Finite Group Databasebeta

← #10694617

Subgroups — #10694617

Order 768.

396 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3167C211yesmaximal, characteristic
2#71836C211yesmaximal, characteristic, complemented
3#71822C211yesmaximal, characteristic, complemented
4#71812C211yesmaximal, characteristic
5#71798C211yesmaximal, characteristic
6#69751C211yesmaximal, characteristic, complemented
7#69732C211yesmaximal, characteristic, complemented
8#9766—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C3×#532C2211yescharacteristic
11C3×#498C411yescharacteristic, complemented
12C3×#492C411yescharacteristic, complemented
13C3×#487C411yescharacteristic, complemented
14C3×#486C411yescharacteristic, complemented
15C3×#465—21no—
16C3×#465—21no—
17C3×C4×SD16—21no—
18C3×C4×SD16—21no—
19C2×C3×C4×C8C2211yescharacteristic
20#4932C2211yescharacteristic
21#4926C2211yescharacteristic
22#4926C2211yescharacteristic
23C2×#803C2211yescharacteristic
24#4550—21no—
25#4550—21no—
26#4543—22no—
27#4540—22no—
28#4538—22no—
29#4537—22no—
30#4458—21no—
31#4457—21no—
32#3167D311yescharacteristic
33#1746—31no—
34#1568—31no—
35#1531—31no—
36#1530—31no—
37#1506—31no—
38#1505—31no—
39C2×C3×#41—21no—
40C2×C3×SD16—21no—
41C2×C3×SD16—21no—
42C22×C3×C8D411yescharacteristic
43C22×C3×C8D411yescharacteristic
44C3×C4⋊Q8C2×C411yescharacteristic
45C3×C4⋊Q8C2×C411yescharacteristic
46C3×C4⋊Q8C2×C411yescharacteristic
47C3×#127C2×C411yescharacteristic
48C3×#125—21no—
49C3×#124—21no—
50C3×C22⋊Q8—21no—
51C3×C22⋊Q8—21no—
52C3×#121—21no—
53C3×#121—21no—
54C3×C4×Q8—21no—
55C3×C4×Q8—21no—
56C3×C4×D4—21no—
57C3×C4×D4—21no—
58C3×#117—21no—
59C2×C3×C42C2311yescharacteristic
60C2×C4×Dic3—21no—
61C22×C3⋊C8—21no—
62C3×#106—21no—
63C3×#106—21no—
64C3×#103—21no—
65C3×#103—21no—
66C3×#102—21no—
67C3×#102—21no—
68C3×C4×C8C2×C411yescharacteristic
69C3×C4×C8C2×C411yescharacteristic
70C3×C4×C8D411yescharacteristic
71C3×C4×C8D411yescharacteristic
72C3⋊#95—21no—
73C3⋊#95D412yes—
74#830—21no—
75#830—21no—
76C4⋊(C3⋊C8)—21no—
77C4⋊(C3⋊C8)—21no—
78#803—21no—
79#803—21no—
80#803—21no—
81#803D412yes—
82#533C2×D311yescharacteristic
83#532C2×D311yescharacteristic
84#498Dic311yescharacteristic, complemented
85#492Dic311yescharacteristic, complemented
86#487Dic311yescharacteristic, complemented
87#486Dic311yescharacteristic, complemented
88#465—21no—
89#465—21no—
90C4×SD16—21no—
91C4×SD16—21no—
92#432—31no—
93#432—31no—
94C2×#97—31no—
95C2×C4×C8C2×D311yescharacteristic
96#376—31no—
97#336—61no—
98#336—61no—
99#335—61no—
100#334—61no—
101#330—62no—
102#329—62no—
103#328—62no—
104#327—62no—
105C3×#41—82no—
106C2×C3×Q8—21no—
107C2×C3×Q8—21no—
108C2×C3×Q8—21no—
109C2×C3×D4—21no—
110C2×C3×D4—21no—
111C2×C3×D4—21no—
112C2×C3×D4—41no—
113C22×C3×C4—41no—
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4#4111yescharacteristic
117C22×Dic3—22no—
118C3×SD16—42no—
119C3×SD16—42no—
120C2×C3×C8—41no—
121C2×C3×C8—41no—
122C2×C3×C8C22⋊C411yescharacteristic
123C2×C3×C8C22⋊C411yescharacteristic
124C2×C3×C8C22⋊C411yescharacteristic
125C2×C3×C8C22⋊C411yescharacteristic
126C3×C4⋊C4—21no—
127C3×C4⋊C4—21no—
128C3×C4⋊C4—21no—
129C3×C4⋊C4—21no—
130C3×C4⋊C4—21no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—41no—
133C3×C4⋊C4—41no—
134C3×C22⋊C4—41no—
135C3×C22⋊C4—41no—
136C3×C22⋊C4—41no—
137C3×C42—41no—
138C3×C42C22×C411yescharacteristic
139C3×C42C2×D411yescharacteristic
140C3×C42C2×D411yescharacteristic
141C3×C42#4111yescharacteristic
142C4×Dic3—41no—
143C4×Dic3—41no—
144C2×C3⋊C8—22no—
145C2×C3⋊C8—41no—
146C2×C3⋊C8—41no—
147C2×C3⋊C8—41no—
148C2×C3⋊C8—41no—
149C2×#41—21no—
150C2×SD16—21no—
151C2×SD16—21no—
152C22×C8—61no—
153C22×C8#6711yescharacteristic
154C22×C8#6711yescharacteristic
155C4⋊Q8C2×Dic311yescharacteristic
156C4⋊Q8C2×Dic311yescharacteristic
157C4⋊Q8C2×Dic311yescharacteristic
158#127C2×Dic311yescharacteristic
159#125—21no—
160#124—21no—
161C22⋊Q8—21no—
162C22⋊Q8—21no—
163#121—21no—
164#121—21no—
165C4×Q8—21no—
166C4×Q8—21no—
167C4×D4—21no—
168C4×D4—21no—
169#117—21no—
170C2×C42—61no—
171C2×C42C22×D311yescharacteristic
172#106—21no—
173#106—21no—
174C4⋊C8—61no—
175C4⋊C8—61no—
176#103—21no—
177#103—21no—
178#102—21no—
179#102—21no—
180C22⋊C8—61no—
181C22⋊C8—61no—
182#97—32no—
183#97—61no—
184#97—61no—
185#97—61no—
186C4×C8C2×Dic311yescharacteristic
187C4×C8C2×Dic311yescharacteristic
188C4×C8#6711yescharacteristic
189C4×C8#6711yescharacteristic
190#95—32no—
191#95—61no—
192C23×C3—41no—
193C23×C3#12111yescharacteristic
194C3×Q8—41no—
195C3×Q8—41no—
196C3×Q8—81no—
197C3×Q8—81no—
198C3×D4—41no—
199C3×D4—41no—
200C3×D4—81no—
201C3×D4—81no—
202C3×D4—82no—
203C2×C3×C4—22no—
204C2×C3×C4—21no—
205C2×C3×C4—21no—
206C2×C3×C4—41no—
207C2×C3×C4—41no—
208C2×C3×C4—41no—
209C2×C3×C4—41no—
210C2×C3×C4—82no—
211C2×C3×C4C2×C22⋊C411yescharacteristic
212C2×C3×C4C4×D411yescharacteristic
213C2×C3×C4C4×D411yescharacteristic
214C2×C3×C4#12111yescharacteristic
215C2×C3×C4#12411yescharacteristic
216C2×C3×C4#12711yescharacteristic
217C2×Dic3—42no—
218C2×Dic3—42no—
219C2×Dic3—42no—
220C3×C8—21no—
221C3×C8—21no—
222C3×C8—21no—
223C3×C8—21no—
224C3⋊C8—42no—
225C3⋊C8—81no—
226C3⋊C8—81no—
227#41—82no—
228C2×Q8—21no—
229C2×Q8—21no—
230C2×Q8—21no—
231C2×D4—21no—
232C2×D4—21no—
233C2×D4—21no—
234C2×D4—41no—
235C22×C4—41no—
236C22×C4—62no—
237C22×C4D3×D411yescharacteristic
238C22×C4#23711yescharacteristic
239C22×C4C2×#6711yescharacteristic
240SD16—42no—
241SD16—42no—
242C2×C8—41no—
243C2×C8—41no—
244C2×C8—62no—
245C2×C8—121no—
246C2×C8—121no—
247C2×C8—121no—
248C2×C8—121no—
249C2×C8#21711yescharacteristic
250C2×C8#21711yescharacteristic
251C2×C8#21711yescharacteristic
252C2×C8#21711yescharacteristic
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C22⋊C4—41no—
262C22⋊C4—41no—
263C22⋊C4—41no—
264C42—41no—
265C42—121no—
266C42—121no—
267C42D3×D411yescharacteristic
268C42#23711yescharacteristic
269C42C22×Dic311yescharacteristic
270C42C2×#6711yescharacteristic
271C22×C3—21no—
272C22×C3—21no—
273C22×C3—81no—
274C22×C3—81no—
275C22×C3—81no—
276C22×C3#39011yescharacteristic
277C22×C3#47112yes—
278C3×C4—21no—
279C3×C4—21no—
280C3×C4—21no—
281C3×C4—21no—
282C3×C4—21no—
283C3×C4—81no—
284C3×C4—81no—
285C3×C4—81no—
286C3×C4#42111yescharacteristic
287C3×C4#42111yescharacteristic
288Dic3—82no—
289C23—41no—
290C23#93811yescharacteristic
291Q8—41no—
292Q8—41no—
293Q8—81no—
294Q8—81no—
295D4—41no—
296D4—41no—
297D4—81no—
298D4—81no—
299D4—82no—
300C2×C4—22no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—82no—
308C2×C4—122no—
309C2×C4—122no—
310C2×C4—122no—
311C2×C4D4×Dic311yescharacteristic
312C2×C4D4×Dic311yescharacteristic
313C2×C4#93611yescharacteristic
314C2×C4#93811yescharacteristic
315C2×C4#94011yescharacteristic
316C2×C4C2×#21711yescharacteristic
317C8—21no—
318C8—21no—
319C8—21no—
320C8—21no—
321C8—122no—
322C8—241no—
323C8—241no—
324C2×C3—22no—
325C2×C3—161no—
326C2×C3#196311yescharacteristic
327C2×C3#197611yescharacteristic
328C2×C3#197711yescharacteristic
329C22—21no—
330C22—21no—
331C22—81no—
332C22—81no—
333C22—81no—
334C22#517712yes—
335C22#522111yescharacteristic
336C4—21no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—21no—
341C4—81no—
342C4—81no—
343C4—81no—
344C4—242no—
345C4#523611yescharacteristic
346C4#523611yescharacteristic
347C3#976611yescharacteristic, complemented, Sylow
348C2—22no—
349C2—161no—
350C2#7181611yescharacteristic
351C2#7205211yescharacteristic
352C2#7208611yescharacteristic
subgroupG/H
centerC22#5221
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#3167C2
socleC22×C3#390
prime radicalC3×#532C22
cosocleC2×C3×C42C23
hypercenter#3167D3
hypocenterC3#9766
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3167

Complements

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

NH ≅ G/Ntype
2. #71836C2semidirect
3. #71822C2semidirect
6. #69751C2semidirect
7. #69732C2semidirect
11. C3×#498C4semidirect
12. C3×#492C4semidirect
13. C3×#487C4semidirect
14. C3×#486C4semidirect
84. #498Dic3semidirect
85. #492Dic3semidirect
86. #487Dic3semidirect
87. #486Dic3semidirect
347. C3#9766semidirect

Maximal subgroups

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

subgroupindex
C3×#31672
#718362
#718222
#718122
#717982
#697512
#697322
#97663