Finite Group Databasebeta

← C2×#82179 #10964478

Subgroups — C2×#82179 #10964478

Order 768.

2918 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×C4⋊C4C211yesmaximal, characteristic
2C22×#924C211yesmaximal, characteristic
3C2×#5211C214yesmaximal
4C2×C4×#217C211yesmaximal, characteristic
5#82179C2116yesmaximal, direct product factor
6C2×#4985C214yesmaximal, complemented
7C2×#4973C212yesmaximal, complemented
8C2×#4977C212yesmaximal, complemented
9C2×#2503—31nomaximal, Sylow
10C24×C3×C4C2212yes—
11C24×C3×C4C2211yescharacteristic
12C23×Dic6C2211yescharacteristic
13C22×C3×C4⋊C4—28no—
14C22×C3×C4⋊C4C2218yes—
15C22×C3×C4⋊C4C2212yes—
16C22×C3×C4⋊C4C2212yes—
17C22×#217C2212yes—
18C22×#217C2212yes—
19C22×C4⋊Dic3C2211yescharacteristic
20C2×#924—28no—
21C2×#924C2218yes—
22C22×Dic3⋊C4C2212yes—
23C22×C4×Dic3C2212yes—
24#5211C22132yes—
25C4×#217C2218yes—
26C2×C3⋊#95C2212yes—
27C2×C3⋊#95C2218yes—
28C2×C3⋊#95C2218yes—
29#4985C22132yescomplemented
30#4977C22116yescomplemented
31#4973C22116yescomplemented
32C22×C22⋊Q8—31no—
33C23×C4⋊C4D311yescharacteristic
34#2503—316no—
35C2×#400—34no—
36C2×#391—32no—
37C2×#382—32no—
38C2×#381—34no—
39C2×C4×C22⋊C4—31no—
40C25×C3C2311yescharacteristic
41C23×C3×C4—28no—
42C23×C3×C4—28no—
43C23×C3×C4—216no—
44C23×C3×C4D418yes—
45C23×C3×C4C2314yes—
46C23×C3×C4C2318yes—
47C23×C3×C4C2311yescharacteristic
48C23×C3×C4C2314yes—
49C23×C3×C4C2311yescharacteristic
50C23×Dic3C2312yes—
51C23×Dic3C2314yes—
52C23×Dic3C2312yes—
53C22×Dic6—24no—
54C22×Dic6—28no—
55C22×Dic6C2314yes—
56C2×C3×C4⋊C4—28no—
57C2×C3×C4⋊C4—232no—
58C2×C3×C4⋊C4—28no—
59C2×C3×C4⋊C4C2318yes—
60C2×C3×C4⋊C4C2318yes—
61C2×#217—28no—
62C2×#217—28no—
63C2×#217C2318yes—
64C2×#217C2318yes—
65C2×C4⋊Dic3—24no—
66C2×C4⋊Dic3C2314yes—
67#924—232no—
68C2×Dic3⋊C4—28no—
69C2×Dic3⋊C4C2318yes—
70C2×C4×Dic3—28no—
71C2×C4×Dic3C2318yes—
72C3⋊#95C2318yes—
73C3⋊#95C23132yes—
74C3⋊#95C23132yes—
75C23×Q8—31no—
76C24×C4C2×D312yes—
77C24×C4C2×D311yescharacteristic
78C2×C22⋊Q8—38no—
79C2×C22⋊Q8—68no—
80C22×C4⋊C4—28no—
81C22×C4⋊C4—32no—
82C22×C4⋊C4—31no—
83C22×C4⋊C4C2×D312yes—
84C22×C4⋊C4C2×D318yes—
85C22×C4⋊C4C2×D312yes—
86C22×C22⋊C4—32no—
87C22×C22⋊C4—32no—
88C22×C42—32no—
89#400—332no—
90#391—316no—
91#382—316no—
92#381—332no—
93C4×C22⋊C4—38no—
94C2×#95—32no—
95C2×#95—38no—
96C2×#95—38no—
97C24×C3—24no—
98C24×C3—24no—
99C24×C3C2×D414yes—
100C24×C3#4118yes—
101C24×C3C2412yes—
102C24×C3C2411yescharacteristic
103C22×C3×C4—24no—
104C22×C3×C4—232no—
105C22×C3×C4—216no—
106C22×C3×C4—216no—
107C22×C3×C4—28no—
108C22×C3×C4—232no—
109C22×C3×C4—216no—
110C22×C3×C4—432no—
111C22×C3×C4C2×D414yes—
112C22×C3×C4C2×D4116yes—
113C22×C3×C4C2×D414yes—
114C22×C3×C4#4118yes—
115C22×C3×C4C2418yes—
116C22×C3×C4C2412yes—
117C22×C3×C4C2412yes—
118C22×Dic3—24no—
119C22×Dic3—24no—
120C22×Dic3—28no—
121C22×Dic3—28no—
122C22×Dic3—28no—
123C22×Dic3—28no—
124C22×Dic3—28no—
125C22×Dic3C2418yes—
126C22×Dic3C2414yes—
127C22×Dic3C2414yes—
128C2×Dic6—28no—
129C2×Dic6—216no—
130C2×Dic6—416no—
131C3×C4⋊C4—216no—
132C3×C4⋊C4—216no—
133#217—216no—
134#217—216no—
135C4⋊Dic3—28no—
136Dic3⋊C4—216no—
137C4×Dic3—216no—
138C25C22×D311yescharacteristic
139C22×Q8—34no—
140C22×Q8—68no—
141C22×Q8—64no—
142C23×C4—28no—
143C23×C4—216no—
144C23×C4—28no—
145C23×C4—32no—
146C23×C4—32no—
147C23×C4—34no—
148C23×C4#6718yes—
149C23×C4C22×D318yes—
150C23×C4C22×D314yes—
151C23×C4C22×D314yes—
152C23×C4C22×D311yescharacteristic
153C23×C4C22×D311yescharacteristic
154C22⋊Q8—632no—
155C2×C4⋊C4—28no—
156C2×C4⋊C4—232no—
157C2×C4⋊C4—28no—
158C2×C4⋊C4—34no—
159C2×C4⋊C4—38no—
160C2×C4⋊C4—68no—
161C2×C4⋊C4—64no—
162C2×C4⋊C4C22×D318yes—
163C2×C4⋊C4C22×D318yes—
164C2×C22⋊C4—38no—
165C2×C22⋊C4—38no—
166C2×C22⋊C4—68no—
167C2×C22⋊C4—68no—
168C2×C42—38no—
169C2×C42—68no—
170#95—38no—
171#95—332no—
172#95—332no—
173C23×C3—28no—
174C23×C3—28no—
175C23×C3—232no—
176C23×C3—24no—
177C23×C3—24no—
178C23×C3#124116yes—
179C23×C3C22×D414yes—
180C23×C3C22×D412yes—
181C23×C3C2×#4114yes—
182C23×C3C2×#4118yes—
183C23×C3C2×#4114yes—
184C23×C32−1+414yes—
185C23×C3C2511yescharacteristic
186C2×C3×C4—28no—
187C2×C3×C4—24no—
188C2×C3×C4—232no—
189C2×C3×C4—28no—
190C2×C3×C4—216no—
191C2×C3×C4—432no—
192C2×C3×C4—416no—
193C2×C3×C4#123116yes—
194C2×C3×C4C22×D414yes—
195C2×C3×C4C22×D414yes—
196C2×C3×C4C2×#4118yes—
197C2×Dic3—24no—
198C2×Dic3—24no—
199C2×Dic3—28no—
200C2×Dic3—28no—
201C2×Dic3—28no—
202C2×Dic3—28no—
203C2×Dic3—28no—
204C2×Dic3—48no—
205C2×Dic3—48no—
206C2×Dic3—416no—
207Dic6—416no—
208C24—24no—
209C24—24no—
210C24#23718yes—
211C24C2×#6714yes—
212C24C23×D312yes—
213C24C23×D311yescharacteristic
214C2×Q8—616no—
215C2×Q8—68no—
216C2×Q8—1216no—
217C22×C4—24no—
218C22×C4—232no—
219C22×C4—216no—
220C22×C4—232no—
221C22×C4—216no—
222C22×C4—216no—
223C22×C4—28no—
224C22×C4—38no—
225C22×C4—34no—
226C22×C4—34no—
227C22×C4—432no—
228C22×C4—68no—
229C22×C4—68no—
230C22×C4—68no—
231C22×C4—68no—
232C22×C4—68no—
233C22×C4—64no—
234C22×C4—64no—
235C22×C4#23518yes—
236C22×C4C2×#6714yes—
237C22×C4C2×#67116yes—
238C22×C4C2×#6714yes—
239C22×C4C23×D318yes—
240C22×C4C23×D312yes—
241C22×C4C23×D312yes—
242C4⋊C4—216no—
243C4⋊C4—216no—
244C4⋊C4—616no—
245C4⋊C4—68no—
246C22⋊C4—616no—
247C22⋊C4—616no—
248C42—616no—
249C22×C3—24no—
250C22×C3—24no—
251C22×C3—216no—
252C22×C3—232no—
253C22×C3C2×#12314yes—
254C22×C3C2×#12418yes—
255C22×C3C2×#12414yes—
256C22×C3#53614yes—
257C22×C3#552116yes—
258C22×C3C23×D411yescharacteristic
259C22×C3C22×#4112yes—
260C22×C3C22×#4112yes—
261C22×C3C2×2−1+412yes—
262C3×C4—24no—
263C3×C4—48no—
264C3×C4C2×#12318yes—
265Dic3—44no—
266Dic3—44no—
267Dic3—48no—
268C23—24no—
269C23—24no—
270C23—232no—
271C23—28no—
272C23—28no—
273C23#936116yes—
274C23C2×#23514yes—
275C23C2×#23718yes—
276C23C2×#23714yes—
277C23#100714yes—
278C23C22×#6714yes—
279C23C22×#6712yes—
280C23C24×D311yescharacteristic
281Q8—1216no—
282C2×C4—24no—
283C2×C4—28no—
284C2×C4—232no—
285C2×C4—216no—
286C2×C4—28no—
287C2×C4—432no—
288C2×C4—416no—
289C2×C4—68no—
290C2×C4—68no—
291C2×C4—68no—
292C2×C4—68no—
293C2×C4—68no—
294C2×C4—64no—
295C2×C4—64no—
296C2×C4—1216no—
297C2×C4—128no—
298C2×C4—128no—
299C2×C4#929116yes—
300C2×C4C2×#23518yes—
301C2×C4C22×#6714yes—
302C2×C4C22×#6714yes—
303C2×C3—28no—
304C2×C3#250318yescomplemented
305C2×C3C22×#12311yescharacteristic
306C2×C3C22×#12411yescharacteristic
307C2×C3C2×#53611yescharacteristic
308C2×C3C2×#55214yes—
309C22—232no—
310C22—24no—
311C22—216no—
312C22—24no—
313C22#5513116yes—
314C22C2×#92914yes—
315C22C2×#93618yes—
316C22C2×#93614yes—
317C22#581614yes—
318C22C22×#23512yes—
319C22C22×#23712yes—
320C22C2×#100712yes—
321C22C23×#6711yescharacteristic
322C4—24no—
323C4—48no—
324C4—124no—
325C4—128no—
326C4—124no—
327C4C2×#92918yes—
328C3C2×#250311yescharacteristic, complemented, Sylow
329C2—28no—
330C2#8217918yesdirect product factor
331C2C2×#551314yes—
332C2C22×#92911yescharacteristic
333C2C22×#93611yescharacteristic
334C2C2×#581611yescharacteristic
subgroupG/H
centerC24C23×D3
commutator, cosocleC23×C3C25
FrattiniC23C24×D3
FittingC23×C3×C4⋊C4C2
socleC24×C3C24
prime radicalC25×C3C23
hypercenterC23×C4⋊C4D3
hypocenterC3C2×#2503
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C23×C4⋊C4

Complements

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

NH ≅ G/Ntype
5. #82179C2direct (N × H)
6. C2×#4985C2semidirect
7. C2×#4973C2semidirect
8. C2×#4977C2semidirect
29. #4985C22semidirect
30. #4977C22semidirect
31. #4973C22semidirect
304. C2×C3#2503semidirect
328. C3C2×#2503semidirect
330. C2#82179direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×C4⋊C42
C22×#9242
C2×#52112
C2×C4×#2172
#821792
C2×#49852
C2×#49732
C2×#49772
C2×#25033