Finite Group Databasebeta

← #99167

Subgroups — #99167

Order 512.

690 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 124 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#13802C211yesmaximal, characteristic
2#8557C211yesmaximal, characteristic
3C4×#323C211yesmaximal, characteristic
4#7028C212yesmaximal, complemented
5#7024C212yesmaximal, complemented
6C2×#380C411yescharacteristic
7C2×C4×C22⋊C4—21no—
8C2×C4×C22⋊C4C2211yescharacteristic
9C22×#95C411yescharacteristic
10#1797C2212yes—
11C4×C22⋊C8C2212yes—
12C2×#323—21no—
13C2×#323—21no—
14#1436—21no—
15#1436—21no—
16#1434C411yescharacteristic
17#1434C2211yescharacteristic
18C22⋊#95—21no—
19C4×#95C411yescharacteristic
20C4×#95C2211yescharacteristic
21#1280—21no—
22#1280—21no—
23C24×C4—21no—
24C24×C4C2×C411yescharacteristic
25C22×C4⋊C4—21no—
26C22×C22⋊C4—21no—
27C22×C22⋊C4D411yescharacteristic
28C22×C22⋊C4D411yescharacteristic
29C22×C42—21no—
30C22×C42—21no—
31C22×C42C2×C411yescharacteristic
32C22×C42C2311yescharacteristic
33C2×C4⋊C8—22no—
34C2×C22⋊C8—22no—
35C2×C22⋊C8—22no—
36C2×C4×C8—22no—
37#380—22no—
38#380C814yescomplemented
39C4×C22⋊C4—22no—
40C4×C22⋊C4—22no—
41C4×C22⋊C4—44no—
42C2×#95—21no—
43C2×#95—22no—
44C2×#95—22no—
45C2×#95—21no—
46C2×#95—21no—
47C2×#95—21no—
48C2×#95—21no—
49C2×#95—21no—
50C2×#95—21no—
51C2×#95C2×C411yescharacteristic
52C2×#95C2×C411yescharacteristic
53C2×#95C2×C411yescharacteristic
54C2×#95C2×C411yescharacteristic
55C2×#95D411yescharacteristic
56C2×#95D411yescharacteristic
57#336—22no—
58#336—22no—
59#323—24no—
60#323—42no—
61C25C22⋊C411yescharacteristic
62C23×C4—21no—
63C23×C4—21no—
64C23×C4—21no—
65C23×C4—21no—
66C23×C4—21no—
67C23×C4—21no—
68C23×C4—21no—
69C23×C4—21no—
70C23×C4—24no—
71C23×C4—42no—
72C23×C4—42no—
73C23×C4—42no—
74C23×C4—42no—
75C23×C4—81no—
76C23×C4—81no—
77C23×C4—81no—
78C23×C4—81no—
79C23×C4C22⋊C411yescharacteristic
80C23×C4C22⋊C411yescharacteristic
81C23×C4C22⋊C411yescharacteristic
82C23×C4C2×C812yes—
83C23×C4M1612yes—
84C23×C4C22×C411yescharacteristic
85C23×C4C2×D411yescharacteristic
86C23×C4C2×D411yescharacteristic
87C22×C8—42no—
88C22×C8—42no—
89C2×C4⋊C4—24no—
90C2×C4⋊C4—81no—
91C2×C4⋊C4—81no—
92C2×C22⋊C4—22no—
93C2×C22⋊C4—22no—
94C2×C22⋊C4—24no—
95C2×C22⋊C4—22no—
96C2×C22⋊C4—22no—
97C2×C22⋊C4—41no—
98C2×C22⋊C4—42no—
99C2×C22⋊C4—42no—
100C2×C22⋊C4SD1612yes—
101C2×C22⋊C4Q1612yes—
102C2×C42—21no—
103C2×C42—21no—
104C2×C42—42no—
105C2×C42—42no—
106C2×C42—42no—
107C2×C42—42no—
108C2×C42—81no—
109C2×C42—81no—
110C2×C42—81no—
111C2×C42—81no—
112C2×C42#4112yes—
113C2×C42#4112yes—
114C4⋊C8—82no—
115C22⋊C8—24no—
116C22⋊C8—42no—
117C22⋊C8—42no—
118C22⋊C8—42no—
119C4×C8—82no—
120#95—24no—
121#95—22no—
122#95—22no—
123#95—21no—
124#95—21no—
125#95—21no—
126#95—21no—
127#95—42no—
128#95—42no—
129#95—42no—
130#95—42no—
131#95—81no—
132#95—81no—
133#95C2×C812yes—
134#95C2×C812yes—
135#95M1612yes—
136#95M1612yes—
137C24—21no—
138C24—21no—
139C24—21no—
140C24—21no—
141C24—41no—
142C24—41no—
143C24—41no—
144C24—41no—
145C24C22⋊C812yes—
146C24#10312yes—
147C24#10412yes—
148C24C2×C22⋊C411yescharacteristic
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—22no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—21no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—48no—
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—48no—
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—81no—
207C22×C4—81no—
208C22×C4—82no—
209C22×C4—82no—
210C22×C4—84no—
211C22×C4—82no—
212C22×C4—82no—
213C22×C4—84no—
214C22×C4—84no—
215C22×C4—84no—
216C22×C4—84no—
217C22×C4C22⋊C812yes—
218C22×C4#10312yes—
219C22×C4#11711yescharacteristic
220C22×C4#11711yescharacteristic
221C22×C4#12312yes—
222C22×C4#12312yes—
223C22×C4C22×C811yescharacteristic
224C22×C4C2×M1611yescharacteristic
225C22×C4C2×M1611yescharacteristic
226C22×C4C2×M1611yescharacteristic
227C22×C4C2×SD1611yescharacteristic
228C22×C4C2×Q1611yescharacteristic
229C22×C4AGL(1,ℤ/8)12yes—
230C2×C8—44no—
231C2×C8—82no—
232C2×C8—82no—
233C2×C8—82no—
234C2×C8—82no—
235C2×C8—82no—
236C4⋊C4—84no—
237C22⋊C4—24no—
238C22⋊C4—24no—
239C22⋊C4—44no—
240C22⋊C4—48no—
241C42—42no—
242C42—42no—
243C42—81no—
244C42—81no—
245C42—84no—
246C42—84no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—22no—
252C23—22no—
253C23—22no—
254C23—22no—
255C23—22no—
256C23—22no—
257C23—42no—
258C23—42no—
259C23—42no—
260C23—42no—
261C23—42no—
262C23—42no—
263C23—42no—
264C23—42no—
265C23—42no—
266C23—42no—
267C23—42no—
268C23—42no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23—41no—
273C23Q8⋊C814yes—
274C23#38111yescharacteristic
275C23C2×C22⋊C811yescharacteristic
276C23#40711yescharacteristic
277C23C2×#10311yescharacteristic
278C23#41811yescharacteristic
279C23C2×#10411yescharacteristic
280C23#42111yescharacteristic
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—24no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—42no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—816no—
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#43111yescharacteristic
348C2×C4#43111yescharacteristic
349C2×C4#43211yescharacteristic
350C2×C4#43211yescharacteristic
351C2×C4#48112yes—
352C2×C4#48412yes—
353C8—84no—
354C8—162no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—24no—
368C22—44no—
369C22—44no—
370C22—44no—
371C22—44no—
372C22—42no—
373C22—42no—
374C22—42no—
375C22—42no—
376C22—42no—
377C22—42no—
378C22C2×Q8⋊C811yescharacteristic
379C22#147811yescharacteristic
380C22#148711yescharacteristic
381C22#149011yescharacteristic
382C22#181711yescharacteristic
383C22#182411yescharacteristic
384C22#182611yescharacteristic
385C22#182711yescharacteristic
386C22#182711yescharacteristic
387C22#182811yescharacteristic
388C22#182911yescharacteristic
389C4—21no—
390C4—21no—
391C4—21no—
392C4—21no—
393C4—22no—
394C4—22no—
395C4—44no—
396C4—42no—
397C4—42no—
398C4—42no—
399C4—82no—
400C4—82no—
401C4—82no—
402C4—81no—
403C2—21no—
404C2—21no—
405C2—21no—
406C2—21no—
407C2—44no—
408C2#870711yescharacteristic
409C2#873111yescharacteristic
410C2#873411yescharacteristic
411C2#876111yescharacteristic
412C2#876511yescharacteristic
413C2#877211yescharacteristic
414C2#877611yescharacteristic
subgroupG/H
center, socleC23#381
commutatorC22×C4C22×C8
Frattini, cosocleC22×C42C23
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7028C2semidirect
5. #7024C2semidirect
38. #380C8semidirect

Maximal subgroups

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

subgroupindex
#138022
#85572
C4×#3232
#70282
#70242