Finite Group Databasebeta

← C2×#10387 #238034

Subgroups — C2×#10387 #238034

Order 512.

1070 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 1070 conjugacy classes of subgroups — too many to draw. Only its 186 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×#2607C211yesmaximal, characteristic
2C2×#1826C212yesmaximal
3C2×#1820C211yesmaximal, characteristic
4#10387C218yesmaximal, direct product factor
5C2×#1306C211yesmaximal, characteristic
6C2×#1284C212yesmaximal
7C22×#117C2211yescharacteristic
8C2×#407C2212yes—
9#2607—22no—
10#2607—24no—
11#2607C2214yes—
12C2×#399—21no—
13C2×#393—21no—
14C2×#391—22no—
15C2×#387—22no—
16C2×#381C2211yescharacteristic
17C2×#381C2211yescharacteristic
18C2×C4×C22⋊C4C2212yes—
19#1826—24no—
20#1826C2218yes—
21#1820—22no—
22#1820C2214yes—
23C2×#342—21no—
24C2×#342—21no—
25C2×#339—22no—
26C2×#339—22no—
27#1306C2214yes—
28#1284C2218yes—
29C23×Q8—21no—
30C24×C4C2311yescharacteristic
31C22×M16—22no—
32C2×#117—21no—
33C2×#117—21no—
34C2×#117—22no—
35C2×#117—22no—
36C2×#117—22no—
37C2×#117—22no—
38C2×#117Q812yescomplemented
39C2×#117C2312yes—
40C22×C4⋊C4—22no—
41C22×C4⋊C4—21no—
42C22×C4⋊C4Q812yescomplemented
43C22×C22⋊C4—22no—
44C22×C22⋊C4—21no—
45C22×C22⋊C4—21no—
46C22×C22⋊C4D411yescharacteristic
47C22×C22⋊C4D411yescharacteristic
48C22×C42—21no—
49C22×C42D412yes—
50C22×C42D412yes—
51#407—22no—
52#407—22no—
53#407—22no—
54#407C2314yes—
55C2×C22⋊C8—22no—
56C2×C22⋊C8—22no—
57#399—24no—
58#399—24no—
59#393—24no—
60#393—24no—
61#391—28no—
62#391—28no—
63#387—28no—
64#387—28no—
65#381—22no—
66#381—22no—
67#381—21no—
68#381D412yes—
69#381C2312yes—
70#381C2312yes—
71C4×C22⋊C4—22no—
72C4×C22⋊C4—22no—
73C4×C22⋊C4D414yes—
74C4×C22⋊C4C2314yes—
75C2×#95—21no—
76C2×#95—22no—
77C2×#95—21no—
78C2×#95—22no—
79C2×#95—22no—
80C2×#95—22no—
81#342—24no—
82#342—24no—
83#339—28no—
84#339—28no—
85C25C2×D411yescharacteristic
86C22×Q8—24no—
87C22×Q8—48no—
88C22×Q8—82no—
89C23×C4—22no—
90C23×C4—22no—
91C23×C4—21no—
92C23×C4—21no—
93C23×C4—22no—
94C23×C4—42no—
95C23×C4—42no—
96C23×C4—41no—
97C23×C4C2×D412yes—
98C23×C4C2×D411yescharacteristic
99C23×C4C2×D412yes—
100C23×C4C2×Q811yescharacteristic
101C23×C4C2×Q811yescharacteristic
102C23×C4#4112yes—
103C23×C4#4112yes—
104C23×C4#4111yescharacteristic
105C23×C4#4111yescharacteristic
106C23×C4C2411yescharacteristic
107C2×M16—24no—
108C2×M16—42no—
109C2×M16—44no—
110C2×M16—44no—
111C22×C8—42no—
112#117—24no—
113#117—24no—
114#117—24no—
115#117—24no—
116#117—42no—
117#117—42no—
118#117—42no—
119#117—42no—
120C2×C4⋊C4—24no—
121C2×C4⋊C4—24no—
122C2×C4⋊C4—28no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—48no—
126C2×C4⋊C4—82no—
127C2×C4⋊C4C2×Q814yescomplemented
128C2×C22⋊C4—24no—
129C2×C22⋊C4—22no—
130C2×C22⋊C4—22no—
131C2×C22⋊C4—24no—
132C2×C22⋊C4—22no—
133C2×C22⋊C4—21no—
134C2×C22⋊C4—21no—
135C2×C22⋊C4—48no—
136C2×C22⋊C4—41no—
137C2×C22⋊C4—41no—
138C2×C22⋊C4—41no—
139C2×C22⋊C4—41no—
140C2×C22⋊C4—42no—
141C2×C22⋊C4—42no—
142C2×C22⋊C4—42no—
143C2×C22⋊C4—42no—
144C2×C22⋊C4—41no—
145C2×C22⋊C4C2×D412yes—
146C2×C22⋊C4C2×D412yes—
147C2×C42—22no—
148C2×C42—22no—
149C2×C42—22no—
150C2×C42—22no—
151C2×C42—42no—
152C2×C42—42no—
153C2×C42—44no—
154C2×C42—44no—
155C2×C42C2×D414yes—
156C2×C42C2×D414yes—
157C22⋊C8—24no—
158C22⋊C8—24no—
159C22⋊C8—42no—
160C22⋊C8—42no—
161#95—24no—
162#95—28no—
163#95—24no—
164#95—24no—
165#95—24no—
166#95—24no—
167#95—24no—
168#95—24no—
169#95—42no—
170C24—22no—
171C24—22no—
172C24—44no—
173C24#12011yescharacteristic
174C24#12011yescharacteristic
175C24C22⋊Q811yescharacteristic
176C24C22⋊Q811yescharacteristic
177C24#12411yescharacteristic
178C24#12411yescharacteristic
179C24C22×D411yescharacteristic
180C2×Q8—416no—
181C2×Q8—84no—
182C2×Q8—816no—
183C22×C4—24no—
184C22×C4—24no—
185C22×C4—28no—
186C22×C4—24no—
187C22×C4—22no—
188C22×C4—24no—
189C22×C4—22no—
190C22×C4—24no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—44no—
199C22×C4—42no—
200C22×C4—44no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—44no—
204C22×C4—44no—
205C22×C4—44no—
206C22×C4—44no—
207C22×C4—44no—
208C22×C4—44no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—44no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—84no—
216C22×C4—82no—
217C22×C4—84no—
218C22×C4#12012yes—
219C22×C4C22⋊Q812yes—
220C22×C4C22⋊Q812yes—
221C22×C4C22⋊Q812yes—
222C22×C4C22⋊Q812yes—
223C22×C4C22⋊Q811yescharacteristic
224C22×C4C22⋊Q811yescharacteristic
225C22×C4#12412yes—
226C22×C4#12412yes—
227C22×C4#12412yes—
228C22×C4#12412yes—
229C22×C4#12412yes—
230C22×C4C22×D412yes—
231C22×C4C22×Q811yescharacteristic
232C22×C4C2×#4112yes—
233C22×C4C2×#4111yescharacteristic
234M16—44no—
235M16—44no—
236M16—84no—
237C2×C8—44no—
238C2×C8—84no—
239C4⋊C4—28no—
240C4⋊C4—416no—
241C4⋊C4—44no—
242C4⋊C4—84no—
243C22⋊C4—48no—
244C22⋊C4—48no—
245C22⋊C4—42no—
246C22⋊C4—42no—
247C22⋊C4—42no—
248C22⋊C4—42no—
249C22⋊C4—42no—
250C22⋊C4—42no—
251C22⋊C4—42no—
252C22⋊C4—42no—
253C22⋊C4—81no—
254C22⋊C4—81no—
255C22⋊C4—82no—
256C42—44no—
257C42—44no—
258C42—44no—
259C42—44no—
260C42—82no—
261C42—82no—
262C23—24no—
263C23—22no—
264C23—22no—
265C23—24no—
266C23—24no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—22no—
272C23—22no—
273C23—42no—
274C23—42no—
275C23—42no—
276C23—42no—
277C23—84no—
278C23—84no—
279C23#39311yescharacteristic
280C23#39311yescharacteristic
281C23#39311yescharacteristic
282C23#39311yescharacteristic
283C23#45812yes—
284C23#45812yes—
285C23C2×#12011yescharacteristic
286C23C2×C22⋊Q811yescharacteristic
287C23C2×#12411yescharacteristic
288Q8—816no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—24no—
294C2×C4—24no—
295C2×C4—24no—
296C2×C4—24no—
297C2×C4—28no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—42no—
308C2×C4—44no—
309C2×C4—416no—
310C2×C4—44no—
311C2×C4—44no—
312C2×C4—44no—
313C2×C4—44no—
314C2×C4—44no—
315C2×C4—44no—
316C2×C4—44no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—88no—
320C2×C4—84no—
321C2×C4—84no—
322C2×C4—84no—
323C2×C4—84no—
324C2×C4—84no—
325C2×C4—82no—
326C2×C4—84no—
327C2×C4—164no—
328C2×C4#39312yes—
329C2×C4#39312yes—
330C2×C4C2×C22⋊Q812yes—
331C2×C4C2×#12412yes—
332C8—84no—
333C22—24no—
334C22—24no—
335C22—22no—
336C22—22no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—22no—
342C22—22no—
343C22—22no—
344C22—22no—
345C22—44no—
346C22—44no—
347C22—81no—
348C22—81no—
349C22—81no—
350C22—81no—
351C22—82no—
352C22—82no—
353C22#203012yes—
354C22#203012yes—
355C22#203114yes—
356C22C2×#39311yescharacteristic
357C22C2×#45811yescharacteristic
358C22C2×#45811yescharacteristic
359C4—22no—
360C4—22no—
361C4—22no—
362C4—22no—
363C4—48no—
364C4—84no—
365C4—82no—
366C4—82no—
367C4—84no—
368C4—162no—
369C2—21no—
370C2—21no—
371C2—22no—
372C2—41no—
373C2—41no—
374C2—41no—
375C2—41no—
376C2#1038714yesdirect product factor
377C2C2×#203011yescharacteristic
378C2C2×#203112yes—
subgroupG/H
center, socleC23#393
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ 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
4. #10387C2direct (N × H)
38. C2×#117Q8semidirect
42. C22×C4⋊C4Q8semidirect
127. C2×C4⋊C4C2×Q8semidirect
376. C2#10387direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#26072
C2×#18262
C2×#18202
#103872
C2×#13062
C2×#12842