Finite Group Databasebeta

← C2×#73716 #10967009

Subgroups — C2×#73716 #10967009

Order 768.

2446 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#870C212yesmaximal
2C2×C3×C4×C4⋊C4C211yesmaximal, characteristic
3C2×#4972C212yesmaximal
4C2×#4975C212yesmaximal
5C2×#4928C212yesmaximal
6C2×#4935C211yesmaximal, characteristic
7C2×C4×C4⋊Dic3C211yesmaximal, characteristic
8#73716C2116yesmaximal, direct product factor
9C2×#4657C214yesmaximal
10C2×#2552—31nomaximal, Sylow
11C22×C3×C4⋊C4C2211yescharacteristic
12C22×C3×C4⋊C4C2211yescharacteristic
13C22×C3×C42C2212yes—
14C22×C3×C42C2211yescharacteristic
15C22×C4⋊Dic3C2212yes—
16C22×C4⋊Dic3C2212yes—
17C22×C4⋊Dic3C2212yes—
18C22×C4⋊Dic3C2211yescharacteristic
19C22×C4⋊Dic3C2211yescharacteristic
20C22×Dic3⋊C4C2214yes—
21C22×Dic3⋊C4C2214yes—
22C22×Dic3⋊C4C2214yes—
23C22×C4×Dic3C2212yes—
24C2×#870—216no—
25C2×#870C22116yes—
26C3×C4×C4⋊C4C2218yes—
27C2×C3×#95C2212yes—
28C2×C3⋊#95C2212yes—
29C2×C3⋊#95C2214yes—
30#4975C22116yes—
31#4972C22116yes—
32#4935C2218yes—
33C4×C4⋊Dic3C2218yes—
34#4928C22116yes—
35#4657C22132yes—
36C22×#125—32no—
37#2552—316no—
38C2×#398—34no—
39C2×#389—32no—
40C2×#389—32no—
41C2×#384—31no—
42C2×#384—32no—
43C2×C4×C4⋊C4—31no—
44C2×C4×C4⋊C4D311yescharacteristic
45C23×C3×C4C2312yes—
46C23×C3×C4C2312yes—
47C23×C3×C4C2311yescharacteristic
48C23×C3×C4C2311yescharacteristic
49C23×C3×C4C2311yescharacteristic
50C23×Dic3C2314yes—
51C23×Dic3C2312yes—
52C23×Dic3C2312yes—
53C2×C3×C4⋊C4—24no—
54C2×C3×C4⋊C4—24no—
55C2×C3×C4⋊C4C2314yes—
56C2×C3×C4⋊C4C2314yes—
57C2×C3×C42—28no—
58C2×C3×C42Q818yes—
59C2×C3×C42C2318yes—
60C2×C3×C42C2314yes—
61C2×C4⋊Dic3—28no—
62C2×C4⋊Dic3—28no—
63C2×C4⋊Dic3—44no—
64C2×C4⋊Dic3D418yes—
65C2×C4⋊Dic3Q818yes—
66C2×C4⋊Dic3C2318yes—
67C2×C4⋊Dic3C2318yes—
68C2×C4⋊Dic3C2318yes—
69C2×C4⋊Dic3C2314yes—
70C2×C4⋊Dic3C2314yes—
71C2×Dic3⋊C4—216no—
72C2×Dic3⋊C4—216no—
73C2×Dic3⋊C4—48no—
74C2×Dic3⋊C4C23116yes—
75C2×Dic3⋊C4C23116yes—
76C2×Dic3⋊C4C23116yes—
77C2×C4×Dic3—28no—
78C2×C4×Dic3C2318yes—
79#870—264no—
80C3×#95C2318yes—
81C3⋊#95C2318yes—
82C3⋊#95C23116yes—
83C2×#125—316no—
84C2×#125—616no—
85C22×C4⋊C4—31no—
86C22×C4⋊C4—31no—
87C22×C4⋊C4—34no—
88C22×C4⋊C4—34no—
89C22×C4⋊C4—32no—
90C22×C4⋊C4—32no—
91C22×C4⋊C4—32no—
92C22×C4⋊C4—34no—
93C22×C4⋊C4C2×D311yescharacteristic
94C22×C4⋊C4C2×D311yescharacteristic
95C22×C42—32no—
96C22×C42C2×D312yes—
97C22×C42C2×D311yescharacteristic
98#398—332no—
99#389—316no—
100#389—316no—
101#384—316no—
102#384—38no—
103C4×C4⋊C4—38no—
104C4×C4⋊C4C2×D318yes—
105C2×#95—32no—
106C2×#95—34no—
107C2×#95C2×D312yes—
108C24×C3C2411yescharacteristic
109C22×C3×C4—22no—
110C22×C3×C4—22no—
111C22×C3×C4—24no—
112C22×C3×C4—24no—
113C22×C3×C4—24no—
114C22×C3×C4—24no—
115C22×C3×C4—24no—
116C22×C3×C4—24no—
117C22×C3×C4C2×D414yes—
118C22×C3×C4C2×Q814yes—
119C22×C3×C4C2×Q814yes—
120C22×C3×C4C2×Q814yes—
121C22×C3×C4C2×Q814yes—
122C22×C3×C4#4114yes—
123C22×C3×C4#4114yes—
124C22×C3×C4C2414yes—
125C22×C3×C4C2414yes—
126C22×C3×C4C2412yes—
127C22×C3×C4C2412yes—
128C22×C3×C4C2412yes—
129C22×Dic3—24no—
130C22×Dic3—24no—
131C22×Dic3—24no—
132C22×Dic3—24no—
133C22×Dic3—28no—
134C22×Dic3—28no—
135C22×Dic3—28no—
136C22×Dic3C2×D418yes—
137C22×Dic3C2×Q818yes—
138C22×Dic3C2418yes—
139C22×Dic3C2414yes—
140C22×Dic3C2414yes—
141C3×C4⋊C4—28no—
142C3×C4⋊C4—28no—
143C3×C42—216no—
144C3×C42C2×Q8116yes—
145C4⋊Dic3—216no—
146C4⋊Dic3—216no—
147C4⋊Dic3—48no—
148C4⋊Dic3C2×D4116yes—
149C4⋊Dic3C2×Q8116yes—
150Dic3⋊C4—232no—
151Dic3⋊C4—232no—
152Dic3⋊C4—416no—
153C4×Dic3—216no—
154C23×C4—32no—
155C23×C4—32no—
156C23×C4—34no—
157C23×C4C22×D312yes—
158C23×C4C22×D312yes—
159C23×C4C22×D311yescharacteristic
160C23×C4C22×D311yescharacteristic
161C23×C4C22×D311yescharacteristic
162#125—664no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—38no—
166C2×C4⋊C4—38no—
167C2×C4⋊C4—34no—
168C2×C4⋊C4—34no—
169C2×C4⋊C4—38no—
170C2×C4⋊C4—316no—
171C2×C4⋊C4—316no—
172C2×C4⋊C4—38no—
173C2×C4⋊C4—316no—
174C2×C4⋊C4—38no—
175C2×C4⋊C4—616no—
176C2×C4⋊C4—616no—
177C2×C4⋊C4—68no—
178C2×C4⋊C4—68no—
179C2×C4⋊C4—128no—
180C2×C4⋊C4—124no—
181C2×C4⋊C4C22×D314yes—
182C2×C4⋊C4C22×D314yes—
183C2×C42—28no—
184C2×C42—38no—
185C2×C42—68no—
186C2×C42Dic618yes—
187C2×C42C22×D318yes—
188C2×C42C22×D314yes—
189#95—38no—
190#95—316no—
191#95C22×D318yes—
192C23×C3C22×D412yes—
193C23×C3C22×Q812yes—
194C23×C3C22×Q812yes—
195C23×C3C2×#4112yes—
196C23×C3C2×#4112yes—
197C23×C3H(5,𝔽2)12yes—
198C23×C32−1+412yes—
199C23×C3C2511yescharacteristic
200C2×C3×C4—28no—
201C2×C3×C4—22no—
202C2×C3×C4—22no—
203C2×C3×C4—24no—
204C2×C3×C4—24no—
205C2×C3×C4—24no—
206C2×C3×C4—24no—
207C2×C3×C4—24no—
208C2×C3×C4—24no—
209C2×C3×C4—48no—
210C2×C3×C4—48no—
211C2×C3×C4#125116yescomplemented
212C2×C3×C4C4⋊Q8116yes—
213C2×C3×C4C22×D414yes—
214C2×C3×C4C22×Q814yes—
215C2×C3×C4C22×Q814yes—
216C2×C3×C4C22×Q814yes—
217C2×C3×C4C22×Q814yes—
218C2×C3×C4C2×#4114yes—
219C2×C3×C4C2×#4114yes—
220C2×Dic3—216no—
221C2×Dic3—216no—
222C2×Dic3—24no—
223C2×Dic3—24no—
224C2×Dic3—24no—
225C2×Dic3—24no—
226C2×Dic3—28no—
227C2×Dic3—28no—
228C2×Dic3—28no—
229C2×Dic3—416no—
230C2×Dic3C22×D418yes—
231C2×Dic3C22×Q818yes—
232C24C23×D311yescharacteristic
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—38no—
242C22×C4—38no—
243C22×C4—34no—
244C22×C4—34no—
245C22×C4—38no—
246C22×C4—64no—
247C22×C4—64no—
248C22×C4—64no—
249C22×C4—64no—
250C22×C4—68no—
251C22×C4—68no—
252C22×C4—68no—
253C22×C4C2×Dic614yes—
254C22×C4C2×Dic614yes—
255C22×C4C2×Dic614yes—
256C22×C4#23514yes—
257C22×C4#23514yes—
258C22×C4D3×D414yes—
259C22×C4D3×Q814yes—
260C22×C4C23×D314yes—
261C22×C4C23×D314yes—
262C22×C4C23×D312yes—
263C22×C4C23×D312yes—
264C22×C4C23×D312yes—
265C4⋊C4—28no—
266C4⋊C4—28no—
267C4⋊C4—316no—
268C4⋊C4—316no—
269C4⋊C4—632no—
270C4⋊C4—632no—
271C4⋊C4—616no—
272C4⋊C4—616no—
273C4⋊C4—1216no—
274C4⋊C4—128no—
275C42—216no—
276C42—616no—
277C42C2×Dic6116yes—
278C22×C3C2×#12514yes—
279C22×C3C2×C4⋊Q814yes—
280C22×C3#54414yes—
281C22×C3#54714yes—
282C22×C3#55014yes—
283C22×C3#55414yes—
284C22×C3#55714yes—
285C22×C3C23×D411yescharacteristic
286C22×C3C23×Q811yescharacteristic
287C22×C3C23×Q811yescharacteristic
288C22×C3C22×#4111yescharacteristic
289C22×C3C22×#4111yescharacteristic
290C22×C3C2×H(5,𝔽2)11yescharacteristic
291C22×C3C2×2−1+411yescharacteristic
292C3×C4—24no—
293C3×C4—44no—
294C3×C4—44no—
295C3×C4C2×#12518yescomplemented
296C3×C4C2×C4⋊Q818yes—
297Dic3—28no—
298Dic3—28no—
299Dic3—48no—
300C23C22×Dic612yes—
301C23C2×#23512yes—
302C23C2×#23512yes—
303C23C2×D3×D412yes—
304C23C2×D3×Q812yes—
305C23#100912yes—
306C23#101012yes—
307C23C24×D311yescharacteristic
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—28no—
311C2×C4—24no—
312C2×C4—24no—
313C2×C4—24no—
314C2×C4—24no—
315C2×C4—24no—
316C2×C4—24no—
317C2×C4—38no—
318C2×C4—38no—
319C2×C4—48no—
320C2×C4—48no—
321C2×C4—64no—
322C2×C4—64no—
323C2×C4—64no—
324C2×C4—64no—
325C2×C4—616no—
326C2×C4—616no—
327C2×C4—68no—
328C2×C4—68no—
329C2×C4—68no—
330C2×C4—1216no—
331C2×C4#870116yescomplemented
332C2×C4#888116yes—
333C2×C4C22×Dic614yes—
334C2×C4C22×Dic614yes—
335C2×C4C22×Dic614yes—
336C2×C4C2×#23514yes—
337C2×C4C2×#23514yes—
338C2×C4C2×D3×D414yes—
339C2×C4C2×D3×Q814yes—
340C2×C3#255218yescomplemented
341C2×C3C22×#12511yescharacteristic
342C2×C3C22×C4⋊Q811yescharacteristic
343C2×C3C2×#54411yescharacteristic
344C2×C3C2×#54711yescharacteristic
345C2×C3C2×#55011yescharacteristic
346C2×C3C2×#55411yescharacteristic
347C2×C3C2×#55711yescharacteristic
348C22C2×#87014yes—
349C22C2×#88814yes—
350C22#551914yes—
351C22#555114yes—
352C22#555314yes—
353C22#556714yes—
354C22#557914yes—
355C22C23×Dic611yescharacteristic
356C22C22×#23511yescharacteristic
357C22C22×#23511yescharacteristic
358C22C22×D3×D411yescharacteristic
359C22C22×D3×Q811yescharacteristic
360C22C2×#100911yescharacteristic
361C22C2×#101011yescharacteristic
362C4—24no—
363C4—44no—
364C4—44no—
365C4—68no—
366C4—68no—
367C4—128no—
368C4C2×#87018yescomplemented
369C4C2×#88818yes—
370C3C2×#255211yescharacteristic, complemented, Sylow
371C2#7371618yesdirect product factor
372C2C22×#87011yescharacteristic
373C2C22×#88811yescharacteristic
374C2C2×#551911yescharacteristic
375C2C2×#555311yescharacteristic
376C2C2×#555111yescharacteristic
377C2C2×#556711yescharacteristic
378C2C2×#557911yescharacteristic
subgroupG/H
centerC24C23×D3
commutator, cosocleC23×C3C25
FrattiniC23C24×D3
FittingC2×C3×C4×C4⋊C4C2
socle, prime radicalC24×C3C24
hypercenterC2×C4×C4⋊C4D3
hypocenterC3C2×#2552
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
8. #73716C2direct (N × H)
211. C2×C3×C4#125semidirect
295. C3×C4C2×#125semidirect
331. C2×C4#870semidirect
340. C2×C3#2552semidirect
368. C4C2×#870semidirect
370. C3C2×#2552semidirect
371. C2#73716direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#8702
C2×C3×C4×C4⋊C42
C2×#49722
C2×#49752
C2×#49282
C2×#49352
C2×C4×C4⋊Dic32
#737162
C2×#46572
C2×#25523