Finite Group Databasebeta

← #10692014

Subgroups — #10692014

Order 768.

428 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3115C211yesmaximal, characteristic
2#82514C211yesmaximal, characteristic, complemented
3#71801C211yesmaximal, characteristic, complemented
4#71798C211yesmaximal, characteristic, complemented
5#71640C211yesmaximal, characteristic, complemented
6#67753C211yesmaximal, characteristic, complemented
7#67751C211yesmaximal, characteristic, complemented
8#9313—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C3×C4×#41C2211yescharacteristic
11C3×#482—21no—
12C3×#475C411yescharacteristic
13C3×#474C411yescharacteristic
14C3×#460C411yescharacteristic
15C3×#459C411yescharacteristic
16C3×#449—21no—
17C3×#442—21no—
18C3×#440—21no—
19C3×#432C2211yescharacteristic
20#5234—21no—
21#5000—22no—
22#4999—21no—
23#4926—22no—
24#4926C2211yescharacteristic
25#4925C2211yescharacteristic
26C4×#208C2211yescharacteristic
27C2×#803C2211yescharacteristic
28#4540—22no—
29#4535—21no—
30D4⋊(C3⋊C8)—21no—
31D4⋊(C3⋊C8)—22no—
32#3115D311yescharacteristic
33#2923—31no—
34#1530—31no—
35#1488—31no—
36#1484—31no—
37#1302—31no—
38#1291—31no—
39C2×C3×#41—21no—
40C2×C3×#41D411yescharacteristic
41C2×C3×SD16—21no—
42C2×C3×D8—21no—
43C3×C4⋊Q8C2×C411yescharacteristic
44C3×#127C2×C411yescharacteristic
45C3×#124—21no—
46C3×#121—21no—
47C3×#121—21no—
48C3×C4×Q8C2×C411yescharacteristic
49C3×C4×Q8D411yescharacteristic
50C3×C4×D4—21no—
51C3×C4×D4—21no—
52C3×C4×D4—21no—
53C3×C4×D4—21no—
54C3×C4×D4C2×C411yescharacteristic
55C3×C4×D4D411yescharacteristic
56C3×#117—21no—
57C3×#117D411yescharacteristic
58C2×C3×C42—21no—
59C2×C3×C42C2311yescharacteristic
60C2×#208—21no—
61C22×C3⋊C8—21no—
62C3×#107—21no—
63C3×#106—21no—
64C3×C4⋊C8C2×C411yescharacteristic
65C3×C4⋊C8C2×C411yescharacteristic
66C3×#103—21no—
67C3×#102—21no—
68C3×#102—21no—
69C3×#102—21no—
70C3×C22⋊C8—21no—
71C3×#97—21no—
72#830—24no—
73#830—21no—
74#830—21no—
75C4⋊(C3⋊C8)—24no—
76C4⋊(C3⋊C8)—21no—
77C4⋊(C3⋊C8)—21no—
78#803—21no—
79#803—21no—
80#803—21no—
81#803D412yes—
82C4×C3⋊C8—21no—
83C4×C3⋊C8D412yes—
84#532C2×D311yescharacteristic
85C4×#41C2×D311yescharacteristic
86#482—21no—
87#475Dic311yescharacteristic
88#474Dic311yescharacteristic
89#460Dic311yescharacteristic
90#459Dic311yescharacteristic
91#449—21no—
92#442—21no—
93#440—21no—
94#433—62no—
95#432—31no—
96#432—62no—
97#432C2×D311yescharacteristic
98#431—31no—
99#424—61no—
100#408—61no—
101C4×M16—31no—
102C2×#97—31no—
103#329—62no—
104Q8⋊C8—61no—
105D4⋊C8—62no—
106D4⋊C8—61no—
107C3×#41—41no—
108C3×#41—41no—
109C3×#41—82no—
110C2×C3×Q8—21no—
111C2×C3×Q8C22⋊C411yescharacteristic
112C2×C3×D4—21no—
113C2×C3×D4—21no—
114C2×C3×D4—21no—
115C2×C3×D4—21no—
116C2×C3×D4—41no—
117C2×C3×D4C22⋊C411yescharacteristic
118C22×C3×C4—21no—
119C22×C3×C4—21no—
120C22×C3×C4—21no—
121C22×C3×C4—41no—
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4#4111yescharacteristic
124C22×C3×C4#4111yescharacteristic
125C3×SD16—42no—
126C3×D8—42no—
127C2×C3×C8—21no—
128C2×C3×C8—21no—
129C3×C4⋊C4—21no—
130C3×C4⋊C4—21no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—21no—
133C3×C4⋊C4C22⋊C411yescharacteristic
134C3×C4⋊C4C22⋊C411yescharacteristic
135C3×C22⋊C4—21no—
136C3×C22⋊C4—21no—
137C3×C22⋊C4—21no—
138C3×C22⋊C4—41no—
139C3×C22⋊C4—41no—
140C3×C42—21no—
141C3×C42—21no—
142C3×C42—21no—
143C3×C42C22×C411yescharacteristic
144C3×C42C2×D411yescharacteristic
145C3×C42C2×D411yescharacteristic
146C3×C42C2×D411yescharacteristic
147#208—41no—
148#208—41no—
149C2×C3⋊C8—22no—
150C2×C3⋊C8—22no—
151C2×C3⋊C8—41no—
152C2×C3⋊C8—41no—
153C2×C3⋊C8—41no—
154C2×C3⋊C8—41no—
155C2×#41—21no—
156C2×#41#6711yescharacteristic
157C2×SD16—21no—
158C2×D8—21no—
159C2×M16—61no—
160C22×C8—61no—
161C4⋊Q8C2×Dic311yescharacteristic
162#127C2×Dic311yescharacteristic
163#124—21no—
164#121—21no—
165#121—21no—
166C4×Q8C2×Dic311yescharacteristic
167C4×Q8#6711yescharacteristic
168C4×D4—21no—
169C4×D4—21no—
170C4×D4—21no—
171C4×D4—21no—
172C4×D4C2×Dic311yescharacteristic
173C4×D4#6711yescharacteristic
174#117—21no—
175#117#6711yescharacteristic
176C2×C42—21no—
177C2×C42C22×D311yescharacteristic
178#107—21no—
179#106—21no—
180C4⋊C8—64no—
181C4⋊C8—61no—
182C4⋊C8—61no—
183C4⋊C8C2×Dic311yescharacteristic
184C4⋊C8C2×Dic311yescharacteristic
185#103—21no—
186#102—21no—
187#102—21no—
188#102—21no—
189C22⋊C8—21no—
190C22⋊C8—64no—
191C22⋊C8—61no—
192C22⋊C8—61no—
193#97—21no—
194#97—32no—
195#97—61no—
196#97—61no—
197#97—61no—
198C4×C8—32no—
199C4×C8—61no—
200C23×C3—21no—
201C23×C3—41no—
202C23×C3#12311yescharacteristic
203C3×Q8—21no—
204C3×Q8—21no—
205C3×Q8—81no—
206C3×D4—21no—
207C3×D4—21no—
208C3×D4—42no—
209C3×D4—42no—
210C3×D4—41no—
211C3×D4—41no—
212C3×D4—81no—
213C3×D4—82no—
214C2×C3×C4—21no—
215C2×C3×C4—21no—
216C2×C3×C4—21no—
217C2×C3×C4—21no—
218C2×C3×C4—21no—
219C2×C3×C4—41no—
220C2×C3×C4—41no—
221C2×C3×C4—42no—
222C2×C3×C4—41no—
223C2×C3×C4—81no—
224C2×C3×C4—81no—
225C2×C3×C4—82no—
226C2×C3×C4C2×C22⋊C411yescharacteristic
227C2×C3×C4C4×D411yescharacteristic
228C2×C3×C4C4×D411yescharacteristic
229C2×C3×C4#12011yescharacteristic
230C2×C3×C4#12111yescharacteristic
231C2×C3×C4#12111yescharacteristic
232C3×C8—41no—
233C3×C8—41no—
234C3⋊C8—42no—
235C3⋊C8—42no—
236C3⋊C8—81no—
237C3⋊C8—81no—
238#41—41no—
239#41—41no—
240#41—82no—
241C2×Q8—21no—
242C2×Q8#21711yescharacteristic
243C2×D4—21no—
244C2×D4—21no—
245C2×D4—21no—
246C2×D4—21no—
247C2×D4—41no—
248C2×D4#21711yescharacteristic
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—41no—
253C22×C4#23711yescharacteristic
254C22×C4#23711yescharacteristic
255C22×C4C2×#6711yescharacteristic
256SD16—42no—
257D8—42no—
258M16—121no—
259M16—121no—
260C2×C8—21no—
261C2×C8—21no—
262C2×C8—62no—
263C2×C8—62no—
264C2×C8—121no—
265C2×C8—121no—
266C2×C8—121no—
267C2×C8—121no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4#21711yescharacteristic
273C4⋊C4#21711yescharacteristic
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—41no—
278C22⋊C4—41no—
279C42—21no—
280C42—21no—
281C42—21no—
282C42D3×D411yescharacteristic
283C42D3×D411yescharacteristic
284C42C22×Dic311yescharacteristic
285C42C2×#6711yescharacteristic
286C22×C3—21no—
287C22×C3—21no—
288C22×C3—21no—
289C22×C3—41no—
290C22×C3—41no—
291C22×C3—41no—
292C22×C3—81no—
293C22×C3—81no—
294C22×C3—81no—
295C22×C3#38611yescharacteristic
296C3×C4—21no—
297C3×C4—21no—
298C3×C4—21no—
299C3×C4—21no—
300C3×C4—21no—
301C3×C4—41no—
302C3×C4—41no—
303C3×C4—41no—
304C3×C4—81no—
305C3×C4#42111yescharacteristic
306C3×C4#42111yescharacteristic
307C23—21no—
308C23—41no—
309C23#93511yescharacteristic
310Q8—21no—
311Q8—21no—
312Q8—81no—
313D4—21no—
314D4—21no—
315D4—42no—
316D4—42no—
317D4—41no—
318D4—41no—
319D4—81no—
320D4—82no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—42no—
329C2×C4—41no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—82no—
333C2×C4D4×Dic311yescharacteristic
334C2×C4D4×Dic311yescharacteristic
335C2×C4#93711yescharacteristic
336C2×C4#93911yescharacteristic
337C2×C4#93911yescharacteristic
338C2×C4C2×#21711yescharacteristic
339C8—41no—
340C8—41no—
341C8—122no—
342C8—122no—
343C8—241no—
344C8—241no—
345C2×C3—41no—
346C2×C3—81no—
347C2×C3—161no—
348C2×C3#188111yescharacteristic
349C2×C3#189711yescharacteristic
350C2×C3#190311yescharacteristic
351C22—21no—
352C22—21no—
353C22—21no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—81no—
358C22—81no—
359C22—81no—
360C22#522011yescharacteristic
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—41no—
367C4—41no—
368C4—41no—
369C4—81no—
370C4#523611yescharacteristic
371C4#523611yescharacteristic
372C3#931311yescharacteristic, complemented, Sylow
373C2—41no—
374C2—81no—
375C2—161no—
376C2#7171211yescharacteristic
377C2#7209011yescharacteristic
378C2#7211611yescharacteristic
subgroupG/H
centerC22#5220
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
Fitting, prime radicalC3×#3115C2
socleC22×C3#386
cosocleC2×C3×C42C23
hypercenter#3115D3
hypocenterC3#9313
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3115

Complements

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

NH ≅ G/Ntype
2. #82514C2semidirect
3. #71801C2semidirect
4. #71798C2semidirect
5. #71640C2semidirect
6. #67753C2semidirect
7. #67751C2semidirect
372. C3#9313semidirect

Maximal subgroups

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

subgroupindex
C3×#31152
#825142
#718012
#717982
#716402
#677532
#677512
#93133