Finite Group Databasebeta

← #124197

Subgroups — #124197

Order 512.

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

normal subgroups only

This group has 602 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#14319C211yesmaximal, characteristic
2C4×#323C211yesmaximal, characteristic
3#7907C211yesmaximal, characteristic
4#7131C212yesmaximal, complemented
5#7123C212yesmaximal, complemented
6C22×C22⋊C8—21no—
7C2×#393C411yescharacteristic
8C2×#391C411yescharacteristic
9C2×C4×C22⋊C4C2211yescharacteristic
10C4×C22⋊C8—21no—
11C2×#323—21no—
12#1446C2211yescharacteristic
13#1444C2211yescharacteristic
14#1443—21no—
15#1439—21no—
16#1439C411yescharacteristic
17#1436—21no—
18#1434C411yescharacteristic
19C22⋊#95—21no—
20#1280—21no—
21#1280C2212yes—
22#1280C2212yes—
23C23×Q8C2×C411yescharacteristic
24C24×C4D411yescharacteristic
25C23×C8—41no—
26C22×C4⋊C4—21no—
27C22×C4⋊C4C2×C411yescharacteristic
28C22×C22⋊C4—21no—
29C22×C22⋊C4—21no—
30C22×C22⋊C4C2×C411yescharacteristic
31C22×C22⋊C4D411yescharacteristic
32C22×C42C2×C411yescharacteristic
33C22×C42D411yescharacteristic
34C2×C22⋊C8—24no—
35C2×C22⋊C8—22no—
36C2×C22⋊C8—22no—
37C2×C22⋊C8—41no—
38C2×C22⋊C8—42no—
39C2×C4×C8—41no—
40#393—22no—
41#393C814yescomplemented
42#391—41no—
43#391—41no—
44C4×C22⋊C4—41no—
45C4×C22⋊C4—41no—
46C2×#95—21no—
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
62C22×Q8—21no—
63C22×Q8—41no—
64C22×Q8—41no—
65C22×Q8—81no—
66C22×Q8—81no—
67C22×Q8C2×C812yescomplemented
68C23×C4—21no—
69C23×C4—21no—
70C23×C4—22no—
71C23×C4—21no—
72C23×C4—21no—
73C23×C4—21no—
74C23×C4—21no—
75C23×C4—81no—
76C23×C4—81no—
77C23×C4C22⋊C411yescharacteristic
78C23×C4C22⋊C411yescharacteristic
79C23×C4C22⋊C411yescharacteristic
80C23×C4SD1612yes—
81C23×C4Q1612yes—
82C23×C4C22×C411yescharacteristic
83C23×C4C2×D411yescharacteristic
84C23×C4C2×D411yescharacteristic
85C22×C8—42no—
86C22×C8—44no—
87C22×C8—42no—
88C22×C8—82no—
89C22×C8—82no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—42no—
92C2×C4⋊C4—81no—
93C2×C4⋊C4—81no—
94C2×C4⋊C4—81no—
95C2×C22⋊C4—22no—
96C2×C22⋊C4—22no—
97C2×C22⋊C4—24no—
98C2×C22⋊C4—41no—
99C2×C22⋊C4—41no—
100C2×C22⋊C4—41no—
101C2×C22⋊C4—42no—
102C2×C22⋊C4—81no—
103C2×C22⋊C4—81no—
104C2×C22⋊C4—81no—
105C2×C22⋊C4—81no—
106C2×C22⋊C4C2×C812yes—
107C2×C22⋊C4M1612yes—
108C2×C42—21no—
109C2×C42—21no—
110C2×C42—21no—
111C2×C42—21no—
112C2×C42—81no—
113C2×C42—81no—
114C22⋊C8—24no—
115C22⋊C8—24no—
116C22⋊C8—44no—
117C22⋊C8—42no—
118C22⋊C8—42no—
119C4×C8—82no—
120#95—24no—
121#95—21no—
122#95—41no—
123#95—41no—
124#95—41no—
125#95—42no—
126#95—41no—
127#95—41no—
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
147C2×Q8—42no—
148C2×Q8—42no—
149C2×Q8—82no—
150C2×Q8—82no—
151C2×Q8—84no—
152C2×Q8—82no—
153C2×Q8—82no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—42no—
174C22×C4—42no—
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—41no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—81no—
212C22×C4—81no—
213C22×C4—82no—
214C22×C4—82no—
215C22×C4—84no—
216C22×C4C22⋊C812yes—
217C22×C4#9911yescharacteristic
218C22×C4#9911yescharacteristic
219C22×C4#10312yes—
220C22×C4#10412yes—
221C22×C4C22×C811yescharacteristic
222C22×C4C2×M1611yescharacteristic
223C22×C4C2×SD1611yescharacteristic
224C22×C4C2×Q1611yescharacteristic
225C22×C4#13512yes—
226C2×C8—44no—
227C2×C8—44no—
228C2×C8—82no—
229C2×C8—82no—
230C2×C8—88no—
231C2×C8—82no—
232C2×C8—82no—
233C4⋊C4—82no—
234C4⋊C4—84no—
235C22⋊C4—42no—
236C22⋊C4—42no—
237C22⋊C4—44no—
238C22⋊C4—84no—
239C22⋊C4—84no—
240C42—42no—
241C42—81no—
242C42—81no—
243C42—81no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—22no—
249C23—22no—
250C23—22no—
251C23—22no—
252C23—22no—
253C23—22no—
254C23—42no—
255C23—42no—
256C23—42no—
257C23—42no—
258C23—42no—
259C23—42no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23—41no—
264C23—81no—
265C23—81no—
266C23—81no—
267C23—81no—
268C23—81no—
269C23—81no—
270C23Q8⋊C814yes—
271C23C2×C22⋊C811yescharacteristic
272C23C2×#9911yescharacteristic
273C23#41311yescharacteristic
274C23C2×#10311yescharacteristic
275C23#41611yescharacteristic
276C23C2×#10411yescharacteristic
277C23C2×#10411yescharacteristic
278Q8—84no—
279Q8—84no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—22no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—22no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—44no—
293C2×C4—44no—
294C2×C4—42no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—42no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—42no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—82no—
319C2×C4—82no—
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×C4—81no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4#32812yes—
348C2×C4#32812yes—
349C8—84no—
350C8—84no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—24no—
364C22—44no—
365C22—42no—
366C22—42no—
367C22—42no—
368C22—42no—
369C22—42no—
370C22—42no—
371C22—82no—
372C22—82no—
373C22—82no—
374C22#145911yescharacteristic
375C22C2×Q8⋊C811yescharacteristic
376C22#147611yescharacteristic
377C22C2×#32811yescharacteristic
378C22#150111yescharacteristic
379C22#152411yescharacteristic
380C22#154211yescharacteristic
381C22#212111yescharacteristic
382C22#212211yescharacteristic
383C22#212711yescharacteristic
384C22#213511yescharacteristic
385C4—22no—
386C4—22no—
387C4—41no—
388C4—41no—
389C4—41no—
390C4—41no—
391C4—42no—
392C4—42no—
393C4—82no—
394C4—82no—
395C4—81no—
396C4—161no—
397C4—161no—
398C2—21no—
399C2—21no—
400C2—21no—
401C2—21no—
402C2—44no—
403C2#756611yescharacteristic
404C2#1147311yescharacteristic
405C2#1148111yescharacteristic
406C2#1171611yescharacteristic
407C2#1172411yescharacteristic
408C2#1173011yescharacteristic
409C2#1173911yescharacteristic
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. #7131C2semidirect
5. #7123C2semidirect
41. #393C8semidirect
67. C22×Q8C2×C8semidirect
134. #95C2×C8semidirect

Maximal subgroups

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

subgroupindex
#143192
C4×#3232
#79072
#71312
#71232