Finite Group Databasebeta

← C2×#8756 #190162

Subgroups — C2×#8756 #190162

Order 512.

1614 conjugacy classes of subgroups; only the 258 normal subgroups are shown.

normal subgroups only

This group has 1614 conjugacy classes of subgroups — too many to draw. Only its 258 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×#2315C211yesmaximal, characteristic
2C2×#1462C212yesmaximal
3#8756C218yesmaximal, direct product factor
4C2×#1305C214yesmaximal, complemented
5C23×C22⋊C4C411yescharacteristic
6C23×C22⋊C4C411yescharacteristic
7C23×C22⋊C4C2211yescharacteristic
8C2×#407C2214yes—
9#2315—24no—
10#2315C414yes—
11#2315C2214yes—
12C2×#385—22no—
13C2×#385—22no—
14C2×#380C412yes—
15C2×#380C2212yes—
16#1462—24no—
17#1462C2218yes—
18C2×#323—22no—
19C2×#323—22no—
20#1305C22116yescomplemented
21C26C2×C411yescharacteristic
22C24×C4—21no—
23C24×C4—21no—
24C24×C4C2×C411yescharacteristic
25C24×C4C2311yescharacteristic
26C22×M16—24no—
27C22×C4⋊C4—22no—
28C22×C4⋊C4—22no—
29C22×C22⋊C4—24no—
30C22×C22⋊C4—24no—
31C22×C22⋊C4—22no—
32C22×C22⋊C4—22no—
33C22×C22⋊C4—21no—
34C22×C22⋊C4—21no—
35C22×C22⋊C4—44no—
36C22×C22⋊C4—44no—
37C22×C22⋊C4C2×C412yes—
38C22×C22⋊C4C2×C414yes—
39C22×C22⋊C4C2×C414yes—
40C22×C22⋊C4C2×C412yes—
41C22×C22⋊C4C2×C412yes—
42C22×C22⋊C4D414yes—
43C22×C22⋊C4D412yes—
44C22×C22⋊C4D412yes—
45C22×C22⋊C4C2312yes—
46#407—24no—
47#407—24no—
48#407—24no—
49#407C2318yes—
50C2×C22⋊C8—24no—
51C2×C22⋊C8—24no—
52#385—28no—
53#385—28no—
54#385—28no—
55#385—28no—
56#380—24no—
57#380—22no—
58#380—22no—
59#380C2×C414yes—
60#380C2×C418yes—
61#380C2314yes—
62C2×#95—22no—
63C2×#95—22no—
64#323—28no—
65#323—28no—
66C25—24no—
67C25—24no—
68C25—22no—
69C25—22no—
70C25—82no—
71C25—82no—
72C25C22⋊C412yes—
73C25C22⋊C411yescharacteristic
74C25C22⋊C411yescharacteristic
75C25C22×C412yes—
76C25C22×C411yescharacteristic
77C23×C4—22no—
78C23×C4—22no—
79C23×C4—22no—
80C23×C4—22no—
81C23×C4—24no—
82C23×C4—24no—
83C23×C4—44no—
84C23×C4—44no—
85C23×C4—82no—
86C23×C4—82no—
87C23×C4—82no—
88C23×C4—82no—
89C23×C4—82no—
90C23×C4—82no—
91C23×C4C22⋊C412yes—
92C23×C4C22⋊C412yes—
93C23×C4C22×C412yes—
94C23×C4C22×C411yescharacteristic
95C23×C4C2×D412yes—
96C23×C4C2×D412yes—
97C23×C4#4114yes—
98C23×C4#4114yes—
99C23×C4C2411yescharacteristic
100C2×M16—28no—
101C2×M16—44no—
102C2×M16—88no—
103C22×C8—44no—
104C2×C4⋊C4—28no—
105C2×C4⋊C4—24no—
106C2×C4⋊C4—24no—
107C2×C4⋊C4—84no—
108C2×C4⋊C4—84no—
109C2×C22⋊C4—28no—
110C2×C22⋊C4—24no—
111C2×C22⋊C4—24no—
112C2×C22⋊C4—24no—
113C2×C22⋊C4—24no—
114C2×C22⋊C4—28no—
115C2×C22⋊C4—28no—
116C2×C22⋊C4—28no—
117C2×C22⋊C4—22no—
118C2×C22⋊C4—22no—
119C2×C22⋊C4—24no—
120C2×C22⋊C4—24no—
121C2×C22⋊C4—44no—
122C2×C22⋊C4—48no—
123C2×C22⋊C4—48no—
124C2×C22⋊C4—44no—
125C2×C22⋊C4—44no—
126C2×C22⋊C4—44no—
127C2×C22⋊C4—48no—
128C2×C22⋊C4—48no—
129C2×C22⋊C4C22⋊C414yes—
130C2×C22⋊C4C22⋊C414yes—
131C2×C22⋊C4C22×C414yes—
132C2×C22⋊C4C22×C414yes—
133C2×C22⋊C4C2×D414yes—
134C2×C22⋊C4C2×D414yes—
135C22⋊C8—28no—
136C22⋊C8—28no—
137C22⋊C8—44no—
138C22⋊C8—44no—
139#95—28no—
140#95—24no—
141#95—24no—
142C24—216no—
143C24—216no—
144C24—21no—
145C24—21no—
146C24—21no—
147C24—21no—
148C24—22no—
149C24—22no—
150C24—22no—
151C24—22no—
152C24—24no—
153C24—24no—
154C24—24no—
155C24—24no—
156C24—24no—
157C24—24no—
158C24—42no—
159C24—42no—
160C24—42no—
161C24—42no—
162C24—42no—
163C24—42no—
164C24—84no—
165C24—84no—
166C24—82no—
167C24—82no—
168C24—82no—
169C24—82no—
170C24—84no—
171C24—84no—
172C24—816no—
173C24—816no—
174C24#10014yes—
175C24#10114yes—
176C24C2×C22⋊C411yescharacteristic
177C24C2×C22⋊C411yescharacteristic
178C24C2×C22⋊C412yes—
179C24C2×C22⋊C412yes—
180C24#11712yes—
181C24#11712yes—
182C24C23×C411yescharacteristic
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—24no—
186C22×C4—24no—
187C22×C4—24no—
188C22×C4—24no—
189C22×C4—44no—
190C22×C4—48no—
191C22×C4—44no—
192C22×C4—48no—
193C22×C4—44no—
194C22×C4—44no—
195C22×C4—44no—
196C22×C4—44no—
197C22×C4—44no—
198C22×C4—44no—
199C22×C4—48no—
200C22×C4—81no—
201C22×C4—82no—
202C22×C4—84no—
203C22×C4—81no—
204C22×C4—84no—
205C22×C4—84no—
206C22×C4—84no—
207C22×C4—84no—
208C22×C4—84no—
209C22×C4—84no—
210C22×C4—82no—
211C22×C4—82no—
212C22×C4—816no—
213C22×C4—816no—
214C22×C4C2×C22⋊C414yes—
215C22×C4C2×C22⋊C412yes—
216C22×C4#11714yes—
217C22×C4#12314yes—
218C22×C4#12314yes—
219C22×C4#12314yes—
220C22×C4#12314yes—
221C22×C4C22×D412yes—
222C22×C4C2×#4114yes—
223M16—88no—
224M16—88no—
225C2×C8—48no—
226C2×C8—88no—
227C4⋊C4—84no—
228C4⋊C4—84no—
229C4⋊C4—88no—
230C22⋊C4—28no—
231C22⋊C4—28no—
232C22⋊C4—28no—
233C22⋊C4—28no—
234C22⋊C4—44no—
235C22⋊C4—44no—
236C22⋊C4—48no—
237C22⋊C4—48no—
238C22⋊C4—44no—
239C22⋊C4—44no—
240C23—22no—
241C23—22no—
242C23—216no—
243C23—216no—
244C23—24no—
245C23—24no—
246C23—24no—
247C23—24no—
248C23—24no—
249C23—24no—
250C23—24no—
251C23—24no—
252C23—21no—
253C23—21no—
254C23—22no—
255C23—21no—
256C23—22no—
257C23—21no—
258C23—22no—
259C23—22no—
260C23—41no—
261C23—41no—
262C23—42no—
263C23—42no—
264C23—41no—
265C23—41no—
266C23—48no—
267C23—42no—
268C23—42no—
269C23—42no—
270C23—42no—
271C23—48no—
272C23—48no—
273C23—48no—
274C23—48no—
275C23—48no—
276C23—42no—
277C23—42no—
278C23—42no—
279C23—42no—
280C23—42no—
281C23—42no—
282C23—42no—
283C23—42no—
284C23—84no—
285C23—84no—
286C23—84no—
287C23—84no—
288C23—88no—
289C23—88no—
290C23—88no—
291C23—88no—
292C23—832no—
293C23—832no—
294C23#38112yes—
295C23#38111yescharacteristic
296C23#38111yescharacteristic
297C23C2×#10012yes—
298C23C2×#10014yes—
299C23C2×#10112yes—
300C23C2×#10114yes—
301C23C22×C22⋊C411yescharacteristic
302C23C2×#11712yes—
303C2×C4—28no—
304C2×C4—24no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—44no—
308C2×C4—44no—
309C2×C4—44no—
310C2×C4—44no—
311C2×C4—44no—
312C2×C4—44no—
313C2×C4—44no—
314C2×C4—44no—
315C2×C4—44no—
316C2×C4—44no—
317C2×C4—44no—
318C2×C4—44no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—84no—
322C2×C4—84no—
323C2×C4—88no—
324C2×C4—88no—
325C2×C4—816no—
326C2×C4—816no—
327C2×C4#38114yes—
328C2×C4C2×#12314yes—
329C8—88no—
330C22—24no—
331C22—24no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—22no—
337C22—22no—
338C22—24no—
339C22—24no—
340C22—22no—
341C22—22no—
342C22—22no—
343C22—22no—
344C22—44no—
345C22—44no—
346C22—44no—
347C22—42no—
348C22—42no—
349C22—48no—
350C22—48no—
351C22—48no—
352C22—42no—
353C22—42no—
354C22—44no—
355C22—48no—
356C22—48no—
357C22—48no—
358C22—42no—
359C22—42no—
360C22—42no—
361C22—42no—
362C22—816no—
363C22—816no—
364C22#182112yes—
365C22#182112yes—
366C22#182812yes—
367C22#182912yes—
368C22C2×#38111yescharacteristic
369C22C22×#10011yescharacteristic
370C22C22×#10111yescharacteristic
371C4—24no—
372C4—24no—
373C4—44no—
374C4—84no—
375C4—84no—
376C2—21no—
377C2—22no—
378C2—21no—
379C2—42no—
380C2—42no—
381C2—44no—
382C2—44no—
383C2#875614yesdirect product factor
384C2C2×#182111yescharacteristic
385C2C2×#182811yescharacteristic
386C2C2×#182911yescharacteristic
subgroupG/H
center, socleC23#381
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC26C2×C4
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #8756C2direct (N × H)
4. C2×#1305C2semidirect
20. #1305C22semidirect
383. C2#8756direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#23152
C2×#14622
#87562
C2×#13052