Finite Group Databasebeta

← C2×#7729 #161701

Subgroups — C2×#7729 #161701

Order 512.

858 conjugacy classes of subgroups.

The subgroup lattice has 361 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2863C211yesmaximal, characteristic
2C2×#1839C211yesmaximal, characteristic
3C2×#1839C211yesmaximal, characteristic
4#7729C218yesmaximal, direct product factor
5C2×#1272C212yesmaximal
6C2×#1271C212yesmaximal
7#2863—22no—
8#2863—22no—
9#2863C414yes—
10#2863C2214yes—
11C2×#391C411yescharacteristic
12C2×#391C411yescharacteristic
13C2×#391C2211yescharacteristic
14C2×#389C411yescharacteristic
15C2×#389C411yescharacteristic
16C2×#389C2211yescharacteristic
17C2×#387—21no—
18C2×#387—21no—
19C2×#387—21no—
20C2×#383—21no—
21C2×C43C2211yescharacteristic
22#1839—22no—
23#1839—22no—
24#1839C2214yes—
25#1839C2214yes—
26C2×#336—21no—
27C2×#336—21no—
28C2×#336—21no—
29C2×#336—21no—
30C2×#336C2212yes—
31C2×#336C2212yes—
32#1272C2218yes—
33#1271C2218yes—
34C23×Q8C2×C411yescharacteristic
35C23×C8—22no—
36C23×C8—22no—
37C22×C4⋊C4—21no—
38C22×C4⋊C4—21no—
39C22×C4⋊C4—21no—
40C22×C4⋊C4C2×C411yescharacteristic
41C22×C4⋊C4C2×C411yescharacteristic
42C22×C4⋊C4C2×C411yescharacteristic
43C22×C42C2×C411yescharacteristic
44C22×C42C2×C411yescharacteristic
45C22×C42D411yescharacteristic
46C22×C42D411yescharacteristic
47C22×C42D411yescharacteristic
48C22×C42D411yescharacteristic
49C22×C42C2311yescharacteristic
50#391—22no—
51#391—22no—
52#391—21no—
53#391—21no—
54#391C2×C412yes—
55#391C2×C414yes—
56#391C2×C414yes—
57#391C2312yes—
58#389—22no—
59#389—22no—
60#389—21no—
61#389—21no—
62#389C2×C412yes—
63#389C2×C414yes—
64#389C2×C414yes—
65#389C2312yes—
66#387—24no—
67#387—24no—
68#387—24no—
69#387—24no—
70#387—24no—
71#387—24no—
72#383—24no—
73#383—24no—
74C2×#95—21no—
75C2×#95—21no—
76C2×#95—21no—
77C2×#95—21no—
78C2×#95—21no—
79C2×#95—21no—
80C2×#95—21no—
81C2×#95—21no—
82C2×#95—21no—
83C43C2×C412yes—
84C43D412yes—
85C43D412yes—
86C43C2312yes—
87#336—22no—
88#336—22no—
89#336—24no—
90#336—24no—
91#336—24no—
92#336—24no—
93#336C2314yes—
94#336C2314yes—
95C22×Q8—21no—
96C22×Q8—24no—
97C22×Q8—42no—
98C22×Q8—42no—
99C22×Q8C22×C412yes—
100C23×C4—21no—
101C23×C4—21no—
102C23×C4—21no—
103C23×C4—21no—
104C23×C4C22⋊C411yescharacteristic
105C23×C4C22⋊C411yescharacteristic
106C23×C4C22⋊C411yescharacteristic
107C23×C4C22⋊C411yescharacteristic
108C23×C4C22×C411yescharacteristic
109C23×C4C2×D411yescharacteristic
110C23×C4C2×D411yescharacteristic
111C22×C8—24no—
112C22×C8—24no—
113C22×C8—44no—
114C22×C8—44no—
115C22×C8—44no—
116C22×C8—44no—
117C22×C8—44no—
118C22×C8—44no—
119C2×C4⋊C4—24no—
120C2×C4⋊C4—24no—
121C2×C4⋊C4—24no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—42no—
128C2×C4⋊C4—42no—
129C2×C4⋊C4—44no—
130C2×C4⋊C4—44no—
131C2×C4⋊C4—44no—
132C2×C4⋊C4C22×C412yes—
133C2×C4⋊C4C22×C412yes—
134C2×C4⋊C4C22×C412yes—
135C2×C42—24no—
136C2×C42—24no—
137C2×C42—24no—
138C2×C42—42no—
139C2×C42—42no—
140C2×C42—42no—
141C2×C42—42no—
142C2×C42C22⋊C412yes—
143C2×C42C22⋊C412yes—
144C2×C42C22⋊C412yes—
145C2×C42C22⋊C412yes—
146C2×C42C22⋊C412yes—
147C2×C42C22⋊C412yes—
148C2×C42C22×C412yes—
149C2×C42C22×C411yescharacteristic
150C2×C42C22×C412yes—
151C2×C42C2×D412yes—
152C2×C42C2×D412yes—
153C2×C42C2×D411yescharacteristic
154C2×C42C2×D411yescharacteristic
155C2×C42C2×D412yes—
156C2×C42C2×D412yes—
157C2×C42C2411yescharacteristic
158#95—24no—
159#95—24no—
160#95—22no—
161#95—24no—
162#95—24no—
163#95—22no—
164#95—24no—
165#95—24no—
166#95—22no—
167#95—22no—
168#95—22no—
169#95—22no—
170C24C2×C22⋊C411yescharacteristic
171C2×Q8—24no—
172C2×Q8—42no—
173C2×Q8—42no—
174C2×Q8—42no—
175C2×Q8—42no—
176C2×Q8—42no—
177C2×Q8—88no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4C2×C22⋊C412yes—
213C22×C4C2×C22⋊C412yes—
214C22×C4C2×C22⋊C412yes—
215C22×C4C2×C22⋊C412yes—
216C22×C4C2×C22⋊C411yescharacteristic
217C22×C4C2×C22⋊C411yescharacteristic
218C22×C4C2×C22⋊C411yescharacteristic
219C22×C4#13114yes—
220C22×C4#13514yes—
221C22×C4#13714yes—
222C22×C4C23×C411yescharacteristic
223C22×C4C22×D411yescharacteristic
224C22×C4C22×D411yescharacteristic
225C2×C8—44no—
226C2×C8—44no—
227C2×C8—44no—
228C2×C8—44no—
229C2×C8—44no—
230C2×C8—44no—
231C2×C8—88no—
232C2×C8—88no—
233C4⋊C4—24no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—48no—
239C4⋊C4—48no—
240C4⋊C4—44no—
241C4⋊C4—42no—
242C4⋊C4—44no—
243C42—24no—
244C42—24no—
245C42—24no—
246C42—24no—
247C42—24no—
248C42—24no—
249C42—42no—
250C42—42no—
251C42—42no—
252C42—41no—
253C42—42no—
254C42—41no—
255C42—41no—
256C42—42no—
257C42—42no—
258C42—41no—
259C23#40812yes—
260C23#41012yes—
261C23#41312yes—
262C23#41612yes—
263C23#41912yes—
264C23#42112yes—
265C23#42112yes—
266C23C22×C22⋊C411yescharacteristic
267Q8—84no—
268Q8—84no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—28no—
282C2×C4—28no—
283C2×C4—28no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—44no—
297C2×C4—44no—
298C2×C4—44no—
299C2×C4—44no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—84no—
313C2×C4—84no—
314C2×C4—84no—
315C2×C4—84no—
316C2×C4#40812yes—
317C2×C4#41612yes—
318C2×C4#41912yes—
319C2×C4C2×#13112yes—
320C2×C4C2×#13512yes—
321C2×C4C2×#13712yes—
322C8—84no—
323C8—84no—
324C22#147314yes—
325C22#148214yes—
326C22#149714yes—
327C22#151514yes—
328C22#152114yes—
329C22#153514yes—
330C22#155414yes—
331C22C2×#40811yescharacteristic
332C22C2×#41011yescharacteristic
333C22C2×#41311yescharacteristic
334C22C2×#41611yescharacteristic
335C22C2×#41911yescharacteristic
336C22C2×#42111yescharacteristic
337C22C2×#42111yescharacteristic
338C4—22no—
339C4—22no—
340C4—22no—
341C4—22no—
342C4—22no—
343C4—22no—
344C4—42no—
345C4—42no—
346C4—42no—
347C4—42no—
348C4—82no—
349C4—82no—
350C4—82no—
351C4—82no—
352C2#772918yesdirect product factor
353C2C2×#147311yescharacteristic
354C2C2×#148211yescharacteristic
355C2C2×#149711yescharacteristic
356C2C2×#151511yescharacteristic
357C2C2×#152111yescharacteristic
358C2C2×#153511yescharacteristic
359C2C2×#155411yescharacteristic
subgroupG/H
center, socle, prime radicalC24C2×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7729C2direct (N × H)
352. C2#7729direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#28632
C2×#18392
#77292
C2×#12722
C2×#12712