Finite Group Databasebeta

← #97028

Subgroups — #97028

Order 512.

1024 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 1024 conjugacy classes of subgroups — too many to draw. Only its 130 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×#2326C211yesmaximal, characteristic
2#8699C212yesmaximal, complemented
3#7595C212yesmaximal, complemented
4#7592C212yesmaximal
5C22×#124C411yescharacteristic
6C22×C4×Q8C411yescharacteristic
7C22×C4×D4C411yescharacteristic
8#2326—24no—
9#2326—24no—
10C2×#391C2211yescharacteristic
11C2×#390C2211yescharacteristic
12C2×#388—22no—
13C2×#385—22no—
14C2×#384C411yescharacteristic
15C2×#376C2211yescharacteristic
16#1817C2214yes—
17#1280—22no—
18#1280—22no—
19#1272—24no—
20#1270—24no—
21C23×Q8C2×C411yescharacteristic
22C23×D4C2×C411yescharacteristic
23C24×C4—21no—
24C2×#124—22no—
25C2×#124—22no—
26C2×#124—22no—
27C2×#124—22no—
28C2×#124—22no—
29C2×#124C814yescomplemented
30C2×C4×Q8—22no—
31C2×C4×Q8—22no—
32C2×C4×Q8—42no—
33C2×C4×Q8—42no—
34C2×C4×D4—22no—
35C2×C4×D4—22no—
36C2×C4×D4—42no—
37C2×C4×D4—42no—
38C22×C4⋊C4—21no—
39C22×C4⋊C4—22no—
40C22×C4⋊C4C2×C411yescharacteristic
41C22×C4⋊C4C2×C411yescharacteristic
42C22×C22⋊C4—22no—
43C22×C22⋊C4—21no—
44C22×C22⋊C4—22no—
45C22×C42—21no—
46C22×C42C2×C411yescharacteristic
47C22×C42C2×C411yescharacteristic
48C22×C42C2311yescharacteristic
49C2×C22⋊C8—24no—
50#391—22no—
51#391D414yes—
52#390—22no—
53#390D414yes—
54#388—216no—
55#385—216no—
56#384—22no—
57#384—22no—
58#376—22no—
59#376—22no—
60C2×#95—22no—
61C2×#95D412yes—
62C2×#95D412yes—
63#336—24no—
64#336—24no—
65C25—21no—
66C22×Q8—21no—
67C22×Q8—21no—
68C22×Q8—21no—
69C22×Q8—41no—
70C22×Q8—41no—
71C22×Q8—41no—
72C22×Q8—41no—
73C22×Q8C22⋊C412yes—
74C22×Q8C2×C812yes—
75C22×Q8M1612yes—
76C22×D4—21no—
77C22×D4—21no—
78C22×D4—21no—
79C22×D4—41no—
80C22×D4—41no—
81C22×D4—41no—
82C22×D4—41no—
83C22×D4C22⋊C412yes—
84C22×D4C2×C812yes—
85C22×D4M1612yes—
86C23×C4—21no—
87C23×C4—21no—
88C23×C4—21no—
89C23×C4—24no—
90C23×C4—42no—
91C23×C4—42no—
92C23×C4—84no—
93C23×C4C22⋊C412yes—
94C23×C4C22⋊C411yescharacteristic
95C23×C4C22⋊C411yescharacteristic
96C23×C4C22×C411yescharacteristic
97C23×C4C2×D412yes—
98C22×C8—44no—
99#124—28no—
100#124—28no—
101#124—28no—
102#124—28no—
103C4×Q8—48no—
104C4×Q8—48no—
105C4×D4—48no—
106C4×D4—48no—
107C2×C4⋊C4—24no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—28no—
111C2×C4⋊C4—41no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4—42no—
116C2×C4⋊C4—42no—
117C2×C4⋊C4—84no—
118C2×C4⋊C4C22⋊C412yes—
119C2×C4⋊C4C22⋊C412yes—
120C2×C22⋊C4—28no—
121C2×C22⋊C4—28no—
122C2×C22⋊C4—28no—
123C2×C22⋊C4—24no—
124C2×C22⋊C4—28no—
125C2×C22⋊C4—42no—
126C2×C22⋊C4—42no—
127C2×C22⋊C4—84no—
128C2×C42—22no—
129C2×C42—22no—
130C2×C42—21no—
131C2×C42—21no—
132C2×C42—21no—
133C2×C42—21no—
134C2×C42—21no—
135C2×C42—21no—
136C2×C42—24no—
137C2×C42—41no—
138C2×C42—41no—
139C2×C42—42no—
140C2×C42—42no—
141C2×C42C2×C812yes—
142C2×C42M1612yes—
143C2×C42C2×D412yes—
144C2×C42C2×D412yes—
145C22⋊C8—44no—
146C22⋊C8—44no—
147#95—28no—
148#95—24no—
149#95—24no—
150C24—22no—
151C24—22no—
152C24—22no—
153C24—44no—
154C24—44no—
155C24—41no—
156C24—41no—
157C24—41no—
158C24—41no—
159C24C2×C22⋊C411yescharacteristic
160C2×Q8—22no—
161C2×Q8—22no—
162C2×Q8—22no—
163C2×Q8—42no—
164C2×Q8—42no—
165C2×Q8—42no—
166C2×Q8—42no—
167C2×Q8—42no—
168C2×Q8—42no—
169C2×Q8—42no—
170C2×Q8—42no—
171C2×Q8—42no—
172C2×Q8—42no—
173C2×Q8—42no—
174C2×Q8—42no—
175C2×Q8C22⋊C814yes—
176C2×D4—22no—
177C2×D4—22no—
178C2×D4—22no—
179C2×D4—42no—
180C2×D4—42no—
181C2×D4—42no—
182C2×D4—42no—
183C2×D4—42no—
184C2×D4—42no—
185C2×D4—42no—
186C2×D4—42no—
187C2×D4—42no—
188C2×D4—42no—
189C2×D4—42no—
190C2×D4—42no—
191C2×D4C22⋊C814yes—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—21no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—48no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—44no—
221C22×C4—44no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—88no—
227C22×C4—88no—
228C22×C4—88no—
229C22×C4C22⋊C814yes—
230C22×C4#10012yes—
231C22×C4#10012yes—
232C22×C4C2×C22⋊C411yescharacteristic
233C22×C4C2×C22⋊C411yescharacteristic
234C22×C4#12014yes—
235C22×C4C22×C811yescharacteristic
236C22×C4C2×M1611yescharacteristic
237C22×C4C2×M1611yescharacteristic
238C22×C4C2×M1611yescharacteristic
239C2×C8—84no—
240C2×C8—84no—
241C2×C8—84no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—48no—
247C4⋊C4—88no—
248C22⋊C4—232no—
249C22⋊C4—48no—
250C22⋊C4—88no—
251C42—24no—
252C42—24no—
253C42—22no—
254C42—22no—
255C42—22no—
256C42—22no—
257C42—42no—
258C42—42no—
259C42—48no—
260C23—22no—
261C23—22no—
262C23—24no—
263C23—48no—
264C23—48no—
265C23—44no—
266C23—44no—
267C23—42no—
268C23—42no—
269C23—42no—
270C23—42no—
271C23—42no—
272C23—42no—
273C23—82no—
274C23—82no—
275C23—84no—
276C23—84no—
277C23—82no—
278C23—82no—
279C23#37911yescharacteristic
280C23C2×C22⋊C811yescharacteristic
281C23#40711yescharacteristic
282C23#41011yescharacteristic
283C23#41011yescharacteristic
284C23C2×#10011yescharacteristic
285C23C2×#10011yescharacteristic
286Q8—44no—
287Q8—44no—
288Q8—44no—
289Q8—44no—
290D4—44no—
291D4—44no—
292D4—44no—
293D4—44no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—24no—
307C2×C4—24no—
308C2×C4—24no—
309C2×C4—24no—
310C2×C4—24no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—42no—
314C2×C4—44no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—44no—
328C2×C4—42no—
329C2×C4—44no—
330C2×C4—44no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4—816no—
336C2×C4—82no—
337C2×C4—82no—
338C2×C4—82no—
339C2×C4—82no—
340C2×C4C2×C22⋊C811yescharacteristic
341C2×C4C2×C22⋊C811yescharacteristic
342C2×C4#40711yescharacteristic
343C2×C4#40711yescharacteristic
344C8—164no—
345C22—22no—
346C22—22no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—22no—
352C22—22no—
353C22—44no—
354C22—48no—
355C22—84no—
356C22—84no—
357C22—84no—
358C22—84no—
359C22—84no—
360C22—82no—
361C22—82no—
362C22#146011yescharacteristic
363C22#146011yescharacteristic
364C22#146611yescharacteristic
365C22#146611yescharacteristic
366C22#177911yescharacteristic
367C22#178211yescharacteristic
368C22#178311yescharacteristic
369C22#179512yes—
370C22#179612yes—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—44no—
376C4—42no—
377C4—42no—
378C4—44no—
379C4—44no—
380C4—82no—
381C4—82no—
382C2—21no—
383C2—21no—
384C2—21no—
385C2—21no—
386C2—84no—
387C2#830211yescharacteristic
388C2#832412yes—
389C2#851912yes—
390C2#852211yescharacteristic
391C2#852411yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC22×C4C22×C8
Frattini, cosocleC22×C42C23
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #8699C2semidirect
3. #7595C2semidirect
29. C2×#124C8semidirect

Maximal subgroups

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

subgroupindex
C2×#23262
#86992
#75952
#75922