Finite Group Databasebeta

← C2×#33802 #7623403

Subgroups — C2×#33802 #7623403

Order 512.

1272 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3170C211yesmaximal, characteristic
2C22×#499C211yesmaximal, characteristic
3#33802C2116yesmaximal, direct product factor
4C2×#2255C212yesmaximal, complemented
5C2×#2245C212yesmaximal, complemented
6C2×#2236C212yesmaximal
7C2×#2235C212yesmaximal
8C2×#2229C214yesmaximal, complemented
9C2×#2163C211yesmaximal, characteristic
10C2×#533—21no—
11C22×#125—21no—
12#3170—24no—
13#3170C2218yes—
14C2×#500C2211yescharacteristic
15C2×#499—21no—
16C2×#499—24no—
17C2×#499—21no—
18C2×#499C2218yes—
19C2×#499C2212yes—
20C2×#499C2212yes—
21C2×#494C2211yescharacteristic
22C2×#487—21no—
23C2×#467—21no—
24C2×#467C2212yes—
25C2×C4×Q16—21no—
26C2×C4×Q16C2212yes—
27C22×#107C2211yescharacteristic
28C22×#107C2211yescharacteristic
29C22×#106—21no—
30C22×C4×C8C2211yescharacteristic
31#2255C22116yescomplemented
32#2245C22116yescomplemented
33#2236C22116yes—
34#2235C22116yes—
35#2229C22132yescomplemented
36#2163C2218yes—
37C2×#367C2214yes—
38C2×#366C2214yes—
39C2×C4⋊C16C2212yes—
40C2×#358C2214yes—
41C2×#358C2214yes—
42C2×C22⋊C16C2212yes—
43C2×#346C2212yes—
44C22×Q16—21no—
45C22×Q16C2312yes—
46C23×C8D411yescharacteristic
47C23×C8C2311yescharacteristic
48#533—28no—
49#533—44no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×#125—22no—
53C2×#125—22no—
54C2×#125—28no—
55C2×#125—41no—
56C2×#125—41no—
57C2×#125—44no—
58C2×#125—41no—
59C2×C22⋊Q8—22no—
60C2×C22⋊Q8—41no—
61C2×C4×Q8—22no—
62C2×C4×Q8—41no—
63C2×#117—41no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—41no—
67C22×C4⋊C4—41no—
68C22×C42D411yescharacteristic
69C22×C16C2314yes—
70#500—22no—
71#500C2314yes—
72#499—28no—
73#499—24no—
74#499—28no—
75#499—24no—
76#499C2318yes—
77#499C2318yes—
78#494—22no—
79#494C2314yes—
80#487—28no—
81#467—24no—
82#467—28no—
83#467C2318yes—
84C4×Q16—24no—
85C4×Q16—28no—
86C4×Q16C2318yes—
87C2×#107—21no—
88C2×#107—21no—
89C2×#107—22no—
90C2×#107—22no—
91C2×#107—21no—
92C2×#107C2312yes—
93C2×#107C2312yes—
94C2×#107C2314yes—
95C2×#107C2314yes—
96C2×#107C2312yes—
97C2×#106—22no—
98C2×#106—28no—
99C2×#106—22no—
100C2×#103—22no—
101C2×#103—22no—
102C2×C4×C8D411yescharacteristic
103C2×C4×C8D414yes—
104C2×C4×C8D411yescharacteristic
105C2×C4×C8C2311yescharacteristic
106C2×C4×C8C2314yes—
107C2×C4×C8C2311yescharacteristic
108#367C23116yes—
109#366C23116yes—
110C4⋊C16C2318yes—
111#358C23116yes—
112#358C23116yes—
113C22⋊C16C2318yes—
114#346C2318yes—
115C22×Q8—22no—
116C22×Q8—41no—
117C23×C4—41no—
118C23×C4—41no—
119C23×C4D811yescharacteristic
120C23×C4D811yescharacteristic
121C23×C4C2×D411yescharacteristic
122C2×Q16—24no—
123C2×Q16—22no—
124C2×Q16—28no—
125C2×Q16—44no—
126C2×Q16C2414yes—
127C22×C8—24no—
128C22×C8—24no—
129C22×C8C2×D411yescharacteristic
130C22×C8C2×D414yes—
131C22×C8C2×D411yescharacteristic
132C22×C8C2×D412yes—
133C22×C8C2×D411yescharacteristic
134C22×C8C2412yes—
135C22×C8C2411yescharacteristic
136C4⋊Q8—24no—
137C4⋊Q8—24no—
138C4⋊Q8—42no—
139C4⋊Q8—42no—
140#125—28no—
141#125—28no—
142#125—48no—
143#125—44no—
144#125—48no—
145#125—44no—
146#125—48no—
147C22⋊Q8—28no—
148C22⋊Q8—44no—
149C22⋊Q8—48no—
150C4×Q8—28no—
151C4×Q8—44no—
152C4×Q8—48no—
153#117—48no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—42no—
160C2×C4⋊C4—42no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—48no—
164C2×C4⋊C4—48no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—41no—
174C2×C22⋊C4—42no—
175C2×C42—42no—
176C2×C42D811yescharacteristic
177C2×C42D811yescharacteristic
178C2×C42D814yes—
179C2×C42C2×D411yescharacteristic
180C2×C42C2×D411yescharacteristic
181C2×C42C2×D414yes—
182C2×C16—28no—
183C2×C16C2418yes—
184#107—24no—
185#107—24no—
186#107—22no—
187#107—22no—
188#107—24no—
189#107—22no—
190#107C2414yes—
191#107C2414yes—
192#107C2414yes—
193#106—28no—
194#106—28no—
195#103—28no—
196#103—28no—
197C4×C8C2×D414yes—
198C4×C8C2×D412yes—
199C4×C8C2×D414yes—
200C4×C8C2×D412yes—
201C4×C8C2412yes—
202C4×C8C2412yes—
203C24C2×D811yescharacteristic
204C2×Q8—24no—
205C2×Q8—44no—
206C2×Q8—42no—
207C2×Q8—44no—
208C2×Q8—44no—
209C2×Q8—84no—
210C22×C4—24no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—42no—
214C22×C4—44no—
215C22×C4—44no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—84no—
221C22×C4—84no—
222C22×C4C2×D812yes—
223C22×C4C2×D814yes—
224C22×C4C2×D811yescharacteristic
225C22×C4C2×D811yescharacteristic
226C22×C4C2×D811yescharacteristic
227C22×C4C2×D811yescharacteristic
228C22×C4C2×D811yescharacteristic
229C22×C4C2×D814yes—
230C22×C4C22×D412yes—
231C22×C4C22×D411yescharacteristic
232Q16—28no—
233Q16—44no—
234Q16—48no—
235C2×C8—22no—
236C2×C8—22no—
237C2×C8—24no—
238C2×C8—28no—
239C2×C8—24no—
240C2×C8#13714yes—
241C2×C8C22×D412yes—
242C2×C8C22×D412yes—
243C2×C8C22×D411yescharacteristic
244C2×C8C22×D412yes—
245C2×C82−1+414yes—
246C2×C8C2511yescharacteristic
247C4⋊C4—24no—
248C4⋊C4—24no—
249C4⋊C4—24no—
250C4⋊C4—44no—
251C4⋊C4—44no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—48no—
256C4⋊C4—48no—
257C4⋊C4—48no—
258C4⋊C4—48no—
259C4⋊C4—44no—
260C4⋊C4—44no—
261C4⋊C4—48no—
262C4⋊C4—48no—
263C4⋊C4—44no—
264C22⋊C4—48no—
265C42—48no—
266C42C2×D814yes—
267C42C2×D812yes—
268C42C2×D814yes—
269C42C2×D812yes—
270C42C22×D412yes—
271C42C22×D412yes—
272C16—28no—
273C23—22no—
274C23—22no—
275C23#51014yes—
276C23C22×D812yes—
277C23C22×D811yescharacteristic
278Q8—44no—
279Q8—44no—
280Q8—88no—
281Q8—84no—
282C2×C4—28no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—24no—
290C2×C4—22no—
291C2×C4—44no—
292C2×C4—44no—
293C2×C4—44no—
294C2×C4—44no—
295C2×C4—44no—
296C2×C4—88no—
297C2×C4—88no—
298C2×C4—84no—
299C2×C4—84no—
300C2×C4—84no—
301C2×C4—84no—
302C2×C4#53612yes—
303C2×C4C22×D812yes—
304C2×C4C22×D812yes—
305C2×C4C22×D812yes—
306C2×C4C22×D812yes—
307C2×C4C22×D812yes—
308C2×C4C22×D811yescharacteristic
309C2×C4C2×#13712yes—
310C2×C4C23×D411yescharacteristic
311C8—22no—
312C8—22no—
313C8#53612yes—
314C8C2×#13714yes—
315C8C2×2−1+412yes—
316C22—28no—
317C22—22no—
318C22—22no—
319C22#322612yes—
320C22C2×#51014yes—
321C22C2×#51012yes—
322C22#341412yes—
323C22C23×D811yescharacteristic
324C4—22no—
325C4—22no—
326C4—22no—
327C4—22no—
328C4—22no—
329C4—84no—
330C4—84no—
331C4—84no—
332C4—84no—
333C4#322612yes—
334C4C2×#53611yescharacteristic
335C4C22×#13711yescharacteristic
336C2—24no—
337C2#3380214yesdirect product factor
338C2C2×#322611yescharacteristic
339C2C22×#51011yescharacteristic
340C2C2×#341411yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC2×C8C25
prime radicalC24C2×D8
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33802C2direct (N × H)
4. C2×#2255C2semidirect
5. C2×#2245C2semidirect
8. C2×#2229C2semidirect
31. #2255C22semidirect
32. #2245C22semidirect
35. #2229C22semidirect
337. C2#33802direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#31702
C22×#4992
#338022
C2×#22552
C2×#22452
C2×#22362
C2×#22352
C2×#22292
C2×#21632