Finite Group Databasebeta

← #124841

Subgroups — #124841

Order 512.

634 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 634 conjugacy classes of subgroups — too many to draw. Only its 86 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#13908C211yesmaximal, characteristic
2#8152C211yesmaximal, characteristic
3#8152C211yesmaximal, characteristic
4#7123C212yesmaximal, complemented
5#7122C212yesmaximal, complemented
6C2×#380—21no—
7C2×#380C411yescharacteristic
8C2×C4×C22⋊C4C2211yescharacteristic
9C22×#95C411yescharacteristic
10#1749—21no—
11#1749—21no—
12C2×#323—21no—
13C2×#323—21no—
14#1444C411yescharacteristic
15#1444C2211yescharacteristic
16#1442C411yescharacteristic
17#1442C2211yescharacteristic
18#1436—21no—
19#1436—21no—
20C22⋊#95—21no—
21#1280C2212yes—
22#1280C2212yes—
23C24×C4—21no—
24C24×C4C2×C411yescharacteristic
25C22×C4⋊C4—21no—
26C22×C4⋊C4—21no—
27C22×C4⋊C4C2×C411yescharacteristic
28C22×C4⋊C4C2×C411yescharacteristic
29C22×C22⋊C4—21no—
30C22×C22⋊C4D411yescharacteristic
31C22×C22⋊C4D411yescharacteristic
32C22×C42D411yescharacteristic
33C22×C42D411yescharacteristic
34C2×C22⋊C8—22no—
35C2×C22⋊C8—22no—
36C2×#97—41no—
37C2×#97—41no—
38#380—44no—
39#380—41no—
40#380—41no—
41C4×C22⋊C4—41no—
42C4×C22⋊C4—41no—
43C2×#95—21no—
44C2×#95—22no—
45C2×#95—22no—
46C2×#95—21no—
47C2×#95—21no—
48C2×#95—21no—
49C2×#95—21no—
50C2×#95—21no—
51C2×#95—21no—
52C2×#95—21no—
53C2×#95C2×C411yescharacteristic
54C2×#95C2×C411yescharacteristic
55C2×#95C2×C411yescharacteristic
56C2×#95C2311yescharacteristic
57#336—41no—
58#336—41no—
59#323—24no—
60#323—24no—
61C25C22⋊C411yescharacteristic
62C23×C4—21no—
63C23×C4—22no—
64C23×C4—21no—
65C23×C4—21no—
66C23×C4—21no—
67C23×C4—21no—
68C23×C4—21no—
69C23×C4—24no—
70C23×C4—21no—
71C23×C4—21no—
72C23×C4—42no—
73C23×C4—42no—
74C23×C4—42no—
75C23×C4—42no—
76C23×C4—81no—
77C23×C4—81no—
78C23×C4—81no—
79C23×C4—81no—
80C23×C4C22⋊C411yescharacteristic
81C23×C4C22⋊C411yescharacteristic
82C23×C4C22⋊C411yescharacteristic
83C23×C4C22×C411yescharacteristic
84C23×C4C2×D411yescharacteristic
85C23×C4C2×D411yescharacteristic
86C22×C8—42no—
87C22×C8—42no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—42no—
91C2×C4⋊C4—42no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4—42no—
94C2×C4⋊C4—81no—
95C2×C4⋊C4—81no—
96C2×C4⋊C4—81no—
97C2×C4⋊C4—81no—
98C2×C22⋊C4—22no—
99C2×C22⋊C4—22no—
100C2×C22⋊C4—22no—
101C2×C22⋊C4—22no—
102C2×C22⋊C4—41no—
103C2×C22⋊C4—41no—
104C2×C22⋊C4—42no—
105C2×C22⋊C4—42no—
106C2×C22⋊C4—42no—
107C2×C42—21no—
108C2×C42—21no—
109C2×C42—21no—
110C2×C42—21no—
111C2×C42—81no—
112C2×C42—81no—
113C22⋊C8—24no—
114C22⋊C8—24no—
115C22⋊C8—42no—
116C22⋊C8—42no—
117#97—82no—
118#97—82no—
119#95—24no—
120#95—21no—
121#95—21no—
122#95—41no—
123#95—41no—
124#95—42no—
125#95—42no—
126#95—42no—
127#95—41no—
128#95—41no—
129#95—42no—
130#95—81no—
131#95—81no—
132#95M1612yes—
133#95M1612yes—
134C24—21no—
135C24—21no—
136C24—21no—
137C24—21no—
138C24—41no—
139C24—41no—
140C24—41no—
141C24—41no—
142C24#10112yes—
143C24#10412yes—
144C24#10412yes—
145C24C2×C22⋊C411yescharacteristic
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—21no—
149C22×C4—22no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—22no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—48no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
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—48no—
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—81no—
207C22×C4—81no—
208C22×C4—81no—
209C22×C4—81no—
210C22×C4—84no—
211C22×C4—82no—
212C22×C4—82no—
213C22×C4—84no—
214C22×C4—84no—
215C22×C4—84no—
216C22×C4—84no—
217C22×C4—84no—
218C22×C4#9911yescharacteristic
219C22×C4#9911yescharacteristic
220C22×C4C2×M1611yescharacteristic
221C22×C4C2×M1611yescharacteristic
222C22×C4AGL(1,ℤ/8)12yes—
223C22×C4#13712yes—
224C2×C8—44no—
225C2×C8—44no—
226C2×C8—82no—
227C2×C8—82no—
228C2×C8—82no—
229C2×C8—82no—
230C4⋊C4—84no—
231C4⋊C4—84no—
232C4⋊C4—82no—
233C4⋊C4—82no—
234C22⋊C4—44no—
235C22⋊C4—44no—
236C22⋊C4—84no—
237C42—42no—
238C42—42no—
239C42—81no—
240C42—81no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—22no—
246C23—22no—
247C23—22no—
248C23—22no—
249C23—22no—
250C23—22no—
251C23—42no—
252C23—42no—
253C23—42no—
254C23—42no—
255C23—42no—
256C23—42no—
257C23—42no—
258C23—42no—
259C23—42no—
260C23—42no—
261C23—42no—
262C23—42no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23#33014yes—
268C23#40711yescharacteristic
269C23C2×#9911yescharacteristic
270C23C2×#10111yescharacteristic
271C23#41811yescharacteristic
272C23#41911yescharacteristic
273C23C2×#10411yescharacteristic
274C23C2×#10411yescharacteristic
275C2×C4—24no—
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—22no—
285C2×C4—22no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—82no—
319C2×C4—82no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4—816no—
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—
343C8—84no—
344C8—84no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—24no—
358C22—44no—
359C22—44no—
360C22—44no—
361C22—44no—
362C22—42no—
363C22—42no—
364C22—42no—
365C22—42no—
366C22—42no—
367C22—42no—
368C22#146311yescharacteristic
369C22#149011yescharacteristic
370C22#149111yescharacteristic
371C22#150711yescharacteristic
372C22#150811yescharacteristic
373C22C2×#33011yescharacteristic
374C22#155611yescharacteristic
375C22#212111yescharacteristic
376C22#212111yescharacteristic
377C22#212611yescharacteristic
378C22#213211yescharacteristic
379C4—44no—
380C4—42no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—41no—
385C4—42no—
386C4—82no—
387C4—81no—
388C4—81no—
389C4—161no—
390C4—161no—
391C2—21no—
392C2—21no—
393C2—21no—
394C2—21no—
395C2—44no—
396C2#762711yescharacteristic
397C2#1151311yescharacteristic
398C2#1151911yescharacteristic
399C2#1177511yescharacteristic
400C2#1178311yescharacteristic
401C2#1178911yescharacteristic
402C2#1179311yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC2×#95C23
prime radicalC25C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7123C2semidirect
5. #7122C2semidirect

Maximal subgroups

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

subgroupindex
#139082
#81522
#71232
#71222