Finite Group Databasebeta

← #100896

Subgroups — #100896

Order 512.

557 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 557 conjugacy classes of subgroups — too many to draw. Only its 84 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
1#25912C211yesmaximal, characteristic
2#14474C211yesmaximal, characteristic
3#9889C211yesmaximal, characteristic
4#9889C211yesmaximal, characteristic
5#8335C211yesmaximal, characteristic
6#7036C211yesmaximal, characteristic
7#7036C211yesmaximal, characteristic
8C22×#125C2211yescharacteristic
9C2×#491C412yescomplemented
10#2857C2211yescharacteristic
11C2×#387—21no—
12#2026—22no—
13#1992C412yescomplemented
14#1872—21no—
15#1872—21no—
16#1749C2211yescharacteristic
17#1749C2211yescharacteristic
18#1749C2211yescharacteristic
19C2×#330—22no—
20#1445—21no—
21#1445—21no—
22#1441—21no—
23#1436—21no—
24C4⋊#95—21no—
25#1434C2211yescharacteristic
26#1434C2211yescharacteristic
27#1294—21no—
28#1294—21no—
29#1294—21no—
30#1294—21no—
31C2×C4⋊Q8C2×C411yescharacteristic
32C2×#125—21no—
33C2×#125—21no—
34C2×#125—44no—
35C2×#125C2×C412yes—
36C2×#125D412yes—
37C2×C22⋊Q8—21no—
38C22×C4⋊C4—21no—
39C22×C4⋊C4—21no—
40C22×C4⋊C4—21no—
41C22×C4⋊C4D411yescharacteristic
42C22×C4⋊C4D411yescharacteristic
43C22×C42—21no—
44C22×C42C2311yescharacteristic
45#491—44no—
46C2×#103—22no—
47C2×#103—22no—
48C2×C22⋊C8—21no—
49C2×C22⋊C8—21no—
50C2×C22⋊C8—21no—
51C2×#97—21no—
52C2×#97—21no—
53C2×#97C2×C412yes—
54#391—21no—
55#389—21no—
56#389—21no—
57#387—44no—
58#384C2×C411yescharacteristic
59#380—21no—
60#380—21no—
61C2×#95—21no—
62C2×#95—21no—
63C2×#95—21no—
64C2×#95—21no—
65C2×#95—21no—
66C2×#95—21no—
67C2×#95—21no—
68C2×#95—21no—
69C2×#95—21no—
70C2×#95—21no—
71C2×#95D411yescharacteristic
72C2×#95D411yescharacteristic
73C2×#95D411yescharacteristic
74C2×#95D411yescharacteristic
75#340—22no—
76#340—22no—
77#336—21no—
78#336—21no—
79#336—21no—
80#330—44no—
81C22×Q8—21no—
82C23×C4—21no—
83C23×C4—21no—
84C23×C4—21no—
85C23×C4—21no—
86C23×C4—21no—
87C23×C4—21no—
88C23×C4C2×D411yescharacteristic
89C23×C4C2×D411yescharacteristic
90C23×C4C2×D411yescharacteristic
91C22×C8—22no—
92C22×C8—41no—
93C22×C8—41no—
94C4⋊Q8—42no—
95#125—24no—
96#125—24no—
97#125—42no—
98#125—42no—
99#125—44no—
100#125—44no—
101C22⋊Q8—42no—
102C22⋊Q8—42no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—24no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—42no—
116C2×C4⋊C4—42no—
117C2×C4⋊C4—42no—
118C2×C4⋊C4—42no—
119C2×C4⋊C4—42no—
120C2×C4⋊C4—42no—
121C2×C4⋊C4—42no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—81no—
124C2×C4⋊C4C22⋊C412yes—
125C2×C4⋊C4C22⋊C412yes—
126C2×C22⋊C4—41no—
127C2×C42—42no—
128C2×C42—42no—
129C2×C42—42no—
130C2×C42—81no—
131C2×C42C22×C411yescharacteristic
132C2×C42C2×D411yescharacteristic
133C2×C42#4111yescharacteristic
134C2×C42#4111yescharacteristic
135#103—44no—
136#103—44no—
137C22⋊C8—22no—
138C22⋊C8—22no—
139C22⋊C8—41no—
140C22⋊C8—41no—
141C22⋊C8—41no—
142C22⋊C8—41no—
143#97—42no—
144#97—42no—
145#97—42no—
146#95—41no—
147#95—41no—
148#95—41no—
149#95—41no—
150#95—42no—
151#95—42no—
152#95—42no—
153#95—42no—
154#95—42no—
155#95—42no—
156#95—42no—
157#95—42no—
158#95—41no—
159#95—41no—
160#95—81no—
161#95—81no—
162C24#12011yescharacteristic
163C2×Q8—42no—
164C2×Q8—42no—
165C2×Q8—42no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4C2×C22⋊C411yescharacteristic
223C22×C4C4×D411yescharacteristic
224C22×C4C4×D411yescharacteristic
225C22×C4#12111yescharacteristic
226C22×C4#12111yescharacteristic
227C22×C4#12311yescharacteristic
228C2×C8—42no—
229C2×C8—42no—
230C2×C8—42no—
231C2×C8—42no—
232C2×C8—42no—
233C2×C8—81no—
234C2×C8—81no—
235C2×C8—81no—
236C2×C8—81no—
237C4⋊C4—24no—
238C4⋊C4—22no—
239C4⋊C4—22no—
240C4⋊C4—22no—
241C4⋊C4—44no—
242C4⋊C4—44no—
243C4⋊C4—44no—
244C4⋊C4—44no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—81no—
248C4⋊C4—81no—
249C4⋊C4—81no—
250C4⋊C4—81no—
251C4⋊C4—82no—
252C4⋊C4—84no—
253C4⋊C4#9914yes—
254C22⋊C4—81no—
255C22⋊C4—81no—
256C42—22no—
257C42—22no—
258C42—41no—
259C42—41no—
260C42—84no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23#38611yescharacteristic
266C23#45411yescharacteristic
267C23#45411yescharacteristic
268C23#45511yescharacteristic
269C23#45511yescharacteristic
270C23#45611yescharacteristic
271C23#45611yescharacteristic
272Q8—84no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—24no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4C2×#9912yes—
345C8—82no—
346C8—82no—
347C8—82no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22#166211yescharacteristic
361C22#166311yescharacteristic
362C22#166411yescharacteristic
363C22#166411yescharacteristic
364C22#188211yescharacteristic
365C22#188211yescharacteristic
366C22#188811yescharacteristic
367C22#189811yescharacteristic
368C22#190611yescharacteristic
369C22#190611yescharacteristic
370C22#191211yescharacteristic
371C4—22no—
372C4—41no—
373C4—41no—
374C4—44no—
375C4—41no—
376C4—41no—
377C4—41no—
378C4—82no—
379C4—82no—
380C4—82no—
381C4—161no—
382C4—161no—
383C4—161no—
384C4—161no—
385C2—21no—
386C2—21no—
387C2—21no—
388C2—21no—
389C2#900411yescharacteristic
390C2#936111yescharacteristic
391C2#936411yescharacteristic
392C2#938011yescharacteristic
393C2#938011yescharacteristic
394C2#944211yescharacteristic
395C2#946611yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#120
LCSG ▷ C2×C42 ▷ 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
9. C2×#491C4semidirect
13. #1992C4semidirect

Maximal subgroups

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

subgroupindex
#259122
#144742
#98892
#83352
#70362