Finite Group Databasebeta

← C2×#8704 #189698

Subgroups — C2×#8704 #189698

Order 512.

2682 conjugacy classes of subgroups; only the 354 normal subgroups are shown.

normal subgroups only

This group has 2682 conjugacy classes of subgroups — too many to draw. Only its 354 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
1C22×#380C211yesmaximal, characteristic
2C2×#1779C212yesmaximal
3#8704C218yesmaximal, direct product factor
4C2×#1280C214yesmaximal, complemented
5C25×C4C2211yescharacteristic
6C23×C4⋊C4C412yes—
7C23×C22⋊C4C412yes—
8C22×C22⋊C8—22no—
9C22×C22⋊C8—22no—
10C22×C22⋊C8C2214yes—
11C2×#380—22no—
12C2×#380—22no—
13C2×#380—24no—
14C2×#380C414yes—
15C2×#380C2214yes—
16C22×#95C2212yes—
17#1779—24no—
18#1779C2218yes—
19#1280C22116yescomplemented
20C26C2×C411yescharacteristic
21C24×C4—28no—
22C24×C4—48no—
23C24×C4C2×C411yescharacteristic
24C24×C4C2×C414yes—
25C24×C4C2×C412yes—
26C24×C4D412yes—
27C24×C4D412yes—
28C24×C4D414yes—
29C24×C4C2312yes—
30C24×C4C2311yescharacteristic
31C23×C8—24no—
32C22×C4⋊C4—24no—
33C22×C4⋊C4—22no—
34C22×C4⋊C4—22no—
35C22×C4⋊C4—48no—
36C22×C4⋊C4C2×C418yes—
37C22×C22⋊C4—24no—
38C22×C22⋊C4—22no—
39C22×C22⋊C4—22no—
40C22×C22⋊C4—48no—
41C22×C22⋊C4C2×C418yes—
42C2×C22⋊C8—216no—
43C2×C22⋊C8—28no—
44C2×C22⋊C8—28no—
45C2×C22⋊C8—24no—
46C2×C22⋊C8—48no—
47C2×C22⋊C8—48no—
48C2×C22⋊C8C2318yes—
49#380—28no—
50#380—28no—
51#380—28no—
52#380—28no—
53C2×#95—24no—
54C2×#95—24no—
55C2×#95C2×C414yes—
56C2×#95C2314yes—
57C25—24no—
58C25—24no—
59C25—24no—
60C25—28no—
61C25—44no—
62C25C22⋊C411yescharacteristic
63C25C22⋊C411yescharacteristic
64C25C22⋊C412yes—
65C25C22×C412yes—
66C25C22×C411yescharacteristic
67C23×C4—24no—
68C23×C4—28no—
69C23×C4—28no—
70C23×C4—28no—
71C23×C4—28no—
72C23×C4—28no—
73C23×C4—28no—
74C23×C4—216no—
75C23×C4—28no—
76C23×C4—24no—
77C23×C4—24no—
78C23×C4—216no—
79C23×C4—48no—
80C23×C4—48no—
81C23×C4—48no—
82C23×C4—48no—
83C23×C4—44no—
84C23×C4—44no—
85C23×C4—44no—
86C23×C4—464no—
87C23×C4—48no—
88C23×C4—48no—
89C23×C4C22⋊C412yes—
90C23×C4C22⋊C414yes—
91C23×C4C22⋊C414yes—
92C23×C4C22⋊C412yes—
93C23×C4C22×C412yes—
94C23×C4C22×C418yes—
95C23×C4C22×C411yescharacteristic
96C23×C4C2×D414yes—
97C23×C4C2×D414yes—
98C23×C4C2×D412yes—
99C23×C4C2×D412yes—
100C23×C4#4118yes—
101C23×C4C2411yescharacteristic
102C22×C8—28no—
103C22×C8—48no—
104C22×C8—48no—
105C22×C8—48no—
106C2×C4⋊C4—28no—
107C2×C4⋊C4—28no—
108C2×C4⋊C4—48no—
109C2×C4⋊C4—416no—
110C2×C4⋊C4—48no—
111C2×C4⋊C4—416no—
112C2×C22⋊C4—28no—
113C2×C22⋊C4—28no—
114C2×C22⋊C4—48no—
115C2×C22⋊C4—416no—
116C2×C22⋊C4—48no—
117C2×C22⋊C4—416no—
118C22⋊C8—216no—
119C22⋊C8—416no—
120C22⋊C8—416no—
121C22⋊C8—48no—
122#95—28no—
123#95—28no—
124C24—24no—
125C24—22no—
126C24—22no—
127C24—232no—
128C24—24no—
129C24—24no—
130C24—22no—
131C24—22no—
132C24—24no—
133C24—24no—
134C24—24no—
135C24—24no—
136C24—24no—
137C24—22no—
138C24—24no—
139C24—24no—
140C24—22no—
141C24—24no—
142C24—28no—
143C24—28no—
144C24—44no—
145C24—44no—
146C24—432no—
147C24—432no—
148C24—416no—
149C24—416no—
150C24#10014yes—
151C24#10114yes—
152C24C2×C22⋊C412yes—
153C24C2×C22⋊C412yes—
154C24C2×C22⋊C411yescharacteristic
155C24C2×C22⋊C411yescharacteristic
156C24#11714yes—
157C24#13118yes—
158C24C23×C411yescharacteristic
159C22×C4—232no—
160C22×C4—24no—
161C22×C4—24no—
162C22×C4—24no—
163C22×C4—24no—
164C22×C4—24no—
165C22×C4—24no—
166C22×C4—28no—
167C22×C4—28no—
168C22×C4—28no—
169C22×C4—28no—
170C22×C4—28no—
171C22×C4—28no—
172C22×C4—28no—
173C22×C4—28no—
174C22×C4—28no—
175C22×C4—28no—
176C22×C4—28no—
177C22×C4—216no—
178C22×C4—44no—
179C22×C4—44no—
180C22×C4—42no—
181C22×C4—44no—
182C22×C4—44no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—416no—
186C22×C4—42no—
187C22×C4—432no—
188C22×C4—432no—
189C22×C4—432no—
190C22×C4—464no—
191C22×C4—432no—
192C22×C4—48no—
193C22×C4—48no—
194C22×C4—48no—
195C22×C4—48no—
196C22×C4—48no—
197C22×C4—48no—
198C22×C4—48no—
199C22×C4—48no—
200C22×C4—832no—
201C22×C4#9918yes—
202C22×C4C2×C22⋊C412yes—
203C22×C4C2×C22⋊C414yes—
204C22×C4#11714yes—
205C22×C4#12318yes—
206C22×C4#12318yes—
207C22×C4C22×D412yes—
208C22×C4C2×#4114yes—
209C2×C8—48no—
210C2×C8—48no—
211C2×C8—48no—
212C2×C8—816no—
213C4⋊C4—416no—
214C4⋊C4—416no—
215C22⋊C4—416no—
216C22⋊C4—416no—
217C23—28no—
218C23—28no—
219C23—22no—
220C23—22no—
221C23—21no—
222C23—21no—
223C23—21no—
224C23—21no—
225C23—216no—
226C23—216no—
227C23—216no—
228C23—216no—
229C23—28no—
230C23—28no—
231C23—24no—
232C23—24no—
233C23—24no—
234C23—24no—
235C23—22no—
236C23—22no—
237C23—216no—
238C23—216no—
239C23—22no—
240C23—22no—
241C23—416no—
242C23—416no—
243C23—416no—
244C23—416no—
245C23—48no—
246C23—48no—
247C23—4128no—
248C23—48no—
249C23—44no—
250C23—44no—
251C23—44no—
252C23—44no—
253C23—48no—
254C23—48no—
255C23—48no—
256C23#38112yes—
257C23#38112yes—
258C23#40814yes—
259C23#40814yes—
260C23C2×#9914yes—
261C23C2×#10014yes—
262C23C2×#10012yes—
263C23C2×#10114yes—
264C23C2×#10112yes—
265C23#433116yes—
266C23C22×C22⋊C411yescharacteristic
267C23C2×#11712yes—
268C23C2×#13114yes—
269C2×C4—216no—
270C2×C4—216no—
271C2×C4—216no—
272C2×C4—216no—
273C2×C4—24no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—24no—
279C2×C4—44no—
280C2×C4—48no—
281C2×C4—44no—
282C2×C4—416no—
283C2×C4—416no—
284C2×C4—416no—
285C2×C4—416no—
286C2×C4—48no—
287C2×C4—48no—
288C2×C4—48no—
289C2×C4—816no—
290C2×C4—816no—
291C2×C4—816no—
292C2×C4#38114yes—
293C2×C4C2×#9918yes—
294C2×C4C2×#12314yes—
295C8—88no—
296C22—232no—
297C22—22no—
298C22—22no—
299C22—22no—
300C22—22no—
301C22—24no—
302C22—24no—
303C22—24no—
304C22—24no—
305C22—24no—
306C22—24no—
307C22—28no—
308C22—28no—
309C22—432no—
310C22—432no—
311C22—416no—
312C22—416no—
313C22—42no—
314C22—42no—
315C22—42no—
316C22—42no—
317C22—44no—
318C22—44no—
319C22#147118yes—
320C22#147218yes—
321C22#181814yes—
322C22#181814yes—
323C22#181914yes—
324C22#182114yes—
325C22C2×#38111yescharacteristic
326C22C2×#40812yes—
327C22C22×#9912yes—
328C22C22×#10011yescharacteristic
329C22C22×#10111yescharacteristic
330C22C2×#43314yes—
331C4—24no—
332C4—28no—
333C4—24no—
334C4—88no—
335C2—24no—
336C2—24no—
337C2—44no—
338C2—44no—
339C2#870418yesdirect product factor
340C2C2×#147112yes—
341C2C2×#147212yes—
342C2C2×#181811yescharacteristic
343C2C2×#181911yescharacteristic
344C2C2×#182111yescharacteristic
subgroupG/H
center, socleC24C2×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC26C2×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C25×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #8704C2direct (N × H)
4. C2×#1280C2semidirect
19. #1280C22semidirect
339. C2#8704direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#3802
C2×#17792
#87042
C2×#12802