Finite Group Databasebeta

← #118221

Subgroups — #118221

Order 512.

569 conjugacy classes of subgroups.

The subgroup lattice has 363 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21371C211yesmaximal, characteristic
2C2×#1531C211yesmaximal, characteristic
3#17590C211yesmaximal, characteristic
4#17501C211yesmaximal, characteristic
5#8029C211yesmaximal, characteristic
6#7978C211yesmaximal, characteristic
7#7919C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#432C2211yescharacteristic
12C2×C4×M16C2211yescharacteristic
13C22×#97C2211yescharacteristic
14#2283C2211yescharacteristic
15#1950—22no—
16#1855—22no—
17#1778—22no—
18#1531—22no—
19#1531C414yescomplemented
20C2×#330—22no—
21#1518—22no—
22#1511—22no—
23#1506—22no—
24#1493—22no—
25#1491—22no—
26#1485—22no—
27#1295—22no—
28#1288—22no—
29#1281—22no—
30C22×M16—21no—
31C23×C8—21no—
32#533—21no—
33#533—21no—
34#533—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533C2×C412yes—
39#533D412yes—
40C2×C4⋊Q8Q811yescharacteristic
41C2×C4⋊Q8Q811yescharacteristic
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#432—21no—
53#432—21no—
54#432—21no—
55#432C2×C412yes—
56#432D412yes—
57#431—21no—
58#431—21no—
59#431—21no—
60#431D412yes—
61C2×#108—24no—
62#423—21no—
63#423—21no—
64C2×C4⋊C8—21no—
65#407—21no—
66#407D412yes—
67C2×C22⋊C8—21no—
68C4×M16—21no—
69C4×M16—21no—
70C4×M16—21no—
71C4×M16—21no—
72C4×M16—21no—
73C4×M16D412yes—
74C2×#97—21no—
75C2×#97—21no—
76C2×#97—21no—
77C2×#97—21no—
78C2×#97—21no—
79C2×#97—21no—
80C2×#97C2×C412yes—
81C2×#97D412yes—
82C2×C4×C8—21no—
83#383—21no—
84#382D412yes—
85#381—21no—
86#380—21no—
87C4×C4⋊C4—21no—
88C4×C22⋊C4—21no—
89#376—21no—
90#341—24no—
91#330—28no—
92#328—28no—
93Q8⋊C8—28no—
94C22×Q8—21no—
95C22×Q8—21no—
96C23×C4C2×D411yescharacteristic
97C23×C4#4111yescharacteristic
98C23×C4#4111yescharacteristic
99C2×M16—22no—
100C2×M16—22no—
101C2×M16—44no—
102C2×M16—41no—
103C2×M16—41no—
104C22×C8—22no—
105C22×C8—22no—
106C22×C8—44no—
107C22×C8—41no—
108C22×C8—41no—
109C22×C8—41no—
110C4⋊Q8—21no—
111C4⋊Q8—21no—
112C4⋊Q8—21no—
113C4⋊Q8—21no—
114C4⋊Q8—21no—
115C4⋊Q8—21no—
116C4⋊Q8C4⋊C412yes—
117C4⋊Q8C4⋊C412yes—
118#125—22no—
119#125—22no—
120#125—41no—
121#125—41no—
122C22⋊Q8—22no—
123C22⋊Q8—22no—
124C22⋊Q8—22no—
125C22⋊Q8—22no—
126C22⋊Q8—41no—
127C22⋊Q8—41no—
128C22⋊Q8—41no—
129C22⋊Q8—41no—
130C4×Q8—22no—
131C4×Q8—22no—
132C4×Q8—22no—
133C4×Q8—22no—
134C4×Q8—41no—
135C4×Q8—41no—
136C4×Q8—41no—
137C4×Q8—41no—
138#117—22no—
139#117—22no—
140#117—41no—
141#117—41no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—41no—
148C2×C4⋊C4—41no—
149C2×C22⋊C4—41no—
150C2×C22⋊C4—41no—
151C2×C42—41no—
152C2×C42—41no—
153C2×C42C22×C411yescharacteristic
154C2×C42C2×D411yescharacteristic
155C2×C42C2×D411yescharacteristic
156C2×C42C2×D411yescharacteristic
157C2×C42C2×D411yescharacteristic
158C2×C42C2×D411yescharacteristic
159C2×C42C2×D411yescharacteristic
160C2×C42C2×Q811yescharacteristic
161C2×C42#4111yescharacteristic
162C2×C42#4111yescharacteristic
163C2×C42#4111yescharacteristic
164C2×C42#4111yescharacteristic
165#108—48no—
166C4⋊C8—22no—
167C4⋊C8—22no—
168C4⋊C8—22no—
169C4⋊C8—41no—
170C4⋊C8—41no—
171C4⋊C8—41no—
172C22⋊C8—22no—
173C22⋊C8—22no—
174C22⋊C8—22no—
175C22⋊C8—22no—
176C22⋊C8—22no—
177C22⋊C8—41no—
178#97—22no—
179#97—22no—
180#97—22no—
181#97—22no—
182#97—41no—
183#97—41no—
184#97—41no—
185#97—41no—
186#97—41no—
187#97C22⋊C414yes—
188C4×C8—22no—
189C4×C8—22no—
190C4×C8—41no—
191C4×C8—41no—
192#95—22no—
193#95—22no—
194#95—41no—
195#95—41no—
196C24#12411yescharacteristic
197C2×Q8—22no—
198C2×Q8—22no—
199C2×Q8—41no—
200C2×Q8—41no—
201C2×Q8—82no—
202C2×Q8—82no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—42no—
209C22×C4#11711yescharacteristic
210C22×C4C4×D411yescharacteristic
211C22×C4C4×D411yescharacteristic
212C22×C4#12011yescharacteristic
213C22×C4#12011yescharacteristic
214C22×C4#12111yescharacteristic
215C22×C4#12111yescharacteristic
216C22×C4#12111yescharacteristic
217C22×C4#12111yescharacteristic
218C22×C4C22⋊Q811yescharacteristic
219C22×C4C22⋊Q811yescharacteristic
220C22×C4C22⋊Q811yescharacteristic
221C22×C4#12311yescharacteristic
222C22×C4#12311yescharacteristic
223C22×C4#12311yescharacteristic
224C22×C4#12411yescharacteristic
225C22×C4#12411yescharacteristic
226C22×C4#12611yescharacteristic
227M16—24no—
228M16—42no—
229M16—44no—
230M16—44no—
231M16—82no—
232C2×C8—24no—
233C2×C8—24no—
234C2×C8—44no—
235C2×C8—42no—
236C2×C8—44no—
237C2×C8—44no—
238C2×C8—44no—
239C2×C8—42no—
240C2×C8—42no—
241C2×C8—42no—
242C2×C8—42no—
243C4⋊C4—22no—
244C4⋊C4—22no—
245C4⋊C4—22no—
246C4⋊C4—22no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—42no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—82no—
256C22⋊C4—42no—
257C22⋊C4—42no—
258C22⋊C4—82no—
259C42—42no—
260C42—42no—
261C42—82no—
262C42C2×C22⋊C411yescharacteristic
263C42C2×C4⋊C411yescharacteristic
264C42C4×D411yescharacteristic
265C42C4×D411yescharacteristic
266C42#12011yescharacteristic
267C42#12111yescharacteristic
268C42#12111yescharacteristic
269C42#12111yescharacteristic
270C42#12111yescharacteristic
271C42C22⋊Q811yescharacteristic
272C42C22⋊Q811yescharacteristic
273C42C22⋊Q811yescharacteristic
274C42#12311yescharacteristic
275C42#12311yescharacteristic
276C42#12611yescharacteristic
277C42#12711yescharacteristic
278C23—22no—
279C23—22no—
280C23—41no—
281C23#38811yescharacteristic
282C23#39311yescharacteristic
283C23#39411yescharacteristic
284Q8—44no—
285Q8—82no—
286Q8—82no—
287Q8—82no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
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—42no—
307C2×C4—42no—
308C2×C4—82no—
309C2×C4—82no—
310C2×C4—82no—
311C2×C4—82no—
312C2×C4—82no—
313C2×C4#38011yescharacteristic
314C2×C4#38511yescharacteristic
315C2×C4#38511yescharacteristic
316C2×C4#38611yescharacteristic
317C2×C4#38811yescharacteristic
318C2×C4#39011yescharacteristic
319C2×C4#39211yescharacteristic
320C2×C4#39411yescharacteristic
321C2×C4#39611yescharacteristic
322C2×C4#39611yescharacteristic
323C2×C4#39711yescharacteristic
324C2×C4#39911yescharacteristic
325C8—44no—
326C8—44no—
327C8—44no—
328C8—82no—
329C8—82no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—44no—
338C22#143911yescharacteristic
339C22#203112yes—
340C22#204912yes—
341C4—21no—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—21no—
350C4—21no—
351C4—21no—
352C4—82no—
353C4—82no—
354C4—162no—
355C4#186112yes—
356C2—22no—
357C2—22no—
358C2—41no—
359C2#1108511yescharacteristic
360C2#1111411yescharacteristic
361C2#1114311yescharacteristic
subgroupG/H
center, socleC22#1439
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#124
LCSG ▷ C2×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. #1531C4semidirect

Maximal subgroups

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

subgroupindex
#213712
C2×#15312
#175902
#175012
#80292
#79782
#79192