Finite Group Databasebeta

← C4×#1325 #128337

Subgroups — C4×#1325 #128337

Order 512.

696 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 158 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
1C4×#384C211yesmaximal, characteristic
2C2×#1325C212yesmaximal
3C4×#324C212yesmaximal
4#7189C212yesmaximal
5C2×#384C411yescharacteristic
6C2×#384C2211yescharacteristic
7C2×C4×C4⋊C4—22no—
8C2×C4×C4⋊C4C411yescharacteristic
9C2×C4×C4⋊C4C2211yescharacteristic
10C2×C43C411yescharacteristic
11C4×C22⋊C8—22no—
12C2×#324C2214yes—
13C4⋊#95—22no—
14C4⋊#95C411yescharacteristic
15C4⋊#95C2211yescharacteristic
16#1325C418yesdirect product factor
17#1280—22no—
18C22×C4⋊C4—22no—
19C22×C4⋊C4—22no—
20C22×C4⋊C4C2×C411yescharacteristic
21C22×C4⋊C4C2×C411yescharacteristic
22C22×C4⋊C4C2×C411yescharacteristic
23C22×C4⋊C4C2311yescharacteristic
24C22×C42—21no—
25C22×C42—21no—
26C22×C42—21no—
27C22×C42C2×C411yescharacteristic
28C22×C42C2×C411yescharacteristic
29C22×C42C2×C411yescharacteristic
30C22×C42D412yes—
31C2×C22⋊C8—24no—
32C2×C4×C8—42no—
33#384—22no—
34#384—22no—
35#384C814yescomplemented
36#384C2×C414yes—
37C4×C4⋊C4—41no—
38C4×C4⋊C4—41no—
39C4×C4⋊C4—48no—
40C4×C4⋊C4—41no—
41C4×C4⋊C4—41no—
42C2×#95—21no—
43C2×#95—22no—
44C2×#95—22no—
45C2×#95D412yes—
46C43—22no—
47C43C814yescomplemented
48#336—42no—
49#324—24no—
50#324C2×C418yes—
51C23×C4—21no—
52C23×C4—21no—
53C23×C4—21no—
54C23×C4—21no—
55C23×C4C22⋊C412yes—
56C23×C4C22⋊C411yescharacteristic
57C23×C4C22⋊C411yescharacteristic
58C23×C4C22×C411yescharacteristic
59C23×C4C2×D412yes—
60C22×C8—44no—
61C2×C4⋊C4—28no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—28no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—41no—
66C2×C4⋊C4—41no—
67C2×C4⋊C4—41no—
68C2×C4⋊C4—41no—
69C2×C4⋊C4—41no—
70C2×C4⋊C4—41no—
71C2×C4⋊C4—41no—
72C2×C4⋊C4—41no—
73C2×C4⋊C4—44no—
74C2×C4⋊C4—44no—
75C2×C4⋊C4—44no—
76C2×C4⋊C4—44no—
77C2×C4⋊C4C4212yes—
78C2×C4⋊C4C2×C812yes—
79C2×C4⋊C4C2×C812yes—
80C2×C4⋊C4M1612yes—
81C2×C4⋊C4C22×C412yes—
82C2×C4⋊C4#4112yes—
83C2×C42—24no—
84C2×C42—24no—
85C2×C42—24no—
86C2×C42—21no—
87C2×C42—21no—
88C2×C42—22no—
89C2×C42—21no—
90C2×C42—21no—
91C2×C42—21no—
92C2×C42—24no—
93C2×C42—22no—
94C2×C42—21no—
95C2×C42—22no—
96C2×C42—41no—
97C2×C42—41no—
98C2×C42—42no—
99C2×C42—42no—
100C2×C42—44no—
101C2×C42—44no—
102C2×C42—41no—
103C2×C42—41no—
104C2×C42C4212yes—
105C2×C42C2×C812yes—
106C2×C42C2×C812yes—
107C2×C42C2×C812yes—
108C2×C42C2×C812yes—
109C2×C42M1612yes—
110C22⋊C8—28no—
111C22⋊C8—44no—
112C4×C8—84no—
113#95—42no—
114#95—44no—
115#95—44no—
116#95—42no—
117C24C2×C22⋊C411yescharacteristic
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—21no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—21no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—41no—
143C22×C4—41no—
144C22×C4—41no—
145C22×C4—41no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4—42no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4—42no—
155C22×C4—42no—
156C22×C4—41no—
157C22×C4—41no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4C22⋊C812yes—
161C22×C4C22⋊C812yes—
162C22×C4C22⋊C814yes—
163C22×C4#9911yescharacteristic
164C22×C4#9911yescharacteristic
165C22×C4#10112yes—
166C22×C4C2×C4211yescharacteristic
167C22×C4#11711yescharacteristic
168C22×C4C4×D414yes—
169C22×C4C22×C811yescharacteristic
170C22×C4C22×C811yescharacteristic
171C22×C4C2×M1611yescharacteristic
172C22×C4C2×M1611yescharacteristic
173C2×C8—48no—
174C2×C8—84no—
175C2×C8—84no—
176C4⋊C4—42no—
177C4⋊C4—42no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—42no—
181C4⋊C4—42no—
182C4⋊C4—416no—
183C4⋊C4—416no—
184C4⋊C4—42no—
185C4⋊C4—42no—
186C42—216no—
187C42—22no—
188C42—22no—
189C42—22no—
190C42—22no—
191C42—22no—
192C42—22no—
193C42—22no—
194C42—42no—
195C42—48no—
196C42—42no—
197C42—44no—
198C42—82no—
199C42—82no—
200C42—82no—
201C42C4×C814yescomplemented
202C23—21no—
203C23—21no—
204C23—21no—
205C23—21no—
206C23C4×C22⋊C411yescharacteristic
207C23C2×C22⋊C811yescharacteristic
208C23C2×C22⋊C811yescharacteristic
209C23C2×#9911yescharacteristic
210C23#41011yescharacteristic
211C23C2×#10111yescharacteristic
212C23#41311yescharacteristic
213C2×C4—24no—
214C2×C4—24no—
215C2×C4—24no—
216C2×C4—24no—
217C2×C4—24no—
218C2×C4—24no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—24no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—24no—
235C2×C4—24no—
236C2×C4—24no—
237C2×C4—22no—
238C2×C4—24no—
239C2×C4—24no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—44no—
255C2×C4—44no—
256C2×C4—44no—
257C2×C4—44no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—82no—
261C2×C4—82no—
262C2×C4—82no—
263C2×C4—82no—
264C2×C4—81no—
265C2×C4—81no—
266C2×C4—81no—
267C2×C4—81no—
268C2×C4#32412yes—
269C2×C4#32412yes—
270C2×C4#35312yes—
271C2×C4#35612yes—
272C2×C4C2×C4×C811yescharacteristic
273C2×C4C4×M1611yescharacteristic
274C2×C4#43111yescharacteristic
275C2×C4#43111yescharacteristic
276C8—88no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—21no—
284C22—21no—
285C22—21no—
286C22—21no—
287C22—21no—
288C22—21no—
289C22C2×#32411yescharacteristic
290C22#145811yescharacteristic
291C22#145911yescharacteristic
292C22#146111yescharacteristic
293C22C4×C22⋊C811yescharacteristic
294C22C4×#9911yescharacteristic
295C22C4×#10111yescharacteristic
296C22C2×#35311yescharacteristic
297C22#212611yescharacteristic
298C22C2×#35611yescharacteristic
299C22#213211yescharacteristic
300C4—28no—
301C4—22no—
302C4—21no—
303C4—21no—
304C4—21no—
305C4—21no—
306C4—44no—
307C4—44no—
308C4—42no—
309C4—44no—
310C4—82no—
311C4—82no—
312C4#132514yesdirect product factor
313C2—21no—
314C2—21no—
315C2—21no—
316C2—21no—
317C2C4×#32411yescharacteristic
318C2C2×#132511yescharacteristic
319C2#1144711yescharacteristic
320C2#1147511yescharacteristic
321C2#1147811yescharacteristic
322C2C4×#35311yescharacteristic
323C2C4×#35611yescharacteristic
subgroupG/H
centerC22×C4#99
commutatorC2×C4C2×C4×C8
Frattini, cosocleC22×C4⋊C4C23
socleC23C2×#99
prime radicalC24C2×C22⋊C4
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C4 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
16. #1325C4direct (N × H)
35. #384C8semidirect
47. C43C8semidirect
201. C42C4×C8semidirect
312. C4#1325direct (N × H)

Maximal subgroups

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

subgroupindex
C4×#3842
C2×#13252
C4×#3242
#71892