Finite Group Databasebeta

← #124357

Subgroups — #124357

Order 512.

650 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 650 conjugacy classes of subgroups — too many to draw. Only its 104 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#13966C211yesmaximal, characteristic
2#8698C211yesmaximal, characteristic
3C2×#1280C211yesmaximal, characteristic
4#7122C212yesmaximal, complemented
5#7122C212yesmaximal, complemented
6C22×C22⋊C8—21no—
7C2×#380C411yescharacteristic
8C2×C4×C22⋊C4C411yescharacteristic
9C22×#95C2211yescharacteristic
10#1779—21no—
11#1442C411yescharacteristic
12#1442C411yescharacteristic
13#1442C2211yescharacteristic
14#1442C2211yescharacteristic
15#1434—21no—
16#1434—21no—
17C22⋊#95—21no—
18C22⋊#95—21no—
19#1280—21no—
20#1280—21no—
21#1280C2212yes—
22#1280C2212yes—
23C24×C4C2×C411yescharacteristic
24C24×C4D411yescharacteristic
25C24×C4D411yescharacteristic
26C23×C8—41no—
27C22×C4⋊C4C2×C411yescharacteristic
28C22×C4⋊C4C2×C411yescharacteristic
29C22×C22⋊C4—21no—
30C22×C22⋊C4—21no—
31C22×C42—21no—
32C2×C22⋊C8—24no—
33C2×C22⋊C8—22no—
34C2×C22⋊C8—22no—
35C2×C22⋊C8—41no—
36C2×C22⋊C8—42no—
37C2×C22⋊C8—41no—
38C2×C22⋊C8—41no—
39#380—41no—
40#380—41no—
41C4×C22⋊C4—22no—
42C4×C22⋊C4C814yescomplemented
43C2×#95—21no—
44C2×#95—21no—
45C2×#95—21no—
46C2×#95—21no—
47C2×#95—21no—
48C2×#95—21no—
49C2×#95—21no—
50C2×#95—21no—
51C2×#95—21no—
52C2×#95—41no—
53C2×#95—41no—
54C2×#95C2×C411yescharacteristic
55C2×#95C2×C411yescharacteristic
56C2×#95C2×C411yescharacteristic
57C2×#95D411yescharacteristic
58C2×#95D411yescharacteristic
59C2×#95C2311yescharacteristic
60C25C22⋊C411yescharacteristic
61C23×C4—21no—
62C23×C4—21no—
63C23×C4—21no—
64C23×C4—21no—
65C23×C4—22no—
66C23×C4—22no—
67C23×C4—21no—
68C23×C4—21no—
69C23×C4—21no—
70C23×C4—21no—
71C23×C4—21no—
72C23×C4—21no—
73C23×C4—22no—
74C23×C4—21no—
75C23×C4—21no—
76C23×C4—41no—
77C23×C4—42no—
78C23×C4—42no—
79C23×C4—81no—
80C23×C4—81no—
81C23×C4—81no—
82C23×C4—81no—
83C23×C4C22⋊C411yescharacteristic
84C23×C4C22⋊C411yescharacteristic
85C23×C4C22⋊C411yescharacteristic
86C23×C4C2×C812yes—
87C23×C4M1612yes—
88C23×C4C22×C411yescharacteristic
89C23×C4C2×D411yescharacteristic
90C23×C4C2×D411yescharacteristic
91C22×C8—44no—
92C22×C8—42no—
93C22×C8—42no—
94C22×C8—82no—
95C22×C8—82no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—81no—
99C2×C4⋊C4—81no—
100C2×C22⋊C4—24no—
101C2×C22⋊C4—42no—
102C2×C22⋊C4—42no—
103C2×C22⋊C4—42no—
104C2×C22⋊C4—42no—
105C2×C22⋊C4—41no—
106C2×C22⋊C4—41no—
107C2×C42—24no—
108C2×C42—81no—
109C2×C42—81no—
110C22⋊C8—24no—
111C22⋊C8—44no—
112C22⋊C8—42no—
113C22⋊C8—42no—
114C22⋊C8—42no—
115C22⋊C8—82no—
116C22⋊C8—82no—
117#95—21no—
118#95—21no—
119#95—41no—
120#95—41no—
121#95—41no—
122#95—41no—
123#95—42no—
124#95—42no—
125#95—42no—
126#95—42no—
127#95—41no—
128#95—41no—
129#95—81no—
130#95—81no—
131#95—81no—
132#95—81no—
133#95C2×C812yescomplemented
134#95C2×C812yescomplemented
135C24—22no—
136C24—22no—
137C24—21no—
138C24—21no—
139C24—21no—
140C24—21no—
141C24—41no—
142C24—41no—
143C24C22⋊C812yes—
144C24#9911yescharacteristic
145C24#9911yescharacteristic
146C24#10112yes—
147C24C2×C22⋊C411yescharacteristic
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—42no—
172C22×C4—42no—
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—42no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—42no—
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—82no—
216C22×C4—84no—
217C22×C4—82no—
218C22×C4—81no—
219C22×C4—81no—
220C22×C4—81no—
221C22×C4—81no—
222C22×C4—81no—
223C22×C4—81no—
224C22×C4—81no—
225C22×C4—81no—
226C22×C4—84no—
227C22×C4—82no—
228C22×C4—82no—
229C22×C4—84no—
230C22×C4—84no—
231C22×C4C22⋊C812yes—
232C22×C4#10412yes—
233C22×C4C22×C811yescharacteristic
234C22×C4C2×M1611yescharacteristic
235C22×C4#13512yes—
236C22×C4AGL(1,ℤ/8)11yescharacteristic
237C22×C4#13711yescharacteristic
238C2×C8—44no—
239C2×C8—88no—
240C2×C8—82no—
241C2×C8—82no—
242C2×C8—82no—
243C2×C8—82no—
244C2×C8—82no—
245C4⋊C4—82no—
246C4⋊C4—82no—
247C22⋊C4—48no—
248C22⋊C4—82no—
249C22⋊C4—82no—
250C42—84no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—22no—
260C23—24no—
261C23—24no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—22no—
267C23—22no—
268C23—21no—
269C23—21no—
270C23—22no—
271C23—42no—
272C23—42no—
273C23—42no—
274C23—42no—
275C23—42no—
276C23—42no—
277C23—42no—
278C23—42no—
279C23—42no—
280C23—42no—
281C23#32412yes—
282C23#32412yes—
283C23#35412yes—
284C23#35512yes—
285C23C2×C22⋊C811yescharacteristic
286C23C2×#9911yescharacteristic
287C23C2×#10111yescharacteristic
288C23#41611yescharacteristic
289C23#41711yescharacteristic
290C23C2×#10411yescharacteristic
291C23#42111yescharacteristic
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—22no—
297C2×C4—21no—
298C2×C4—22no—
299C2×C4—44no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—44no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—42no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—41no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4—82no—
338C2×C4—82no—
339C2×C4—84no—
340C2×C4—84no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C8—84no—
365C8—162no—
366C22—21no—
367C22—21no—
368C22—22no—
369C22—21no—
370C22—21no—
371C22—22no—
372C22—22no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—22no—
378C22—22no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—22no—
384C22—22no—
385C22—22no—
386C22—22no—
387C22—44no—
388C22—44no—
389C22—44no—
390C22—44no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22#132414yes—
396C22C2×#32411yescharacteristic
397C22#147611yescharacteristic
398C22#147711yescharacteristic
399C22#150411yescharacteristic
400C22#151611yescharacteristic
401C22#153111yescharacteristic
402C22#155211yescharacteristic
403C22#212111yescharacteristic
404C22#212111yescharacteristic
405C22C2×#35411yescharacteristic
406C22C2×#35511yescharacteristic
407C4—42no—
408C4—42no—
409C4—41no—
410C4—41no—
411C4—41no—
412C4—42no—
413C4—41no—
414C4—82no—
415C4—82no—
416C4—81no—
417C4—161no—
418C4—161no—
419C2—21no—
420C2—21no—
421C2—21no—
422C2—24no—
423C2—21no—
424C2—42no—
425C2#757911yescharacteristic
426C2C2×#132411yescharacteristic
427C2#1145311yescharacteristic
428C2#1175911yescharacteristic
429C2#1176611yescharacteristic
430C2#1184511yescharacteristic
431C2#1185211yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C22×C8
Frattini, cosocleC2×#95C23
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C25 ◁ C22×#95 ◁ G
LECSG ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7122C2semidirect
5. #7122C2semidirect
42. C4×C22⋊C4C8semidirect
133. #95C2×C8semidirect
134. #95C2×C8semidirect

Maximal subgroups

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

subgroupindex
#139662
#86982
C2×#12802
#71222