Finite Group Databasebeta

← #110212

Subgroups — #110212

Order 512.

595 conjugacy classes of subgroups.

The subgroup lattice has 359 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21837C211yesmaximal, characteristic
2#17501C211yesmaximal, characteristic
3#17479C211yesmaximal, characteristic
4#17314C211yesmaximal, characteristic
5#8006C211yesmaximal, characteristic
6#7973C211yesmaximal, characteristic
7#7857C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×C4×M16C2211yescharacteristic
14#2283C2211yescharacteristic
15#1991—22no—
16#1988C412yes—
17#1987C412yes—
18#1825—22no—
19#1497C414yescomplemented
20#1493—22no—
21#1493—22no—
22#1491—22no—
23#1485C414yescomplemented
24#1479—22no—
25#1450—22no—
26C8×M16—22no—
27C8×M16—22no—
28#1293—22no—
29#1288—22no—
30#1274—22no—
31C22×M16—21no—
32#533—21no—
33#533—21no—
34#533—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533C2×C412yes—
39#533C2×C412yes—
40C2×C4⋊Q8C2×C411yescharacteristic
41C2×C4⋊Q8C2×C411yescharacteristic
42C2×#125—21no—
43C2×C22⋊Q8—21no—
44C2×C22⋊Q8—21no—
45C2×C4×Q8—21no—
46C2×C4×Q8—21no—
47C2×#117—21no—
48C22×C42C2311yescharacteristic
49#432—21no—
50#432—21no—
51#432—21no—
52#432C2×C412yes—
53#431—21no—
54#431—21no—
55#431—21no—
56#431—21no—
57#431—21no—
58#431—21no—
59#431—21no—
60#431—21no—
61#431—21no—
62#431C2×C412yes—
63#431C2×C412yes—
64#431D412yes—
65#423—21no—
66#423—21no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8C2×C412yes—
69#407—21no—
70#407—21no—
71C2×C22⋊C8—21no—
72C2×C22⋊C8D412yes—
73C4×M16—21no—
74C4×M16—21no—
75C4×M16—21no—
76C4×M16—21no—
77C4×M16C2×C412yes—
78C4×M16D412yes—
79C2×#97—21no—
80C2×C4×C8D412yes—
81C2×C4×C8D412yes—
82C2×C4×C8D412yes—
83#383—21no—
84#382—21no—
85#381—21no—
86#380—21no—
87C4×C4⋊C4C2×C412yes—
88C4×C22⋊C4—21no—
89#376—21no—
90#340—24no—
91#340—24no—
92Q8⋊C8—28no—
93Q8⋊C8—28no—
94#322—24no—
95C82—24no—
96C22×Q8—21no—
97C22×Q8—21no—
98C23×C4C2×D411yescharacteristic
99C23×C4#4111yescharacteristic
100C23×C4#4111yescharacteristic
101C2×M16—28no—
102C2×M16—22no—
103C2×M16—22no—
104C2×M16—41no—
105C2×M16—41no—
106C22×C8—22no—
107C22×C8—22no—
108C22×C8C22⋊C414yes—
109C4⋊Q8—21no—
110C4⋊Q8—21no—
111C4⋊Q8—21no—
112C4⋊Q8—21no—
113C4⋊Q8—21no—
114C4⋊Q8—21no—
115C4⋊Q8C4212yes—
116C4⋊Q8C4212yes—
117#125—22no—
118#125—22no—
119#125—41no—
120#125—41no—
121C22⋊Q8—22no—
122C22⋊Q8—22no—
123C22⋊Q8—22no—
124C22⋊Q8—22no—
125C22⋊Q8—41no—
126C22⋊Q8—41no—
127C22⋊Q8—41no—
128C22⋊Q8—41no—
129C4×Q8—22no—
130C4×Q8—22no—
131C4×Q8—22no—
132C4×Q8—22no—
133C4×Q8—41no—
134C4×Q8—41no—
135C4×Q8—41no—
136C4×Q8—41no—
137#117—22no—
138#117—22no—
139#117—41no—
140#117—41no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—41no—
147C2×C4⋊C4—41no—
148C2×C22⋊C4—41no—
149C2×C22⋊C4—41no—
150C2×C42—22no—
151C2×C42—41no—
152C2×C42C22×C411yescharacteristic
153C2×C42C22×C411yescharacteristic
154C2×C42C22×C411yescharacteristic
155C2×C42C2×D411yescharacteristic
156C2×C42C2×D411yescharacteristic
157C2×C42C2×D411yescharacteristic
158C2×C42C2×D411yescharacteristic
159C2×C42C2×D411yescharacteristic
160C2×C42#4111yescharacteristic
161C2×C42#4111yescharacteristic
162C2×C42#4111yescharacteristic
163C2×C42#4111yescharacteristic
164C4⋊C8—22no—
165C4⋊C8—22no—
166C4⋊C8—22no—
167C4⋊C8—22no—
168C4⋊C8—22no—
169C4⋊C8—22no—
170C4⋊C8—41no—
171C4⋊C8—41no—
172C22⋊C8—22no—
173C22⋊C8—22no—
174C22⋊C8—22no—
175C22⋊C8—22no—
176C22⋊C8—22no—
177C22⋊C8—22no—
178C22⋊C8—41no—
179C22⋊C8—41no—
180#97—22no—
181#97—22no—
182#97—22no—
183#97—41no—
184C4×C8—22no—
185C4×C8—22no—
186C4×C8—22no—
187C4×C8—22no—
188C4×C8—22no—
189C4×C8—41no—
190C4×C8C22⋊C414yes—
191C4×C8C22⋊C414yes—
192#95—22no—
193#95—41no—
194#95—41no—
195#95—41no—
196C24#12411yescharacteristic
197C2×Q8—22no—
198C2×Q8—22no—
199C2×Q8—41no—
200C2×Q8—41no—
201C2×Q8—44no—
202C2×Q8—82no—
203C22×C4—24no—
204C22×C4—42no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—42no—
208C22×C4—81no—
209C22×C4C2×C22⋊C411yescharacteristic
210C22×C4#11711yescharacteristic
211C22×C4#11711yescharacteristic
212C22×C4C4×D411yescharacteristic
213C22×C4C4×D411yescharacteristic
214C22×C4C4×D411yescharacteristic
215C22×C4C4×D411yescharacteristic
216C22×C4C4×D411yescharacteristic
217C22×C4C4×D411yescharacteristic
218C22×C4#12111yescharacteristic
219C22×C4#12111yescharacteristic
220C22×C4#12111yescharacteristic
221C22×C4#12111yescharacteristic
222C22×C4#12311yescharacteristic
223C22×C4#12311yescharacteristic
224C22×C4#12611yescharacteristic
225C22×C4#12611yescharacteristic
226C22×C4#12711yescharacteristic
227M16—28no—
228M16—28no—
229M16—82no—
230M16—82no—
231C2×C8—28no—
232C2×C8—24no—
233C2×C8—24no—
234C2×C8—24no—
235C2×C8—44no—
236C2×C8—42no—
237C2×C8—42no—
238C2×C8—82no—
239C4⋊C4—22no—
240C4⋊C4—22no—
241C4⋊C4—22no—
242C4⋊C4—22no—
243C4⋊C4—44no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C22⋊C4—42no—
253C22⋊C4—42no—
254C22⋊C4—82no—
255C42—42no—
256C42—42no—
257C42—44no—
258C42C2×C4211yescharacteristic
259C42C2×C22⋊C411yescharacteristic
260C42C2×C22⋊C411yescharacteristic
261C42#11711yescharacteristic
262C42C4×D411yescharacteristic
263C42C4×D411yescharacteristic
264C42C4×D411yescharacteristic
265C42C4×D411yescharacteristic
266C42C4×D411yescharacteristic
267C42C4×D411yescharacteristic
268C42#12011yescharacteristic
269C42#12111yescharacteristic
270C42#12111yescharacteristic
271C42#12311yescharacteristic
272C42#12411yescharacteristic
273C42#12411yescharacteristic
274C23—22no—
275C23—41no—
276C23—41no—
277C23#38811yescharacteristic
278C23#38811yescharacteristic
279C23#39011yescharacteristic
280Q8—44no—
281Q8—44no—
282Q8—82no—
283Q8—82no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
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—42no—
302C2×C4—42no—
303C2×C4—82no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4C4×C22⋊C411yescharacteristic
309C2×C4C4×C22⋊C411yescharacteristic
310C2×C4C4×C22⋊C411yescharacteristic
311C2×C4#38611yescharacteristic
312C2×C4#38611yescharacteristic
313C2×C4#38611yescharacteristic
314C2×C4#38811yescharacteristic
315C2×C4#38811yescharacteristic
316C2×C4#38811yescharacteristic
317C2×C4#38811yescharacteristic
318C2×C4#39011yescharacteristic
319C2×C4#39011yescharacteristic
320C2×C4#44314yes—
321C8—28no—
322C8—44no—
323C8—44no—
324C8—82no—
325C8—82no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—82no—
335C22#143711yescharacteristic
336C22#197412yes—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—82no—
347C4—82no—
348C4—162no—
349C4#176412yes—
350C4#190912yes—
351C4#193712yes—
352C2—22no—
353C2—41no—
354C2—41no—
355C2#1092911yescharacteristic
356C2#1094711yescharacteristic
357C2#1095011yescharacteristic
subgroupG/H
center, socleC22#1437
commutatorC42C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24#124
LCSG ▷ C42 ▷ 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
19. #1497C4semidirect
23. #1485C4semidirect

Maximal subgroups

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

subgroupindex
#218372
#175012
#174792
#173142
#80062
#79732
#78572