Finite Group Databasebeta

← #124195

Subgroups — #124195

Order 512.

586 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 586 conjugacy classes of subgroups — too many to draw. Only its 108 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#14317C211yesmaximal, characteristic
2C4×#323C211yesmaximal, characteristic
3#7907C211yesmaximal, characteristic
4#7132C212yesmaximal, complemented
5#7122C212yesmaximal, complemented
6C22×C22⋊C8—21no—
7C2×#399C411yescharacteristic
8C2×#389C411yescharacteristic
9C2×C4×C22⋊C4C2211yescharacteristic
10C4×C22⋊C8—21no—
11C2×#323—21no—
12#1445C2211yescharacteristic
13#1442C2211yescharacteristic
14#1441—21no—
15#1439—21no—
16#1439C411yescharacteristic
17#1436—21no—
18#1434C411yescharacteristic
19C22⋊#95—21no—
20#1280—21no—
21#1280C2212yes—
22#1280C2212yes—
23C24×C4D411yescharacteristic
24C23×C8—41no—
25C22×C4⋊C4—21no—
26C22×C4⋊C4—21no—
27C22×C4⋊C4C2×C411yescharacteristic
28C22×C4⋊C4C2×C411yescharacteristic
29C22×C22⋊C4—21no—
30C22×C22⋊C4—21no—
31C22×C22⋊C4C2×C411yescharacteristic
32C22×C22⋊C4D411yescharacteristic
33C22×C42C2×C411yescharacteristic
34C22×C42D411yescharacteristic
35C2×C22⋊C8—24no—
36C2×C22⋊C8—22no—
37C2×C22⋊C8—22no—
38C2×C22⋊C8—41no—
39C2×C22⋊C8—42no—
40C2×C4×C8—41no—
41#399—22no—
42#399C814yescomplemented
43#389—41no—
44#389—41no—
45C4×C22⋊C4—41no—
46C4×C22⋊C4—41no—
47C2×#95—21no—
48C2×#95—21no—
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—22no—
65C23×C4—21no—
66C23×C4—21no—
67C23×C4—21no—
68C23×C4—21no—
69C23×C4—81no—
70C23×C4—81no—
71C23×C4C22⋊C411yescharacteristic
72C23×C4C22⋊C411yescharacteristic
73C23×C4C22⋊C411yescharacteristic
74C23×C4SD1612yes—
75C23×C4Q1612yes—
76C23×C4C22×C411yescharacteristic
77C23×C4C2×D411yescharacteristic
78C23×C4C2×D411yescharacteristic
79C22×C8—42no—
80C22×C8—44no—
81C22×C8—42no—
82C22×C8—82no—
83C22×C8—82no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—24no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4—42no—
90C2×C4⋊C4—81no—
91C2×C4⋊C4—81no—
92C2×C4⋊C4—81no—
93C2×C4⋊C4—81no—
94C2×C4⋊C4—81no—
95C2×C4⋊C4C2×C812yescomplemented
96C2×C22⋊C4—22no—
97C2×C22⋊C4—22no—
98C2×C22⋊C4—24no—
99C2×C22⋊C4—41no—
100C2×C22⋊C4—41no—
101C2×C22⋊C4—41no—
102C2×C22⋊C4—42no—
103C2×C22⋊C4—81no—
104C2×C22⋊C4—81no—
105C2×C22⋊C4—81no—
106C2×C22⋊C4—81no—
107C2×C22⋊C4C2×C812yes—
108C2×C22⋊C4M1612yes—
109C2×C42—21no—
110C2×C42—21no—
111C2×C42—21no—
112C2×C42—21no—
113C2×C42—81no—
114C2×C42—81no—
115C22⋊C8—24no—
116C22⋊C8—24no—
117C22⋊C8—44no—
118C22⋊C8—42no—
119C22⋊C8—42no—
120C4×C8—82no—
121#95—21no—
122#95—41no—
123#95—41no—
124#95—41no—
125#95—41no—
126#95—41no—
127#95—42no—
128#95—42no—
129#95—81no—
130#95—81no—
131#95—81no—
132#95—81no—
133#95—81no—
134#95C2×C812yescomplemented
135C24—21no—
136C24—21no—
137C24—21no—
138C24—21no—
139C24—41no—
140C24—41no—
141C24—41no—
142C24—41no—
143C24C22⋊C812yes—
144C24#10312yes—
145C24#10412yes—
146C24C2×C22⋊C411yescharacteristic
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
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—22no—
160C22×C4—22no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—41no—
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—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—81no—
205C22×C4—81no—
206C22×C4—82no—
207C22×C4—82no—
208C22×C4—84no—
209C22×C4C22⋊C812yes—
210C22×C4#9911yescharacteristic
211C22×C4#9911yescharacteristic
212C22×C4#10312yes—
213C22×C4#10412yes—
214C22×C4C22×C811yescharacteristic
215C22×C4C2×M1611yescharacteristic
216C22×C4C2×SD1611yescharacteristic
217C22×C4C2×Q1611yescharacteristic
218C22×C4#13512yes—
219C2×C8—44no—
220C2×C8—44no—
221C2×C8—82no—
222C2×C8—82no—
223C2×C8—88no—
224C2×C8—82no—
225C2×C8—82no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—84no—
229C4⋊C4—82no—
230C4⋊C4—84no—
231C22⋊C4—42no—
232C22⋊C4—42no—
233C22⋊C4—44no—
234C22⋊C4—84no—
235C22⋊C4—84no—
236C42—42no—
237C42—81no—
238C42—81no—
239C42—81no—
240C23—21no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—22no—
245C23—22no—
246C23—22no—
247C23—22no—
248C23—22no—
249C23—22no—
250C23—42no—
251C23—42no—
252C23—42no—
253C23—42no—
254C23—42no—
255C23—42no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—81no—
261C23—81no—
262C23—81no—
263C23—81no—
264C23—81no—
265C23—81no—
266C23Q8⋊C814yes—
267C23C2×C22⋊C811yescharacteristic
268C23C2×#9911yescharacteristic
269C23#41311yescharacteristic
270C23C2×#10311yescharacteristic
271C23#41611yescharacteristic
272C23C2×#10411yescharacteristic
273C23C2×#10411yescharacteristic
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—22no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—22no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—44no—
287C2×C4—44no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—42no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—41no—
312C2×C4—82no—
313C2×C4—82no—
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#32812yes—
342C2×C4#32812yes—
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—42no—
360C22—42no—
361C22—42no—
362C22—42no—
363C22—42no—
364C22—42no—
365C22—82no—
366C22—82no—
367C22—82no—
368C22#145911yescharacteristic
369C22C2×Q8⋊C811yescharacteristic
370C22#147611yescharacteristic
371C22C2×#32811yescharacteristic
372C22#150111yescharacteristic
373C22#152411yescharacteristic
374C22#154211yescharacteristic
375C22#212111yescharacteristic
376C22#212311yescharacteristic
377C22#212711yescharacteristic
378C22#213411yescharacteristic
379C4—22no—
380C4—22no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—41no—
385C4—42no—
386C4—42no—
387C4—82no—
388C4—82no—
389C4—81no—
390C4—161no—
391C4—161no—
392C2—21no—
393C2—21no—
394C2—21no—
395C2—21no—
396C2—44no—
397C2#756611yescharacteristic
398C2#1147411yescharacteristic
399C2#1147911yescharacteristic
400C2#1171611yescharacteristic
401C2#1172311yescharacteristic
402C2#1173211yescharacteristic
403C2#1173711yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C22×C8
Frattini, cosocleC2×#95C23
prime radicalC25C22⋊C4
LCSG ▷ C22×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. #7122C2semidirect
42. #399C8semidirect
95. C2×C4⋊C4C2×C8semidirect
134. #95C2×C8semidirect

Maximal subgroups

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

subgroupindex
#143172
C4×#3232
#79072
#71322
#71222