Finite Group Databasebeta

← #98540

Subgroups — #98540

Order 512.

1486 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 1486 conjugacy classes of subgroups — too many to draw. Only its 142 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×#2434C211yesmaximal, characteristic
2#8559C212yesmaximal
3#8342C212yesmaximal, complemented
4#7643C212yesmaximal, complemented
5C24×Q8C411yescharacteristic
6C22×C4⋊Q8C411yescharacteristic
7C22×C4⋊Q8C2211yescharacteristic
8C22×C22⋊Q8C411yescharacteristic
9#2434—24no—
10#2434—24no—
11C2×#395—22no—
12C2×#391C411yescharacteristic
13C2×#391C2211yescharacteristic
14C2×#385—22no—
15C2×C4×C22⋊C4C2211yescharacteristic
16#1797C2214yes—
17C2×#333—24no—
18C2×#323—22no—
19#1280—22no—
20#1272—24no—
21C23×Q8—22no—
22C23×Q8—22no—
23C23×Q8—21no—
24C23×Q8—42no—
25C23×Q8—42no—
26C23×Q8—44no—
27C23×Q8—44no—
28C23×Q8C2×C411yescharacteristic
29C23×Q8C2×C411yescharacteristic
30C23×Q8C2×C411yescharacteristic
31C24×C4—21no—
32C24×C4C2×C411yescharacteristic
33C2×C4⋊Q8—22no—
34C2×C4⋊Q8—22no—
35C2×C4⋊Q8—22no—
36C2×C4⋊Q8—44no—
37C2×C4⋊Q8—44no—
38C2×C4⋊Q8D414yes—
39C2×C22⋊Q8—22no—
40C2×C22⋊Q8—22no—
41C2×C22⋊Q8—42no—
42C2×C22⋊Q8—42no—
43C22×C4⋊C4—21no—
44C22×C4⋊C4—22no—
45C22×C4⋊C4—22no—
46C22×C4⋊C4C2×C411yescharacteristic
47C22×C22⋊C4—21no—
48C22×C22⋊C4D412yes—
49C22×C42C2×C411yescharacteristic
50C22×C42C2311yescharacteristic
51C2×C4⋊C8—24no—
52C2×C22⋊C8—24no—
53#395—216no—
54#391—22no—
55#391—22no—
56#391—22no—
57#391D414yes—
58#385—216no—
59C4×C22⋊C4—22no—
60C4×C22⋊C4—22no—
61C2×#95—22no—
62C2×#95—22no—
63C2×#95D412yes—
64#336—24no—
65#333—48no—
66#323—44no—
67C25C22⋊C411yescharacteristic
68C22×Q8—24no—
69C22×Q8—24no—
70C22×Q8—24no—
71C22×Q8—21no—
72C22×Q8—21no—
73C22×Q8—21no—
74C22×Q8—42no—
75C22×Q8—42no—
76C22×Q8—42no—
77C22×Q8—44no—
78C22×Q8—44no—
79C22×Q8—48no—
80C22×Q8—48no—
81C22×Q8—42no—
82C22×Q8—42no—
83C22×Q8—41no—
84C22×Q8—41no—
85C22×Q8—41no—
86C22×Q8—41no—
87C22×Q8—41no—
88C22×Q8—41no—
89C22×Q8—48no—
90C22×Q8—48no—
91C22×Q8—44no—
92C22×Q8—44no—
93C22×Q8—432no—
94C22×Q8—432no—
95C22×Q8—44no—
96C22×Q8—44no—
97C22×Q8—42no—
98C22×Q8C22⋊C412yes—
99C22×Q8C22⋊C412yes—
100C22×Q8C22⋊C412yes—
101C23×C4—22no—
102C23×C4—21no—
103C23×C4—21no—
104C23×C4—22no—
105C23×C4—21no—
106C23×C4—21no—
107C23×C4—24no—
108C23×C4—21no—
109C23×C4—21no—
110C23×C4—44no—
111C23×C4—44no—
112C23×C4—42no—
113C23×C4—42no—
114C23×C4—42no—
115C23×C4—42no—
116C23×C4C22⋊C411yescharacteristic
117C23×C4C22⋊C412yes—
118C23×C4C22×C411yescharacteristic
119C23×C4C2×D412yes—
120C22×C8—44no—
121C4⋊Q8—28no—
122C4⋊Q8—44no—
123C4⋊Q8—48no—
124C4⋊Q8—48no—
125C4⋊Q8—44no—
126C4⋊Q8—44no—
127C22⋊Q8—48no—
128C22⋊Q8—48no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—28no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—28no—
133C2×C4⋊C4—41no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4—42no—
136C2×C4⋊C4—48no—
137C2×C4⋊C4—48no—
138C2×C4⋊C4—42no—
139C2×C4⋊C4C22⋊C412yes—
140C2×C22⋊C4—24no—
141C2×C22⋊C4—24no—
142C2×C22⋊C4—24no—
143C2×C22⋊C4—42no—
144C2×C22⋊C4—42no—
145C2×C22⋊C4SD1614yes—
146C2×C22⋊C4Q1614yes—
147C2×C42—21no—
148C2×C42—21no—
149C2×C42—42no—
150C2×C42—42no—
151C2×C42C2×D412yes—
152C2×C42C2×D412yes—
153C4⋊C8—48no—
154C22⋊C8—44no—
155C22⋊C8—44no—
156#95—28no—
157#95—28no—
158#95—24no—
159C24—21no—
160C24—21no—
161C24—22no—
162C24—41no—
163C24—41no—
164C24—42no—
165C24#10012yes—
166C24#10314yes—
167C24C2×C22⋊C411yescharacteristic
168C2×Q8—24no—
169C2×Q8—24no—
170C2×Q8—22no—
171C2×Q8—22no—
172C2×Q8—24no—
173C2×Q8—48no—
174C2×Q8—48no—
175C2×Q8—48no—
176C2×Q8—48no—
177C2×Q8—42no—
178C2×Q8—42no—
179C2×Q8—48no—
180C2×Q8—48no—
181C2×Q8—48no—
182C2×Q8—42no—
183C2×Q8—42no—
184C2×Q8—42no—
185C2×Q8—42no—
186C2×Q8—42no—
187C2×Q8—42no—
188C2×Q8—42no—
189C2×Q8—42no—
190C2×Q8—432no—
191C2×Q8—432no—
192C2×Q8—432no—
193C2×Q8—432no—
194C2×Q8—416no—
195C2×Q8—416no—
196C2×Q8—42no—
197C2×Q8—42no—
198C2×Q8—42no—
199C2×Q8—84no—
200C2×Q8—84no—
201C2×Q8#9914yes—
202C2×Q8#10014yes—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—24no—
208C22×C4—24no—
209C22×C4—24no—
210C22×C4—24no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—48no—
222C22×C4—48no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—48no—
226C22×C4—48no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—48no—
230C22×C4—416no—
231C22×C4—416no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—44no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4#10012yes—
250C22×C4#10314yes—
251C22×C4C2×C22⋊C411yescharacteristic
252C22×C4C2×C22⋊C411yescharacteristic
253C22×C4#12014yes—
254C22×C4C2×SD1612yes—
255C22×C4C2×Q1612yes—
256C22×C4#13714yes—
257C2×C8—84no—
258C2×C8—84no—
259C2×C8—84no—
260C4⋊C4—416no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—48no—
264C4⋊C4—416no—
265C22⋊C4—28no—
266C22⋊C4—28no—
267C22⋊C4—48no—
268C42—24no—
269C42—42no—
270C42—42no—
271C42—42no—
272C23—22no—
273C23—21no—
274C23—21no—
275C23—24no—
276C23—22no—
277C23—22no—
278C23—24no—
279C23—42no—
280C23—42no—
281C23—44no—
282C23—44no—
283C23—42no—
284C23—42no—
285C23—44no—
286C23—44no—
287C23—42no—
288C23—41no—
289C23—41no—
290C23#33314yescomplemented
291C23#37911yescharacteristic
292C23C2×#10011yescharacteristic
293C23C2×#10011yescharacteristic
294C23C2×#10312yes—
295C23#41912yes—
296Q8—44no—
297Q8—44no—
298Q8—48no—
299Q8—44no—
300Q8—48no—
301Q8—44no—
302Q8—432no—
303Q8—84no—
304Q8—84no—
305C2×C4—24no—
306C2×C4—24no—
307C2×C4—24no—
308C2×C4—24no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—24no—
314C2×C4—22no—
315C2×C4—24no—
316C2×C4—24no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—44no—
320C2×C4—48no—
321C2×C4—48no—
322C2×C4—44no—
323C2×C4—44no—
324C2×C4—44no—
325C2×C4—44no—
326C2×C4—44no—
327C2×C4—42no—
328C2×C4—432no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—41no—
337C2×C4—42no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—82no—
345C2×C4—82no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4C2×#9912yes—
349C2×C4C2×#10012yes—
350C2×C4#44814yes—
351C2×C4#45114yes—
352C8—164no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—22no—
358C22—22no—
359C22—22no—
360C22—22no—
361C22—24no—
362C22—44no—
363C22—48no—
364C22—44no—
365C22—42no—
366C22—42no—
367C22—44no—
368C22—44no—
369C22C2×#33311yescharacteristic
370C22#154812yes—
371C22#155811yescharacteristic
372C22#178111yescharacteristic
373C22#178311yescharacteristic
374C22#178311yescharacteristic
375C22#178712yes—
376C22#179612yes—
377C4—22no—
378C4—22no—
379C4—24no—
380C4—24no—
381C4—44no—
382C4—44no—
383C4—42no—
384C4—44no—
385C4—82no—
386C4—82no—
387C2—21no—
388C2—22no—
389C2—21no—
390C2—44no—
391C2#834211yescharacteristic
392C2#837912yes—
393C2#839112yes—
394C2#852411yescharacteristic
395C2#854711yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC25C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ 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
3. #8342C2semidirect
4. #7643C2semidirect
290. C23#333semidirect

Maximal subgroups

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

subgroupindex
C2×#24342
#85592
#83422
#76432