Finite Group Databasebeta

← #99158

Subgroups — #99158

Order 512.

576 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 86 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#14347C211yesmaximal, characteristic
2#8698C211yesmaximal, characteristic
3#8698C211yesmaximal, characteristic
4#7032C212yesmaximal
5#7032C212yesmaximal
6C2×#380C411yescharacteristic
7C2×#380C411yescharacteristic
8C22×#95C2211yescharacteristic
9#1779C2212yes—
10#1779C2212yes—
11#1442—21no—
12#1442—21no—
13#1442—21no—
14#1442—21no—
15C22⋊#95C411yescharacteristic
16C22⋊#95C411yescharacteristic
17C22⋊#95C2211yescharacteristic
18C22⋊#95C2211yescharacteristic
19#1280—21no—
20#1280—21no—
21#1280—21no—
22#1280—21no—
23C24×C4C2×C411yescharacteristic
24C24×C4C2×C411yescharacteristic
25C24×C4C2311yescharacteristic
26C22×C4⋊C4—21no—
27C22×C4⋊C4—21no—
28C22×C22⋊C4C2×C411yescharacteristic
29C22×C22⋊C4C2×C411yescharacteristic
30C22×C22⋊C4C2×C411yescharacteristic
31C22×C22⋊C4C2×C411yescharacteristic
32C2×C22⋊C8—22no—
33C2×C22⋊C8—22no—
34C2×C22⋊C8—22no—
35C2×C22⋊C8—22no—
36C2×C22⋊C8—22no—
37C2×C22⋊C8—22no—
38#380—42no—
39#380—42no—
40C2×#95—22no—
41C2×#95—22no—
42C2×#95—21no—
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×#95D411yescharacteristic
53C2×#95D411yescharacteristic
54C2×#95D411yescharacteristic
55C2×#95D411yescharacteristic
56C25C22×C411yescharacteristic
57C23×C4—21no—
58C23×C4—21no—
59C23×C4—22no—
60C23×C4—22no—
61C23×C4—21no—
62C23×C4—21no—
63C23×C4—22no—
64C23×C4—22no—
65C23×C4—21no—
66C23×C4—21no—
67C23×C4—21no—
68C23×C4—21no—
69C23×C4—41no—
70C23×C4—41no—
71C23×C4—81no—
72C23×C4—81no—
73C23×C4—81no—
74C23×C4—81no—
75C23×C4—81no—
76C23×C4—81no—
77C23×C4C22⋊C411yescharacteristic
78C23×C4C22⋊C411yescharacteristic
79C23×C4C22⋊C411yescharacteristic
80C23×C4C22⋊C411yescharacteristic
81C23×C4C2×D411yescharacteristic
82C23×C4C2×D411yescharacteristic
83C23×C4#4112yes—
84C23×C4#4112yes—
85C22×C8—42no—
86C22×C8—42no—
87C2×C4⋊C4—42no—
88C2×C4⋊C4—42no—
89C2×C4⋊C4—81no—
90C2×C4⋊C4—81no—
91C2×C4⋊C4—81no—
92C2×C4⋊C4—81no—
93C2×C22⋊C4—22no—
94C2×C22⋊C4—22no—
95C2×C22⋊C4—22no—
96C2×C22⋊C4—22no—
97C2×C22⋊C4—41no—
98C2×C22⋊C4—41no—
99C2×C22⋊C4—41no—
100C2×C22⋊C4—41no—
101C2×C22⋊C4—41no—
102C2×C22⋊C4—41no—
103C2×C22⋊C4—41no—
104C2×C22⋊C4—41no—
105C22⋊C8—42no—
106C22⋊C8—42no—
107C22⋊C8—42no—
108C22⋊C8—42no—
109C22⋊C8—82no—
110C22⋊C8—82no—
111C22⋊C8—82no—
112C22⋊C8—82no—
113#95—22no—
114#95—22no—
115#95—22no—
116#95—22no—
117#95—42no—
118#95—42no—
119#95—81no—
120#95—81no—
121#95—81no—
122#95—81no—
123#95—81no—
124#95—81no—
125#95—81no—
126#95—81no—
127C24—21no—
128C24—21no—
129C24—21no—
130C24—21no—
131C24—41no—
132C24—41no—
133C24—41no—
134C24—41no—
135C24C2×C22⋊C411yescharacteristic
136C24#11711yescharacteristic
137C24#11711yescharacteristic
138C24#13112yes—
139C24#13112yes—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—24no—
145C22×C4—24no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—41no—
157C22×C4—41no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—42no—
171C22×C4—42no—
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—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—81no—
205C22×C4—81no—
206C22×C4—81no—
207C22×C4—81no—
208C22×C4—81no—
209C22×C4—81no—
210C22×C4—81no—
211C22×C4—81no—
212C22×C4—84no—
213C22×C4—84no—
214C22×C4—84no—
215C22×C4—84no—
216C22×C4—84no—
217C22×C4—84no—
218C22×C4#10012yes—
219C22×C4#10112yes—
220C22×C4#12312yes—
221C22×C4#12312yes—
222C2×C8—82no—
223C2×C8—82no—
224C2×C8—82no—
225C2×C8—82no—
226C2×C8—82no—
227C2×C8—82no—
228C4⋊C4—84no—
229C4⋊C4—84no—
230C22⋊C4—82no—
231C22⋊C4—82no—
232C22⋊C4—82no—
233C22⋊C4—82no—
234C23—21no—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—21no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—22no—
251C23—22no—
252C23—22no—
253C23—22no—
254C23—42no—
255C23—42no—
256C23—42no—
257C23—42no—
258C23—42no—
259C23—42no—
260C23—42no—
261C23—42no—
262C23—42no—
263C23—42no—
264C23—42no—
265C23—42no—
266C23#38111yescharacteristic
267C23#40811yescharacteristic
268C23#40811yescharacteristic
269C23#41011yescharacteristic
270C23#41011yescharacteristic
271C23C2×#10011yescharacteristic
272C23C2×#10111yescharacteristic
273C23#43312yes—
274C23#43312yes—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—84no—
302C2×C4—84no—
303C2×C4—84no—
304C2×C4—84no—
305C2×C4—84no—
306C2×C4—84no—
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—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—
339C8—162no—
340C8—162no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—22no—
354C22—22no—
355C22—22no—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—44no—
360C22—44no—
361C22—44no—
362C22—44no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22#146911yescharacteristic
372C22#146911yescharacteristic
373C22#147011yescharacteristic
374C22#147011yescharacteristic
375C22#181811yescharacteristic
376C22#182011yescharacteristic
377C22#182111yescharacteristic
378C22#182811yescharacteristic
379C22#182811yescharacteristic
380C22#182911yescharacteristic
381C22#183011yescharacteristic
382C4—42no—
383C4—42no—
384C4—42no—
385C4—42no—
386C4—81no—
387C4—81no—
388C4—161no—
389C4—161no—
390C4—161no—
391C4—161no—
392C2—21no—
393C2—21no—
394C2—21no—
395C2—21no—
396C2—42no—
397C2—42no—
398C2#870311yescharacteristic
399C2#873611yescharacteristic
400C2#873711yescharacteristic
401C2#875011yescharacteristic
402C2#875511yescharacteristic
403C2#875611yescharacteristic
404C2#875711yescharacteristic
subgroupG/H
center, socleC23#381
commutator, prime radicalC25C22×C4
Frattini, cosocleC24×C4C23
LCSG ▷ C25 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C23 ▷ C1

Maximal subgroups

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

subgroupindex
#143472
#86982
#70322