Finite Group Databasebeta

← #10691900

Subgroups — #10691900

Order 768.

454 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2331C211yesmaximal, characteristic
2#83222C211yesmaximal, characteristic
3#71774C211yesmaximal, characteristic
4#69546C211yesmaximal, characteristic
5#69441C211yesmaximal, characteristic
6#67631C211yesmaximal, characteristic
7#67614C211yesmaximal, characteristic
8#9297—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10#5470C2211yescharacteristic
11C3×#391—22no—
12C3×#389—21no—
13C3×#387—22no—
14C3×#382C2211yescharacteristic
15C3×C4×C4⋊C4—21no—
16C3×#376C2211yescharacteristic
17#5228—22no—
18#5092C412yescomplemented
19#5090—22no—
20#5079C412yescomplemented
21#4925C2211yescharacteristic
22#4925C2211yescharacteristic
23#4924C2211yescharacteristic
24#4546—22no—
25#4523—21no—
26#4523—21no—
27#4452—22no—
28#3209—31no—
29#2331D311yescharacteristic
30#1995—31no—
31#1548—31no—
32#1518—31no—
33#1286—31no—
34#1274—31no—
35C22×C3×Q8D412yes—
36C3×C4⋊Q8—21no—
37C3×C4⋊Q8—21no—
38C3×C4⋊Q8C2×C412yes—
39C3×C4⋊Q8D412yes—
40C2×C3×C4⋊C4—21no—
41C2×C3×C4⋊C4—21no—
42C2×C3×C4⋊C4—21no—
43C2×C3×C4⋊C4—22no—
44C2×C3×C42—22no—
45C2×C3×C42D411yescharacteristic
46C2×C3×C42D411yescharacteristic
47C2×C3×C42C2311yescharacteristic
48C2×#216—24no—
49#926—21no—
50#924—21no—
51#924—21no—
52C2×#208—21no—
53#870—21no—
54#869C2×C411yescharacteristic
55#869C2×C411yescharacteristic
56C4×Dic6—21no—
57C4×Dic6—21no—
58C3×#95—22no—
59C3×#95—22no—
60C3×#95—21no—
61C3×#95—21no—
62C3×#95—21no—
63C3×#95D412yes—
64#835—24no—
65#830—21no—
66#830—21no—
67C4⋊(C3⋊C8)—21no—
68C4⋊(C3⋊C8)—21no—
69C4⋊(C3⋊C8)—21no—
70C4⋊(C3⋊C8)—21no—
71C4×C3⋊C8—21no—
72C4×C3⋊C8C2×C412yes—
73#533—31no—
74C2×C4⋊Q8C2×D311yescharacteristic
75#494—32no—
76#487—32no—
77#462—62no—
78#451—62no—
79#431—31no—
80#431—31no—
81#423—31no—
82#391—22no—
83#389—21no—
84#387—22no—
85#382C2×D311yescharacteristic
86C4×C4⋊C4—21no—
87#376C2×D311yescharacteristic
88#339—61no—
89#339—61no—
90#333—62no—
91#328—62no—
92C2×C3×Q8—22no—
93C2×C3×Q8—82no—
94C2×C3×Q8C22⋊C414yes—
95C22×C3×C4—21no—
96C22×C3×C4—22no—
97C22×C3×C4—21no—
98C22×C3×C4—22no—
99C22×C3×C4C2×D411yescharacteristic
100C22×C3×C4C2×D411yescharacteristic
101C22×C3×C4C2×D411yescharacteristic
102C2×Dic6—21no—
103C2×Dic6—21no—
104C3×C4⋊C4—22no—
105C3×C4⋊C4—22no—
106C3×C4⋊C4—22no—
107C3×C4⋊C4—22no—
108C3×C4⋊C4—41no—
109C3×C4⋊C4—41no—
110C3×C4⋊C4—82no—
111C3×C42—21no—
112C3×C42—21no—
113C3×C42—21no—
114C3×C42—82no—
115C3×C42Q1612yescomplemented
116C3×C42C22×C411yescharacteristic
117C3×C42C2×D411yescharacteristic
118C3×C42#4111yescharacteristic
119C3×C42#4111yescharacteristic
120#217—41no—
121#216—84no—
122C4⋊Dic3—21no—
123C4⋊Dic3—21no—
124C4⋊Dic3—21no—
125C4⋊Dic3—21no—
126Dic3⋊C4—41no—
127Dic3⋊C4—41no—
128C4×Dic3—41no—
129#208—41no—
130#208—41no—
131C2×C3⋊C8—22no—
132C2×C3⋊C8—41no—
133C2×C3⋊C8—41no—
134C22×Q8#6712yes—
135C2×Q16—64no—
136C2×M16—61no—
137C4⋊Q8—21no—
138C4⋊Q8—21no—
139C4⋊Q8—31no—
140C4⋊Q8—31no—
141C4⋊Q8C4×D312yes—
142C4⋊Q8D1212yes—
143#125—61no—
144C22⋊Q8—61no—
145C22⋊Q8—61no—
146C4×Q8—61no—
147C4×Q8—61no—
148#117—61no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C42—22no—
154C2×C42#6711yescharacteristic
155C2×C42#6711yescharacteristic
156C2×C42C22×D311yescharacteristic
157C4⋊C8—61no—
158C4⋊C8—61no—
159C4⋊C8—61no—
160C4⋊C8—61no—
161#103—64no—
162C22⋊C8—61no—
163C22⋊C8—61no—
164C4×C8—32no—
165C4×C8—61no—
166#95—22no—
167#95—22no—
168#95—21no—
169#95—21no—
170#95—21no—
171#95#6712yes—
172C23×C3#12011yescharacteristic
173C3×Q8—44no—
174C3×Q8—82no—
175C2×C3×C4—24no—
176C2×C3×C4—22no—
177C2×C3×C4—41no—
178C2×C3×C4—41no—
179C2×C3×C4—41no—
180C2×C3×C4—42no—
181C2×C3×C4—42no—
182C2×C3×C4—41no—
183C2×C3×C4—42no—
184C2×C3×C4—41no—
185C2×C3×C4—41no—
186C2×C3×C4—42no—
187C2×C3×C4—41no—
188C2×C3×C4—42no—
189C2×C3×C4C2×C22⋊C411yescharacteristic
190C2×C3×C4C4×D411yescharacteristic
191C2×C3×C4C4×D411yescharacteristic
192C2×C3×C4#12111yescharacteristic
193C2×C3×C4#12111yescharacteristic
194C2×C3×C4#12311yescharacteristic
195C2×C3×C4C2×Q1611yescharacteristic
196C2×C3×C4#13511yescharacteristic
197C2×C3×C4#13711yescharacteristic
198C2×C3×C4#13711yescharacteristic
199C2×Dic3—41no—
200C2×Dic3—41no—
201Dic6—42no—
202Dic6—81no—
203C3⋊C8—42no—
204C3⋊C8—81no—
205C3⋊C8—81no—
206C2×Q8—22no—
207C2×Q8—61no—
208C2×Q8—61no—
209C2×Q8—82no—
210C2×Q8(C2×D3)⋊C414yes—
211C22×C4—21no—
212C22×C4—22no—
213C22×C4—21no—
214C22×C4—22no—
215C22×C4C2×#6711yescharacteristic
216C22×C4C2×#6711yescharacteristic
217C22×C4C2×#6711yescharacteristic
218Q16—244no—
219M16—121no—
220M16—121no—
221C2×C8—62no—
222C2×C8—121no—
223C2×C8—121no—
224C4⋊C4—22no—
225C4⋊C4—22no—
226C4⋊C4—22no—
227C4⋊C4—22no—
228C4⋊C4—41no—
229C4⋊C4—41no—
230C4⋊C4—61no—
231C4⋊C4—61no—
232C4⋊C4—61no—
233C4⋊C4—61no—
234C4⋊C4—82no—
235C4⋊C4—121no—
236C4⋊C4—121no—
237C22⋊C4—121no—
238C42—21no—
239C42—21no—
240C42—21no—
241C42—82no—
242C42—121no—
243C42#21612yescomplemented
244C42C2×C4×D311yescharacteristic
245C42C2×D1211yescharacteristic
246C42#23511yescharacteristic
247C42#23511yescharacteristic
248C22×C3—21no—
249C22×C3—21no—
250C22×C3#38611yescharacteristic
251C22×C3#45111yescharacteristic
252C22×C3#45211yescharacteristic
253C3×C4—21no—
254C3×C4—21no—
255C3×C4—21no—
256C3×C4—21no—
257C3×C4—42no—
258C3×C4—44no—
259C3×C4—82no—
260C3×C4—81no—
261C3×C4C4×Q1611yescharacteristic
262C3×C4#44111yescharacteristic
263C3×C4#46211yescharacteristic
264C3×C4#46411yescharacteristic
265Dic3—81no—
266Dic3—81no—
267C23#95311yescharacteristic
268Q8—44no—
269Q8—82no—
270Q8—122no—
271Q8—241no—
272C2×C4—24no—
273C2×C4—22no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—41no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—42no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—121no—
287C2×C4—121no—
288C2×C4C2×(C2×D3)⋊C411yescharacteristic
289C2×C4C4×#6711yescharacteristic
290C2×C4C4×#6711yescharacteristic
291C2×C4#92911yescharacteristic
292C2×C4#93011yescharacteristic
293C2×C4#93011yescharacteristic
294C2×C4#94211yescharacteristic
295C2×C4C2×#21611yescharacteristic
296C2×C4#95011yescharacteristic
297C2×C4#95111yescharacteristic
298C8—122no—
299C8—241no—
300C8—241no—
301C2×C3—21no—
302C2×C3—21no—
303C2×C3#187811yescharacteristic
304C2×C3#190511yescharacteristic
305C2×C3#191111yescharacteristic
306C22—21no—
307C22—21no—
308C22#521311yescharacteristic
309C22#522811yescharacteristic
310C22#523911yescharacteristic
311C4—21no—
312C4—21no—
313C4—21no—
314C4—21no—
315C4—42no—
316C4—44no—
317C4—82no—
318C4—81no—
319C4—241no—
320C4—241no—
321C4#502811yescharacteristic
322C4C4×#21611yescharacteristic
323C4#503011yescharacteristic
324C4#503111yescharacteristic
325C3#929711yescharacteristic, complemented, Sylow
326C2—21no—
327C2—21no—
328C2#7163211yescharacteristic
329C2#7165211yescharacteristic
330C2#7166211yescharacteristic
subgroupG/H
centerC22#5213
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2331C2
socleC22×C3#386
prime radicalC23×C3#120
cosocleC2×C3×C42C23
hypercenter#2331D3
hypocenterC3#9297
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2331

Complements

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

NH ≅ G/Ntype
18. #5092C4semidirect
20. #5079C4semidirect
115. C3×C42Q16semidirect
243. C42#216semidirect
325. C3#9297semidirect

Maximal subgroups

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

subgroupindex
C3×#23312
#832222
#717742
#695462
#694412
#676312
#676142
#92973