Finite Group Databasebeta

← C2×#7326 #128163

Subgroups — C2×#7326 #128163

Order 512.

994 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 154 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×#1463C212yesmaximal
2C2×#1445C211yesmaximal, characteristic
3#7326C214yesmaximal, direct product factor
4C23×C4⋊C4C2211yescharacteristic
5C2×#407—22no—
6C22×#95—22no—
7C22×#95C411yescharacteristic
8C22×#95C411yescharacteristic
9#1463—22no—
10#1463—22no—
11#1463C414yescomplemented
12#1463C2214yes—
13C2×#324—22no—
14C2×#324C414yescomplemented
15#1445—22no—
16#1445C412yes—
17#1445C2212yes—
18C24×C4—21no—
19C24×C4—21no—
20C24×C4C2×C411yescharacteristic
21C24×C4D412yes—
22C22×M16—42no—
23C22×C4⋊C4—41no—
24C22×C4⋊C4—41no—
25C22×C4⋊C4—41no—
26C22×C4⋊C4—41no—
27C22×C4⋊C4C2×C411yescharacteristic
28C22×C4⋊C4C2×C412yes—
29C22×C4⋊C4C2×C412yes—
30C22×C4⋊C4D412yes—
31C22×C4⋊C4D411yescharacteristic
32C22×C4⋊C4D411yescharacteristic
33C22×C4⋊C4Q811yescharacteristic
34C22×C4⋊C4Q811yescharacteristic
35C22×C4⋊C4C2311yescharacteristic
36#407—24no—
37#407—24no—
38#407—42no—
39#407—42no—
40C2×C22⋊C8—24no—
41C2×C22⋊C8—42no—
42C2×#95—21no—
43C2×#95—21no—
44C2×#95—24no—
45C2×#95—24no—
46C2×#95—41no—
47C2×#95—41no—
48C2×#95—41no—
49C2×#95—41no—
50C2×#95—48no—
51C2×#95C2×C412yes—
52C2×#95C2×C412yes—
53#324—24no—
54#324—24no—
55#324—24no—
56#324C2×C418yescomplemented
57C25C22⋊C411yescharacteristic
58C23×C4—22no—
59C23×C4—21no—
60C23×C4—21no—
61C23×C4—21no—
62C23×C4—21no—
63C23×C4—22no—
64C23×C4—22no—
65C23×C4—22no—
66C23×C4—24no—
67C23×C4—24no—
68C23×C4—22no—
69C23×C4—22no—
70C23×C4—44no—
71C23×C4—44no—
72C23×C4—44no—
73C23×C4—44no—
74C23×C4—44no—
75C23×C4—42no—
76C23×C4—42no—
77C23×C4—44no—
78C23×C4—44no—
79C23×C4—41no—
80C23×C4—41no—
81C23×C4—41no—
82C23×C4—41no—
83C23×C4C4211yescharacteristic
84C23×C4C4211yescharacteristic
85C23×C4C22⋊C412yes—
86C23×C4C22⋊C411yescharacteristic
87C23×C4C22⋊C412yes—
88C23×C4C22⋊C412yes—
89C23×C4C4⋊C411yescharacteristic
90C23×C4C4⋊C412yes—
91C23×C4C4⋊C412yes—
92C23×C4C4⋊C411yescharacteristic
93C23×C4C22×C411yescharacteristic
94C23×C4C2×D412yes—
95C2×M16—44no—
96C2×M16—44no—
97C2×M16—84no—
98C2×M16—84no—
99C22×C8—44no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4—42no—
103C2×C4⋊C4—42no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4—42no—
106C2×C4⋊C4—42no—
107C2×C4⋊C4—42no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4—41no—
110C2×C4⋊C4—41no—
111C2×C4⋊C4—41no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4—41no—
116C2×C4⋊C4—41no—
117C2×C4⋊C4—41no—
118C2×C4⋊C4C4212yes—
119C2×C4⋊C4C22⋊C412yes—
120C2×C4⋊C4C22⋊C412yes—
121C2×C4⋊C4C4⋊C412yes—
122C2×C4⋊C4C4⋊C412yes—
123C2×C4⋊C4C4⋊C411yescharacteristic
124C2×C4⋊C4C4⋊C411yescharacteristic
125C2×C4⋊C4C22×C412yes—
126C2×C4⋊C4C2×D411yescharacteristic
127C2×C4⋊C4C2×Q811yescharacteristic
128C22⋊C8—28no—
129C22⋊C8—44no—
130C22⋊C8—44no—
131C22⋊C8—44no—
132#95—42no—
133#95—42no—
134#95—42no—
135#95—42no—
136#95—48no—
137#95—48no—
138C24—24no—
139C24—22no—
140C24—22no—
141C24—22no—
142C24—22no—
143C24#9511yescharacteristic
144C24#9511yescharacteristic
145C24#9911yescharacteristic
146C24#9911yescharacteristic
147C24#9911yescharacteristic
148C24#9911yescharacteristic
149C24C2×C22⋊C411yescharacteristic
150C22×C4—22no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—22no—
162C22×C4—24no—
163C22×C4—24no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—24no—
177C22×C4—24no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—44no—
183C22×C4—44no—
184C22×C4—44no—
185C22×C4—44no—
186C22×C4—44no—
187C22×C4—44no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—44no—
192C22×C4—44no—
193C22×C4—44no—
194C22×C4—44no—
195C22×C4—42no—
196C22×C4—44no—
197C22×C4—416no—
198C22×C4—416no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—44no—
202C22×C4—44no—
203C22×C4—44no—
204C22×C4—44no—
205C22×C4—44no—
206C22×C4—44no—
207C22×C4—44no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—82no—
211C22×C4—82no—
212C22×C4—84no—
213C22×C4—84no—
214C22×C4—88no—
215C22×C4—88no—
216C22×C4—82no—
217C22×C4—82no—
218C22×C4—82no—
219C22×C4—82no—
220C22×C4—82no—
221C22×C4—82no—
222C22×C4—82no—
223C22×C4—82no—
224C22×C4#9511yescharacteristic
225C22×C4#9511yescharacteristic
226C22×C4#9512yes—
227C22×C4#9512yes—
228C22×C4C2×C4211yescharacteristic
229C22×C4C2×C22⋊C412yes—
230C22×C4C2×C4⋊C411yescharacteristic
231C22×C4C2×C4⋊C412yes—
232M16—84no—
233M16—84no—
234M16—88no—
235C2×C8—48no—
236C2×C8—88no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—41no—
241C4⋊C4—42no—
242C4⋊C4—41no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C23—28no—
248C23—28no—
249C23—28no—
250C23—28no—
251C23—22no—
252C23—22no—
253C23—22no—
254C23—22no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—22no—
264C23—22no—
265C23—22no—
266C23—22no—
267C23—22no—
268C23—22no—
269C23—22no—
270C23—22no—
271C23—22no—
272C23—22no—
273C23#34211yescharacteristic
274C23#34211yescharacteristic
275C23#34211yescharacteristic
276C23#34211yescharacteristic
277C23#34414yes—
278C23#35312yes—
279C23#35412yes—
280C23#35512yes—
281C23#35612yes—
282C23C2×#9511yescharacteristic
283C23C2×#9911yescharacteristic
284C23C2×#9911yescharacteristic
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—22no—
292C2×C4—21no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
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—42no—
313C2×C4—44no—
314C2×C4—44no—
315C2×C4—44no—
316C2×C4—44no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—48no—
322C2×C4—48no—
323C2×C4—44no—
324C2×C4—44no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4—82no—
331C2×C4—84no—
332C2×C4—88no—
333C2×C4—88no—
334C2×C4—84no—
335C2×C4—88no—
336C2×C4—88no—
337C2×C4—82no—
338C2×C4—82no—
339C2×C4—82no—
340C2×C4—82no—
341C8—88no—
342C22—22no—
343C22—22no—
344C22—24no—
345C22—24no—
346C22—22no—
347C22—24no—
348C22—24no—
349C22—22no—
350C22—28no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—24no—
360C22—24no—
361C22—28no—
362C22—24no—
363C22—24no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22#130612yes—
369C22#130612yes—
370C22#139312yes—
371C22#139312yes—
372C22#139412yes—
373C22#139412yes—
374C22C2×#34211yescharacteristic
375C22C2×#34412yes—
376C22C2×#35311yescharacteristic
377C22C2×#35411yescharacteristic
378C22C2×#35511yescharacteristic
379C22C2×#35611yescharacteristic
380C4—42no—
381C4—42no—
382C4—42no—
383C4—42no—
384C4—41no—
385C4—41no—
386C4—41no—
387C4—41no—
388C4—84no—
389C4—84no—
390C2—21no—
391C2—21no—
392C2—22no—
393C2—21no—
394C2—21no—
395C2—22no—
396C2—22no—
397C2—22no—
398C2#732614yesdirect product factor
399C2C2×#130611yescharacteristic
400C2C2×#139311yescharacteristic
401C2C2×#139411yescharacteristic
subgroupG/H
center, socleC23#342
commutatorC22×C4C2×C42
Frattini, cosocleC22×C4⋊C4C23
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C25 ◁ C23×C4⋊C4 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7326C2direct (N × H)
11. #1463C4semidirect
14. C2×#324C4semidirect
56. #324C2×C4semidirect
398. C2#7326direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#14632
C2×#14452
#73262