Finite Group Databasebeta

← #99262

Subgroups — #99262

Order 512.

534 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 98 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#13948C211yesmaximal, characteristic
2#8559C211yesmaximal, characteristic
3#8559C211yesmaximal, characteristic
4#7039C212yesmaximal, complemented
5#7039C212yesmaximal, complemented
6C2×#388—21no—
7C2×C4×C22⋊C4C2211yescharacteristic
8#1797C2212yes—
9#1797C2212yes—
10C2×#323—21no—
11C2×#323—21no—
12#1442—21no—
13#1439—21no—
14#1436—21no—
15C4⋊#95C2211yescharacteristic
16C4⋊#95C2211yescharacteristic
17#1434C411yescharacteristic
18C22⋊#95C411yescharacteristic
19C22⋊#95C411yescharacteristic
20C4×#95C411yescharacteristic
21#1280—21no—
22#1280—21no—
23C24×C4C2×C411yescharacteristic
24C22×C4⋊C4—21no—
25C22×C4⋊C4—21no—
26C22×C22⋊C4—21no—
27C22×C22⋊C4—21no—
28C22×C22⋊C4D411yescharacteristic
29C22×C22⋊C4D411yescharacteristic
30C22×C42—21no—
31C22×C42C2×C411yescharacteristic
32C22×C42C2311yescharacteristic
33C2×C4⋊C8—22no—
34C2×C4⋊C8—22no—
35C2×C22⋊C8—22no—
36C2×C22⋊C8—22no—
37#388—42no—
38#388—42no—
39C4×C22⋊C4—22no—
40C4×C22⋊C4—22no—
41C2×#95—21no—
42C2×#95—21no—
43C2×#95—21no—
44C2×#95—21no—
45C2×#95—21no—
46C2×#95—21no—
47C2×#95—21no—
48C2×#95C2×C411yescharacteristic
49C2×#95C2×C411yescharacteristic
50C2×#95C2×C411yescharacteristic
51C2×#95C2×C411yescharacteristic
52C2×#95D411yescharacteristic
53C2×#95D411yescharacteristic
54#336—22no—
55#336—22no—
56#323—42no—
57#323—42no—
58C25C22⋊C411yescharacteristic
59C23×C4—21no—
60C23×C4—21no—
61C23×C4—21no—
62C23×C4—21no—
63C23×C4—21no—
64C23×C4—21no—
65C23×C4—41no—
66C23×C4—41no—
67C23×C4—81no—
68C23×C4—81no—
69C23×C4C22⋊C411yescharacteristic
70C23×C4C22⋊C411yescharacteristic
71C23×C4C22⋊C411yescharacteristic
72C23×C4C22×C411yescharacteristic
73C23×C4C2×D411yescharacteristic
74C23×C4C2×D411yescharacteristic
75C22×C8—42no—
76C22×C8—42no—
77C2×C4⋊C4—42no—
78C2×C4⋊C4—42no—
79C2×C4⋊C4—41no—
80C2×C4⋊C4—41no—
81C2×C4⋊C4—41no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—81no—
84C2×C4⋊C4—81no—
85C2×C22⋊C4—22no—
86C2×C22⋊C4—22no—
87C2×C22⋊C4—22no—
88C2×C22⋊C4—22no—
89C2×C22⋊C4—21no—
90C2×C22⋊C4—21no—
91C2×C22⋊C4—42no—
92C2×C22⋊C4—42no—
93C2×C22⋊C4—42no—
94C2×C22⋊C4—42no—
95C2×C22⋊C4—81no—
96C2×C22⋊C4—81no—
97C2×C22⋊C4D812yes—
98C2×C22⋊C4SD1612yes—
99C2×C42—21no—
100C2×C42—21no—
101C2×C42—42no—
102C2×C42—42no—
103C2×C42—81no—
104C2×C42—81no—
105C2×C42—81no—
106C2×C42#4112yes—
107C2×C42#4112yes—
108C4⋊C8—82no—
109C4⋊C8—82no—
110C22⋊C8—42no—
111C22⋊C8—42no—
112C22⋊C8—42no—
113C22⋊C8—42no—
114#95—22no—
115#95—22no—
116#95—41no—
117#95—41no—
118#95—41no—
119#95—41no—
120#95—42no—
121#95—42no—
122#95—41no—
123#95—41no—
124#95—42no—
125#95—81no—
126#95—81no—
127#95—81no—
128C24—21no—
129C24—21no—
130C24—21no—
131C24—21no—
132C24—21no—
133C24—21no—
134C24—41no—
135C24—41no—
136C24—41no—
137C24—41no—
138C24#10212yes—
139C24C2×C22⋊C411yescharacteristic
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—41no—
153C22×C4—41no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—41no—
157C22×C4—41no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—41no—
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—81no—
193C22×C4—81no—
194C22×C4—81no—
195C22×C4—81no—
196C22×C4—84no—
197C22×C4—84no—
198C22×C4#10212yes—
199C22×C4#11711yescharacteristic
200C22×C4#11711yescharacteristic
201C22×C4#12312yes—
202C22×C4#12312yes—
203C22×C4C2×D811yescharacteristic
204C22×C4C2×SD1611yescharacteristic
205C22×C4#13512yes—
206C22×C4AGL(1,ℤ/8)11yescharacteristic
207C22×C4#13712yes—
208C22×C4#13711yescharacteristic
209C2×C8—82no—
210C2×C8—82no—
211C2×C8—82no—
212C2×C8—82no—
213C2×C8—82no—
214C2×C8—82no—
215C4⋊C4—82no—
216C4⋊C4—82no—
217C4⋊C4—82no—
218C4⋊C4—82no—
219C22⋊C4—24no—
220C22⋊C4—24no—
221C22⋊C4—42no—
222C22⋊C4—42no—
223C22⋊C4—84no—
224C22⋊C4—84no—
225C42—42no—
226C42—42no—
227C42—81no—
228C42—81no—
229C42—84no—
230C23—21no—
231C23—21no—
232C23—21no—
233C23—21no—
234C23—22no—
235C23—22no—
236C23—22no—
237C23—42no—
238C23—42no—
239C23—42no—
240C23—42no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23—81no—
250C23—81no—
251C23—81no—
252C23—81no—
253C23—81no—
254C23—81no—
255C23—81no—
256C23—81no—
257C23#38111yescharacteristic
258C23#41311yescharacteristic
259C23#41311yescharacteristic
260C23C2×#10211yescharacteristic
261C23#41611yescharacteristic
262C23#41711yescharacteristic
263C23#41911yescharacteristic
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—84no—
300C2×C4—84no—
301C2×C4—81no—
302C2×C4—81no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—81no—
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#48012yes—
328C2×C4#48212yes—
329C2×C4#48312yes—
330C2×C4#48512yes—
331C8—162no—
332C8—162no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—42no—
346C22—42no—
347C22—42no—
348C22—82no—
349C22—82no—
350C22—82no—
351C22—82no—
352C22—81no—
353C22—81no—
354C22—81no—
355C22—81no—
356C22#154011yescharacteristic
357C22#154711yescharacteristic
358C22#155411yescharacteristic
359C22#155911yescharacteristic
360C22#182211yescharacteristic
361C22#182311yescharacteristic
362C22#182511yescharacteristic
363C22#182711yescharacteristic
364C22#182711yescharacteristic
365C22#182811yescharacteristic
366C22#182911yescharacteristic
367C4—22no—
368C4—22no—
369C4—22no—
370C4—22no—
371C4—41no—
372C4—41no—
373C4—42no—
374C4—82no—
375C4—82no—
376C4—81no—
377C4—81no—
378C4—161no—
379C4—161no—
380C2—21no—
381C2—21no—
382C2—21no—
383C2—21no—
384C2—82no—
385C2#872611yescharacteristic
386C2#874411yescharacteristic
387C2#875911yescharacteristic
388C2#876011yescharacteristic
389C2#876211yescharacteristic
390C2#876411yescharacteristic
391C2#877111yescharacteristic
subgroupG/H
center, socleC23#381
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC25C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ 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. #7039C2semidirect
5. #7039C2semidirect

Maximal subgroups

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

subgroupindex
#139482
#85592
#70392