Finite Group Databasebeta

← #110431

Subgroups — #110431

Order 512.

634 conjugacy classes of subgroups.

The subgroup lattice has 387 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1991C211yesmaximal, characteristic
2C2×#1528C211yesmaximal, characteristic
3C2×#1470C211yesmaximal, characteristic
4C2×#1451C211yesmaximal, characteristic
5#8033C211yesmaximal, characteristic
6#7914C211yesmaximal, characteristic
7#7875C211yesmaximal, characteristic
8C22×#125C2211yescharacteristic
9C2×#432C2211yescharacteristic
10C2×#432C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×C4×M16C2211yescharacteristic
13C22×C4×C8C2211yescharacteristic
14C2×#382C2211yescharacteristic
15#1991—21no—
16#1991—21no—
17#1991—22no—
18C2×#340C412yes—
19C2×#340C412yes—
20#1528—22no—
21#1528C414yescomplemented
22C2×#330—22no—
23#1470—22no—
24#1470C414yescomplemented
25C2×#324—22no—
26#1451—22no—
27#1451—22no—
28C2×#322—21no—
29C2×#322—21no—
30#1295—22no—
31#1280—22no—
32#1277—22no—
33C22×M16—21no—
34C23×C8C2×C412yes—
35C2×#125—21no—
36C2×#125—21no—
37C2×#125—21no—
38C2×#125—21no—
39C2×#125—21no—
40C2×#125—21no—
41C2×#125—21no—
42C2×#125—21no—
43C2×#125—21no—
44C2×#125—21no—
45C2×#125C2×C412yes—
46C2×#125C2×C412yes—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4C2×C411yescharacteristic
50C22×C4⋊C4C2×C411yescharacteristic
51C22×C4⋊C4C2×C412yes—
52C22×C42C2311yescharacteristic
53#432—21no—
54#432—21no—
55#432—21no—
56#432—21no—
57#432—21no—
58#432—21no—
59#432C2×C412yes—
60#432C2×C412yes—
61#431—21no—
62#431—21no—
63#431C2×C412yes—
64#431D412yes—
65C2×C4⋊C8—21no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8—21no—
68C2×C22⋊C8—21no—
69C2×C22⋊C8—21no—
70C2×C22⋊C8D412yes—
71C4×M16—21no—
72C4×M16—21no—
73C4×M16C2×C412yes—
74C4×M16D412yes—
75C2×#97—21no—
76C2×#97—21no—
77C2×#97D412yes—
78C2×C4×C8—21no—
79C2×C4×C8—21no—
80C2×C4×C8—21no—
81C2×C4×C8—21no—
82C2×C4×C8—21no—
83C2×C4×C8—21no—
84C2×C4×C8D411yescharacteristic
85C2×C4×C8D411yescharacteristic
86C2×C4×C8D412yes—
87#382—21no—
88#382—21no—
89#382—21no—
90#382—21no—
91C2×#95—21no—
92C2×#95—21no—
93#340—22no—
94#340—22no—
95#340—22no—
96#340—22no—
97#330—28no—
98#324—28no—
99#322—24no—
100#322—24no—
101C23×C4—21no—
102C23×C4—21no—
103C23×C4—22no—
104C23×C4C22×C411yescharacteristic
105C23×C4C2×D411yescharacteristic
106C23×C4#4111yescharacteristic
107C2×M16—22no—
108C2×M16—22no—
109C2×M16—44no—
110C22×C8—22no—
111C22×C8—22no—
112C22×C8—22no—
113C22×C8—22no—
114C22×C8—24no—
115C22×C8—42no—
116C22×C8—41no—
117C22×C8—41no—
118C22×C8C22⋊C412yes—
119C22×C8C22⋊C412yes—
120#125—22no—
121#125—22no—
122#125—22no—
123#125—22no—
124#125—22no—
125#125—22no—
126#125—22no—
127#125—22no—
128#125—41no—
129#125—41no—
130#125—41no—
131#125—41no—
132#125—41no—
133#125—41no—
134#125—41no—
135#125—41no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—44no—
161C2×C4⋊C4C4212yes—
162C2×C4⋊C4C4212yes—
163C2×C42C22×C411yescharacteristic
164C2×C42C22×C411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C42C2×D411yescharacteristic
168C2×C42C2×D411yescharacteristic
169C2×C42C2×D411yescharacteristic
170C2×C42#4111yescharacteristic
171C2×C42#4111yescharacteristic
172C2×C42#4111yescharacteristic
173C2×C42#4111yescharacteristic
174C2×C42#4111yescharacteristic
175C4⋊C8—22no—
176C4⋊C8—22no—
177C4⋊C8—22no—
178C4⋊C8—22no—
179C4⋊C8—41no—
180C4⋊C8—41no—
181C22⋊C8—22no—
182C22⋊C8—22no—
183C22⋊C8—22no—
184C22⋊C8—41no—
185C22⋊C8—41no—
186C22⋊C8C22⋊C414yes—
187#97—22no—
188#97—22no—
189#97—22no—
190#97—41no—
191#97—41no—
192#97C22⋊C414yes—
193C4×C8—22no—
194C4×C8—22no—
195C4×C8—21no—
196C4×C8—21no—
197C4×C8—22no—
198C4×C8—22no—
199C4×C8—22no—
200C4×C8—22no—
201C4×C8—21no—
202C4×C8—21no—
203#95—41no—
204#95—41no—
205#95—41no—
206#95—41no—
207C24C4×D411yescharacteristic
208C22×C4—24no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—42no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—44no—
219C22×C4—42no—
220C22×C4—44no—
221C22×C4—82no—
222C22×C4—82no—
223C22×C4—82no—
224C22×C4C2×C4211yescharacteristic
225C22×C4C2×C22⋊C411yescharacteristic
226C22×C4C2×C22⋊C411yescharacteristic
227C22×C4#11711yescharacteristic
228C22×C4#11711yescharacteristic
229C22×C4C4×D411yescharacteristic
230C22×C4C4×D411yescharacteristic
231C22×C4C4×D411yescharacteristic
232C22×C4C4×D411yescharacteristic
233C22×C4C4×D411yescharacteristic
234C22×C4#12111yescharacteristic
235C22×C4#12111yescharacteristic
236C22×C4#12111yescharacteristic
237C22×C4#12311yescharacteristic
238C22×C4#12411yescharacteristic
239C22×C4#12411yescharacteristic
240C22×C4#12611yescharacteristic
241C22×C4#12711yescharacteristic
242M16—44no—
243M16—44no—
244C2×C8—22no—
245C2×C8—22no—
246C2×C8—22no—
247C2×C8—24no—
248C2×C8—24no—
249C2×C8—22no—
250C2×C8—22no—
251C2×C8—44no—
252C2×C8—44no—
253C2×C8—44no—
254C2×C8—44no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—82no—
258C2×C8—82no—
259C2×C8#10114yes—
260C4⋊C4—22no—
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—44no—
277C4⋊C4—82no—
278C42C2×C22⋊C411yescharacteristic
279C42#11711yescharacteristic
280C42C4×D411yescharacteristic
281C42C4×D411yescharacteristic
282C42C4×D411yescharacteristic
283C42C4×D411yescharacteristic
284C42C4×D411yescharacteristic
285C42C4×D411yescharacteristic
286C42#12011yescharacteristic
287C42#12111yescharacteristic
288C42#12111yescharacteristic
289C42#12111yescharacteristic
290C42#12311yescharacteristic
291C42#12311yescharacteristic
292C42#12411yescharacteristic
293C42#12611yescharacteristic
294C23—21no—
295C23—21no—
296C23—22no—
297C23C4×C22⋊C411yescharacteristic
298C23#38811yescharacteristic
299C23#39011yescharacteristic
300C23#44414yes—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—44no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4—164no—
331C2×C4C4×C22⋊C411yescharacteristic
332C2×C4C4×C22⋊C411yescharacteristic
333C2×C4#38611yescharacteristic
334C2×C4#38611yescharacteristic
335C2×C4#38611yescharacteristic
336C2×C4#38811yescharacteristic
337C2×C4#38811yescharacteristic
338C2×C4#38811yescharacteristic
339C2×C4#38811yescharacteristic
340C2×C4#38811yescharacteristic
341C2×C4#39011yescharacteristic
342C2×C4#39011yescharacteristic
343C2×C4C2×#10112yes—
344C2×C4#41312yes—
345C8—24no—
346C8—44no—
347C8—44no—
348C8—42no—
349C8—82no—
350C8—82no—
351C22—24no—
352C22—24no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22#143711yescharacteristic
358C22#176512yes—
359C22#193812yes—
360C22#196611yescharacteristic
361C22#196711yescharacteristic
362C22#197512yes—
363C22#197612yes—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—82no—
375C4—82no—
376C4—162no—
377C4C4×#10112yes—
378C4#190112yes—
379C2—22no—
380C2—22no—
381C2#979012yes—
382C2#979212yes—
383C2#1093011yescharacteristic
384C2#1094411yescharacteristic
385C2#1096311yescharacteristic
subgroupG/H
center, socleC23#390
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24C4×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
21. #1528C4semidirect
24. #1470C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19912
C2×#15282
C2×#14702
C2×#14512
#80332
#79142
#78752