Finite Group Databasebeta

← #120808

Subgroups — #120808

Order 512.

618 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21949C211yesmaximal, characteristic
2#8112C211yesmaximal, characteristic
3#8108C211yesmaximal, characteristic
4#8020C211yesmaximal, characteristic
5#8020C211yesmaximal, characteristic
6#7946C211yesmaximal, characteristic
7#7909C211yesmaximal, characteristic
8C2×#407C2211yescharacteristic
9C2×#407C2211yescharacteristic
10C2×#407C2211yescharacteristic
11C2×#407C2211yescharacteristic
12#2280C2211yescharacteristic
13#2280C2211yescharacteristic
14C22×#95C2211yescharacteristic
15#1855—22no—
16#1855—22no—
17#1811—22no—
18#1306—22no—
19#1305C414yescomplemented
20#1294—22no—
21#1294—22no—
22#1284—22no—
23#1280—22no—
24C24×C4—21no—
25C24×C4C2311yescharacteristic
26C22×M16—21no—
27C22×M16—21no—
28C22×M16—21no—
29C22×M16—21no—
30C2×#117—21no—
31C2×#117Q812yes—
32C2×#108—24no—
33#407—21no—
34#407—21no—
35#407—21no—
36#407—21no—
37#407—21no—
38#407—21no—
39#407—21no—
40#407—21no—
41#407—21no—
42#407—21no—
43#407—21no—
44#407C2×C412yes—
45#407C2×C412yes—
46#407D412yes—
47#407D412yes—
48#407Q812yes—
49C2×C22⋊C8—21no—
50C2×C22⋊C8—21no—
51C2×C22⋊C8—21no—
52C2×C22⋊C8—21no—
53C2×C22⋊C8—21no—
54C2×C22⋊C8—21no—
55C2×C22⋊C8—21no—
56C2×C22⋊C8D412yes—
57#381—21no—
58#381—21no—
59#380—21no—
60#380—21no—
61#380—21no—
62#380C2×C412yes—
63#379—21no—
64#379—21no—
65C4×C22⋊C4—21no—
66C4×C22⋊C4—21no—
67C4×C22⋊C4—21no—
68C4×C22⋊C4—21no—
69C2×#95—21no—
70C2×#95—21no—
71C2×#95—21no—
72C2×#95—21no—
73C2×#95—21no—
74C2×#95D412yes—
75#341—24no—
76C25#4111yescharacteristic
77C23×C4—22no—
78C23×C4—44no—
79C23×C4—41no—
80C23×C4—41no—
81C23×C4—82no—
82C23×C4—82no—
83C23×C4—82no—
84C23×C4C22×C411yescharacteristic
85C23×C4C2×D411yescharacteristic
86C23×C4C2×D411yescharacteristic
87C23×C4C2×D411yescharacteristic
88C23×C4C2×D411yescharacteristic
89C23×C4C2×Q811yescharacteristic
90C23×C4C2×Q811yescharacteristic
91C23×C4#4111yescharacteristic
92C23×C4#4111yescharacteristic
93C23×C4#4111yescharacteristic
94C23×C4#4111yescharacteristic
95C23×C4#4111yescharacteristic
96C23×C4#4111yescharacteristic
97C23×C4#4111yescharacteristic
98C2×M16—22no—
99C2×M16—22no—
100C2×M16—22no—
101C2×M16—22no—
102C2×M16—22no—
103C2×M16—41no—
104C2×M16—41no—
105C2×M16—41no—
106C2×M16—82no—
107C2×M16—82no—
108C2×M16—82no—
109C2×M16—82no—
110C22×C8—22no—
111C22×C8—41no—
112C22×C8—41no—
113C22×C8—41no—
114#117—82no—
115#117—82no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C22⋊C4—22no—
119C2×C22⋊C4—22no—
120C2×C22⋊C4—41no—
121C2×C22⋊C4—41no—
122C2×C22⋊C4—41no—
123C2×C22⋊C4—41no—
124C2×C42—22no—
125C2×C42—41no—
126#108—84no—
127C22⋊C8—22no—
128C22⋊C8—22no—
129C22⋊C8—22no—
130C22⋊C8—22no—
131C22⋊C8—22no—
132C22⋊C8—22no—
133C22⋊C8—22no—
134C22⋊C8—22no—
135C22⋊C8—22no—
136C22⋊C8—41no—
137C22⋊C8—41no—
138C22⋊C8—41no—
139C22⋊C8—41no—
140C22⋊C8—41no—
141C22⋊C8—41no—
142C22⋊C8C4⋊C414yes—
143#95—22no—
144#95—22no—
145#95—22no—
146#95—41no—
147#95—41no—
148#95—41no—
149#95—41no—
150#95—41no—
151C24—42no—
152C24—41no—
153C24—42no—
154C24—41no—
155C24#11711yescharacteristic
156C24C22⋊Q811yescharacteristic
157C24#12311yescharacteristic
158C24#12411yescharacteristic
159C24#12411yescharacteristic
160C24#12511yescharacteristic
161C24#12611yescharacteristic
162C22×C4—24no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—44no—
171C22×C4—44no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—81no—
176C22×C4—81no—
177C22×C4—81no—
178C22×C4—81no—
179C22×C4—81no—
180C22×C4—81no—
181C22×C4—81no—
182C22×C4—81no—
183C22×C4—82no—
184C22×C4—82no—
185C22×C4—82no—
186C22×C4—82no—
187C22×C4—816no—
188C22×C4—82no—
189C22×C4—82no—
190C22×C4C2×C4⋊C411yescharacteristic
191C22×C4#11711yescharacteristic
192C22×C4C4×D411yescharacteristic
193C22×C4C4×D411yescharacteristic
194C22×C4C4×D411yescharacteristic
195C22×C4C4×Q811yescharacteristic
196C22×C4#12011yescharacteristic
197C22×C4#12111yescharacteristic
198C22×C4#12111yescharacteristic
199C22×C4#12111yescharacteristic
200C22×C4C22⋊Q811yescharacteristic
201C22×C4C22⋊Q811yescharacteristic
202C22×C4C22⋊Q811yescharacteristic
203C22×C4C22⋊Q811yescharacteristic
204C22×C4#12311yescharacteristic
205C22×C4#12311yescharacteristic
206C22×C4#12311yescharacteristic
207C22×C4#12311yescharacteristic
208C22×C4#12411yescharacteristic
209C22×C4#12411yescharacteristic
210C22×C4#12511yescharacteristic
211C22×C4#12511yescharacteristic
212C22×C4#12511yescharacteristic
213C22×C4#12511yescharacteristic
214C22×C4#12611yescharacteristic
215C22×C4#12611yescharacteristic
216C22×C4#12611yescharacteristic
217C22×C4C4⋊Q811yescharacteristic
218M16—44no—
219M16—44no—
220M16—82no—
221M16—82no—
222M16—82no—
223M16—82no—
224M16—82no—
225M16—82no—
226C2×C8—24no—
227C2×C8—44no—
228C2×C8—42no—
229C2×C8—42no—
230C2×C8—42no—
231C2×C8—82no—
232C2×C8—82no—
233C2×C8—82no—
234C4⋊C4—82no—
235C4⋊C4—82no—
236C22⋊C4—81no—
237C22⋊C4—81no—
238C22⋊C4—81no—
239C22⋊C4—81no—
240C22⋊C4—82no—
241C22⋊C4—82no—
242C22⋊C4—82no—
243C22⋊C4—82no—
244C42—82no—
245C42—82no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—22no—
253C23—22no—
254C23—22no—
255C23—22no—
256C23—22no—
257C23—22no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23—48no—
271C23—48no—
272C23#38211yescharacteristic
273C23#38711yescharacteristic
274C23#38811yescharacteristic
275C23#39311yescharacteristic
276C23#39711yescharacteristic
277C23#39811yescharacteristic
278C23#40011yescharacteristic
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—44no—
308C2×C4—82no—
309C2×C4—84no—
310C2×C4—88no—
311C2×C4—88no—
312C2×C4—82no—
313C2×C4—84no—
314C2×C4—82no—
315C2×C4—82no—
316C2×C4—82no—
317C2×C4—82no—
318C2×C4—82no—
319C2×C4—82no—
320C2×C4#38511yescharacteristic
321C2×C4#38711yescharacteristic
322C2×C4#38811yescharacteristic
323C2×C4#38911yescharacteristic
324C2×C4#39411yescharacteristic
325C2×C4#39811yescharacteristic
326C2×C4#40011yescharacteristic
327C2×C4#40011yescharacteristic
328C8—44no—
329C8—82no—
330C8—82no—
331C8—82no—
332C22—21no—
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—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—42no—
351C22—42no—
352C22—44no—
353C22—42no—
354C22—42no—
355C22—42no—
356C22—44no—
357C22—44no—
358C22—42no—
359C22—44no—
360C22#144211yescharacteristic
361C22#193212yes—
362C22#203212yes—
363C22#208111yescharacteristic
364C22#208211yescharacteristic
365C4—21no—
366C4—21no—
367C4—21no—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—84no—
374C4—162no—
375C4—162no—
376C2—21no—
377C2—22no—
378C2—22no—
379C2—21no—
380C2—41no—
381C2—41no—
382C2—41no—
383C2—41no—
384C2#1127911yescharacteristic
385C2#1128311yescharacteristic
386C2#1128911yescharacteristic
subgroupG/H
center, socleC22#1442
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25#41
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
19. #1305C4semidirect

Maximal subgroups

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

subgroupindex
#219492
#81122
#81082
#80202
#79462
#79092