Finite Group Databasebeta

← #124809

Subgroups — #124809

Order 512.

620 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 82 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
1C4×#399C211yesmaximal, characteristic
2#8700C211yesmaximal, characteristic
3#8152C211yesmaximal, characteristic
4#7132C212yesmaximal, complemented
5#7123C212yesmaximal, complemented
6C2×#399C411yescharacteristic
7C2×C4×C4⋊C4—21no—
8C2×C4×C4⋊C4C411yescharacteristic
9C2×C4×C22⋊C4—21no—
10C2×C4×C22⋊C4C2211yescharacteristic
11#1779—21no—
12#1749—21no—
13C2×#323—21no—
14#1445—21no—
15#1445C2211yescharacteristic
16#1444C2211yescharacteristic
17#1439—21no—
18#1439C411yescharacteristic
19C4×#95C411yescharacteristic
20#1280—21no—
21#1280C2212yes—
22#1280C2212yes—
23C24×C4D411yescharacteristic
24C22×C4⋊C4—21no—
25C22×C4⋊C4—21no—
26C22×C4⋊C4C2×C411yescharacteristic
27C22×C4⋊C4C2×C411yescharacteristic
28C22×C22⋊C4—21no—
29C22×C22⋊C4—21no—
30C22×C22⋊C4C2×C411yescharacteristic
31C22×C22⋊C4D411yescharacteristic
32C22×C42—21no—
33C22×C42—21no—
34C22×C42C2×C411yescharacteristic
35C22×C42D411yescharacteristic
36C2×C22⋊C8—22no—
37C2×C22⋊C8—22no—
38C2×C22⋊C8—41no—
39C2×C22⋊C8—41no—
40C2×#97—41no—
41#399—22no—
42#399—22no—
43C4×C4⋊C4—44no—
44C4×C4⋊C4—41no—
45C4×C4⋊C4—41no—
46C4×C22⋊C4—41no—
47C4×C22⋊C4—41no—
48C4×C22⋊C4—44no—
49C2×#95—21no—
50C2×#95—21no—
51C2×#95—21no—
52C2×#95—21no—
53C2×#95—21no—
54C2×#95—21no—
55C2×#95C2×C411yescharacteristic
56C2×#95C2×C411yescharacteristic
57C2×#95D411yescharacteristic
58C2×#95C2311yescharacteristic
59#336—41no—
60#323—24no—
61C25C22⋊C411yescharacteristic
62C23×C4—21no—
63C23×C4—21no—
64C23×C4—21no—
65C23×C4—21no—
66C23×C4—22no—
67C23×C4—21no—
68C23×C4—21no—
69C23×C4—41no—
70C23×C4—42no—
71C23×C4—42no—
72C23×C4C22⋊C411yescharacteristic
73C23×C4C22⋊C411yescharacteristic
74C23×C4C22⋊C411yescharacteristic
75C23×C4C22×C411yescharacteristic
76C23×C4C2×D411yescharacteristic
77C23×C4C2×D411yescharacteristic
78C22×C8—42no—
79C22×C8—42no—
80C2×C4⋊C4—24no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—42no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4—42no—
88C2×C4⋊C4—42no—
89C2×C4⋊C4—42no—
90C2×C4⋊C4—42no—
91C2×C4⋊C4M1612yes—
92C2×C22⋊C4—24no—
93C2×C22⋊C4—22no—
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—22no—
97C2×C22⋊C4—41no—
98C2×C22⋊C4—41no—
99C2×C22⋊C4—41no—
100C2×C22⋊C4—42no—
101C2×C22⋊C4—42no—
102C2×C22⋊C4—42no—
103C2×C22⋊C4—42no—
104C2×C22⋊C4—42no—
105C2×C42—24no—
106C2×C42—21no—
107C2×C42—21no—
108C2×C42—21no—
109C2×C42—21no—
110C2×C42—42no—
111C2×C42—42no—
112C2×C42—42no—
113C2×C42—42no—
114C2×C42—42no—
115C2×C42—42no—
116C2×C42—41no—
117C2×C42—41no—
118C22⋊C8—24no—
119C22⋊C8—42no—
120C22⋊C8—42no—
121C22⋊C8—42no—
122C22⋊C8—82no—
123C22⋊C8—82no—
124#97—82no—
125#95—21no—
126#95—41no—
127#95—41no—
128#95—41no—
129#95—42no—
130#95—42no—
131#95—42no—
132#95—42no—
133#95—42no—
134#95—42no—
135#95M1612yes—
136C24—21no—
137C24—21no—
138C24—21no—
139C24—21no—
140C24—21no—
141C24—21no—
142C24—41no—
143C24—41no—
144C24—41no—
145C24—41no—
146C24#10412yes—
147C24C2×C22⋊C411yescharacteristic
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—21no—
164C22×C4—22no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—48no—
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—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—42no—
204C22×C4—42no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—81no—
208C22×C4—81no—
209C22×C4—84no—
210C22×C4#9911yescharacteristic
211C22×C4#9911yescharacteristic
212C22×C4C2×M1611yescharacteristic
213C22×C4C2×M1611yescharacteristic
214C22×C4#13512yes—
215C22×C4#13712yes—
216C2×C8—44no—
217C2×C8—82no—
218C2×C8—82no—
219C2×C8—82no—
220C2×C8—82no—
221C2×C8—82no—
222C4⋊C4—42no—
223C4⋊C4—48no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—48no—
227C4⋊C4—42no—
228C22⋊C4—44no—
229C22⋊C4—48no—
230C22⋊C4—42no—
231C22⋊C4—42no—
232C22⋊C4—48no—
233C42—48no—
234C42—42no—
235C42—42no—
236C42—42no—
237C42—81no—
238C42—82no—
239C42—82no—
240C23—21no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—22no—
245C23—22no—
246C23—22no—
247C23—42no—
248C23—42no—
249C23—42no—
250C23—42no—
251C23—42no—
252C23—42no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23—81no—
262C23—81no—
263C23—81no—
264C23—81no—
265C23—81no—
266C23—81no—
267C23#40711yescharacteristic
268C23C2×#9911yescharacteristic
269C23#41311yescharacteristic
270C23#41611yescharacteristic
271C23#41911yescharacteristic
272C23C2×#10411yescharacteristic
273C23#42111yescharacteristic
274C2×C4—24no—
275C2×C4—24no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—24no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—84no—
319C2×C4—84no—
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#35212yes—
338C2×C4#35312yes—
339C8—84no—
340C8—162no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—44no—
354C22—42no—
355C22—42no—
356C22—42no—
357C22—82no—
358C22—82no—
359C22—82no—
360C22—81no—
361C22—81no—
362C22—81no—
363C22—81no—
364C22#146511yescharacteristic
365C22#148811yescharacteristic
366C22#149111yescharacteristic
367C22#150111yescharacteristic
368C22#151111yescharacteristic
369C22#153211yescharacteristic
370C22#155411yescharacteristic
371C22C2×#35211yescharacteristic
372C22#212011yescharacteristic
373C22C2×#35311yescharacteristic
374C22#213211yescharacteristic
375C4—22no—
376C4—22no—
377C4—44no—
378C4—41no—
379C4—41no—
380C4—42no—
381C4—42no—
382C4—82no—
383C4—81no—
384C4—82no—
385C4—82no—
386C4—81no—
387C2—21no—
388C2—21no—
389C2—21no—
390C2—21no—
391C2—82no—
392C2#762311yescharacteristic
393C2#1152611yescharacteristic
394C2#1152911yescharacteristic
395C2#1172611yescharacteristic
396C2#1173511yescharacteristic
397C2#1180911yescharacteristic
398C2#1181611yescharacteristic
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. #7132C2semidirect
5. #7123C2semidirect

Maximal subgroups

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

subgroupindex
C4×#3992
#87002
#81522
#71322
#71232