Finite Group Databasebeta

← C2×#83042 #10962114

Subgroups — C2×#83042 #10962114

Order 768.

2286 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
1C22×C3×#125C211yesmaximal, characteristic
2C22×C4×Dic6C211yesmaximal, characteristic
3#83042C2116yesmaximal, direct product factor
4C2×#4985C212yesmaximal
5C2×#4986C212yesmaximal
6C2×#4983C212yesmaximal
7C2×#4979C212yesmaximal
8C2×#4978C214yesmaximal
9C2×#4937C211yesmaximal, characteristic
10C2×#2634—31nomaximal, Sylow
11C23×Dic6C2211yescharacteristic
12C2×C3×#125—28no—
13C2×C3×#125C2218yes—
14C22×C3×C4⋊C4C2212yes—
15C22×C3×C4⋊C4C2212yes—
16C22×C3×C4⋊C4C2211yescharacteristic
17C22×C3×C4⋊C4C2211yescharacteristic
18C22×C3×C42C2211yescharacteristic
19C22×C4⋊Dic3C2211yescharacteristic
20C22×Dic3⋊C4C2212yes—
21C22×Dic3⋊C4C2214yes—
22C22×C4×Dic3C2212yes—
23C2×C4×Dic6—28no—
24C2×C4×Dic6C2218yes—
25C2×C3⋊#95C2212yes—
26C2×C3⋊#95C2212yes—
27C2×C3⋊#95C2212yes—
28C2×C3⋊#95C2214yes—
29C2×C3⋊#95C2214yes—
30C2×C3⋊#95C2214yes—
31#4986C22116yes—
32#4985C22116yes—
33#4983C22116yes—
34#4979C22116yes—
35#4978C22132yes—
36#4937C2218yes—
37C22×#125D311yescharacteristic
38C22×C4×Q8—31no—
39#2634—316no—
40C2×#400—32no—
41C2×#400—32no—
42C2×#398—32no—
43C2×#395—32no—
44C2×#387—34no—
45C2×#383—31no—
46C23×C3×C4C2312yes—
47C23×C3×C4C2312yes—
48C23×C3×C4C2311yescharacteristic
49C23×C3×C4C2311yescharacteristic
50C23×C3×C4C2311yescharacteristic
51C23×Dic3C2314yes—
52C23×Dic3C2312yes—
53C23×Dic3C2312yes—
54C22×Dic6—24no—
55C22×Dic6—28no—
56C22×Dic6C2314yes—
57C3×#125—232no—
58C2×C3×C4⋊C4—28no—
59C2×C3×C4⋊C4—24no—
60C2×C3×C4⋊C4—24no—
61C2×C3×C4⋊C4—28no—
62C2×C3×C4⋊C4C2318yes—
63C2×C3×C4⋊C4C2318yes—
64C2×C3×C4⋊C4C2314yes—
65C2×C3×C4⋊C4C2314yes—
66C2×C3×C42D418yes—
67C2×C3×C42C2314yes—
68C2×C4⋊Dic3—24no—
69C2×C4⋊Dic3C2314yes—
70C2×Dic3⋊C4—28no—
71C2×Dic3⋊C4—216no—
72C2×Dic3⋊C4C2318yes—
73C2×Dic3⋊C4C23116yes—
74C2×C4×Dic3—28no—
75C2×C4×Dic3C2318yes—
76C4×Dic6—232no—
77C3⋊#95C2318yes—
78C3⋊#95C2318yes—
79C3⋊#95C2318yes—
80C3⋊#95C23116yes—
81C3⋊#95C23116yes—
82C3⋊#95C23116yes—
83C23×Q8—31no—
84C2×#125—28no—
85C2×#125C2×D318yes—
86C2×C4×Q8—38no—
87C2×C4×Q8—68no—
88C22×C4⋊C4—34no—
89C22×C4⋊C4—32no—
90C22×C4⋊C4—31no—
91C22×C4⋊C4C2×D312yes—
92C22×C4⋊C4C2×D311yescharacteristic
93C22×C4⋊C4C2×D311yescharacteristic
94C22×C4⋊C4C2×D312yes—
95C22×C42—32no—
96C22×C42C2×D311yescharacteristic
97#400—316no—
98#400—316no—
99#398—316no—
100#395—316no—
101#387—332no—
102#383—38no—
103C2×#95—32no—
104C2×#95—34no—
105C2×#95—32no—
106C2×#95—34no—
107C2×#95—34no—
108C2×#95—32no—
109C24×C3C2411yescharacteristic
110C22×C3×C4—24no—
111C22×C3×C4—24no—
112C22×C3×C4—24no—
113C22×C3×C4—24no—
114C22×C3×C4—24no—
115C22×C3×C4—28no—
116C22×C3×C4—28no—
117C22×C3×C4C2×D414yes—
118C22×C3×C4C2×D414yes—
119C22×C3×C4C2×D414yes—
120C22×C3×C4C2414yes—
121C22×C3×C4C2414yes—
122C22×C3×C4C2412yes—
123C22×C3×C4C2412yes—
124C22×C3×C4C2412yes—
125C22×Dic3—24no—
126C22×Dic3—24no—
127C22×Dic3—24no—
128C22×Dic3—28no—
129C22×Dic3—28no—
130C22×Dic3—28no—
131C22×Dic3—28no—
132C22×Dic3#4118yes—
133C22×Dic3C2418yes—
134C22×Dic3C2414yes—
135C22×Dic3C2414yes—
136C2×Dic6—232no—
137C2×Dic6—28no—
138C2×Dic6—216no—
139C3×C4⋊C4—216no—
140C3×C4⋊C4—28no—
141C3×C4⋊C4—28no—
142C3×C4⋊C4—216no—
143C3×C42C2×D4116yes—
144C4⋊Dic3—28no—
145Dic3⋊C4—216no—
146Dic3⋊C4—232no—
147C4×Dic3—216no—
148C22×Q8—34no—
149C22×Q8—68no—
150C22×Q8—64no—
151C23×C4—32no—
152C23×C4—32no—
153C23×C4—34no—
154C23×C4C22×D312yes—
155C23×C4C22×D312yes—
156C23×C4C22×D311yescharacteristic
157C23×C4C22×D311yescharacteristic
158C23×C4C22×D311yescharacteristic
159#125—232no—
160C4×Q8—632no—
161C2×C4⋊C4—28no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—28no—
165C2×C4⋊C4—38no—
166C2×C4⋊C4—34no—
167C2×C4⋊C4—316no—
168C2×C4⋊C4—616no—
169C2×C4⋊C4—68no—
170C2×C4⋊C4—64no—
171C2×C4⋊C4C22×D318yes—
172C2×C4⋊C4C22×D318yes—
173C2×C4⋊C4C22×D314yes—
174C2×C4⋊C4C22×D314yes—
175C2×C42—38no—
176C2×C42—68no—
177C2×C42#6718yes—
178C2×C42C22×D314yes—
179#95—38no—
180#95—38no—
181#95—38no—
182#95—316no—
183#95—316no—
184#95—316no—
185C23×C3C22×D412yes—
186C23×C3C2×#4114yes—
187C23×C32−1+414yes—
188C23×C32−1+414yes—
189C23×C3C2511yescharacteristic
190C2×C3×C4—28no—
191C2×C3×C4—28no—
192C2×C3×C4—28no—
193C2×C3×C4—24no—
194C2×C3×C4—24no—
195C2×C3×C4—24no—
196C2×C3×C4—24no—
197C2×C3×C4—24no—
198C2×C3×C4—28no—
199C2×C3×C4—28no—
200C2×C3×C4—48no—
201C2×C3×C4—48no—
202C2×C3×C4C22×D414yes—
203C2×C3×C4C22×D414yes—
204C2×C3×C4C22×D414yes—
205C2×Dic3—216no—
206C2×Dic3—24no—
207C2×Dic3—24no—
208C2×Dic3—24no—
209C2×Dic3—28no—
210C2×Dic3—28no—
211C2×Dic3—28no—
212C2×Dic3—28no—
213C2×Dic3—48no—
214C2×Dic3—416no—
215C2×Dic3C2×#4118yes—
216Dic6—232no—
217C24C23×D311yescharacteristic
218C2×Q8—616no—
219C2×Q8—632no—
220C2×Q8—68no—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—24no—
224C22×C4—28no—
225C22×C4—28no—
226C22×C4—24no—
227C22×C4—24no—
228C22×C4—38no—
229C22×C4—34no—
230C22×C4—38no—
231C22×C4—34no—
232C22×C4—64no—
233C22×C4—64no—
234C22×C4—68no—
235C22×C4—68no—
236C22×C4—68no—
237C22×C4—68no—
238C22×C4—64no—
239C22×C4C2×#6714yes—
240C22×C4C2×#6714yes—
241C22×C4C2×#6714yes—
242C22×C4C23×D314yes—
243C22×C4C23×D314yes—
244C22×C4C23×D312yes—
245C22×C4C23×D312yes—
246C22×C4C23×D312yes—
247C4⋊C4—216no—
248C4⋊C4—28no—
249C4⋊C4—28no—
250C4⋊C4—216no—
251C4⋊C4—632no—
252C4⋊C4—616no—
253C4⋊C4—68no—
254C42—616no—
255C42C2×#67116yes—
256C22×C3#52514yes—
257C22×C3#53614yes—
258C22×C3#53614yes—
259C22×C3#541116yes—
260C22×C3C23×D411yescharacteristic
261C22×C3C22×#4112yes—
262C22×C3C2×2−1+412yes—
263C22×C3C2×2−1+412yes—
264C3×C4—24no—
265C3×C4—24no—
266C3×C4—24no—
267C3×C4—44no—
268C3×C4—44no—
269Dic3—28no—
270Dic3—44no—
271Dic3—48no—
272C23#100714yes—
273C23D3×#4114yes—
274C23#101014yes—
275C23C22×#6712yes—
276C23C24×D311yescharacteristic
277Q8—632no—
278C2×C4—24no—
279C2×C4—24no—
280C2×C4—24no—
281C2×C4—28no—
282C2×C4—28no—
283C2×C4—28no—
284C2×C4—28no—
285C2×C4—28no—
286C2×C4—24no—
287C2×C4—24no—
288C2×C4—38no—
289C2×C4—48no—
290C2×C4—48no—
291C2×C4—64no—
292C2×C4—64no—
293C2×C4—616no—
294C2×C4—68no—
295C2×C4—68no—
296C2×C4—68no—
297C2×C4—68no—
298C2×C4—64no—
299C2×C4—1216no—
300C2×C4—128no—
301C2×C4C22×#6714yes—
302C2×C4C22×#6714yes—
303C2×C4C22×#6714yes—
304C2×C3#263418yescomplemented
305C2×C3C2×#52511yescharacteristic
306C2×C3C2×#53611yescharacteristic
307C2×C3C2×#53611yescharacteristic
308C2×C3C2×#54114yes—
309C22#5696116yes—
310C22#581614yes—
311C22#582514yes—
312C22#582814yes—
313C22C2×#100712yes—
314C22C2×D3×#4112yes—
315C22C2×#101012yes—
316C22C23×#6711yescharacteristic
317C4—24no—
318C4—24no—
319C4—24no—
320C4—44no—
321C4—44no—
322C4—68no—
323C4—128no—
324C4—124no—
325C3C2×#263411yescharacteristic, complemented, Sylow
326C2#8304218yesdirect product factor
327C2C2×#569614yes—
328C2C2×#581611yescharacteristic
329C2C2×#582511yescharacteristic
330C2C2×#582811yescharacteristic
subgroupG/H
centerC24C23×D3
commutator, cosocleC23×C3C25
FrattiniC23C24×D3
FittingC22×C3×#125C2
socle, prime radicalC24×C3C24
hypercenterC22×#125D3
hypocenterC3C2×#2634
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C22×#125

Complements

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

NH ≅ G/Ntype
3. #83042C2direct (N × H)
304. C2×C3#2634semidirect
325. C3C2×#2634semidirect
326. C2#83042direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#1252
C22×C4×Dic62
#830422
C2×#49852
C2×#49862
C2×#49832
C2×#49792
C2×#49782
C2×#49372
C2×#26343