×D3)⋊C4—81no—
143(C2×D3)⋊C4—81no—
144(C2×D3)⋊C4—81no—
145(C2×D3)⋊C4—81no—
146C4⋊Dic3—41no—
147C4⋊Dic3—41no—
148Dic3⋊C4—81no—
149Dic3⋊C4—81no—
150C2×C3⋊C8—22no—
151C2×C3⋊C8—22no—
152C2×C3⋊C8—41no—
153C2×C3⋊C8—41no—
154C2×C3⋊C8—41no—
155C2×C3⋊C8—41no—
156C22×Q8—31no—
157C22×D4—31no—
158C22×C8—21no—
159C22×C8—61no—
160C22×C8—61no—
161#125—21no—
162#125—21no—
163#125—21no—
164#125—21no—
165#124—61no—
166#124—61no—
167#124—61no—
168#124—61no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—31no—
175C2×C4⋊C4—31no—
176C2×C4⋊C4—61no—
177C2×C4⋊C4—61no—
178C2×C4⋊C4#6711yescharacteristic
179C2×C4⋊C4#6711yescharacteristic
180C2×C22⋊C4—61no—
181C2×C22⋊C4—61no—
182C2×C22⋊C4—61no—
183C2×C22⋊C4—61no—
184C2×C22⋊C4—61no—
185C2×C22⋊C4—61no—
186C2×C42C22×D311yescharacteristic
187#103—122no—
188#102—122no—
189#97—21no—
190#97—21no—
191#97—32no—
192#97—61no—
193#97—61no—
194#97—61no—
195#95—61no—
196#95—61no—
197#95—61no—
198#95—61no—
199C23×C3#12011yescharacteristic
200C22×D3—41no—
201C22×D3—41no—
202C22×D3—41no—
203C22×D3—41no—
204C22×D3—41no—
205C22×D3—41no—
206C22×D3—41no—
207C2×C3×C4—21no—
208C2×C3×C4—21no—
209C2×C3×C4—21no—
210C2×C3×C4—21no—
211C2×C3×C4—21no—
212C2×C3×C4—21no—
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—81no—
220C2×C3×C4—81no—
221C2×C3×C4—81no—
222C2×C3×C4#12111yescharacteristic
223C2×C3×C4#12111yescharacteristic
224C2×C3×C4#12111yescharacteristic
225C2×C3×C4#12311yescharacteristic
226C2×C3×C4#12311yescharacteristic
227C2×C3×C4#12411yescharacteristic
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—
237D12—42no—
238D12—81no—
239Dic6—42no—
240Dic6—81no—
241C3×C8—42no—
242C3⋊C8—42no—
243C3⋊C8—42no—
244C24—61no—
245C2×Q8—61no—
246C2×Q8—61no—
247C2×Q8—121no—
248C2×Q8—121no—
249C2×D4—61no—
250C2×D4—61no—
251C2×D4—121no—
252C2×D4—121no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—41no—
256C22×C4—61no—
257C22×C4—61no—
258C22×C4—61no—
259C22×C4D3×D411yescharacteristic
260C22×C4D3×D411yescharacteristic
261C22×C4C2×#6711yescharacteristic
262C2×C8—22no—
263C2×C8—41no—
264C2×C8—41no—
265C2×C8—62no—
266C2×C8—62no—
267C2×C8—121no—
268C2×C8—121no—
269C2×C8—121no—
270C2×C8—121no—
271C4⋊C4—21no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—121no—
284C4⋊C4—121no—
285C4⋊C4—241no—
286C4⋊C4—241no—
287C22⋊C4—121no—
288C22⋊C4—121no—
289C22⋊C4—121no—
290C22⋊C4—121no—
291C22⋊C4—241no—
292C22⋊C4—241no—
293C22⋊C4—241no—
294C22⋊C4—241no—
295C42#23511yescharacteristic
296C42#23511yescharacteristic
297C42D3×D411yescharacteristic
298C42#23711yescharacteristic
299C22×C3#39411yescharacteristic
300C22×C3#45311yescharacteristic
301C22×C3#45411yescharacteristic
302C22×C3#45411yescharacteristic
303C22×C3#45511yescharacteristic
304C22×C3#45511yescharacteristic
305C22×C3#45611yescharacteristic
306C2×D3—81no—
307C2×D3—81no—
308C2×D3—81no—
309C2×D3—81no—
310C2×D3—81no—
311C2×D3—81no—
312C2×D3—81no—
313C3×C4—21no—
314C3×C4—21no—
315C3×C4—21no—
316C3×C4—21no—
317C3×C4—21no—
318C3×C4—21no—
319C3×C4—81no—
320C3×C4—81no—
321C3×C4—161no—
322Dic3—81no—
323Dic3—81no—
324Dic3—81no—
325C23—121no—
326C23—121no—
327C23—121no—
328C23—121no—
329C23—121no—
330C23—121no—
331C23—121no—
332C23#93711yescharacteristic
333Q8—122no—
334Q8—241no—
335D4—122no—
336D4—241no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—121no—
353C2×C4—121no—
354C2×C4—121no—
355C2×C4—121no—
356C2×C4—121no—
357C2×C4—121no—
358C2×C4—121no—
359C2×C4—121no—
360C2×C4—121no—
361C2×C4#88311yescharacteristic
362C2×C4#88411yescharacteristic
363C2×C4#88411yescharacteristic
364C2×C4#88511yescharacteristic
365C2×C4#92911yescharacteristic
366C2×C4#93911yescharacteristic
367C8—42no—
368C8—122no—
369C8—122no—
370C2×C3#165711yescharacteristic
371C2×C3#166111yescharacteristic
372C2×C3#166211yescharacteristic
373C2×C3#166411yescharacteristic
374C2×C3#203711yescharacteristic
375C2×C3#203811yescharacteristic
376C2×C3#204011yescharacteristic
377D3—161no—
378C22—241no—
379C22—241no—
380C22—241no—
381C22—241no—
382C22—241no—
383C22—241no—
384C22—241no—
385C22#495911yescharacteristic
386C22#506211yescharacteristic
387C22#509511yescharacteristic
388C22#519811yescharacteristic
389C22#520111yescharacteristic
390C22#520211yescharacteristic
391C22#520211yescharacteristic
392C4—21no—
393C4—21no—
394C4—21no—
395C4—21no—
396C4—21no—
397C4—21no—
398C4—81no—
399C4—81no—
400C4—161no—
401C4—241no—
402C4—241no—
403C4—241no—
404C3#1042611yescharacteristic, complemented, Sylow
405C2—481no—
406C2#6969511yescharacteristic
407C2#7030611yescharacteristic
408C2#7031211yescharacteristic
409C2#7124911yescharacteristic
410C2#7125111yescharacteristic
411C2#7139211yescharacteristic
412C2#7139411yescharacteristic
subgroupG/H
centerC23#937
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1992C2
socleC23×C3#120
prime radicalC22×D12D4
hypercenter#1992D3
hypocenterC3#10426
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1992

Complements

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

NH ≅ G/Ntype
2. C3×#1992C2semidirect
3. C2×#4543C2semidirect
6. #69575C2semidirect
7. #67659C2semidirect
31. #1992D3semidirect
404. C3#10426semidirect

Maximal subgroups

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

subgroupindex
#830712
C3×#19922
C2×#45432
C2×#44642
#695812
#695752
#676592
#104263