Finite Group Databasebeta

← C2×#7417 #150236

Subgroups — C2×#7417 #150236

Order 512.

557 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 557 conjugacy classes of subgroups — too many to draw. Only its 82 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×#2039C211yesmaximal, characteristic
2C2×#1325C211yesmaximal, characteristic, complemented
3C2×#1296C211yesmaximal, characteristic, complemented
4#7417C214yesmaximal, direct product factor
5C2×#537C411yescharacteristic
6C22×C4⋊C8—21no—
7C22×#104—21no—
8C2×#417—21no—
9C2×#384C2211yescharacteristic
10#2039—21no—
11#2039—21no—
12#2039C412yes—
13#2039C2212yes—
14C2×#343C411yescharacteristic
15C2×#324—21no—
16#1325—21no—
17#1325C2212yescomplemented
18#1296—21no—
19#1296C2212yescomplemented
20C22×#41—21no—
21C22×M16—21no—
22C23×C8—41no—
23#537—21no—
24#537—41no—
25#537—42no—
26#537C2×C412yes—
27C2×C22⋊Q8—41no—
28C2×#121—41no—
29C2×#121—41no—
30C22×C4⋊C4—41no—
31C22×C4⋊C4C2×C411yescharacteristic
32C22×C4⋊C4D411yescharacteristic
33C22×C42D411yescharacteristic
34C2×C4⋊C8—24no—
35C2×C4⋊C8—22no—
36C2×C4⋊C8—41no—
37C2×C4⋊C8—42no—
38C2×#104—22no—
39C2×#104—22no—
40C2×#104—42no—
41C2×#104—42no—
42#417—22no—
43#417—22no—
44#417—42no—
45C2×#103—41no—
46C2×#102—41no—
47C2×C22⋊C8—41no—
48#384—41no—
49#384C2×C411yescharacteristic
50#384D411yescharacteristic
51#384D411yescharacteristic
52#384C2311yescharacteristic
53#343—21no—
54#343C2×C412yes—
55#324—22no—
56#324—41no—
57C2×#41—22no—
58C2×#41—22no—
59C2×#41—41no—
60C2×#41—42no—
61C2×#41—42no—
62C2×#41—41no—
63C2×#41—42no—
64C2×#41—42no—
65C22×Q8—21no—
66C22×D4—21no—
67C22×D4—41no—
68C22×D4—41no—
69C23×C4—21no—
70C23×C4—21no—
71C23×C4—21no—
72C23×C4—41no—
73C23×C4C22⋊C411yescharacteristic
74C2×M16—22no—
75C2×M16—41no—
76C2×M16—42no—
77C2×M16—42no—
78C22×C8—44no—
79C22×C8—42no—
80C22×C8—41no—
81C22×C8—82no—
82C22×C8—82no—
83C22×C8—81no—
84C22⋊Q8—42no—
85C22⋊Q8—44no—
86C22⋊Q8—42no—
87#121—42no—
88#121—42no—
89#121—42no—
90#121—42no—
91#121—42no—
92#121—44no—
93#121—42no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4—41no—
103C2×C4⋊C4—41no—
104C2×C4⋊C4—41no—
105C2×C4⋊C4—82no—
106C2×C4⋊C4—82no—
107C2×C4⋊C4C22⋊C411yescharacteristic
108C2×C4⋊C4C22⋊C411yescharacteristic
109C2×C4⋊C4C22×C411yescharacteristic
110C2×C4⋊C4C2×D411yescharacteristic
111C2×C22⋊C4—42no—
112C2×C42—21no—
113C2×C42—22no—
114C2×C42C22⋊C411yescharacteristic
115C2×C42D812yes—
116C2×C42SD1612yes—
117C2×C42C2×D411yescharacteristic
118C4⋊C8—24no—
119C4⋊C8—44no—
120C4⋊C8—42no—
121#104—44no—
122#104—42no—
123#104—42no—
124#103—42no—
125#103—42no—
126#102—42no—
127#102—42no—
128C22⋊C8—42no—
129C22⋊C8—81no—
130C24—41no—
131C24#9911yescharacteristic
132#41—44no—
133#41—44no—
134#41—42no—
135#41—42no—
136#41—42no—
137#41—42no—
138#41—82no—
139#41—82no—
140#41—82no—
141#41—82no—
142C2×Q8—22no—
143C2×Q8—41no—
144C2×Q8—42no—
145C2×Q8—42no—
146C2×D4—22no—
147C2×D4—42no—
148C2×D4—42no—
149C2×D4—41no—
150C2×D4—42no—
151C2×D4—42no—
152C2×D4—42no—
153C2×D4—42no—
154C2×D4—82no—
155C2×D4—82no—
156C2×D4—162no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—22no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—82no—
180C22×C4—82no—
181C22×C4—82no—
182C22×C4—82no—
183C22×C4#9911yescharacteristic
184C22×C4#10212yes—
185C22×C4#10412yes—
186C22×C4C2×C22⋊C411yescharacteristic
187M16—42no—
188M16—42no—
189M16—82no—
190C2×C8—42no—
191C2×C8—44no—
192C2×C8—82no—
193C2×C8—82no—
194C2×C8—88no—
195C2×C8—82no—
196C2×C8—82no—
197C2×C8—162no—
198C4⋊C4—41no—
199C4⋊C4—41no—
200C4⋊C4—41no—
201C4⋊C4—41no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—81no—
206C4⋊C4—81no—
207C4⋊C4—81no—
208C4⋊C4—82no—
209C4⋊C4—82no—
210C4⋊C4—81no—
211C4⋊C4—81no—
212C22⋊C4—44no—
213C22⋊C4—44no—
214C42—22no—
215C42—22no—
216C42—21no—
217C42—21no—
218C42#10212yes—
219C42C2×D811yescharacteristic
220C42C2×SD1611yescharacteristic
221C23—22no—
222C23—22no—
223C23—42no—
224C23—82no—
225C23—82no—
226C23—82no—
227C23#32712yes—
228C23AΣL(1,𝔽16)11yescharacteristic
229C23AΣL(1,𝔽16)11yescharacteristic
230C23#35512yes—
231C23C2×#9911yescharacteristic
232Q8—42no—
233Q8—42no—
234Q8—82no—
235D4—42no—
236D4—42no—
237D4—82no—
238D4—84no—
239D4—82no—
240D4—82no—
241D4—162no—
242D4—162no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—24no—
248C2×C4—22no—
249C2×C4—21no—
250C2×C4—24no—
251C2×C4—24no—
252C2×C4—24no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—42no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—82no—
270C2×C4—84no—
271C2×C4—82no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4—82no—
275C2×C4—81no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4—161no—
280C2×C4—161no—
281C2×C4#32712yes—
282C2×C4C2×#10211yescharacteristic
283C2×C4C2×#10411yescharacteristic
284C8—84no—
285C8—82no—
286C8—162no—
287C22—21no—
288C22—21no—
289C22—21no—
290C22—21no—
291C22—24no—
292C22—24no—
293C22—82no—
294C22—82no—
295C22—82no—
296C22—164no—
297C22#134512yes—
298C22#134512yes—
299C22#140212yes—
300C22#140512yes—
301C22C2×#32711yescharacteristic
302C22C2×AΣL(1,𝔽16)11yescharacteristic
303C22C2×#35511yescharacteristic
304C4—21no—
305C4—21no—
306C4—24no—
307C4—21no—
308C4—21no—
309C4—21no—
310C4—22no—
311C4—21no—
312C4—82no—
313C4—81no—
314C4—81no—
315C4—81no—
316C4—81no—
317C2—21no—
318C2—22no—
319C2—21no—
320C2—162no—
321C2#741714yesdirect product factor
322C2C2×#134511yescharacteristic
323C2C2×#140211yescharacteristic
324C2C2×#140511yescharacteristic
subgroupG/H
center, socleC23AΣL(1,𝔽16)
commutatorC2×C4⋊C4C22×C4
Frattini, cosocle#384C23
prime radicalC2×#537C4
LCSG ▷ C2×C4⋊C4 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24 ◁ C23×C4 ◁ C2×#384 ◁ G
LECSG ▷ #384 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1325C2semidirect
3. C2×#1296C2semidirect
4. #7417C2direct (N × H)
17. #1325C22semidirect
19. #1296C22semidirect
321. C2#7417direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#20392
C2×#13252
C2×#12962
#74172