Finite Group Databasebeta

← C4⋊C4×GO+(4,𝔽2) #11805472

Subgroups — C4⋊C4×GO+(4,𝔽2) #11805472

Order 1152.

1194 conjugacy classes of subgroups; only the 207 normal subgroups are shown.

normal subgroups only

This group has 1194 conjugacy classes of subgroups — too many to draw. Only its 207 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×GO+(4,𝔽2)C212yesmaximal
2C2×C4×GO+(4,𝔽2)C211yesmaximal, characteristic
3D32×C4⋊C4C212yesmaximal, complemented
4C4⋊C4×#157C211yesmaximal, characteristic, complemented
5#10593413C211yesmaximal, characteristic, complemented
6#10593410C212yesmaximal, complemented
7#10593403C212yesmaximal, complemented
8#10593398C214yesmaximal, complemented
9C22×GO+(4,𝔽2)C2211yescharacteristic
10C2×C4×D32C2214yes—
11C2×C4×D32C2212yes—
12C2×C4⋊#157C2211yescharacteristic
13C2×C4×#157C2212yes—
14C2×C4×#157C2211yescharacteristic
15C2×#3773C2212yes—
16C2×#3773C2212yes—
17C2×#3773C2211yescharacteristic
18C2×D32⋊C4C2214yes—
19C2×D32⋊C4C2212yes—
20C4×GO+(4,𝔽2)—28no—
21C4×GO+(4,𝔽2)C418yescomplemented
22C4⋊C4×C3⋊D3C2211yescharacteristic
23C3×D3×C4⋊C4—22no—
24#63900C2212yes—
25D3×C4⋊Dic3—22no—
26#63883C2214yes—
27D3×Dic3⋊C4—24no—
28#63781C2212yes—
29#63737C2214yes—
30C2×D3×C4⋊C4—62no—
31C22×D32C2312yes—
32C22×#157C2312yes—
33C22×#157C2311yescharacteristic
34C22×#157C2311yescharacteristic
35C2×GO+(4,𝔽2)C2×C414yes—
36C2×GO+(4,𝔽2)D414yes—
37C2×GO+(4,𝔽2)Q814yes—
38C2×C4×C3⋊D3C2312yes—
39C2×C4×C3⋊D3C2311yescharacteristic
40C2×C3×C4×D3—22no—
41C2×C3×C4×D3—24no—
42C2×#624C2314yes—
43C2×#624C2312yes—
44C2×D3×Dic3—22no—
45C2×D3×Dic3—24no—
46C4×D32—24no—
47C4×D32—28no—
48C4×D32—44no—
49C4×D32C2×C418yescomplemented
50C4⋊#157—22no—
51C4×#157—24no—
52C4×#157C2×C414yescomplemented
53#3773—24no—
54#3773—42no—
55#3773C2×C414yescomplemented
56D32⋊C4—28no—
57D32⋊C4C2×C418yescomplemented
58C32×C4⋊C4D411yescharacteristic, complemented
59C4⋊(C3⋊Dic3)D411yescharacteristic, complemented
60#3750D412yescomplemented
61C3×C4⋊Dic3—22no—
62C3×Dic3⋊C4—24no—
63#3724—22no—
64Dic3⋊Dic3—24no—
65D4×C4⋊C4—91nomaximal, Sylow
66C22×C4×D3—62no—
67C22×C4×D3—64no—
68C2×C3×C4⋊C4—62no—
69C2×C4⋊Dic3—62no—
70C2×Dic3⋊C4—64no—
71D3×C4⋊C4—22no—
72D3×C4⋊C4—64no—
73D3×C4⋊C4—62no—
74D3×C4⋊C4—62no—
75D3×C4⋊C4—64no—
76D3×C4⋊C4—62no—
77C22×C3⋊D3C2411yescharacteristic
78C22×C3×D3—22no—
79C2×D32—22no—
80C2×D32—22no—
81C2×D32—22no—
82C2×D32C22×C414yes—
83C2×D32C2×D414yes—
84C2×D32C2×Q814yes—
85C2×#157—21no—
86C2×#157—21no—
87C2×#157—22no—
88C2×#157—22no—
89C2×#157—22no—
90C2×#157C22×C412yes—
91C2×#157C22×C412yes—
92C2×#157C2×D412yes—
93C2×#157C2×Q812yes—
94GO+(4,𝔽2)C4⋊C4116yesdirect product factor
95C2×C32×C4C2×D412yes—
96C2×C32×C4C2×D411yescharacteristic
97C2×C3⋊Dic3C2×D412yes—
98C2×C3⋊Dic3C2×D411yescharacteristic
99C4×C3⋊D3—22no—
100C4×C3⋊D3—22no—
101C4×C3⋊D3C22×C412yes—
102C4×C3⋊D3#4112yes—
103C2×C3×Dic3—22no—
104C2×C3×Dic3—24no—
105C3×C4×D3—28no—
106C3×C4×D3—48no—
107#624—22no—
108#624—24no—
109#624—24no—
110#624C22×C414yes—
111D3×Dic3—28no—
112D3×Dic3—48no—
113C2×C4×D4—91no—
114C2×C4×D4—92no—
115C22×C4⋊C4—92no—
116#389—92no—
117#385—94no—
118#384—91no—
119#380—92no—
120C4×C4⋊C4—91no—
121C23×D3—62no—
122C22×C3×C4—62no—
123C22×C3×C4—64no—
124C22×Dic3—62no—
125C22×Dic3—64no—
126C2×C4×D3—22no—
127C2×C4×D3—24no—
128C2×C4×D3—68no—
129C2×C4×D3—62no—
130C2×C4×D3—62no—
131C2×C4×D3—64no—
132C2×C4×D3—64no—
133C2×C4×D3—68no—
134C2×C4×D3—62no—
135C2×C4×D3—64no—
136C2×C4×D3—128no—
137C2×C4×D3—128no—
138C3×C4⋊C4—22no—
139C3×C4⋊C4—64no—
140C3×C4⋊C4—62no—
141C4⋊Dic3—22no—
142C4⋊Dic3—62no—
143C4⋊Dic3—64no—
144Dic3⋊C4—24no—
145Dic3⋊C4—64no—
146Dic3⋊C4—64no—
147Dic3⋊C4—64no—
148C22×C32C22×D411yescharacteristic
149C2×C3⋊D3C23×C411yescharacteristic
150C2×C3⋊D3C22×D411yescharacteristic
151C2×C3⋊D3C22×Q811yescharacteristic
152C2×C3⋊D3C2×#4111yescharacteristic
153C2×C3⋊D3H(5,𝔽2)11yescharacteristic
154C2×C3⋊D32−1+411yescharacteristic
155C2×C3×D3—24no—
156C2×C3×D3—24no—
157C2×C3×D3—24no—
158D32—24no—
159D32—24no—
160D32—24no—
161D32C2×C4⋊C418yescomplemented
162#157—22no—
163#157—42no—
164#157C2×C4⋊C414yescomplemented
165C32×C4—22no—
166C32×C4C4×D412yescomplemented
167C3⋊Dic3—22no—
168C3⋊Dic3C4×D412yescomplemented
169C3×Dic3—24no—
170C3×Dic3—44no—
171C22×D4—91no—
172C23×C4—92no—
173C23×C4—94no—
174C4×D4—98no—
175C4×D4—188no—
176C2×C4⋊C4—62no—
177C2×C4⋊C4—92no—
178C2×C4⋊C4—94no—
179C2×C4⋊C4—91no—
180C2×C4⋊C4—91no—
181C2×C4⋊C4—92no—
182C2×C4⋊C4—91no—
183C2×C4⋊C4—92no—
184C2×C4⋊C4—184no—
185C2×C4⋊C4—182no—
186C2×C22⋊C4—92no—
187C2×C22⋊C4—94no—
188C2×C42—91no—
189C2×C42—92no—
190#95—92no—
191#95—94no—
192C23×C3—62no—
193C22×D3—22no—
194C22×D3—64no—
195C22×D3—64no—
196C22×D3—64no—
197C22×D3—62no—
198C22×D3—64no—
199C22×D3—64no—
200C22×D3—64no—
201C2×C3×C4—22no—
202C2×C3×C4—24no—
203C2×C3×C4—68no—
204C2×C3×C4—62no—
205C2×C3×C4—64no—
206C2×C3×C4—128no—
207C2×Dic3—22no—
208C2×Dic3—24no—
209C2×Dic3—62no—
210C2×Dic3—64no—
211C2×Dic3—68no—
212C2×Dic3—128no—
213C4×D3—28no—
214C4×D3—48no—
215C4×D3—64no—
216C4×D3—64no—
217C4×D3—64no—
218C4×D3—64no—
219C4×D3—68no—
220C4×D3—124no—
221C4×D3—124no—
222C4×D3—124no—
223C4×D3—124no—
224C4×D3—128no—
225C2×C32C2×C4×D411yescharacteristic
226C2×C32D4211yescharacteristic
227C2×C32D4×Q811yescharacteristic
228C3⋊D3C22×C4⋊C411yescharacteristic
229C3⋊D3#52011yescharacteristic
230C3⋊D3#54711yescharacteristic
231C3⋊D3#55411yescharacteristic
232C3×D3—28no—
233C24—92no—
234C2×D4—94no—
235C2×D4—94no—
236C2×D4—94no—
237C22×C4—62no—
238C22×C4—64no—
239C22×C4—92no—
240C22×C4—98no—
241C22×C4—94no—
242C22×C4—91no—
243C22×C4—91no—
244C22×C4—91no—
245C22×C4—92no—
246C22×C4—92no—
247C22×C4—184no—
248C22×C4—182no—
249C22×C4—184no—
250C22×C4—188no—
251C22×C4—364no—
252C4⋊C4—64no—
253C4⋊C4—62no—
254C4⋊C4—94no—
255C4⋊C4—92no—
256C4⋊C4—91no—
257C4⋊C4—182no—
258C4⋊C4—184no—
259C4⋊C4—184no—
260C4⋊C4—182no—
261C4⋊C4—362no—
262C4⋊C4GO+(4,𝔽2)11yescharacteristic, direct product factor
263C22⋊C4—98no—
264C22⋊C4—188no—
265C42—94no—
266C42—184no—
267C22×C3—22no—
268C22×C3—64no—
269C22×C3—64no—
270C22×C3—64no—
271C2×D3—24no—
272C2×D3—24no—
273C2×D3—24no—
274C2×D3—68no—
275C2×D3—62no—
276C2×D3—62no—
277C2×D3—62no—
278C2×D3—62no—
279C2×D3—62no—
280C2×D3—62no—
281C2×D3—68no—
282C2×D3—68no—
283C2×D3—68no—
284C3×C4—24no—
285C3×C4—44no—
286C3×C4—64no—
287C3×C4—124no—
288Dic3—24no—
289Dic3—44no—
290Dic3—64no—
291Dic3—124no—
292C32D4×C4⋊C411yescharacteristic, complemented, Sylow
293C23—62no—
294C23—94no—
295C23—94no—
296C23—94no—
297C23—91no—
298C23—182no—
299C23—182no—
300C23—182no—
301D4—916no—
302C2×C4—62no—
303C2×C4—68no—
304C2×C4—64no—
305C2×C4—94no—
306C2×C4—92no—
307C2×C4—92no—
308C2×C4—92no—
309C2×C4—91no—
310C2×C4—92no—
311C2×C4—92no—
312C2×C4—92no—
313C2×C4—92no—
314C2×C4—128no—
315C2×C4—182no—
316C2×C4—182no—
317C2×C4—182no—
318C2×C4—188no—
319C2×C4—184no—
320C2×C4—184no—
321C2×C4—181no—
322C2×C4—181no—
323C2×C4—182no—
324C2×C4—182no—
325C2×C4—182no—
326C2×C4—368no—
327C2×C4C2×GO+(4,𝔽2)12yes—
328C2×C4C2×GO+(4,𝔽2)11yescharacteristic
329C2×C3—22no—
330C2×C3—22no—
331C2×C3—22no—
332C2×C3—68no—
333D3—28no—
334D3—62no—
335D3—62no—
336D3—62no—
337D3—62no—
338C22—64no—
339C22—64no—
340C22—64no—
341C22—98no—
342C22—91no—
343C22—91no—
344C22—91no—
345C22—91no—
346C22—91no—
347C22—91no—
348C22—184no—
349C22—184no—
350C22—184no—
351C22C22×GO+(4,𝔽2)11yescharacteristic
352C4—22no—
353C4—64no—
354C4—94no—
355C4—92no—
356C4—124no—
357C4—182no—
358C4—182no—
359C4—362no—
360C4C4×GO+(4,𝔽2)12yescomplemented
361C3—22no—
362C2—68no—
363C2—91no—
364C2—91no—
365C2—91no—
366C2—91no—
367C2C2×C4×GO+(4,𝔽2)11yescharacteristic
368C2D4×GO+(4,𝔽2)11yescharacteristic
369C2Q8×GO+(4,𝔽2)11yescharacteristic
subgroupG/H
center, FrattiniC22C22×GO+(4,𝔽2)
commutatorC2×C3⋊D3C23×C4
FittingC32×C4⋊C4D4
socleC22×C32C22×D4
prime radicalC22×GO+(4,𝔽2)C22
cosocleC22×C3⋊D3C24
hypercenterC4⋊C4GO+(4,𝔽2)
hypocenterC32D4×C4⋊C4
LCSG ▷ C2×C3⋊D3 ▷ C32
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
3. D32×C4⋊C4C2semidirect
4. C4⋊C4×#157C2semidirect
5. #10593413C2semidirect
6. #10593410C2semidirect
7. #10593403C2semidirect
8. #10593398C2semidirect
21. C4×GO+(4,𝔽2)C4semidirect
49. C4×D32C2×C4semidirect
52. C4×#157C2×C4semidirect
55. #3773C2×C4semidirect
57. D32⋊C4C2×C4semidirect
58. C32×C4⋊C4D4semidirect
59. C4⋊(C3⋊Dic3)D4semidirect
60. #3750D4semidirect
94. GO+(4,𝔽2)C4⋊C4direct (N × H)
161. D32C2×C4⋊C4semidirect
164. #157C2×C4⋊C4semidirect
166. C32×C4C4×D4semidirect
168. C3⋊Dic3C4×D4semidirect
262. C4⋊C4GO+(4,𝔽2)direct (N × H)
292. C32D4×C4⋊C4semidirect
360. C4C4×GO+(4,𝔽2)semidirect

Maximal subgroups

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

subgroupindex
C2×C4×GO+(4,𝔽2)2
D32×C4⋊C42
C4⋊C4×#1572
#105934132
#105934102
#105934032
#105933982
D4×C4⋊C49