Finite Group Databasebeta

← #10691893

Subgroups — #10691893

Order 768.

470 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2327C211yesmaximal, characteristic
2#83223C211yesmaximal, characteristic
3#71773C211yesmaximal, characteristic
4#69546C211yesmaximal, characteristic
5#69441C211yesmaximal, characteristic
6#67631C211yesmaximal, characteristic
7#67614C211yesmaximal, characteristic
8#9296—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×C3×C4×Q8—21no—
11#5470C2211yescharacteristic
12C3×#391—21no—
13C3×#387—22no—
14C3×#387—22no—
15C3×#382C2211yescharacteristic
16C3×#376C2211yescharacteristic
17#5094—22no—
18#5093C412yescomplemented
19#5080C412yescomplemented
20#4971—22no—
21#4925C2211yescharacteristic
22#4925C2211yescharacteristic
23#4924C2211yescharacteristic
24#4547—22no—
25#4523—21no—
26#4523—21no—
27#4452—22no—
28#3229—31no—
29#2327D311yescharacteristic
30#1995—31no—
31#1546—31no—
32#1518—31no—
33#1286—31no—
34#1274—31no—
35C22×C3×Q8—21no—
36C3×C4⋊Q8—21no—
37C3×C4⋊Q8—21no—
38C3×C4⋊Q8C2×C412yes—
39C3×C4⋊Q8D412yes—
40C3×C4×Q8—42no—
41C3×C4×Q8—42no—
42C2×C3×C4⋊C4—21no—
43C2×C3×C4⋊C4—22no—
44C2×C3×C4⋊C4—21no—
45C2×C3×C4⋊C4D412yes—
46C2×C3×C42—22no—
47C2×C3×C42D411yescharacteristic
48C2×C3×C42D411yescharacteristic
49C2×C3×C42C2311yescharacteristic
50#926—21no—
51#924—21no—
52#924—21no—
53C2×#208—21no—
54#870—21no—
55#869C2×C411yescharacteristic
56#869C2×C411yescharacteristic
57C4×Dic6—21no—
58C4×Dic6—21no—
59C3×#95—24no—
60C3×#95—21no—
61C3×#95—21no—
62C3×#95—21no—
63C3×#95D412yes—
64#830—21no—
65#830—21no—
66#810—24no—
67#808—24no—
68C4⋊(C3⋊C8)—21no—
69C4⋊(C3⋊C8)—21no—
70C4⋊(C3⋊C8)—21no—
71C4⋊(C3⋊C8)—21no—
72C4×C3⋊C8—21no—
73C4×C3⋊C8C2×C412yes—
74#533—31no—
75C2×C4⋊Q8C2×D311yescharacteristic
76C2×C4×Q8—21no—
77#498—32no—
78#487—32no—
79#483—62no—
80#475—62no—
81#431—31no—
82#431—31no—
83#423—31no—
84#391—21no—
85#387—22no—
86#387—22no—
87#382C2×D311yescharacteristic
88#376C2×D311yescharacteristic
89#339—61no—
90#339—61no—
91#332—62no—
92#328—62no—
93C2×C3×Q8—22no—
94C2×C3×Q8—22no—
95C2×C3×Q8—41no—
96C2×C3×Q8—41no—
97C2×C3×Q8—41no—
98C2×C3×Q8—41no—
99C22×C3×C4—21no—
100C22×C3×C4—22no—
101C22×C3×C4—21no—
102C22×C3×C4—22no—
103C22×C3×C4C2×D411yescharacteristic
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C2×Dic6—21no—
107C2×Dic6—21no—
108C3×C4⋊C4—22no—
109C3×C4⋊C4—22no—
110C3×C4⋊C4—22no—
111C3×C4⋊C4—44no—
112C3×C4⋊C4—41no—
113C3×C4⋊C4—41no—
114C3×C4⋊C4C22⋊C414yes—
115C3×C42—21no—
116C3×C42—21no—
117C3×C42—21no—
118C3×C42—44no—
119C3×C42SD1612yes—
120C3×C42C22×C411yescharacteristic
121C3×C42C2×D411yescharacteristic
122C3×C42#4111yescharacteristic
123C3×C42#4111yescharacteristic
124#217—41no—
125C4⋊Dic3—21no—
126C4⋊Dic3—21no—
127C4⋊Dic3—21no—
128C4⋊Dic3—21no—
129Dic3⋊C4—41no—
130Dic3⋊C4—41no—
131C4×Dic3—41no—
132#208—41no—
133#208—41no—
134C2×C3⋊C8—22no—
135C2×C3⋊C8—41no—
136C2×C3⋊C8—41no—
137C22×Q8—21no—
138C2×M16—61no—
139C4⋊Q8—21no—
140C4⋊Q8—21no—
141C4⋊Q8—31no—
142C4⋊Q8—31no—
143C4⋊Q8C4×D312yes—
144C4⋊Q8D1212yes—
145#125—61no—
146C22⋊Q8—61no—
147C22⋊Q8—61no—
148C4×Q8—42no—
149C4×Q8—42no—
150C4×Q8—61no—
151C4×Q8—61no—
152#117—61no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4#6712yes—
157C2×C42—22no—
158C2×C42#6711yescharacteristic
159C2×C42#6711yescharacteristic
160C2×C42C22×D311yescharacteristic
161#106—64no—
162C4⋊C8—61no—
163C4⋊C8—61no—
164C4⋊C8—61no—
165C4⋊C8—61no—
166#103—64no—
167C22⋊C8—61no—
168C22⋊C8—61no—
169C4×C8—32no—
170C4×C8—61no—
171#95—24no—
172#95—21no—
173#95—21no—
174#95—21no—
175#95#6712yes—
176C23×C3#12011yescharacteristic
177C3×Q8—42no—
178C3×Q8—42no—
179C3×Q8—42no—
180C3×Q8—42no—
181C2×C3×C4—24no—
182C2×C3×C4—22no—
183C2×C3×C4—41no—
184C2×C3×C4—41no—
185C2×C3×C4—41no—
186C2×C3×C4—41no—
187C2×C3×C4—42no—
188C2×C3×C4—42no—
189C2×C3×C4—41no—
190C2×C3×C4—42no—
191C2×C3×C4—41no—
192C2×C3×C4—42no—
193C2×C3×C4—42no—
194C2×C3×C4—41no—
195C2×C3×C4C2×C22⋊C411yescharacteristic
196C2×C3×C4C4×D411yescharacteristic
197C2×C3×C4C4×D411yescharacteristic
198C2×C3×C4#12111yescharacteristic
199C2×C3×C4#12111yescharacteristic
200C2×C3×C4#12311yescharacteristic
201C2×C3×C4C2×SD1611yescharacteristic
202C2×C3×C4#13511yescharacteristic
203C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
204C2×C3×C4#13711yescharacteristic
205C2×Dic3—41no—
206C2×Dic3—41no—
207Dic6—42no—
208Dic6—81no—
209C3⋊C8—42no—
210C3⋊C8—81no—
211C3⋊C8—81no—
212C2×Q8—22no—
213C2×Q8—22no—
214C2×Q8—41no—
215C2×Q8—41no—
216C2×Q8—41no—
217C2×Q8—41no—
218C2×Q8—61no—
219C2×Q8—61no—
220C22×C4—21no—
221C22×C4—22no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4C2×#6711yescharacteristic
225C22×C4C2×#6711yescharacteristic
226C22×C4C2×#6711yescharacteristic
227M16—121no—
228M16—121no—
229C2×C8—62no—
230C2×C8—121no—
231C2×C8—121no—
232C4⋊C4—22no—
233C4⋊C4—22no—
234C4⋊C4—22no—
235C4⋊C4—44no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—61no—
239C4⋊C4—61no—
240C4⋊C4—61no—
241C4⋊C4—61no—
242C4⋊C4—121no—
243C4⋊C4—121no—
244C4⋊C4(C2×D3)⋊C414yes—
245C22⋊C4—121no—
246C42—21no—
247C42—21no—
248C42—21no—
249C42—44no—
250C42—121no—
251C42#21412yes—
252C42C2×C4×D311yescharacteristic
253C42C2×D1211yescharacteristic
254C42#23511yescharacteristic
255C42#23511yescharacteristic
256C22×C3—21no—
257C22×C3—21no—
258C22×C3#38611yescharacteristic
259C22×C3#45011yescharacteristic
260C22×C3#45211yescharacteristic
261C3×C4—21no—
262C3×C4—21no—
263C3×C4—21no—
264C3×C4—21no—
265C3×C4—42no—
266C3×C4—44no—
267C3×C4—82no—
268C3×C4—81no—
269C3×C4C4×SD1611yescharacteristic
270C3×C4#44011yescharacteristic
271C3×C4#46111yescharacteristic
272C3×C4#46311yescharacteristic
273Dic3—81no—
274Dic3—81no—
275C23#95311yescharacteristic
276Q8—42no—
277Q8—42no—
278Q8—42no—
279Q8—42no—
280Q8—122no—
281Q8—241no—
282C2×C4—24no—
283C2×C4—22no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—41no—
296C2×C4—121no—
297C2×C4—121no—
298C2×C4C2×(C2×D3)⋊C411yescharacteristic
299C2×C4C4×#6711yescharacteristic
300C2×C4C4×#6711yescharacteristic
301C2×C4#92911yescharacteristic
302C2×C4#93011yescharacteristic
303C2×C4#93011yescharacteristic
304C2×C4#93211yescharacteristic
305C2×C4C2×#21411yescharacteristic
306C2×C4#95011yescharacteristic
307C2×C4#95111yescharacteristic
308C8—122no—
309C8—241no—
310C8—241no—
311C2×C3—21no—
312C2×C3—21no—
313C2×C3#187711yescharacteristic
314C2×C3#190511yescharacteristic
315C2×C3#191211yescharacteristic
316C22—21no—
317C22—21no—
318C22#521311yescharacteristic
319C22#521811yescharacteristic
320C22#523911yescharacteristic
321C4—21no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—42no—
326C4—44no—
327C4—82no—
328C4—81no—
329C4—241no—
330C4—241no—
331C4#501611yescharacteristic
332C4C4×#21411yescharacteristic
333C4#501811yescharacteristic
334C4#501911yescharacteristic
335C3#929611yescharacteristic, complemented, Sylow
336C2—21no—
337C2—21no—
338C2#7161211yescharacteristic
339C2#7166211yescharacteristic
340C2#7166311yescharacteristic
subgroupG/H
centerC22#5213
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2327C2
socleC22×C3#386
prime radicalC23×C3#120
cosocleC2×C3×C42C23
hypercenter#2327D3
hypocenterC3#9296
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2327

Complements

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

NH ≅ G/Ntype
18. #5093C4semidirect
19. #5080C4semidirect
335. C3#9296semidirect

Maximal subgroups

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

subgroupindex
C3×#23272
#832232
#717732
#695462
#694412
#676312
#676142
#92963