Finite Group Databasebeta

← #496319

Subgroups — #496319

Order 512.

1148 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 1148 conjugacy classes of subgroups — too many to draw. Only its 202 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×#2545C211yesmaximal, characteristic
2#20727C212yesmaximal, complemented
3#13228C218yesmaximal, complemented
4#12700C212yesmaximal, complemented
5#12699C212yesmaximal
6C22×C4⋊Q8C2211yescharacteristic
7C22×#124C411yescharacteristic
8C22×#124C2211yescharacteristic
9C22×C4×Q8C2211yescharacteristic
10#2920—22no—
11#2545—22no—
12#2545—22no—
13#2545C414yes—
14#2545C2214yes—
15C2×#396—22no—
16C2×#391C411yescharacteristic
17C2×#390C411yescharacteristic
18C2×#388—22no—
19C2×C4×C4⋊C4C411yescharacteristic
20C2×#356C2218yescomplemented
21C2×#355C2218yes—
22#1821—22no—
23#1801—22no—
24#1801C2214yescomplemented
25#1784C2214yes—
26C4×#101—22no—
27C4×#101C2214yescomplemented
28C23×Q8C2×C411yescharacteristic
29C23×Q8C2311yescharacteristic
30C23×D4C2×C411yescharacteristic
31C2×C4⋊Q8—22no—
32C2×C4⋊Q8—44no—
33C2×C4⋊Q8C2×C412yes—
34C2×C4⋊Q8C2312yes—
35C2×#124—22no—
36C2×#124—22no—
37C2×#124—42no—
38C2×#124—41no—
39C2×#124—41no—
40C2×#124—44no—
41C2×#124C2×C414yes—
42C2×#124C2×C412yes—
43C2×#124C2312yes—
44C2×C4×Q8—41no—
45C2×C4×Q8—42no—
46C2×C4×Q8—41no—
47C2×C4×Q8C2×C412yes—
48C2×C4×Q8D412yes—
49C2×C4×Q8D412yes—
50C2×C4×Q8C2312yes—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4C2×C411yescharacteristic
54C22×C4⋊C4D412yes—
55C22×C22⋊C4—21no—
56C22×C22⋊C4—22no—
57C22×C22⋊C4—21no—
58C22×C22⋊C4—22no—
59C22×C42C2×C411yescharacteristic
60C22×C42C2×C411yescharacteristic
61C22×C42C2×C411yescharacteristic
62C22×C42D412yes—
63#432—42no—
64#432—42no—
65#431—42no—
66#431—42no—
67#423—24no—
68C2×#101—24no—
69C2×#101C2318yes—
70#407—24no—
71C4×M16—24no—
72#396—24no—
73#396—24no—
74#396—24no—
75#396—24no—
76#391—21no—
77#391—21no—
78#391C2×C414yes—
79#390—21no—
80#390—21no—
81#390C2×C414yes—
82#388—24no—
83#388—24no—
84#388—24no—
85#388—24no—
86C4×C4⋊C4—22no—
87C4×C4⋊C4C2×C414yes—
88C2×#95—22no—
89C2×#95—21no—
90#356—48no—
91#355—48no—
92#343—24no—
93#343—24no—
94C25—21no—
95C22×Q8—24no—
96C22×Q8—21no—
97C22×Q8—41no—
98C22×Q8—41no—
99C22×Q8—42no—
100C22×Q8—42no—
101C22×Q8—42no—
102C22×Q8—41no—
103C22×Q8—41no—
104C22×Q8C22×C412yes—
105C22×Q8C22×C411yescharacteristic
106C22×Q8C2×D412yes—
107C22×Q8C2411yescharacteristic
108C22×D4—21no—
109C22×D4—42no—
110C22×D4—41no—
111C22×D4—41no—
112C22×D4—42no—
113C22×D4C22×C412yes—
114C23×C4—21no—
115C23×C4—21no—
116C23×C4—21no—
117C23×C4C22⋊C411yescharacteristic
118C23×C4C22⋊C412yes—
119C23×C4C22⋊C411yescharacteristic
120C23×C4C22×C411yescharacteristic
121C23×C4C2×D412yes—
122C2×M16—28no—
123C4⋊Q8—28no—
124C4⋊Q8—44no—
125C4⋊Q8—48no—
126#124—28no—
127#124—44no—
128#124—44no—
129#124—44no—
130#124—44no—
131#124—44no—
132#124—48no—
133C4×Q8—42no—
134C4×Q8—42no—
135C4×Q8—42no—
136C4×Q8—42no—
137C4×Q8—42no—
138C4×Q8—42no—
139C4×Q8—42no—
140C4×Q8—42no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4—42no—
147C2×C4⋊C4—42no—
148C2×C4⋊C4—44no—
149C2×C4⋊C4—48no—
150C2×C4⋊C4C22⋊C411yescharacteristic
151C2×C4⋊C4C22⋊C411yescharacteristic
152C2×C4⋊C4C22⋊C412yes—
153C2×C4⋊C4C22⋊C412yes—
154C2×C4⋊C4C22×C412yes—
155C2×C4⋊C4C2×D412yes—
156C2×C4⋊C4C2×D412yes—
157C2×C22⋊C4—24no—
158C2×C22⋊C4—24no—
159C2×C22⋊C4—24no—
160C2×C22⋊C4—24no—
161C2×C22⋊C4—24no—
162C2×C22⋊C4—24no—
163C2×C22⋊C4—48no—
164C2×C22⋊C4—42no—
165C2×C22⋊C4—42no—
166C2×C22⋊C4—48no—
167C2×C22⋊C4—42no—
168C2×C22⋊C4—41no—
169C2×C22⋊C4—41no—
170C2×C42—22no—
171C2×C42—21no—
172C2×C42—21no—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—21no—
178C2×C42—21no—
179C2×C42—42no—
180C2×C42—42no—
181C2×C42C22⋊C412yes—
182C2×C42C22⋊C412yes—
183C2×C42C22⋊C412yes—
184C2×C42C22×C412yes—
185C2×C42C22×C412yes—
186C2×C42C22×C412yes—
187C2×C42C2×D412yes—
188C2×C42C2×D412yes—
189C4⋊C8—44no—
190C4⋊C8—44no—
191#101—216no—
192#101—48no—
193C22⋊C8—44no—
194C22⋊C8—44no—
195#97—44no—
196C4×C8—44no—
197#95—24no—
198#95—24no—
199#95—24no—
200C24—22no—
201C24—42no—
202C24—44no—
203C24—42no—
204C24—44no—
205C24—41no—
206C24—41no—
207C24C2×C22⋊C411yescharacteristic
208C2×Q8—22no—
209C2×Q8—24no—
210C2×Q8—24no—
211C2×Q8—24no—
212C2×Q8—41no—
213C2×Q8—42no—
214C2×Q8—41no—
215C2×Q8—41no—
216C2×Q8—41no—
217C2×Q8—48no—
218C2×Q8—48no—
219C2×Q8—42no—
220C2×Q8—42no—
221C2×Q8—42no—
222C2×Q8—42no—
223C2×Q8—42no—
224C2×Q8—84no—
225C2×Q8—84no—
226C2×Q8H(5,𝔽2)14yes—
227C2×D4—24no—
228C2×D4—42no—
229C2×D4—42no—
230C2×D4—42no—
231C2×D4—42no—
232C2×D4—42no—
233C2×D4—84no—
234C2×D4—84no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—24no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—24no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—44no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4#9912yes—
264C22×C4#9911yescharacteristic
265C22×C4#9911yescharacteristic
266C22×C4C2×C22⋊C412yes—
267C22×C4C2×C22⋊C411yescharacteristic
268C22×C4C2×C22⋊C412yes—
269C22×C4C2×C22⋊C412yes—
270C22×C4C2×C22⋊C412yes—
271C22×C4C2×C22⋊C412yes—
272C22×C4C23×C411yescharacteristic
273C22×C4C22×D412yes—
274M16—48no—
275M16—48no—
276C2×C8—48no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—48no—
285C4⋊C4—48no—
286C4⋊C4—48no—
287C22⋊C4—48no—
288C22⋊C4—48no—
289C22⋊C4—48no—
290C22⋊C4—48no—
291C22⋊C4—48no—
292C22⋊C4—44no—
293C22⋊C4—44no—
294C42—22no—
295C42—22no—
296C42—22no—
297C42—22no—
298C42—22no—
299C42—22no—
300C42—22no—
301C42—22no—
302C42—22no—
303C42—22no—
304C42—22no—
305C42—22no—
306C42—24no—
307C42—24no—
308C42—42no—
309C42—42no—
310C23—21no—
311C23—21no—
312C23—22no—
313C23—22no—
314C23—44no—
315C23—44no—
316C23—42no—
317C23—42no—
318C23—42no—
319C23—82no—
320C23—82no—
321C23—88no—
322C23—88no—
323C23—82no—
324C23—82no—
325C23—82no—
326C23—82no—
327C23C2×#9911yescharacteristic
328C23C2×#9911yescharacteristic
329C23C22×C22⋊C411yescharacteristic
330Q8—44no—
331Q8—48no—
332Q8—44no—
333Q8—84no—
334Q8—84no—
335D4—84no—
336D4—84no—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—22no—
344C2×C4—21no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—42no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—48no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—44no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—82no—
374C2×C4—84no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4C2×#9912yes—
378C2×C4C2×#9912yes—
379C2×C4#41012yes—
380C2×C4#51812yes—
381C2×C4#53514yes—
382C8—88no—
383C22—24no—
384C22—24no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—84no—
392C22—84no—
393C22—82no—
394C22—82no—
395C22—82no—
396C22—164no—
397C22—164no—
398C22#213212yes—
399C22#213312yes—
400C22#230911yescharacteristic
401C22C22×#9911yescharacteristic
402C22C2×#41011yescharacteristic
403C4—24no—
404C4—22no—
405C4—22no—
406C4—22no—
407C4—22no—
408C4—44no—
409C4—42no—
410C4—42no—
411C4—42no—
412C4—42no—
413C4—82no—
414C4—82no—
415C2—21no—
416C2—21no—
417C2—22no—
418C2—22no—
419C2—162no—
420C2#2179311yescharacteristic
421C2C2×#213211yescharacteristic
422C2C2×#213311yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×Q8C24
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20727C2semidirect
3. #13228C2semidirect
4. #12700C2semidirect
20. C2×#356C22semidirect
24. #1801C22semidirect
27. C4×#101C22semidirect

Maximal subgroups

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

subgroupindex
C2×#25452
#207272
#132282
#127002
#126992