Finite Group Databasebeta

← C2×#7717 #161679

Subgroups — C2×#7717 #161679

Order 512.

870 conjugacy classes of subgroups.

The subgroup lattice has 365 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2864C211yesmaximal, characteristic
2C2×#1839C211yesmaximal, characteristic
3C2×#1831C211yesmaximal, characteristic
4#7717C218yesmaximal, direct product factor
5C2×#1271C212yesmaximal
6C2×#1271C212yesmaximal
7C22×C4⋊C8—21no—
8#2864—22no—
9#2864—22no—
10#2864C414yes—
11#2864C2214yes—
12C2×#389C411yescharacteristic
13C2×#389C411yescharacteristic
14C2×#389C2211yescharacteristic
15C2×#389C2211yescharacteristic
16C2×#387—21no—
17C2×#387—21no—
18C2×#387—21no—
19C2×#387—21no—
20C2×#382C411yescharacteristic
21C2×#382C411yescharacteristic
22C2×C43C2211yescharacteristic
23#1839—22no—
24#1839C2214yes—
25#1831—22no—
26#1831C2214yes—
27C2×#336—21no—
28C2×#336—21no—
29C2×#336—21no—
30C2×#336C2212yes—
31C2×#336C2212yes—
32#1271C2218yes—
33#1271C2218yes—
34C23×C8—22no—
35C23×C8—22no—
36C22×C4⋊C4—21no—
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×C4⋊C4C2×C411yescharacteristic
44C22×C42C2×C411yescharacteristic
45C22×C42C2×C411yescharacteristic
46C22×C42D411yescharacteristic
47C22×C42D411yescharacteristic
48C22×C42D411yescharacteristic
49C22×C42D411yescharacteristic
50C22×C42C2311yescharacteristic
51C2×C4⋊C8—24no—
52C2×C4⋊C8—44no—
53#389—22no—
54#389—22no—
55#389—21no—
56#389—21no—
57#389—21no—
58#389—21no—
59#389C2×C412yes—
60#389C2×C412yes—
61#389C2×C414yes—
62#389C2×C414yes—
63#389C2312yes—
64#389C2312yes—
65#387—24no—
66#387—24no—
67#387—24no—
68#387—24no—
69#387—24no—
70#387—24no—
71#387—24no—
72#387—24no—
73#382—22no—
74#382—22no—
75#382C2×C414yes—
76#382C2×C414yes—
77C2×#95—21no—
78C2×#95—21no—
79C2×#95—21no—
80C2×#95—21no—
81C2×#95—21no—
82C2×#95—21no—
83C2×#95—21no—
84C2×#95—21no—
85C43C2×C412yes—
86C43D412yes—
87C43D412yes—
88C43C2312yes—
89#336—22no—
90#336—24no—
91#336—22no—
92#336—24no—
93#336—24no—
94#336C2314yes—
95#336C2314yes—
96C23×C4—21no—
97C23×C4—21no—
98C23×C4—21no—
99C23×C4—21no—
100C23×C4C22⋊C411yescharacteristic
101C23×C4C22⋊C411yescharacteristic
102C23×C4C22⋊C411yescharacteristic
103C23×C4C22⋊C411yescharacteristic
104C23×C4C22×C411yescharacteristic
105C23×C4C2×D411yescharacteristic
106C23×C4C2×D411yescharacteristic
107C22×C8—24no—
108C22×C8—24no—
109C22×C8—44no—
110C22×C8—44no—
111C22×C8—44no—
112C22×C8—44no—
113C22×C8—44no—
114C22×C8—44no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4—24no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4—24no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—22no—
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⋊C4—42no—
133C2×C4⋊C4—42no—
134C2×C4⋊C4C22×C412yes—
135C2×C4⋊C4C22×C412yes—
136C2×C4⋊C4C22×C412yes—
137C2×C4⋊C4C22×C412yes—
138C2×C42—24no—
139C2×C42—24no—
140C2×C42—24no—
141C2×C42—22no—
142C2×C42—22no—
143C2×C42—42no—
144C2×C42—42no—
145C2×C42—42no—
146C2×C42C22⋊C412yes—
147C2×C42C22⋊C412yes—
148C2×C42C22⋊C412yes—
149C2×C42C22⋊C412yes—
150C2×C42C22⋊C412yes—
151C2×C42C22⋊C412yes—
152C2×C42C22×C412yes—
153C2×C42C22×C412yes—
154C2×C42C22×C411yescharacteristic
155C2×C42C2×D412yes—
156C2×C42C2×D412yes—
157C2×C42C2×D411yescharacteristic
158C2×C42C2×D411yescharacteristic
159C2×C42C2×D412yes—
160C2×C42C2×D412yes—
161C2×C42C2411yescharacteristic
162C4⋊C8—48no—
163#95—24no—
164#95—24no—
165#95—24no—
166#95—24no—
167#95—24no—
168#95—24no—
169#95—22no—
170#95—22no—
171#95—22no—
172#95—22no—
173C24C2×C22⋊C411yescharacteristic
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—22no—
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—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4#10114yes—
208C22×C4C2×C22⋊C412yes—
209C22×C4C2×C22⋊C412yes—
210C22×C4C2×C22⋊C412yes—
211C22×C4C2×C22⋊C411yescharacteristic
212C22×C4C2×C22⋊C412yes—
213C22×C4C2×C22⋊C411yescharacteristic
214C22×C4C2×C22⋊C411yescharacteristic
215C22×C4#13114yes—
216C22×C4#13514yes—
217C22×C4AGL(1,ℤ/8)12yes—
218C22×C4#13712yes—
219C22×C4C23×C411yescharacteristic
220C22×C4C22×D411yescharacteristic
221C22×C4C22×D411yescharacteristic
222C2×C8—44no—
223C2×C8—44no—
224C2×C8—44no—
225C2×C8—44no—
226C2×C8—44no—
227C2×C8—44no—
228C2×C8—88no—
229C2×C8—88no—
230C4⋊C4—24no—
231C4⋊C4—24no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4—48no—
237C4⋊C4—48no—
238C4⋊C4—44no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—44no—
244C4⋊C4—44no—
245C42—24no—
246C42—24no—
247C42—22no—
248C42—22no—
249C42—24no—
250C42—24no—
251C42—24no—
252C42—24no—
253C42—24no—
254C42—42no—
255C42—42no—
256C42—41no—
257C42—41no—
258C42—41no—
259C42—42no—
260C42—42no—
261C42—41no—
262C23#40812yes—
263C23#41012yes—
264C23C2×#10112yes—
265C23#41612yes—
266C23#41712yes—
267C23#42112yes—
268C23#42112yes—
269C23C22×C22⋊C411yescharacteristic
270C2×C4—28no—
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—22no—
280C2×C4—22no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—28no—
286C2×C4—24no—
287C2×C4—24no—
288C2×C4—28no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—44no—
299C2×C4—44no—
300C2×C4—44no—
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—42no—
313C2×C4—84no—
314C2×C4—84no—
315C2×C4—84no—
316C2×C4—84no—
317C2×C4#40812yes—
318C2×C4C2×#10114yes—
319C2×C4#41612yes—
320C2×C4#41712yes—
321C2×C4C2×#13112yes—
322C2×C4C2×#13512yes—
323C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
324C2×C4C2×#13711yescharacteristic
325C8—84no—
326C8—84no—
327C22#147014yes—
328C22#148214yes—
329C22#149514yes—
330C22#151214yes—
331C22#152114yes—
332C22#153714yes—
333C22#155214yes—
334C22C2×#40811yescharacteristic
335C22C2×#41011yescharacteristic
336C22C22×#10111yescharacteristic
337C22C2×#41611yescharacteristic
338C22C2×#41711yescharacteristic
339C22C2×#42111yescharacteristic
340C22C2×#42111yescharacteristic
341C4—24no—
342C4—21no—
343C4—21no—
344C4—22no—
345C4—22no—
346C4—22no—
347C4—22no—
348C4—22no—
349C4—42no—
350C4—42no—
351C4—42no—
352C4—82no—
353C4—82no—
354C4—82no—
355C4—82no—
356C2#771718yesdirect product factor
357C2C2×#147011yescharacteristic
358C2C2×#148211yescharacteristic
359C2C2×#149511yescharacteristic
360C2C2×#151211yescharacteristic
361C2C2×#152111yescharacteristic
362C2C2×#153711yescharacteristic
363C2C2×#155211yescharacteristic
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. #7717C2direct (N × H)
356. C2#7717direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#28642
C2×#18392
C2×#18312
#77172
C2×#12712