Finite Group Databasebeta

← #10694590

Subgroups — #10694590

Order 768.

440 conjugacy classes of subgroups.

The subgroup lattice has 358 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#69749C211yesmaximal, characteristic, complemented
7#69732C211yesmaximal, characteristic, complemented
8#9761—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
20C2×#815—21no—
21C2×C8×Dic3—21no—
22#4932C2211yescharacteristic
23#4926C2211yescharacteristic
24#4926C2211yescharacteristic
25C2×#803C2211yescharacteristic
26#4543—22no—
27#4540—22no—
28#4538—22no—
29#4537—22no—
30#4458—21no—
31#4457—21no—
32#3167D311yescharacteristic
33#1726—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—
76#815—24no—
77C8×Dic3—24no—
78C4⋊(C3⋊C8)—21no—
79C4⋊(C3⋊C8)—21no—
80#803—21no—
81#803—21no—
82#803—21no—
83#803D412yes—
84#533C2×D311yescharacteristic
85#532C2×D311yescharacteristic
86#498Dic311yescharacteristic, complemented
87#492Dic311yescharacteristic, complemented
88#487Dic311yescharacteristic, complemented
89#486Dic311yescharacteristic, complemented
90#465—21no—
91#465—21no—
92C4×SD16—21no—
93C4×SD16—21no—
94#432—31no—
95#432—31no—
96C2×#97—31no—
97C2×#97—61no—
98C2×C4×C8—61no—
99C2×C4×C8C2×D311yescharacteristic
100#376—31no—
101#335—61no—
102#334—61no—
103#330—62no—
104#329—62no—
105#328—62no—
106#327—62no—
107C3×#41—82no—
108C2×C3×Q8—21no—
109C2×C3×Q8—21no—
110C2×C3×Q8—21no—
111C2×C3×D4—21no—
112C2×C3×D4—21no—
113C2×C3×D4—21no—
114C2×C3×D4—41no—
115C22×C3×C4—41no—
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4#4111yescharacteristic
119C22×Dic3—22no—
120C3×SD16—42no—
121C3×SD16—42no—
122C2×C3×C8—22no—
123C2×C3×C8—22no—
124C2×C3×C8C22⋊C411yescharacteristic
125C2×C3×C8C22⋊C411yescharacteristic
126C2×C3×C8C22⋊C411yescharacteristic
127C2×C3×C8C22⋊C411yescharacteristic
128C3×C4⋊C4—21no—
129C3×C4⋊C4—21no—
130C3×C4⋊C4—21no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—21no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4—41no—
135C3×C4⋊C4—41no—
136C3×C22⋊C4—41no—
137C3×C22⋊C4—41no—
138C3×C22⋊C4—41no—
139C3×C42—41no—
140C3×C42C22×C411yescharacteristic
141C3×C42C2×D411yescharacteristic
142C3×C42C2×D411yescharacteristic
143C3×C42#4111yescharacteristic
144C4×Dic3—22no—
145C4×Dic3—22no—
146C2×C3⋊C8—22no—
147C2×C3⋊C8—22no—
148C2×C3⋊C8—22no—
149C2×C3⋊C8—41no—
150C2×C3⋊C8—41no—
151C2×#41—21no—
152C2×SD16—21no—
153C2×SD16—21no—
154C22×C8—61no—
155C22×C8#6711yescharacteristic
156C22×C8#6711yescharacteristic
157C4⋊Q8C2×Dic311yescharacteristic
158C4⋊Q8C2×Dic311yescharacteristic
159C4⋊Q8C2×Dic311yescharacteristic
160#127C2×Dic311yescharacteristic
161#125—21no—
162#124—21no—
163C22⋊Q8—21no—
164C22⋊Q8—21no—
165#121—21no—
166#121—21no—
167C4×Q8—21no—
168C4×Q8—21no—
169C4×D4—21no—
170C4×D4—21no—
171#117—21no—
172C2×C42—61no—
173C2×C42C22×D311yescharacteristic
174#106—21no—
175#106—21no—
176C4⋊C8—61no—
177C4⋊C8—61no—
178#103—21no—
179#103—21no—
180#102—21no—
181#102—21no—
182C22⋊C8—61no—
183C22⋊C8—61no—
184#97—32no—
185#97—64no—
186#97—61no—
187#97—61no—
188#97—61no—
189C4×C8—64no—
190C4×C8C2×Dic311yescharacteristic
191C4×C8C2×Dic311yescharacteristic
192C4×C8#6711yescharacteristic
193C4×C8#6711yescharacteristic
194#95—32no—
195#95—61no—
196C23×C3—41no—
197C23×C3#12111yescharacteristic
198C3×Q8—41no—
199C3×Q8—41no—
200C3×Q8—81no—
201C3×Q8—81no—
202C3×D4—41no—
203C3×D4—41no—
204C3×D4—81no—
205C3×D4—81no—
206C3×D4—82no—
207C2×C3×C4—22no—
208C2×C3×C4—21no—
209C2×C3×C4—21no—
210C2×C3×C4—41no—
211C2×C3×C4—41no—
212C2×C3×C4—41no—
213C2×C3×C4—41no—
214C2×C3×C4—82no—
215C2×C3×C4C2×C22⋊C411yescharacteristic
216C2×C3×C4C4×D411yescharacteristic
217C2×C3×C4C4×D411yescharacteristic
218C2×C3×C4#12111yescharacteristic
219C2×C3×C4#12411yescharacteristic
220C2×C3×C4#12711yescharacteristic
221C2×Dic3—24no—
222C2×Dic3—42no—
223C2×Dic3—42no—
224C3×C8—21no—
225C3×C8—21no—
226C3×C8#10412yes—
227C3×C8#10412yes—
228C3⋊C8—24no—
229C3⋊C8—81no—
230C3⋊C8—81no—
231#41—82no—
232C2×Q8—21no—
233C2×Q8—21no—
234C2×Q8—21no—
235C2×D4—21no—
236C2×D4—21no—
237C2×D4—21no—
238C2×D4—41no—
239C22×C4—41no—
240C22×C4—62no—
241C22×C4D3×D411yescharacteristic
242C22×C4#23711yescharacteristic
243C22×C4C2×#6711yescharacteristic
244SD16—42no—
245SD16—42no—
246C2×C8—22no—
247C2×C8—22no—
248C2×C8—62no—
249C2×C8—62no—
250C2×C8—62no—
251C2×C8—121no—
252C2×C8—121no—
253C2×C8#21711yescharacteristic
254C2×C8#21711yescharacteristic
255C2×C8#21711yescharacteristic
256C2×C8#21711yescharacteristic
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—21no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C22⋊C4—41no—
266C22⋊C4—41no—
267C22⋊C4—41no—
268C42—41no—
269C42—62no—
270C42—62no—
271C42D3×D411yescharacteristic
272C42#23711yescharacteristic
273C42C22×Dic311yescharacteristic
274C42C2×#6711yescharacteristic
275C22×C3—21no—
276C22×C3—21no—
277C22×C3—81no—
278C22×C3—81no—
279C22×C3—81no—
280C22×C3#39011yescharacteristic
281C22×C3#47112yes—
282C3×C4—21no—
283C3×C4—21no—
284C3×C4—21no—
285C3×C4—21no—
286C3×C4—21no—
287C3×C4—81no—
288C3×C4—81no—
289C3×C4—81no—
290C3×C4C2×#10411yescharacteristic
291C3×C4C2×#10411yescharacteristic
292Dic3—44no—
293C23—41no—
294C23#93811yescharacteristic
295Q8—41no—
296Q8—41no—
297Q8—81no—
298Q8—81no—
299D4—41no—
300D4—41no—
301D4—81no—
302D4—81no—
303D4—82no—
304C2×C4—22no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—64no—
312C2×C4—82no—
313C2×C4—122no—
314C2×C4—122no—
315C2×C4D4×Dic311yescharacteristic
316C2×C4D4×Dic311yescharacteristic
317C2×C4#93611yescharacteristic
318C2×C4#93811yescharacteristic
319C2×C4#94011yescharacteristic
320C2×C4C2×#21711yescharacteristic
321C8—21no—
322C8—21no—
323C8—64no—
324C8—241no—
325C8—241no—
326C8#83712yes—
327C8#83712yes—
328C2×C3—22no—
329C2×C3—161no—
330C2×C3#196211yescharacteristic
331C2×C3#197611yescharacteristic
332C2×C3#197711yescharacteristic
333C22—21no—
334C22—21no—
335C22—81no—
336C22—81no—
337C22—81no—
338C22#517712yes—
339C22#522111yescharacteristic
340C4—21no—
341C4—21no—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—81no—
346C4—81no—
347C4—81no—
348C4—124no—
349C4C2×#83711yescharacteristic
350C4C2×#83711yescharacteristic
351C3#976111yescharacteristic, complemented, Sylow
352C2—22no—
353C2—161no—
354C2#7181511yescharacteristic
355C2#7205211yescharacteristic
356C2#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#9761
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. #69749C2semidirect
7. #69732C2semidirect
11. C3×#498C4semidirect
12. C3×#492C4semidirect
13. C3×#487C4semidirect
14. C3×#486C4semidirect
86. #498Dic3semidirect
87. #492Dic3semidirect
88. #487Dic3semidirect
89. #486Dic3semidirect
351. C3#9761semidirect

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
#697492
#697322
#97613