Finite Group Databasebeta

← C4×GL(2,ℤ/4) #85484

Subgroups — C4×GL(2,ℤ/4) #85484

Order 384.

497 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 497 conjugacy classes of subgroups — too many to draw. Only its 58 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
1C22×C4×A4C211yesmaximal, characteristic, complemented
2C2×GL(2,ℤ/4)C211yesmaximal, characteristic
3C2×C4×S4C211yesmaximal, characteristic, complemented
4#5432C211yesmaximal, characteristic, complemented
5(C2×S4)⋊C4C211yesmaximal, characteristic, complemented
6SL(2,ℤ/4)⋊C4C211yesmaximal, characteristic, complemented
7C4×SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
8C4×#120—31nomaximal, Sylow
9C23×A4C2211yescharacteristic
10C22×S4C2211yescharacteristic
11C2×C4×A4—22no—
12C2×C4×A4C2211yescharacteristic
13C2×C4×A4C2211yescharacteristic
14GL(2,ℤ/4)C414yesdirect product factor
15C2×SL(2,ℤ/4)C2211yescharacteristic
16C2×SL(2,ℤ/4)C2211yescharacteristic
17C2×SL(2,ℤ/4)C2211yescharacteristic
18C4×S4—22no—
19C4×#67—41nomaximal
20C24×C4D311yescharacteristic, complemented
21C2×#120—31no—
22C2×C4×D4—31no—
23C2×C4×D4—31no—
24C2×C4×D4—31no—
25#386—31no—
26#386—31no—
27#386—31no—
28#385—31no—
29#385—31no—
30#385—31no—
31#379—31no—
32C4×C22⋊C4—31no—
33C4×C22⋊C4—31no—
34C4×C22⋊C4—31no—
35C22×A4—21no—
36C22×A4—21no—
37C22×A4C2×C412yescomplemented
38C22×A4C2311yescharacteristic
39C2×S4—21no—
40C2×S4—21no—
41C2×S4C2×C412yescomplemented
42C22×C3×C4—41no—
43C2×#67—41no—
44C2×C4×D3—41no—
45C4×A4—21no—
46C4×A4D412yescomplemented
47SL(2,ℤ/4)—21no—
48SL(2,ℤ/4)—21no—
49SL(2,ℤ/4)C2×C412yescomplemented
50#217—41no—
51(C2×D3)⋊C4—41no—
52Dic3⋊C4—41no—
53C4×Dic3—41no—
54C25C2×D311yescharacteristic
55C22×D4—31no—
56C22×D4—31no—
57C22×D4—31no—
58C23×C4—22no—
59C23×C4—31no—
60C23×C4—31no—
61C23×C4—31no—
62C23×C4—31no—
63C23×C4—31no—
64C23×C4—62no—
65C23×C4C2×D311yescharacteristic
66C23×C4C2×D311yescharacteristic
67#120—34no—
68#120—34no—
69C4×D4—62no—
70C4×D4—64no—
71C4×D4—61no—
72C4×D4—61no—
73C4×D4—61no—
74C4×D4—62no—
75C4×D4—61no—
76C2×C4⋊C4—31no—
77C2×C4⋊C4—31no—
78C2×C4⋊C4—31no—
79C2×C22⋊C4—31no—
80C2×C22⋊C4—31no—
81C2×C22⋊C4—31no—
82C2×C22⋊C4—31no—
83C2×C22⋊C4—31no—
84C2×C22⋊C4—31no—
85C2×C22⋊C4—31no—
86C2×C22⋊C4—31no—
87C2×C22⋊C4—31no—
88C2×C22⋊C4—31no—
89C2×C22⋊C4—31no—
90C2×C22⋊C4—31no—
91C2×C42—31no—
92C2×C42—31no—
93C2×C42—31no—
94#95—31no—
95#95—31no—
96#95—31no—
97#95—31no—
98#95—31no—
99#95—31no—
100C23×C3—41no—
101C22×D3—41no—
102C2×A4—22no—
103C2×A4C22×C411yescharacteristic
104C2×A4C2×D411yescharacteristic
105C2×A4#4111yescharacteristic
106S4—22no—
107C2×C3×C4—41no—
108C2×C3×C4—41no—
109C2×C3×C4—82no—
110#67—44no—
111C2×Dic3—41no—
112C2×Dic3—41no—
113C2×Dic3—41no—
114C4×D3—82no—
115C24—21no—
116C24—21no—
117C24—31no—
118C24—31no—
119C24—32no—
120C24—31no—
121C24—61no—
122C24—61no—
123C24C4×D312yescomplemented
124C24C22×D311yescharacteristic
125C2×D4—34no—
126C2×D4—32no—
127C2×D4—34no—
128C2×D4—32no—
129C2×D4—61no—
130C2×D4—62no—
131C2×D4—61no—
132C2×D4—61no—
133C2×D4—62no—
134C2×D4—62no—
135C2×D4—61no—
136C2×D4—61no—
137C2×D4—61no—
138C22×C4—21no—
139C22×C4—31no—
140C22×C4—31no—
141C22×C4—31no—
142C22×C4—32no—
143C22×C4—31no—
144C22×C4—31no—
145C22×C4—31no—
146C22×C4—34no—
147C22×C4—31no—
148C22×C4—31no—
149C22×C4—31no—
150C22×C4—31no—
151C22×C4—31no—
152C22×C4—31no—
153C22×C4—31no—
154C22×C4—31no—
155C22×C4—62no—
156C22×C4—62no—
157C22×C4—61no—
158C22×C4—61no—
159C22×C4—62no—
160C22×C4—62no—
161C22×C4—62no—
162C22×C4—62no—
163C22×C4—61no—
164C22×C4—62no—
165C22×C4—62no—
166C22×C4—62no—
167C22×C4—61no—
168C22×C4#6712yescomplemented
169C22×C4S411yescharacteristic, complemented
170C4⋊C4—62no—
171C4⋊C4—62no—
172C4⋊C4—61no—
173C4⋊C4—61no—
174C22⋊C4—34no—
175C22⋊C4—32no—
176C22⋊C4—34no—
177C22⋊C4—32no—
178C22⋊C4—62no—
179C22⋊C4—62no—
180C22⋊C4—61no—
181C22⋊C4—61no—
182C22⋊C4—62no—
183C22⋊C4—62no—
184C22⋊C4—61no—
185C22⋊C4—62no—
186C22⋊C4—61no—
187C22⋊C4—61no—
188C22⋊C4—61no—
189C22⋊C4—61no—
190C22⋊C4—61no—
191C42—61no—
192C42—62no—
193C42—62no—
194C42—61no—
195C22×C3—42no—
196C22×C3—41no—
197C22×C3—81no—
198C22×C3—81no—
199C2×D3—42no—
200C2×D3—81no—
201C2×D3—81no—
202A4C4×D411yescharacteristic, complemented
203C3×C4—42no—
204C3×C4—81no—
205Dic3—42no—
206Dic3—81no—
207Dic3—81no—
208C23—22no—
209C23—31no—
210C23—32no—
211C23—31no—
212C23—31no—
213C23—31no—
214C23—32no—
215C23—32no—
216C23—32no—
217C23—32no—
218C23—31no—
219C23—32no—
220C23—61no—
221C23—61no—
222C23—61no—
223C23—61no—
224C23—62no—
225C23—61no—
226C23—61no—
227C23—62no—
228C23—62no—
229C23—61no—
230C23—64no—
231C23—61no—
232C23—61no—
233C23—61no—
234C23—61no—
235C23—61no—
236C23—61no—
237C23C2×C4×D311yescharacteristic
238C23#23511yescharacteristic
239C23C2×#6711yescharacteristic
240C23C2×S411yescharacteristic
241D4—62no—
242D4—62no—
243D4—62no—
244D4—68no—
245D4—64no—
246D4—62no—
247D4—64no—
248C2×C4—22no—
249C2×C4—31no—
250C2×C4—32no—
251C2×C4—32no—
252C2×C4—32no—
253C2×C4—32no—
254C2×C4—32no—
255C2×C4—32no—
256C2×C4—32no—
257C2×C4—31no—
258C2×C4—62no—
259C2×C4—61no—
260C2×C4—61no—
261C2×C4—61no—
262C2×C4—62no—
263C2×C4—64no—
264C2×C4—61no—
265C2×C4—61no—
266C2×C4—61no—
267C2×C4—61no—
268C2×C4—61no—
269C2×C4—61no—
270C2×C4—62no—
271C2×C4—61no—
272C2×C4—61no—
273C2×C4—61no—
274C2×C4—61no—
275C2×C4—61no—
276C2×C4—61no—
277C2×C4—61no—
278C2×C4—61no—
279C2×C4—61no—
280C2×C4—61no—
281C2×C4—61no—
282C2×C4—61no—
283C2×C4—61no—
284C2×C4—122no—
285C2×C4—122no—
286C2×C4C2×S411yescharacteristic
287C2×C4C2×S411yescharacteristic
288C2×C3—41no—
289C2×C3—41no—
290C2×C3—41no—
291C2×C3—82no—
292D3—82no—
293C22—21no—
294C22—21no—
295C22—31no—
296C22—31no—
297C22—31no—
298C22—31no—
299C22—31no—
300C22—31no—
301C22—31no—
302C22—31no—
303C22—31no—
304C22—34no—
305C22—32no—
306C22—62no—
307C22—62no—
308C22—62no—
309C22—62no—
310C22—61no—
311C22—62no—
312C22—62no—
313C22—62no—
314C22—62no—
315C22—62no—
316C22—61no—
317C22—61no—
318C22—62no—
319C22—121no—
320C22—121no—
321C22—121no—
322C22—121no—
323C22C4×#6711yescharacteristic, complemented
324C22C4×S412yescomplemented
325C22C22×S411yescharacteristic
326C4—21no—
327C4—32no—
328C4—62no—
329C4—62no—
330C4—62no—
331C4—61no—
332C4—121no—
333C4—121no—
334C4—121no—
335C4—121no—
336C4GL(2,ℤ/4)12yesdirect product factor
337C3—41noSylow
338C2—22no—
339C2—31no—
340C2—31no—
341C2—31no—
342C2—31no—
343C2—62no—
344C2—122no—
345C2C2×C4×S411yescharacteristic
346C2#591211yescharacteristic
347C2C2×GL(2,ℤ/4)11yescharacteristic
subgroupG/H
centerC2×C4C2×S4
commutatorC2×A4C22×C4
FrattiniC22C22×S4
FittingC24×C4D3
socleC24C22×D3
prime radicalC2×GL(2,ℤ/4)C2
cosocleC22×A4C23
hypercenterC22×C4S4
hypocenterA4C4×D4
LCSG ▷ C2×A4 ▷ A4
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×C4×A4C2semidirect
3. C2×C4×S4C2semidirect
4. #5432C2semidirect
5. (C2×S4)⋊C4C2semidirect
6. SL(2,ℤ/4)⋊C4C2semidirect
7. C4×SL(2,ℤ/4)C2semidirect
14. GL(2,ℤ/4)C4direct (N × H)
20. C24×C4D3semidirect
37. C22×A4C2×C4semidirect
41. C2×S4C2×C4semidirect
46. C4×A4D4semidirect
49. SL(2,ℤ/4)C2×C4semidirect
123. C24C4×D3semidirect
168. C22×C4#67semidirect
169. C22×C4S4semidirect
202. A4C4×D4semidirect
323. C22C4×#67semidirect
324. C22C4×S4semidirect
336. C4GL(2,ℤ/4)direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×A42
C2×GL(2,ℤ/4)2
C2×C4×S42
#54322
(C2×S4)⋊C42
SL(2,ℤ/4)⋊C42
C4×SL(2,ℤ/4)2
C4×#1203
C4×#674