Finite Group Databasebeta

← C4⋊C4×#627 #11805569

Subgroups — C4⋊C4×#627 #11805569

Order 1152.

1233 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#627C212yesmaximal
2C2×C4×#627C211yesmaximal, characteristic
3C4⋊C4×C3⋊Dic3C211yesmaximal, characteristic
4C3×Dic3×C4⋊C4C212yesmaximal
5#10591359C211yesmaximal, characteristic
6#10591349C212yesmaximal
7#10591328C212yesmaximal
8#10591327C214yesmaximal
9C4⋊C4×Dic6—32nomaximal
10C22×#627C2211yescharacteristic
11C2×C32×C4⋊C4C2211yescharacteristic
12C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
13C2×C4×C3⋊Dic3C2212yes—
14C2×C4×C3⋊Dic3C2211yescharacteristic
15C2×C3×C4⋊Dic3C2212yes—
16C2×C3×C4×Dic3C2214yes—
17C2×C3×C4×Dic3C2212yes—
18C2×#3724C2212yes—
19C2×#3724C2212yes—
20C2×#3724C2211yescharacteristic
21C2×Dic3⋊Dic3C2214yes—
22C2×Dic3⋊Dic3C2214yes—
23C2×Dic3⋊Dic3C2212yes—
24C4×#627—28no—
25C4×#627C418yescomplemented
26#63658C2212yes—
27C3×C3⋊#95C2214yes—
28C2×C4×Dic6—32no—
29C2×C4×Dic6—34no—
30C3×C4×C4⋊C4—32no—
31#4975—32no—
32#4975—32no—
33Dic3×C4⋊C4—32no—
34Dic3×C4⋊C4—61no—
35Dic3×C4⋊C4D312yes—
36#4935—32no—
37#4928—34no—
38#4647—34no—
39#4647—34no—
40C22×C32×C4C2312yes—
41C22×C32×C4C2311yescharacteristic
42C22×C3⋊Dic3C2312yes—
43C22×C3⋊Dic3C2311yescharacteristic
44C22×C3⋊Dic3C2311yescharacteristic
45C22×C3×Dic3C2314yes—
46C22×C3×Dic3C2312yes—
47C22×C3×Dic3C2312yes—
48C2×#627C2×C414yes—
49C2×#627D414yes—
50C2×#627Q814yes—
51C32×C4⋊C4Q814yescomplemented
52C4⋊(C3⋊Dic3)—22no—
53C4×C3⋊Dic3—24no—
54C4×C3⋊Dic3C2×C414yes—
55C3×C4⋊Dic3—24no—
56C3×C4×Dic3—28no—
57C3×C4×Dic3C2×C418yes—
58#3724—24no—
59#3724—42no—
60#3724C2×C414yes—
61Dic3⋊Dic3—28no—
62Dic3⋊Dic3—44no—
63Dic3⋊Dic3C2×C418yes—
64Q8×C4⋊C4—91noSylow
65C22×Dic6—32no—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4—32no—
68C2×C3×C4⋊C4C2×D312yes—
69C2×C3×C42—32no—
70C2×C3×C42—34no—
71C2×C4⋊Dic3—32no—
72C2×C4⋊Dic3—32no—
73C2×C4⋊Dic3—34no—
74C2×C4⋊Dic3—34no—
75C2×C4⋊Dic3—61no—
76C2×C4⋊Dic3C2×D312yes—
77C2×Dic3⋊C4—32no—
78C2×Dic3⋊C4—32no—
79C2×Dic3⋊C4—34no—
80C2×Dic3⋊C4—34no—
81C2×Dic3⋊C4—34no—
82C2×Dic3⋊C4—34no—
83C2×C4×Dic3—32no—
84C2×C4×Dic3—34no—
85C2×C4×Dic3—61no—
86C2×C4×Dic3—62no—
87C2×C4×Dic3C2×D314yes—
88C2×C4×Dic3C2×D312yes—
89C4×Dic6—316no—
90C4×Dic6—616no—
91C3×#95—34no—
92C3⋊#95—34no—
93C3⋊#95—62no—
94C3⋊#95C2×D314yes—
95C23×C32C2411yescharacteristic
96C2×C32×C4—22no—
97C2×C32×C4—22no—
98C2×C32×C4C22×C412yes—
99C2×C32×C4C2×Q814yes—
100C2×C32×C4C2×Q812yes—
101C2×C32×C4#4112yes—
102C2×C3⋊Dic3—22no—
103C2×C3⋊Dic3—21no—
104C2×C3⋊Dic3—21no—
105C2×C3⋊Dic3—22no—
106C2×C3⋊Dic3—22no—
107C2×C3⋊Dic3C22×C412yes—
108C2×C3⋊Dic3C22×C412yes—
109C2×C3⋊Dic3C2×D412yes—
110C2×C3⋊Dic3C2×Q812yes—
111C2×C3×Dic3—22no—
112C2×C3×Dic3—22no—
113C2×C3×Dic3—24no—
114C2×C3×Dic3—24no—
115C2×C3×Dic3—24no—
116C2×C3×Dic3C22×C414yes—
117C2×C3×Dic3C22×C414yes—
118C2×C3×Dic3C2×D414yes—
119C2×C3×Dic3C2×Q814yes—
120#627C4⋊C4116yesdirect product factor
121C2×C4×Q8—91no—
122C2×C4×Q8—92no—
123#389—92no—
124#389—94no—
125#384—92no—
126#384—91no—
127C4×C4⋊C4—32no—
128C4×C4⋊C4—91no—
129C22×C3×C4—21no—
130C22×C3×C4—22no—
131C22×C3×C4—32no—
132C22×C3×C4—32no—
133C22×C3×C4—34no—
134C22×C3×C4C22×D314yes—
135C22×C3×C4C22×D312yes—
136C22×Dic3—32no—
137C22×Dic3—32no—
138C22×Dic3—34no—
139C22×Dic3—61no—
140C22×Dic3—61no—
141C22×Dic3—62no—
142C22×Dic3C22×D314yes—
143C22×Dic3C22×D312yes—
144C22×Dic3C22×D312yes—
145C2×Dic6—38no—
146C2×Dic6—38no—
147C2×Dic6—38no—
148C3×C4⋊C4—24no—
149C3×C4⋊C4—64no—
150C3×C4⋊C4Dic618yescomplemented
151C3×C42—38no—
152C3×C42—68no—
153C4⋊Dic3—24no—
154C4⋊Dic3—38no—
155C4⋊Dic3—68no—
156C4⋊Dic3—64no—
157C4⋊Dic3—64no—
158C4⋊Dic3—124no—
159Dic3⋊C4—38no—
160Dic3⋊C4—38no—
161Dic3⋊C4—68no—
162Dic3⋊C4—68no—
163Dic3⋊C4—124no—
164Dic3⋊C4—124no—
165C4×Dic3—28no—
166C4×Dic3—38no—
167C4×Dic3—64no—
168C4×Dic3—68no—
169C4×Dic3—124no—
170C4×Dic3C4×D318yes—
171C22×C32C23×C411yescharacteristic
172C22×C32C22×D411yescharacteristic
173C22×C32C22×Q811yescharacteristic
174C22×C32C22×Q811yescharacteristic
175C22×C32C2×#4111yescharacteristic
176C22×C32H(5,𝔽2)11yescharacteristic
177C22×C322−1+411yescharacteristic
178C32×C4—24no—
179C32×C4C4×Q814yescomplemented
180C3⋊Dic3—22no—
181C3⋊Dic3—42no—
182C3⋊Dic3C2×C4⋊C414yes—
183C3×Dic3—24no—
184C3×Dic3—44no—
185C3×Dic3C2×C4⋊C418yes—
186C22×Q8—91no—
187C4×Q8—98no—
188C4×Q8—188no—
189C2×C4⋊C4—32no—
190C2×C4⋊C4—91no—
191C2×C4⋊C4—92no—
192C2×C4⋊C4—92no—
193C2×C4⋊C4—94no—
194C2×C4⋊C4—91no—
195C2×C4⋊C4—92no—
196C2×C4⋊C4—94no—
197C2×C4⋊C4D3211yescharacteristic
198C2×C42—32no—
199C2×C42—34no—
200C2×C42—91no—
201C2×C42—92no—
202#95—34no—
203#95—92no—
204C23×C3—21no—
205C23×C3C23×D312yes—
206C2×C3×C4—24no—
207C2×C3×C4—24no—
208C2×C3×C4—22no—
209C2×C3×C4—22no—
210C2×C3×C4—22no—
211C2×C3×C4—24no—
212C2×C3×C4—34no—
213C2×C3×C4—34no—
214C2×C3×C4—34no—
215C2×C3×C4—34no—
216C2×C3×C4—42no—
217C2×C3×C4—42no—
218C2×C3×C4—62no—
219C2×C3×C4—62no—
220C2×C3×C4—64no—
221C2×C3×C4—64no—
222C2×C3×C4—64no—
223C2×C3×C4C2×Dic618yes—
224C2×C3×C4C2×Dic614yes—
225C2×C3×C4C2×C4×D314yes—
226C2×C3×C4#23514yes—
227C2×Dic3—22no—
228C2×Dic3—22no—
229C2×Dic3—24no—
230C2×Dic3—24no—
231C2×Dic3—24no—
232C2×Dic3—34no—
233C2×Dic3—34no—
234C2×Dic3—34no—
235C2×Dic3—34no—
236C2×Dic3—62no—
237C2×Dic3—62no—
238C2×Dic3—62no—
239C2×Dic3—62no—
240C2×Dic3—62no—
241C2×Dic3—62no—
242C2×Dic3—64no—
243C2×Dic3—64no—
244C2×Dic3—64no—
245C2×Dic3—64no—
246C2×Dic3—122no—
247C2×Dic3—121no—
248C2×Dic3—121no—
249C2×Dic3—122no—
250C2×Dic3C2×C4×D314yes—
251C2×Dic3C2×C4×D314yes—
252C2×Dic3D3×D414yes—
253C2×Dic3D3×Q814yes—
254Dic6—332no—
255C2×C32C22×C4⋊C411yescharacteristic
256C2×C32C2×C4×Q811yescharacteristic
257C2×C32#52011yescharacteristic
258C2×C32D4×Q811yescharacteristic
259C2×C32#55011yescharacteristic
260C2×C32#55711yescharacteristic
261C2×C32Q8211yescharacteristic
262C2×Q8—94no—
263C2×Q8—94no—
264C2×Q8—94no—
265C22×C4—32no—
266C22×C4—32no—
267C22×C4—34no—
268C22×C4—91no—
269C22×C4—91no—
270C22×C4—92no—
271C22×C4C2×D3212yes—
272C22×C4C2×D3211yescharacteristic
273C4⋊C4—64no—
274C4⋊C4—94no—
275C4⋊C4—98no—
276C4⋊C4—188no—
277C4⋊C4—184no—
278C4⋊C4—182no—
279C4⋊C4—362no—
280C4⋊C4—364no—
281C4⋊C4#62714yesdirect product factor
282C42—38no—
283C42—68no—
284C42—94no—
285C42—184no—
286C22×C3—21no—
287C22×C3—21no—
288C22×C3—21no—
289C22×C3—21no—
290C22×C3—21no—
291C22×C3—21no—
292C22×C3—21no—
293C22×C3C22×Dic612yes—
294C22×C3C22×C4×D312yes—
295C22×C3C2×#23512yes—
296C22×C3C2×D3×D412yes—
297C22×C3C2×D3×Q812yes—
298C22×C3#100912yes—
299C22×C3#101012yes—
300C3×C4—28no—
301C3×C4—24no—
302C3×C4—38no—
303C3×C4—44no—
304C3×C4—64no—
305C3×C4—124no—
306C3×C4C4×Dic618yescomplemented
307Dic3—24no—
308Dic3—38no—
309Dic3—44no—
310Dic3—64no—
311Dic3—64no—
312Dic3—122no—
313Dic3—124no—
314Dic3—124no—
315Dic3D3×C4⋊C418yes—
316C32Q8×C4⋊C411yescharacteristic, complemented, Sylow
317C23C22×D3211yescharacteristic
318Q8—916no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—34no—
322C2×C4—34no—
323C2×C4—34no—
324C2×C4—34no—
325C2×C4—62no—
326C2×C4—62no—
327C2×C4—64no—
328C2×C4—64no—
329C2×C4—64no—
330C2×C4—92no—
331C2×C4—92no—
332C2×C4—92no—
333C2×C4—92no—
334C2×C4—181no—
335C2×C4—181no—
336C2×C4—182no—
337C2×C4—182no—
338C2×C4—182no—
339C2×C4#379812yes—
340C2×C4C4×D3212yes—
341C2×C4C2×#62714yes—
342C2×C4C2×#62712yes—
343C2×C3—21no—
344C2×C3—21no—
345C2×C3—21no—
346C2×C3—21no—
347C2×C3—21no—
348C2×C3—21no—
349C2×C3—21no—
350C2×C3C2×C4×Dic612yes—
351C2×C3C2×D3×C4⋊C412yes—
352C2×C3#552312yes—
353C2×C3D4×Dic612yes—
354C2×C3#555312yes—
355C2×C3Q8×Dic612yes—
356C2×C3#556712yes—
357C22C2×#379811yescharacteristic
358C22C2×C4×D3211yescharacteristic
359C22#6430911yescharacteristic
360C22D32×D411yescharacteristic
361C22D32×Q811yescharacteristic
362C22#6432011yescharacteristic
363C22C22×#62711yescharacteristic
364C4—24no—
365C4—38no—
366C4—64no—
367C4—94no—
368C4—124no—
369C4—182no—
370C4—362no—
371C4C4×#62714yescomplemented
372C3—21no—
373C3C4⋊C4×Dic612yescomplemented
374C2#1059451111yescharacteristic
375C2#1059452011yescharacteristic
376C2#1059459511yescharacteristic
377C2D32×C4⋊C411yescharacteristic
378C2C2×C4×#62711yescharacteristic
379C2D4×#62711yescharacteristic
380C2Q8×#62711yescharacteristic
subgroupG/H
center, FrattiniC23C22×D32
commutatorC22×C32C23×C4
FittingC2×C32×C4⋊C4C22
socle, prime radical, cosocleC23×C32C24
hypercenterC2×C4⋊C4D32
hypocenterC32Q8×C4⋊C4
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
25. C4×#627C4semidirect
51. C32×C4⋊C4Q8semidirect
120. #627C4⋊C4direct (N × H)
150. C3×C4⋊C4Dic6semidirect
179. C32×C4C4×Q8semidirect
281. C4⋊C4#627direct (N × H)
306. C3×C4C4×Dic6semidirect
316. C32Q8×C4⋊C4semidirect
371. C4C4×#627semidirect
373. C3C4⋊C4×Dic6semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#6272
C4⋊C4×C3⋊Dic32
C3×Dic3×C4⋊C42
#105913592
#105913492
#105913282
#105913272
C4⋊C4×Dic63