Finite Group Databasebeta

← #116138

Subgroups — #116138

Order 512.

559 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20811C211yesmaximal, characteristic
2#17672C211yesmaximal, characteristic
3#17596C211yesmaximal, characteristic
4#17545C211yesmaximal, characteristic
5#8052C211yesmaximal, characteristic
6#7998C211yesmaximal, characteristic
7#7977C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#432C2211yescharacteristic
12C2×#432C2211yescharacteristic
13#2283C2211yescharacteristic
14C4×#117C2211yescharacteristic
15#1843—21no—
16#1832—21no—
17#1798—22no—
18#1767—22no—
19#1537—22no—
20#1531—22no—
21#1528—22no—
22#1520—22no—
23#1515—22no—
24#1508—22no—
25#1506—21no—
26#1506—21no—
27#1506—22no—
28#1503C414yescomplemented
29#1297—22no—
30#1291—22no—
31#1288—22no—
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—
48C2×#117—21no—
49C22×C42C2311yescharacteristic
50#432—21no—
51#432—21no—
52#432—21no—
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#431—21no—
63#431—21no—
64#431—21no—
65#431C2×C412yes—
66#423—21no—
67#423—21no—
68C2×C4⋊C8—24no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—21no—
71#407—21no—
72#407—21no—
73C2×C22⋊C8—21no—
74C2×C22⋊C8D412yes—
75C4×M16—21no—
76C4×M16D412yes—
77C2×#97—21no—
78C2×#97D412yes—
79#383D412yes—
80#382—21no—
81#381—21no—
82#380—21no—
83C4×C4⋊C4—21no—
84C4×C4⋊C4—21no—
85C4×C4⋊C4C2×C412yes—
86C4×C22⋊C4—21no—
87C4×C22⋊C4—21no—
88C4×C22⋊C4D412yes—
89#376—21no—
90#376D411yescharacteristic
91#376D411yescharacteristic
92C43—21no—
93#336—24no—
94#330—28no—
95#328—28no—
96#328—24no—
97#328—24no—
98C22×Q8—21no—
99C22×Q8—21no—
100C23×C4C2×D411yescharacteristic
101C23×C4C2×D411yescharacteristic
102C23×C4#4111yescharacteristic
103C2×M16—22no—
104C2×M16—41no—
105C22×C8—22no—
106C22×C8—22no—
107C4⋊Q8—21no—
108C4⋊Q8—21no—
109C4⋊Q8—21no—
110C4⋊Q8—21no—
111C4⋊Q8—21no—
112C4⋊Q8—21no—
113C4⋊Q8C4⋊C412yes—
114C4⋊Q8C4⋊C412yes—
115#125—22no—
116#125—22no—
117#125—41no—
118#125—41no—
119C22⋊Q8—22no—
120C22⋊Q8—22no—
121C22⋊Q8—22no—
122C22⋊Q8—22no—
123C22⋊Q8—41no—
124C22⋊Q8—41no—
125C22⋊Q8—41no—
126C22⋊Q8—41no—
127C4×Q8—22no—
128C4×Q8—22no—
129C4×Q8—22no—
130C4×Q8—22no—
131C4×Q8—41no—
132C4×Q8—41no—
133C4×Q8—41no—
134C4×Q8—41no—
135#117—22no—
136#117—22no—
137#117—44no—
138#117—41no—
139#117—41no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4—41no—
147C2×C4⋊C4—41no—
148C2×C22⋊C4—22no—
149C2×C22⋊C4—41no—
150C2×C22⋊C4—41no—
151C2×C42—21no—
152C2×C42—21no—
153C2×C42—22no—
154C2×C42—22no—
155C2×C42—41no—
156C2×C42—41no—
157C2×C42C22×C411yescharacteristic
158C2×C42C2×D411yescharacteristic
159C2×C42C2×D411yescharacteristic
160C2×C42C2×D411yescharacteristic
161C2×C42C2×D411yescharacteristic
162C2×C42C2×D411yescharacteristic
163C2×C42C2×Q811yescharacteristic
164C2×C42#4111yescharacteristic
165C2×C42#4111yescharacteristic
166C2×C42#4111yescharacteristic
167C2×C42#4111yescharacteristic
168C2×C42#4111yescharacteristic
169C4⋊C8—22no—
170C4⋊C8—48no—
171C4⋊C8—41no—
172C4⋊C8—41no—
173C4⋊C8—41no—
174C4⋊C8—41no—
175C4⋊C8—41no—
176C4⋊C8—41no—
177C4⋊C8—41no—
178C22⋊C8—22no—
179C22⋊C8—22no—
180C22⋊C8—22no—
181C22⋊C8—41no—
182C22⋊C8—41no—
183C22⋊C8—41no—
184C22⋊C8—41no—
185C22⋊C8—41no—
186#97—22no—
187#97—22no—
188#97—22no—
189#97—41no—
190#97—41no—
191#97—41no—
192C4×C8—22no—
193C4×C8—22no—
194#95—21no—
195#95—21no—
196#95—22no—
197#95—22no—
198#95—22no—
199#95—41no—
200#95C22⋊C412yes—
201#95C22⋊C412yes—
202C24#12111yescharacteristic
203C2×Q8—22no—
204C2×Q8—22no—
205C2×Q8—41no—
206C2×Q8—41no—
207C2×Q8—82no—
208C2×Q8—82no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—42no—
216C22×C4—81no—
217C22×C4C2×C22⋊C411yescharacteristic
218C22×C4C4×D411yescharacteristic
219C22×C4C4×D411yescharacteristic
220C22×C4#12011yescharacteristic
221C22×C4#12011yescharacteristic
222C22×C4#12111yescharacteristic
223C22×C4#12111yescharacteristic
224C22×C4#12111yescharacteristic
225C22×C4#12111yescharacteristic
226C22×C4#12111yescharacteristic
227C22×C4C22⋊Q811yescharacteristic
228C22×C4C22⋊Q811yescharacteristic
229C22×C4C22⋊Q811yescharacteristic
230C22×C4#12311yescharacteristic
231C22×C4#12311yescharacteristic
232C22×C4#12311yescharacteristic
233C22×C4#12411yescharacteristic
234C22×C4#12611yescharacteristic
235M16—82no—
236M16—82no—
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—44no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—41no—
252C4⋊C4—42no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—82no—
257C22⋊C4—44no—
258C22⋊C4—42no—
259C22⋊C4—42no—
260C22⋊C4—82no—
261C42—24no—
262C42—42no—
263C42—42no—
264C42—42no—
265C42—82no—
266C42—82no—
267C42—82no—
268C42C2×C4⋊C411yescharacteristic
269C42#11711yescharacteristic
270C42C4×D411yescharacteristic
271C42C4×D411yescharacteristic
272C42#12011yescharacteristic
273C42#12111yescharacteristic
274C42#12111yescharacteristic
275C42C22⋊Q811yescharacteristic
276C42C22⋊Q811yescharacteristic
277C42C22⋊Q811yescharacteristic
278C42#12311yescharacteristic
279C42#12311yescharacteristic
280C42#12411yescharacteristic
281C42#12411yescharacteristic
282C42#12611yescharacteristic
283C42#12711yescharacteristic
284C23—22no—
285C23—41no—
286C23—41no—
287C23#38611yescharacteristic
288C23#39411yescharacteristic
289C23#39611yescharacteristic
290Q8—82no—
291Q8—82no—
292Q8—82no—
293Q8—82no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—24no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4—82no—
325C2×C4#38011yescharacteristic
326C2×C4#38511yescharacteristic
327C2×C4#38511yescharacteristic
328C2×C4#38811yescharacteristic
329C2×C4#38811yescharacteristic
330C2×C4#39011yescharacteristic
331C2×C4#39211yescharacteristic
332C2×C4#39311yescharacteristic
333C2×C4#39411yescharacteristic
334C2×C4#39611yescharacteristic
335C2×C4#39711yescharacteristic
336C2×C4#39911yescharacteristic
337C8—82no—
338C8—82no—
339C8—82no—
340C8—82no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—82no—
350C22#143911yescharacteristic
351C22#190212yes—
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—44no—
364C4—82no—
365C4—82no—
366C4—82no—
367C4—162no—
368C4#203112yes—
369C2—22no—
370C2—41no—
371C2—41no—
372C2#1105911yescharacteristic
373C2#1108111yescharacteristic
374C2#1109411yescharacteristic
subgroupG/H
center, socleC22#1439
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#121
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
28. #1503C4semidirect

Maximal subgroups

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

subgroupindex
#208112
#176722
#175962
#175452
#80522
#79982
#79772