Finite Group Databasebeta

← #10694540

Subgroups — #10694540

Order 768.

428 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3131C211yesmaximal, characteristic
2#71835C211yesmaximal, characteristic, complemented
3#71824C211yesmaximal, characteristic, complemented
4#71811C211yesmaximal, characteristic
5#71800C211yesmaximal, characteristic
6C8×C4⋊Dic3C211yesmaximal, characteristic, complemented
7#69729C211yesmaximal, characteristic, complemented
8#9751—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C3×#532C2211yescharacteristic
11C3×#488—21no—
12C3×#488—21no—
13C3×#487C411yescharacteristic, complemented
14C3×#487C411yescharacteristic, complemented
15C3×#486C411yescharacteristic, complemented
16C3×#486C411yescharacteristic, complemented
17C3×#416—21no—
18C3×#416—21no—
19C2×C3×C4×C8C2211yescharacteristic
20C2×C8×Dic3—21no—
21C4×C4⋊Dic3C2211yescharacteristic
22#4925C2211yescharacteristic
23#4925C2211yescharacteristic
24C2×C4⋊(C3⋊C8)C2211yescharacteristic
25#4550—21no—
26#4538—22no—
27#4537—22no—
28#4535—22no—
29D4⋊(C3⋊C8)—22no—
30#4458—21no—
31#4457—21no—
32#3131D311yescharacteristic
33C8×C4⋊C4—31no—
34#1564—31no—
35#1503—31no—
36#1502—31no—
37#1481—31no—
38#1480—31no—
39C2×C3×#41—21no—
40C22×C3×C8D411yescharacteristic
41C22×C3×C8D411yescharacteristic
42C3×C4⋊Q8C2×C411yescharacteristic
43C3×C4⋊Q8C2×C411yescharacteristic
44C3×C4⋊Q8C2×C411yescharacteristic
45C3×#127C2×C411yescharacteristic
46C3×#125—21no—
47C3×#124—21no—
48C3×C22⋊Q8—21no—
49C3×C22⋊Q8—21no—
50C3×#121—21no—
51C3×#121—21no—
52C3×C4×Q8—21no—
53C3×C4×Q8—21no—
54C3×C4×D4—21no—
55C3×C4×D4—21no—
56C3×#117—21no—
57C2×C3×C42C2311yescharacteristic
58C2×C4⋊Dic3—21no—
59C2×C4×Dic3—21no—
60C22×C3⋊C8—21no—
61C3×#103—21no—
62C3×#103—21no—
63C3×#103—21no—
64C3×#103—21no—
65C3×#102—21no—
66C3×#102—21no—
67C3×#102—21no—
68C3×#102—21no—
69C3×C4×C8C2×C411yescharacteristic
70C3×C4×C8C2×C411yescharacteristic
71C3×C4×C8D411yescharacteristic
72C3×C4×C8D411yescharacteristic
73C3⋊#95D412yes—
74#830—21no—
75#830—21no—
76C8×Dic3—42no—
77C4⋊(C3⋊C8)—21no—
78C4⋊(C3⋊C8)—21no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)D412yes—
81C4×C3⋊C8—21no—
82C4×C3⋊C8—21no—
83#533C2×D311yescharacteristic
84#532C2×D311yescharacteristic
85#488—21no—
86#488—21no—
87#487Dic311yescharacteristic, complemented
88#487Dic311yescharacteristic, complemented
89#486Dic311yescharacteristic, complemented
90#486Dic311yescharacteristic, complemented
91#431—31no—
92#431—31no—
93C2×C4⋊C8—31no—
94#416—21no—
95#416—21no—
96C2×C4×C8—61no—
97C2×C4×C8C2×D311yescharacteristic
98C4×C4⋊C4—31no—
99#336—61no—
100#335—61no—
101#334—61no—
102#328—62no—
103#327—62no—
104Q8⋊C8—62no—
105D4⋊C8—62no—
106C3×#41—41no—
107C3×#41—41no—
108C3×#41—41no—
109C3×#41—41no—
110C2×C3×Q8—21no—
111C2×C3×Q8—21no—
112C2×C3×Q8—21no—
113C2×C3×D4—21no—
114C2×C3×D4—21no—
115C2×C3×D4—21no—
116C2×C3×D4—41no—
117C22×C3×C4—41no—
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4#4111yescharacteristic
121C22×Dic3—22no—
122C2×C3×C8—21no—
123C2×C3×C8—21no—
124C2×C3×C8—21no—
125C2×C3×C8—21no—
126C2×C3×C8C22⋊C411yescharacteristic
127C2×C3×C8C22⋊C411yescharacteristic
128C2×C3×C8C22⋊C411yescharacteristic
129C2×C3×C8C22⋊C411yescharacteristic
130C3×C4⋊C4—21no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—21no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4—41no—
137C3×C4⋊C4—41no—
138C3×C22⋊C4—41no—
139C3×C22⋊C4—41no—
140C3×C22⋊C4—41no—
141C3×C42—41no—
142C3×C42C22×C411yescharacteristic
143C3×C42C2×D411yescharacteristic
144C3×C42C2×D411yescharacteristic
145C3×C42#4111yescharacteristic
146C4⋊Dic3—24no—
147C4×Dic3—41no—
148C4×Dic3—41no—
149C2×C3⋊C8—22no—
150C2×C3⋊C8—41no—
151C2×C3⋊C8—41no—
152C2×C3⋊C8—41no—
153C2×C3⋊C8—41no—
154C2×#41—21no—
155C22×C8—61no—
156C22×C8#6711yescharacteristic
157C22×C8#6711yescharacteristic
158C4⋊Q8C2×Dic311yescharacteristic
159C4⋊Q8C2×Dic311yescharacteristic
160C4⋊Q8C2×Dic311yescharacteristic
161#127C2×Dic311yescharacteristic
162#125—21no—
163#124—21no—
164C22⋊Q8—21no—
165C22⋊Q8—21no—
166#121—21no—
167#121—21no—
168C4×Q8—21no—
169C4×Q8—21no—
170C4×D4—21no—
171C4×D4—21no—
172#117—21no—
173C2×C4⋊C4—61no—
174C2×C42—61no—
175C2×C42C22×D311yescharacteristic
176C4⋊C8—32no—
177C4⋊C8—61no—
178C4⋊C8—61no—
179C4⋊C8—61no—
180#103—21no—
181#103—21no—
182#103—21no—
183#103—21no—
184#102—21no—
185#102—21no—
186#102—21no—
187#102—21no—
188C22⋊C8—61no—
189C22⋊C8—61no—
190C4×C8—61no—
191C4×C8—61no—
192C4×C8—122no—
193C4×C8C2×Dic311yescharacteristic
194C4×C8C2×Dic311yescharacteristic
195C4×C8#6711yescharacteristic
196C4×C8#6711yescharacteristic
197#95—32no—
198C23×C3—41no—
199C23×C3#12111yescharacteristic
200C3×Q8—41no—
201C3×Q8—41no—
202C3×Q8—42no—
203C3×Q8—81no—
204C3×D4—41no—
205C3×D4—41no—
206C3×D4—42no—
207C3×D4—81no—
208C3×D4—81no—
209C3×D4—81no—
210C2×C3×C4—22no—
211C2×C3×C4—41no—
212C2×C3×C4—41no—
213C2×C3×C4—41no—
214C2×C3×C4—41no—
215C2×C3×C4—81no—
216C2×C3×C4—81no—
217C2×C3×C4C2×C22⋊C411yescharacteristic
218C2×C3×C4C4×D411yescharacteristic
219C2×C3×C4C4×D411yescharacteristic
220C2×C3×C4#12111yescharacteristic
221C2×C3×C4#12411yescharacteristic
222C2×C3×C4#12711yescharacteristic
223C2×C3×C4#13511yescharacteristic
224C2×C3×C4#13511yescharacteristic
225C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
226C2×C3×C4#13711yescharacteristic
227C2×Dic3—24no—
228C2×Dic3—42no—
229C2×Dic3—42no—
230C3×C8—21no—
231C3×C8—21no—
232C3×C8—21no—
233C3×C8—21no—
234C3⋊C8—42no—
235C3⋊C8—81no—
236C3⋊C8—81no—
237#41—41no—
238#41—41no—
239#41—41no—
240#41—41no—
241C2×Q8—21no—
242C2×Q8—21no—
243C2×Q8—21no—
244C2×D4—21no—
245C2×D4—21no—
246C2×D4—21no—
247C2×D4—41no—
248C22×C4—41no—
249C22×C4—62no—
250C22×C4D3×D411yescharacteristic
251C22×C4#23711yescharacteristic
252C22×C4C2×#6711yescharacteristic
253C2×C8—21no—
254C2×C8—21no—
255C2×C8—21no—
256C2×C8—21no—
257C2×C8—62no—
258C2×C8—121no—
259C2×C8—121no—
260C2×C8—121no—
261C2×C8—121no—
262C2×C8#21711yescharacteristic
263C2×C8#21711yescharacteristic
264C2×C8#21711yescharacteristic
265C2×C8#21711yescharacteristic
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—64no—
275C22⋊C4—41no—
276C22⋊C4—41no—
277C22⋊C4—41no—
278C42—41no—
279C42—121no—
280C42—121no—
281C42D3×D411yescharacteristic
282C42#23711yescharacteristic
283C42C22×Dic311yescharacteristic
284C42C2×#6711yescharacteristic
285C22×C3—21no—
286C22×C3—21no—
287C22×C3—81no—
288C22×C3—81no—
289C22×C3—81no—
290C22×C3#39011yescharacteristic
291C22×C3#46311yescharacteristic
292C22×C3#46411yescharacteristic
293C3×C4—21no—
294C3×C4—21no—
295C3×C4—21no—
296C3×C4—81no—
297C3×C4—81no—
298C3×C4—81no—
299C3×C4#41611yescharacteristic
300C3×C4#41711yescharacteristic
301C3×C4#44312yes—
302C3×C4#46311yescharacteristic
303C3×C4#46411yescharacteristic
304Dic3—44no—
305C23—41no—
306C23#93811yescharacteristic
307Q8—42no—
308Q8—41no—
309Q8—41no—
310Q8—81no—
311D4—42no—
312D4—41no—
313D4—41no—
314D4—81no—
315D4—81no—
316D4—81no—
317C2×C4—22no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—64no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—122no—
326C2×C4—122no—
327C2×C4D4×Dic311yescharacteristic
328C2×C4D4×Dic311yescharacteristic
329C2×C4#93611yescharacteristic
330C2×C4#93811yescharacteristic
331C2×C4#94011yescharacteristic
332C2×C4#94911yescharacteristic
333C2×C4#95011yescharacteristic
334C2×C4#95011yescharacteristic
335C2×C4#95111yescharacteristic
336C2×C4C2×#21711yescharacteristic
337C8—21no—
338C8—21no—
339C8—21no—
340C8—21no—
341C8—122no—
342C8—241no—
343C8—241no—
344C2×C3—21no—
345C2×C3—21no—
346C2×C3—161no—
347C2×C3#197411yescharacteristic
348C2×C3#197611yescharacteristic
349C2×C3#198311yescharacteristic
350C22—21no—
351C22—21no—
352C22—81no—
353C22—81no—
354C22—81no—
355C22#505411yescharacteristic
356C22#505711yescharacteristic
357C22#522111yescharacteristic
358C4—21no—
359C4—21no—
360C4—21no—
361C4—81no—
362C4—81no—
363C4—81no—
364C4—124no—
365C4#505411yescharacteristic
366C4#505711yescharacteristic
367C4#519612yes—
368C4#523211yescharacteristic
369C4#523311yescharacteristic
370C3#975111yescharacteristic, complemented, Sylow
371C2—21no—
372C2—21no—
373C2—161no—
374C2#7172111yescharacteristic
375C2#7208411yescharacteristic
376C2#7208611yescharacteristic
subgroupG/H
centerC2×C4#938
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#3131C2
socleC22×C3#390
prime radicalC3×#532C22
cosocleC2×C3×C42C23
hypercenter#3131D3
hypocenterC3#9751
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3131

Complements

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

NH ≅ G/Ntype
2. #71835C2semidirect
3. #71824C2semidirect
6. C8×C4⋊Dic3C2semidirect
7. #69729C2semidirect
13. C3×#487C4semidirect
14. C3×#487C4semidirect
15. C3×#486C4semidirect
16. C3×#486C4semidirect
87. #487Dic3semidirect
88. #487Dic3semidirect
89. #486Dic3semidirect
90. #486Dic3semidirect
370. C3#9751semidirect

Maximal subgroups

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

subgroupindex
C3×#31312
#718352
#718242
#718112
#718002
C8×C4⋊Dic32
#697292
#97513