Finite Group Databasebeta

← C2×#83024 #10954098

Subgroups — C2×#83024 #10954098

Order 768.

2310 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#125C211yesmaximal, characteristic
2#83024C2116yesmaximal, direct product factor
3C2×#4984C212yesmaximal
4C2×#4980C212yesmaximal
5C2×#4980C212yesmaximal
6C2×#4980C212yesmaximal
7C2×#4980C212yesmaximal
8C2×Dic3×C4⋊C4C212yesmaximal
9C2×#4937C211yesmaximal, characteristic
10C2×C4×C4⋊Dic3C211yesmaximal, characteristic
11C2×#2336—31nomaximal, Sylow
12C2×C3×#125C4116yescomplemented
13C2×C3×#125C2218yes—
14C22×C3×C4⋊C4C2212yes—
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×C3×C4⋊C4C2211yescharacteristic
17C22×C3×C4⋊C4C2212yes—
18C22×C3×C42C2211yescharacteristic
19C22×C4⋊Dic3C2212yes—
20C22×C4⋊Dic3C2212yes—
21C22×C4⋊Dic3C2212yes—
22C22×C4×Dic3C2212yes—
23C22×C4×Dic3C2211yescharacteristic
24C22×C4×Dic3C2211yescharacteristic
25C22×C4×Dic3C2212yes—
26C2×C3⋊#95C2212yes—
27C2×C3⋊#95C2212yes—
28C2×C3⋊#95C2211yescharacteristic
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2211yescharacteristic
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2212yes—
35#4984C22116yes—
36#4980C22116yes—
37#4980C22116yes—
38#4980C22116yes—
39#4980C22116yes—
40Dic3×C4⋊C4C22116yes—
41#4937C2218yes—
42C4×C4⋊Dic3C2218yes—
43C22×#125D311yescharacteristic
44#2336—316no—
45C2×#389—32no—
46C2×#387—32no—
47C2×#387—32no—
48C2×#387—32no—
49C2×#387—32no—
50C2×#383—31no—
51C2×C4×C4⋊C4—31no—
52C2×C4×C4⋊C4—32no—
53C23×C3×C4C2311yescharacteristic
54C23×C3×C4C2311yescharacteristic
55C23×C3×C4C2312yes—
56C23×C3×C4C2312yes—
57C23×C3×C4C2311yescharacteristic
58C23×Dic3C2312yes—
59C23×Dic3C2312yes—
60C23×Dic3C2312yes—
61C23×Dic3C2312yes—
62C3×#125C2×C4164yescomplemented
63C2×C3×C4⋊C4C2×C4116yes—
64C2×C3×C4⋊C4C2×C418yes—
65C2×C3×C4⋊C4C2×C418yes—
66C2×C3×C4⋊C4C2×C4116yes—
67C2×C3×C4⋊C4C2318yes—
68C2×C3×C4⋊C4C2314yes—
69C2×C3×C4⋊C4C2314yes—
70C2×C3×C4⋊C4C2318yes—
71C2×C3×C42C2×C418yes—
72C2×C3×C42C2314yes—
73C2×C4⋊Dic3—28no—
74C2×C4⋊Dic3—28no—
75C2×C4⋊Dic3—44no—
76C2×C4⋊Dic3C2318yes—
77C2×C4⋊Dic3C2318yes—
78C2×C4⋊Dic3C2318yes—
79C2×C4×Dic3—28no—
80C2×C4×Dic3—42no—
81C2×C4×Dic3—44no—
82C2×C4×Dic3—42no—
83C2×C4×Dic3C2318yes—
84C2×C4×Dic3C2314yes—
85C2×C4×Dic3C2314yes—
86C2×C4×Dic3C2318yes—
87C3⋊#95C2318yes—
88C3⋊#95C2318yes—
89C3⋊#95C2314yes—
90C3⋊#95C2318yes—
91C3⋊#95C2314yes—
92C3⋊#95C2318yes—
93C3⋊#95C2318yes—
94C3⋊#95C2318yes—
95C3⋊#95C2318yes—
96C2×#125Dic3116yescomplemented
97C2×#125C2×D318yes—
98C22×C4⋊C4—32no—
99C22×C4⋊C4—32no—
100C22×C4⋊C4—32no—
101C22×C4⋊C4C2×D312yes—
102C22×C4⋊C4C2×D311yescharacteristic
103C22×C4⋊C4C2×D311yescharacteristic
104C22×C4⋊C4C2×D312yes—
105C22×C42—32no—
106C22×C42—31no—
107C22×C42—31no—
108C22×C42—32no—
109C22×C42C2×D311yescharacteristic
110#389—316no—
111#387—316no—
112#387—316no—
113#387—316no—
114#387—316no—
115#383—38no—
116C4×C4⋊C4—316no—
117C4×C4⋊C4—38no—
118C2×#95—32no—
119C2×#95—32no—
120C2×#95—32no—
121C2×#95—32no—
122C2×#95—31no—
123C2×#95—31no—
124C2×#95—32no—
125C2×#95—32no—
126C2×#95—32no—
127C24×C3C2411yescharacteristic
128C22×C3×C4—24no—
129C22×C3×C4—24no—
130C22×C3×C4—24no—
131C22×C3×C4—22no—
132C22×C3×C4—24no—
133C22×C3×C4—24no—
134C22×C3×C4—22no—
135C22×C3×C4—24no—
136C22×C3×C4C22×C414yes—
137C22×C3×C4C22×C414yes—
138C22×C3×C4C22×C414yes—
139C22×C3×C4C22×C418yes—
140C22×C3×C4C22×C418yes—
141C22×C3×C4C2412yes—
142C22×C3×C4C2412yes—
143C22×C3×C4C2414yes—
144C22×C3×C4C2414yes—
145C22×C3×C4C2412yes—
146C22×Dic3—24no—
147C22×Dic3—24no—
148C22×Dic3—24no—
149C22×Dic3—24no—
150C22×Dic3—24no—
151C22×Dic3—24no—
152C22×Dic3—24no—
153C22×Dic3—24no—
154C22×Dic3—24no—
155C22×Dic3—24no—
156C22×Dic3—24no—
157C22×Dic3#4118yes—
158C22×Dic3C2414yes—
159C22×Dic3C2414yes—
160C22×Dic3C2414yes—
161C22×Dic3C2414yes—
162C3×C4⋊C4C22×C4132yes—
163C3×C4⋊C4C22×C4116yes—
164C3×C4⋊C4C22×C4116yes—
165C3×C4⋊C4C22×C4132yes—
166C3×C42C22×C4116yes—
167C4⋊Dic3—216no—
168C4⋊Dic3—216no—
169C4⋊Dic3—48no—
170C4×Dic3—216no—
171C4×Dic3—44no—
172C4×Dic3—48no—
173C4×Dic3—44no—
174C23×C4—32no—
175C23×C4—32no—
176C23×C4—32no—
177C23×C4—32no—
178C23×C4C22×D311yescharacteristic
179C23×C4C22×D312yes—
180C23×C4C22×D312yes—
181C23×C4C22×D311yescharacteristic
182C23×C4C22×D311yescharacteristic
183#125C2×Dic3164yescomplemented
184C2×C4⋊C4—38no—
185C2×C4⋊C4—38no—
186C2×C4⋊C4—38no—
187C2×C4⋊C4—68no—
188C2×C4⋊C4—68no—
189C2×C4⋊C4—124no—
190C2×C4⋊C4C2×Dic3116yes—
191C2×C4⋊C4C2×Dic318yes—
192C2×C4⋊C4C2×Dic318yes—
193C2×C4⋊C4C2×Dic3116yes—
194C2×C4⋊C4C22×D318yes—
195C2×C4⋊C4C22×D314yes—
196C2×C4⋊C4C22×D314yes—
197C2×C4⋊C4C22×D318yes—
198C2×C42—38no—
199C2×C42—34no—
200C2×C42—34no—
201C2×C42—38no—
202C2×C42—68no—
203C2×C42—122no—
204C2×C42—122no—
205C2×C42—124no—
206C2×C42C2×Dic318yes—
207C2×C42C22×D314yes—
208#95—38no—
209#95—38no—
210#95—38no—
211#95—34no—
212#95—38no—
213#95—34no—
214#95—38no—
215#95—38no—
216#95—38no—
217C23×C3C23×C412yes—
218C23×C3C2×#4114yes—
219C23×C3H(5,𝔽2)12yes—
220C23×C32−1+414yes—
221C23×C32−1+412yes—
222C23×C3C2511yescharacteristic
223C2×C3×C4—216no—
224C2×C3×C4—28no—
225C2×C3×C4—28no—
226C2×C3×C4—28no—
227C2×C3×C4—216no—
228C2×C3×C4—24no—
229C2×C3×C4—24no—
230C2×C3×C4—24no—
231C2×C3×C4—22no—
232C2×C3×C4—24no—
233C2×C3×C4—24no—
234C2×C3×C4—22no—
235C2×C3×C4—24no—
236C2×C3×C4C23×C414yes—
237C2×C3×C4C23×C414yes—
238C2×C3×C4C23×C414yes—
239C2×C3×C4C23×C418yes—
240C2×C3×C4C23×C418yes—
241C2×Dic3—216no—
242C2×Dic3—24no—
243C2×Dic3—24no—
244C2×Dic3—24no—
245C2×Dic3—24no—
246C2×Dic3—24no—
247C2×Dic3—24no—
248C2×Dic3—24no—
249C2×Dic3—24no—
250C2×Dic3—24no—
251C2×Dic3—24no—
252C2×Dic3—24no—
253C2×Dic3—48no—
254C2×Dic3—48no—
255C2×Dic3—48no—
256C2×Dic3C2×#4118yes—
257C24C23×D311yescharacteristic
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—22no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—22no—
266C22×C4—38no—
267C22×C4—34no—
268C22×C4—34no—
269C22×C4—34no—
270C22×C4—34no—
271C22×C4—64no—
272C22×C4—64no—
273C22×C4—64no—
274C22×C4—64no—
275C22×C4—64no—
276C22×C4—64no—
277C22×C4—64no—
278C22×C4—64no—
279C22×C4—64no—
280C22×C4—64no—
281C22×C4—64no—
282C22×C4C22×Dic314yes—
283C22×C4C22×Dic314yes—
284C22×C4C22×Dic314yes—
285C22×C4C22×Dic318yes—
286C22×C4C22×Dic318yes—
287C22×C4C23×D312yes—
288C22×C4C23×D312yes—
289C22×C4C23×D314yes—
290C22×C4C23×D314yes—
291C22×C4C23×D312yes—
292C4⋊C4—616no—
293C4⋊C4—616no—
294C4⋊C4—128no—
295C4⋊C4C22×Dic3132yes—
296C4⋊C4C22×Dic3116yes—
297C4⋊C4C22×Dic3116yes—
298C4⋊C4C22×Dic3132yes—
299C42—616no—
300C42—124no—
301C42—124no—
302C42—128no—
303C42C22×Dic3116yes—
304C22×C3C4×#4114yes—
305C22×C3#51914yes—
306C22×C3#52014yes—
307C22×C3#53918yes—
308C22×C3#54118yes—
309C22×C3C24×C411yescharacteristic
310C22×C3C22×#4112yes—
311C22×C3C2×H(5,𝔽2)11yescharacteristic
312C22×C3C2×2−1+412yes—
313C22×C3C2×2−1+411yescharacteristic
314C3×C4—28no—
315C3×C4—24no—
316C3×C4—24no—
317C3×C4—24no—
318C3×C4—28no—
319Dic3—28no—
320Dic3—44no—
321Dic3—44no—
322Dic3—44no—
323C23#100714yes—
324C23D3×#4114yes—
325C23#100912yes—
326C23#101012yes—
327C23C23×Dic312yes—
328C23C24×D311yescharacteristic
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—24no—
332C2×C4—24no—
333C2×C4—22no—
334C2×C4—24no—
335C2×C4—22no—
336C2×C4—24no—
337C2×C4—28no—
338C2×C4—216no—
339C2×C4—28no—
340C2×C4—28no—
341C2×C4—216no—
342C2×C4—38no—
343C2×C4—616no—
344C2×C4—64no—
345C2×C4—64no—
346C2×C4—64no—
347C2×C4—64no—
348C2×C4—64no—
349C2×C4—64no—
350C2×C4—64no—
351C2×C4—64no—
352C2×C4—64no—
353C2×C4—64no—
354C2×C4—64no—
355C2×C4—128no—
356C2×C4—128no—
357C2×C4—128no—
358C2×C4C23×Dic314yes—
359C2×C4C23×Dic314yes—
360C2×C4C23×Dic314yes—
361C2×C4C23×Dic318yes—
362C2×C4C23×Dic318yes—
363C2×C3#233618yescomplemented
364C2×C3C2×C4×#4111yescharacteristic
365C2×C3C2×#51911yescharacteristic
366C2×C3C2×#52011yescharacteristic
367C2×C3C2×#53912yes—
368C2×C3C2×#54112yes—
369C22#569218yes—
370C22#569618yes—
371C22#581214yes—
372C22Dic3×#4114yes—
373C22#582714yes—
374C22C2×#100712yes—
375C22C2×D3×#4112yes—
376C22C2×#100911yescharacteristic
377C22C2×#101011yescharacteristic
378C22C24×Dic311yescharacteristic
379C4—24no—
380C4—28no—
381C4—24no—
382C4—24no—
383C4—28no—
384C4—68no—
385C4—124no—
386C4—124no—
387C4—124no—
388C3C2×#233611yescharacteristic, complemented, Sylow
389C2#8302418yesdirect product factor
390C2C2×#569612yes—
391C2C2×#569212yes—
392C2C2×#581211yescharacteristic
393C2C2×Dic3×#4111yescharacteristic
394C2C2×#582711yescharacteristic
subgroupG/H
centerC24C23×D3
commutatorC22×C3C24×C4
FrattiniC23C24×D3
FittingC22×C3×#125C2
socle, prime radicalC24×C3C24
cosocleC23×C3C25
hypercenterC22×#125D3
hypocenterC3C2×#2336
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C24 ◁ C22×#125

Complements

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

NH ≅ G/Ntype
2. #83024C2direct (N × H)
12. C2×C3×#125C4semidirect
62. C3×#125C2×C4semidirect
96. C2×#125Dic3semidirect
183. #125C2×Dic3semidirect
363. C2×C3#2336semidirect
388. C3C2×#2336semidirect
389. C2#83024direct (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
#830242
C2×#49842
C2×#49802
C2×Dic3×C4⋊C42
C2×#49372
C2×C4×C4⋊Dic32
C2×#23363