Finite Group Databasebeta

← C2×#34209 #7625521

Subgroups — C2×#34209 #7625521

Order 512.

10914 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
1C23×C4⋊Q8C211yesmaximal, characteristic
2C22×#519C212yesmaximal
3#34209C2132yesmaximal, direct product factor
4C2×#2365C218yesmaximal, complemented
5C2×#2344C218yesmaximal
6C2×#2327C214yesmaximal, complemented
7C2×#2310C212yesmaximal
8C2×C4×C4⋊Q8C214yesmaximal, complemented
9C2×#2285C211yesmaximal, characteristic
10C2×C4×#117C211yesmaximal, characteristic
11C24×Q8C2212yes—
12C22×C4⋊Q8C4132yescomplemented
13C22×C4⋊Q8C2212yes—
14C22×C4⋊Q8C2214yes—
15C22×C4⋊Q8C2212yes—
16C22×C4⋊Q8C22116yes—
17C2×#519—232no—
18C2×#519C22132yes—
19C22×C4×Q8C2218yes—
20C22×C4×Q8C2218yes—
21C22×#117C2218yes—
22C22×#117C2212yes—
23C22×#117C2211yescharacteristic
24C22×#117C2211yescharacteristic
25C23×C4⋊C4C2214yes—
26C23×C42C2211yescharacteristic
27#2365C221128yescomplemented
28#2344C221128yes—
29#2327C22164yescomplemented
30#2310C22132yes—
31C4×C4⋊Q8C22164yescomplemented
32C2×#391C2218yes—
33C2×#391C2218yes—
34C2×#389C22116yes—
35C2×#387C22132yes—
36#2285C22116yes—
37C2×#384C2212yes—
38C2×#382C2212yes—
39C2×#381C2218yes—
40C2×#380C2214yes—
41C4×#117C22116yes—
42C2×C4×C4⋊C4C22116yes—
43C2×C4×C4⋊C4C2214yes—
44C2×C4×C4⋊C4C2214yes—
45C2×C4×C22⋊C4C2214yes—
46C2×#376C2212yes—
47C2×C43C2212yes—
48C23×Q8—216no—
49C23×Q8—216no—
50C23×Q8C2×C4132yes—
51C23×Q8C23116yes—
52C23×Q8C2314yes—
53C23×Q8C2314yes—
54C24×C4C2314yes—
55C24×C4C2312yes—
56C24×C4C2311yescharacteristic
57C2×C4⋊Q8C2×C4132yes—
58C2×C4⋊Q8C2×C4164yes—
59C2×C4⋊Q8C2×C4132yes—
60C2×C4⋊Q8C2×C41256yescomplemented
61C2×C4⋊Q8C23116yes—
62C2×C4⋊Q8C23116yes—
63C2×C4⋊Q8C23132yes—
64#519—2256no—
65C2×C4×Q8—264no—
66C2×C4×Q8—264no—
67C2×C4×Q8C23164yes—
68C2×C4×Q8C23164yes—
69C2×#117—216no—
70C2×#117—28no—
71C2×#117—264no—
72C2×#117D4116yes—
73C2×#117C2318yes—
74C2×#117C23116yes—
75C2×#117C23164yes—
76C2×#117C2318yes—
77C22×C4⋊C4C2×C4164yes—
78C22×C4⋊C4C2314yes—
79C22×C4⋊C4C23116yes—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2318yes—
82C22×C4⋊C4C23132yes—
83C22×C4⋊C4C2318yes—
84C22×C4⋊C4C2318yes—
85C22×C4⋊C4C2312yes—
86C22×C22⋊C4C2314yes—
87C22×C22⋊C4C2312yes—
88C22×C22⋊C4C2312yes—
89C22×C42C2×C4116yes—
90C22×C42C23116yes—
91C22×C42C2314yes—
92C22×C42C2312yes—
93C22×C42C2318yes—
94C22×C42C2311yescharacteristic
95C22×C42C2311yescharacteristic
96C22×C42C2312yes—
97C22×C42C2318yes—
98C22×C42C2312yes—
99#391C23164yes—
100#391C23164yes—
101#389C231128yes—
102#387C231256yes—
103#384C23116yes—
104#382C23116yes—
105#381C23164yes—
106#380C23132yes—
107C4×C4⋊C4C231128yes—
108C4×C4⋊C4C23132yes—
109C4×C4⋊C4C23132yes—
110C4×C22⋊C4C23132yes—
111#376C23116yes—
112C2×#95C2318yes—
113C2×#95C2318yes—
114C2×#95C23132yes—
115C43C23116yes—
116C25C2411yescharacteristic
117C22×Q8—2256no—
118C22×Q8—216no—
119C22×Q8—216no—
120C22×Q8—264no—
121C22×Q8—216no—
122C22×Q8—216no—
123C22×Q8—264no—
124C22×Q8C22×C4132yes—
125C22×Q8C22×C4132yes—
126C22×Q8C22×C41128yes—
127C22×Q8C24116yes—
128C22×Q8C24116yes—
129C23×C4—216no—
130C23×C4—216no—
131C23×C4—216no—
132C23×C4C22×C4132yes—
133C23×C4C22×C4116yes—
134C23×C4C22×C418yes—
135C23×C4C2×D418yes—
136C23×C4#4118yes—
137C23×C4C2418yes—
138C23×C4C2414yes—
139C23×C4C24116yes—
140C23×C4C2418yes—
141C23×C4C2418yes—
142C23×C4C2412yes—
143C23×C4C2414yes—
144C23×C4C2414yes—
145C23×C4C2411yescharacteristic
146C23×C4C2411yescharacteristic
147C23×C4C2412yes—
148C4⋊Q8C22×C41128yes—
149C4⋊Q8C22×C41128yes—
150C4⋊Q8C22×C41256yes—
151C4×Q8—2256no—
152C4×Q8—2256no—
153#117—264no—
154#117—232no—
155#117—2256no—
156#117C2×D4164yes—
157C2×C4⋊C4—264no—
158C2×C4⋊C4—216no—
159C2×C4⋊C4—28no—
160C2×C4⋊C4—232no—
161C2×C4⋊C4C22×C4164yes—
162C2×C4⋊C4C22×C4164yes—
163C2×C4⋊C4C22×C41256yes—
164C2×C4⋊C4C2×D4116yes—
165C2×C4⋊C4C24116yes—
166C2×C4⋊C4C24164yes—
167C2×C4⋊C4C2418yes—
168C2×C4⋊C4C24132yes—
169C2×C4⋊C4C24132yes—
170C2×C4⋊C4C24132yes—
171C2×C4⋊C4C2418yes—
172C2×C22⋊C4—216no—
173C2×C22⋊C4—28no—
174C2×C22⋊C4C2×D4116yes—
175C2×C22⋊C4C24116yes—
176C2×C22⋊C4C2418yes—
177C2×C22⋊C4C2418yes—
178C2×C42—264no—
179C2×C42—216no—
180C2×C42—28no—
181C2×C42—232no—
182C2×C42C22×C418yes—
183C2×C42C22×C4116yes—
184C2×C42C22×C418yes—
185C2×C42C22×C4164yes—
186C2×C42C2×D4116yes—
187C2×C42C24164yes—
188C2×C42C24116yes—
189C2×C42C2418yes—
190C2×C42C24132yes—
191C2×C42C2414yes—
192C2×C42C2414yes—
193C2×C42C2418yes—
194C2×C42C2418yes—
195#95C24132yes—
196#95C24132yes—
197#95C241128yes—
198C24—28no—
199C24C23×C414yes—
200C24C22×D414yes—
201C24C2×#4114yes—
202C24C2512yes—
203C24C2511yescharacteristic
204C2×Q8—2128no—
205C2×Q8—2128no—
206C2×Q8—2512no—
207C2×Q8—232no—
208C2×Q8—232no—
209C2×Q8—232no—
210C2×Q8—232no—
211C2×Q8C23×C4164yes—
212C2×Q8C23×C4164yes—
213C22×C4—2128no—
214C22×C4—216no—
215C22×C4—232no—
216C22×C4—232no—
217C22×C4—216no—
218C22×C4—216no—
219C22×C4—28no—
220C22×C4—264no—
221C22×C4—232no—
222C22×C4—216no—
223C22×C4—28no—
224C22×C4—216no—
225C22×C4—232no—
226C22×C4C4×D4164yes—
227C22×C4C23×C418yes—
228C22×C4C23×C4132yes—
229C22×C4C23×C4164yes—
230C22×C4C23×C4132yes—
231C22×C4C23×C4116yes—
232C22×C4C23×C414yes—
233C22×C4C23×C414yes—
234C22×C4C23×C418yes—
235C22×C4C22×D4116yes—
236C22×C4C22×D414yes—
237C22×C4C22×D414yes—
238C22×C4C22×D4116yes—
239C22×C4C2×#4114yes—
240C22×C4C2×#4114yes—
241C22×C4C2×#41116yes—
242C22×C42−1+4116yes—
243C22×C4C2518yes—
244C22×C4C25116yes—
245C22×C4C2514yes—
246C22×C4C25116yes—
247C22×C4C2518yes—
248C22×C4C2512yes—
249C22×C4C2512yes—
250C22×C4C2514yes—
251C4⋊C4—2128no—
252C4⋊C4—232no—
253C4⋊C4—216no—
254C4⋊C4—264no—
255C4⋊C4C23×C41128yes—
256C4⋊C4C23×C41128yes—
257C4⋊C4C22×D4132yes—
258C22⋊C4—232no—
259C22⋊C4—216no—
260C22⋊C4C22×D4132yes—
261C42—2128no—
262C42—232no—
263C42—216no—
264C42—264no—
265C42C23×C4116yes—
266C42C23×C4132yes—
267C42C23×C4116yes—
268C42C22×D4132yes—
269C23—28no—
270C23—216no—
271C23—216no—
272C23—216no—
273C23C2×C4×D4116yes—
274C23C24×C414yes—
275C23C24×C412yes—
276C23C23×D414yes—
277C23C23×D412yes—
278C23C22×#4114yes—
279C23C22×#4112yes—
280C23C2×2−1+418yes—
281C23C2611yescharacteristic
282Q8—2128no—
283Q8—2128no—
284C2×C4—216no—
285C2×C4—216no—
286C2×C4—216no—
287C2×C4—28no—
288C2×C4—2128no—
289C2×C4—264no—
290C2×C4—216no—
291C2×C4—232no—
292C2×C4—264no—
293C2×C4—216no—
294C2×C4—28no—
295C2×C4—216no—
296C2×C4—232no—
297C2×C4—432no—
298C2×C4—416no—
299C2×C4C2×C4×D4164yes—
300C2×C4C2×C4×D4116yes—
301C2×C4C2×C4×D4116yes—
302C2×C4#519132yescomplemented
303C2×C4C24×C418yes—
304C2×C4C24×C4132yes—
305C2×C4C24×C414yes—
306C2×C4C24×C414yes—
307C2×C4C24×C418yes—
308C2×C4C23×D4116yes—
309C2×C4C23×D414yes—
310C2×C4C23×D414yes—
311C2×C4C22×#4114yes—
312C2×C4C22×#4114yes—
313C2×C4C2×2−1+4116yes—
314C22—28no—
315C22—28no—
316C22—232no—
317C22—28no—
318C22C22×C4×D418yes—
319C22C22×C4×D414yes—
320C22C2×#51918yes—
321C22#349918yes—
322C22#350918yes—
323C22C25×C411yescharacteristic
324C22C24×D411yescharacteristic
325C22C23×#4111yescharacteristic
326C22C22×2−1+414yes—
327C4—216no—
328C4—232no—
329C4—28no—
330C4—416no—
331C4—48no—
332C4C22×C4×D418yes—
333C4C22×C4×D418yes—
334C4C2×#519116yescomplemented
335C2—28no—
336C2#3420918yesdirect product factor
337C2C23×C4×D411yescharacteristic
338C2C22×#51912yes—
339C2C2×#349912yes—
340C2C2×#350912yes—
subgroupG/H
center, socleC24C25
commutatorC22C25×C4
Frattini, cosocleC23C26
prime radicalC25C24
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #34209C2direct (N × H)
4. C2×#2365C2semidirect
6. C2×#2327C2semidirect
8. C2×C4×C4⋊Q8C2semidirect
12. C22×C4⋊Q8C4semidirect
27. #2365C22semidirect
29. #2327C22semidirect
31. C4×C4⋊Q8C22semidirect
60. C2×C4⋊Q8C2×C4semidirect
302. C2×C4#519semidirect
334. C4C2×#519semidirect
336. C2#34209direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C4⋊Q82
C22×#5192
#342092
C2×#23652
C2×#23442
C2×#23272
C2×#23102
C2×C4×C4⋊Q82
C2×#22852
C2×C4×#1172