Finite Group Databasebeta

← #124359

Subgroups — #124359

Order 512.

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

normal subgroups only

This group has 722 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#13965C211yesmaximal, characteristic
2#8698C211yesmaximal, characteristic
3C2×#1280C211yesmaximal, characteristic
4#7123C212yesmaximal, complemented
5#7123C212yesmaximal, complemented
6C22×C22⋊C8—21no—
7C2×#385—21no—
8C2×#385—21no—
9C2×#380C411yescharacteristic
10C2×C4×C22⋊C4C411yescharacteristic
11C22×#95C2211yescharacteristic
12#1779—21no—
13#1444C411yescharacteristic
14#1444C411yescharacteristic
15#1444C2211yescharacteristic
16#1444C2211yescharacteristic
17C4⋊#95—21no—
18C4⋊#95—21no—
19#1280—21no—
20#1280—21no—
21#1280C2212yes—
22#1280C2212yes—
23C24×C4C2×C411yescharacteristic
24C24×C4D411yescharacteristic
25C24×C4D411yescharacteristic
26C23×C8—41no—
27C22×C4⋊C4—21no—
28C22×C4⋊C4—21no—
29C22×C4⋊C4—21no—
30C22×C4⋊C4—21no—
31C22×C4⋊C4C2×C411yescharacteristic
32C22×C4⋊C4C2×C411yescharacteristic
33C22×C22⋊C4—21no—
34C22×C22⋊C4—21no—
35C22×C42—21no—
36C2×C22⋊C8—24no—
37C2×C22⋊C8—22no—
38C2×C22⋊C8—22no—
39C2×C22⋊C8—41no—
40C2×C22⋊C8—42no—
41C2×C22⋊C8—41no—
42C2×C22⋊C8—41no—
43#385—44no—
44#385—44no—
45#380—41no—
46#380—41no—
47C4×C22⋊C4—22no—
48C4×C22⋊C4C814yescomplemented
49C2×#95—21no—
50C2×#95—21no—
51C2×#95—21no—
52C2×#95—21no—
53C2×#95—21no—
54C2×#95—41no—
55C2×#95—41no—
56C2×#95C2×C411yescharacteristic
57C2×#95C2×C411yescharacteristic
58C2×#95C2×C411yescharacteristic
59C2×#95D411yescharacteristic
60C2×#95D411yescharacteristic
61C2×#95C2311yescharacteristic
62C25C22⋊C411yescharacteristic
63C23×C4—21no—
64C23×C4—21no—
65C23×C4—21no—
66C23×C4—21no—
67C23×C4—21no—
68C23×C4—21no—
69C23×C4—22no—
70C23×C4—22no—
71C23×C4—21no—
72C23×C4—21no—
73C23×C4—21no—
74C23×C4—21no—
75C23×C4—22no—
76C23×C4—21no—
77C23×C4—21no—
78C23×C4—41no—
79C23×C4—42no—
80C23×C4—42no—
81C23×C4—81no—
82C23×C4—81no—
83C23×C4—81no—
84C23×C4—81no—
85C23×C4C22⋊C411yescharacteristic
86C23×C4C22⋊C411yescharacteristic
87C23×C4C22⋊C411yescharacteristic
88C23×C4C2×C812yes—
89C23×C4M1612yes—
90C23×C4C22×C411yescharacteristic
91C23×C4C2×D411yescharacteristic
92C23×C4C2×D411yescharacteristic
93C22×C8—44no—
94C22×C8—42no—
95C22×C8—42no—
96C22×C8—82no—
97C22×C8—82no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4—42no—
102C2×C4⋊C4—42no—
103C2×C4⋊C4—42no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4—42no—
106C2×C4⋊C4—42no—
107C2×C4⋊C4—41no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4—81no—
110C2×C4⋊C4—81no—
111C2×C4⋊C4—81no—
112C2×C4⋊C4—81no—
113C2×C4⋊C4—81no—
114C2×C4⋊C4—81no—
115C2×C22⋊C4—24no—
116C2×C22⋊C4—24no—
117C2×C22⋊C4—24no—
118C2×C22⋊C4—42no—
119C2×C22⋊C4—42no—
120C2×C22⋊C4—41no—
121C2×C22⋊C4—41no—
122C2×C42—24no—
123C2×C42—81no—
124C2×C42—81no—
125C22⋊C8—24no—
126C22⋊C8—44no—
127C22⋊C8—42no—
128C22⋊C8—42no—
129C22⋊C8—42no—
130C22⋊C8—82no—
131C22⋊C8—82no—
132#95—21no—
133#95—21no—
134#95—41no—
135#95—41no—
136#95—41no—
137#95—41no—
138#95—42no—
139#95—42no—
140#95—42no—
141#95—42no—
142#95—42no—
143#95C2×C812yescomplemented
144#95C2×C812yescomplemented
145C24—22no—
146C24—22no—
147C24—21no—
148C24—21no—
149C24—21no—
150C24—21no—
151C24—41no—
152C24—41no—
153C24C22⋊C812yes—
154C24#9911yescharacteristic
155C24#9911yescharacteristic
156C24#10112yes—
157C24C2×C22⋊C411yescharacteristic
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—42no—
192C22×C4—42no—
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—42no—
210C22×C4—42no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—82no—
226C22×C4—84no—
227C22×C4—82no—
228C22×C4—81no—
229C22×C4—81no—
230C22×C4—81no—
231C22×C4—81no—
232C22×C4—81no—
233C22×C4—81no—
234C22×C4—81no—
235C22×C4—81no—
236C22×C4—84no—
237C22×C4—82no—
238C22×C4—82no—
239C22×C4—84no—
240C22×C4—84no—
241C22×C4C22⋊C812yes—
242C22×C4#10412yes—
243C22×C4C22×C811yescharacteristic
244C22×C4C2×M1611yescharacteristic
245C22×C4#13512yes—
246C22×C4AGL(1,ℤ/8)11yescharacteristic
247C22×C4#13711yescharacteristic
248C2×C8—44no—
249C2×C8—88no—
250C2×C8—82no—
251C2×C8—82no—
252C2×C8—82no—
253C2×C8—82no—
254C2×C8—82no—
255C4⋊C4—84no—
256C4⋊C4—84no—
257C4⋊C4—82no—
258C4⋊C4—82no—
259C4⋊C4—84no—
260C4⋊C4—82no—
261C4⋊C4—82no—
262C22⋊C4—216no—
263C22⋊C4—82no—
264C22⋊C4—82no—
265C42—84no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—22no—
275C23—24no—
276C23—24no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—22no—
282C23—22no—
283C23—21no—
284C23—21no—
285C23—22no—
286C23—42no—
287C23—42no—
288C23—42no—
289C23—42no—
290C23—42no—
291C23—42no—
292C23—42no—
293C23—42no—
294C23—42no—
295C23—42no—
296C23#32412yes—
297C23#32412yes—
298C23#35312yes—
299C23#35612yes—
300C23C2×C22⋊C811yescharacteristic
301C23C2×#9911yescharacteristic
302C23C2×#10111yescharacteristic
303C23#41611yescharacteristic
304C23#41711yescharacteristic
305C23C2×#10411yescharacteristic
306C23#42111yescharacteristic
307C2×C4—24no—
308C2×C4—24no—
309C2×C4—22no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—22no—
314C2×C4—21no—
315C2×C4—22no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—44no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—44no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—84no—
363C2×C4—84no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C8—84no—
380C8—162no—
381C22—21no—
382C22—21no—
383C22—22no—
384C22—21no—
385C22—21no—
386C22—22no—
387C22—22no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—22no—
393C22—21no—
394C22—21no—
395C22—22no—
396C22—21no—
397C22—21no—
398C22—22no—
399C22—22no—
400C22—22no—
401C22—22no—
402C22—44no—
403C22—44no—
404C22—44no—
405C22—44no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22—41no—
410C22#132514yes—
411C22C2×#32411yescharacteristic
412C22#147611yescharacteristic
413C22#147711yescharacteristic
414C22#150411yescharacteristic
415C22#151611yescharacteristic
416C22#153111yescharacteristic
417C22#155211yescharacteristic
418C22#212111yescharacteristic
419C22#212111yescharacteristic
420C22C2×#35311yescharacteristic
421C22C2×#35611yescharacteristic
422C4—44no—
423C4—44no—
424C4—42no—
425C4—42no—
426C4—41no—
427C4—41no—
428C4—41no—
429C4—42no—
430C4—41no—
431C4—81no—
432C4—161no—
433C4—161no—
434C2—21no—
435C2—21no—
436C2—21no—
437C2—24no—
438C2—21no—
439C2—42no—
440C2#757911yescharacteristic
441C2C2×#132511yescharacteristic
442C2#1145311yescharacteristic
443C2#1175711yescharacteristic
444C2#1176811yescharacteristic
445C2#1184311yescharacteristic
446C2#1185411yescharacteristic
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. #7123C2semidirect
5. #7123C2semidirect
48. C4×C22⋊C4C8semidirect
143. #95C2×C8semidirect
144. #95C2×C8semidirect

Maximal subgroups

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

subgroupindex
#139652
#86982
C2×#12802
#71232