Finite Group Databasebeta

← #110549

Subgroups — #110549

Order 512.

601 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#20469C211yesmaximal, characteristic
2#17748C211yesmaximal, characteristic
3C2×#1506C211yesmaximal, characteristic
4#17385C211yesmaximal, characteristic
5#8072C211yesmaximal, characteristic
6#7995C211yesmaximal, characteristic
7#7922C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#432C2211yescharacteristic
12C2×#432C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#376C2211yescharacteristic
15#1768—22no—
16#1749—21no—
17C4×C22⋊C8—21no—
18#1746—21no—
19#1726—21no—
20#1558C412yescomplemented
21#1556C412yescomplemented
22#1554—22no—
23#1550—22no—
24#1506—22no—
25#1506—21no—
26#1506—21no—
27C2×#328C414yes—
28#1472—21no—
29#1472—21no—
30#1470—22no—
31#1461C414yescomplemented
32#1300—22no—
33#1291—22no—
34#1281—22no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533—21no—
42#533—21no—
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×#125—21no—
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8C2×C412yes—
48C2×C4×Q8—21no—
49C2×C4×Q8C2×C412yes—
50C2×#117—21no—
51C22×C42—21no—
52C22×C42C2311yescharacteristic
53#432—21no—
54#432—21no—
55#432—21no—
56#432—21no—
57#432—21no—
58#432—21no—
59#432—21no—
60#432—21no—
61#432—21no—
62#432C2×C412yes—
63#432C2×C412yes—
64#432D412yes—
65#431—21no—
66#431—21no—
67#431—21no—
68#431—21no—
69#431—21no—
70#431C2×C412yes—
71#431C2×C412yes—
72#431D412yes—
73#423—21no—
74#423—21no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8D412yes—
78#407—21no—
79#407—21no—
80C2×C22⋊C8—21no—
81C2×C22⋊C8C2×C412yes—
82C2×C22⋊C8D411yescharacteristic
83C2×C22⋊C8D411yescharacteristic
84C4×M16—21no—
85C4×M16—21no—
86C2×#97—21no—
87C2×#97—22no—
88C2×#97—21no—
89C2×C4×C8—21no—
90C2×C4×C8—22no—
91#376—21no—
92#376—21no—
93#376D412yes—
94#376D411yescharacteristic
95#376D411yescharacteristic
96C2×#95—21no—
97C2×#95C2×C412yes—
98#336—22no—
99#336—22no—
100#333—24no—
101#332—24no—
102#328—24no—
103#328—24no—
104#324—24no—
105#324—24no—
106C22×Q8—21no—
107C22×Q8C4212yes—
108C23×C4—22no—
109C23×C4C22×C411yescharacteristic
110C23×C4C2×D411yescharacteristic
111C23×C4#4111yescharacteristic
112C2×M16—41no—
113C2×M16—41no—
114C22×C8—21no—
115C22×C8—22no—
116C22×C8—21no—
117C22×C8—22no—
118C4⋊Q8—21no—
119C4⋊Q8—21no—
120C4⋊Q8—21no—
121C4⋊Q8—21no—
122C4⋊Q8—21no—
123C4⋊Q8—21no—
124C4⋊Q8—21no—
125C4⋊Q8—21no—
126#125—41no—
127#125—41no—
128#125—41no—
129#125—41no—
130C22⋊Q8—22no—
131C22⋊Q8—22no—
132C22⋊Q8—22no—
133C22⋊Q8—22no—
134C22⋊Q8—41no—
135C22⋊Q8—41no—
136C22⋊Q8—41no—
137C22⋊Q8—41no—
138C4×Q8—22no—
139C4×Q8—22no—
140C4×Q8—22no—
141C4×Q8—22no—
142C4×Q8—41no—
143C4×Q8—41no—
144C4×Q8—41no—
145C4×Q8—41no—
146#117—41no—
147#117—41no—
148#117—41no—
149#117—41no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4C4212yes—
156C2×C22⋊C4—22no—
157C2×C42—21no—
158C2×C42—22no—
159C2×C42—21no—
160C2×C42—22no—
161C2×C42—44no—
162C2×C42C22×C411yescharacteristic
163C2×C42C22×C411yescharacteristic
164C2×C42C2×D411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C42C2×D411yescharacteristic
168C2×C42C2×D411yescharacteristic
169C2×C42#4111yescharacteristic
170C2×C42#4111yescharacteristic
171C2×C42#4111yescharacteristic
172C2×C42#4111yescharacteristic
173C2×C42#4111yescharacteristic
174C4⋊C8—22no—
175C4⋊C8—22no—
176C4⋊C8—22no—
177C4⋊C8—22no—
178C4⋊C8—22no—
179C4⋊C8—41no—
180C4⋊C8—41no—
181C4⋊C8—41no—
182C4⋊C8—41no—
183C4⋊C8C22⋊C414yes—
184C22⋊C8—22no—
185C22⋊C8—22no—
186C22⋊C8—21no—
187C22⋊C8—21no—
188C22⋊C8—22no—
189C22⋊C8—22no—
190C22⋊C8—22no—
191C22⋊C8—22no—
192C22⋊C8—41no—
193C22⋊C8—41no—
194C22⋊C8C22⋊C412yes—
195C22⋊C8C22⋊C412yes—
196#97—22no—
197#97—22no—
198#97—22no—
199#97—44no—
200#97—41no—
201#97—41no—
202#97—41no—
203C4×C8—22no—
204C4×C8—22no—
205C4×C8—22no—
206C4×C8—44no—
207C4×C8—41no—
208#95—22no—
209#95—22no—
210#95—21no—
211#95—21no—
212#95C22⋊C412yes—
213#95C22⋊C412yes—
214C24C4×D411yescharacteristic
215C2×Q8—22no—
216C2×Q8—22no—
217C2×Q8—44no—
218C2×Q8—41no—
219C2×Q8—41no—
220C2×Q8—82no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—44no—
229C22×C4—44no—
230C22×C4—82no—
231C22×C4C2×C4211yescharacteristic
232C22×C4C2×C22⋊C411yescharacteristic
233C22×C4C2×C22⋊C411yescharacteristic
234C22×C4#11711yescharacteristic
235C22×C4#11711yescharacteristic
236C22×C4C4×D411yescharacteristic
237C22×C4C4×D411yescharacteristic
238C22×C4C4×D411yescharacteristic
239C22×C4C4×D411yescharacteristic
240C22×C4C4×D411yescharacteristic
241C22×C4#12011yescharacteristic
242C22×C4#12111yescharacteristic
243C22×C4#12111yescharacteristic
244C22×C4#12111yescharacteristic
245C22×C4#12311yescharacteristic
246C22×C4#12311yescharacteristic
247C22×C4#12411yescharacteristic
248C22×C4#12611yescharacteristic
249M16—82no—
250M16—82no—
251C2×C8—22no—
252C2×C8—22no—
253C2×C8—24no—
254C2×C8—42no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—44no—
259C2×C8—42no—
260C2×C8—42no—
261C2×C8—42no—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—42no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—42no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C22⋊C4—42no—
275C22⋊C4—42no—
276C42—24no—
277C42—42no—
278C42—44no—
279C42—42no—
280C42—42no—
281C42C2×C22⋊C411yescharacteristic
282C42#11711yescharacteristic
283C42C4×D411yescharacteristic
284C42C4×D411yescharacteristic
285C42C4×D411yescharacteristic
286C42C4×D411yescharacteristic
287C42C4×D411yescharacteristic
288C42C4×D411yescharacteristic
289C42#12111yescharacteristic
290C42#12111yescharacteristic
291C42#12111yescharacteristic
292C42#12311yescharacteristic
293C42#12411yescharacteristic
294C42#12411yescharacteristic
295C42#12611yescharacteristic
296C42#12711yescharacteristic
297C23—22no—
298C23—22no—
299C23—41no—
300C23C4×C22⋊C411yescharacteristic
301C23#38611yescharacteristic
302C23#38811yescharacteristic
303Q8—82no—
304Q8—82no—
305Q8—82no—
306Q8—82no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—24no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—44no—
334C2×C4—88no—
335C2×C4—82no—
336C2×C4C4×C22⋊C411yescharacteristic
337C2×C4C4×C22⋊C411yescharacteristic
338C2×C4#38611yescharacteristic
339C2×C4#38611yescharacteristic
340C2×C4#38811yescharacteristic
341C2×C4#38811yescharacteristic
342C2×C4#38811yescharacteristic
343C2×C4#38811yescharacteristic
344C2×C4#38811yescharacteristic
345C2×C4#39011yescharacteristic
346C2×C4#39011yescharacteristic
347C2×C4#39011yescharacteristic
348C8—42no—
349C8—42no—
350C8—44no—
351C8—82no—
352C8—82no—
353C8—82no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—44no—
362C22#143711yescharacteristic
363C22#175712yes—
364C22#190212yes—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—44no—
377C4—82no—
378C4—82no—
379C4—82no—
380C4#190612yes—
381C2—22no—
382C2—22no—
383C2—41no—
384C2#1094511yescharacteristic
385C2#1095211yescharacteristic
386C2#1096111yescharacteristic
subgroupG/H
center, socleC22#1437
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24C4×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
20. #1558C4semidirect
21. #1556C4semidirect
31. #1461C4semidirect

Maximal subgroups

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

subgroupindex
#204692
#177482
C2×#15062
#173852
#80722
#79952
#79222