Finite Group Databasebeta

← #110618

Subgroups — #110618

Order 512.

664 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21311C211yesmaximal, characteristic
2#20543C211yesmaximal, characteristic
3#17545C211yesmaximal, characteristic
4#17464C211yesmaximal, characteristic
5#8048C211yesmaximal, characteristic
6#7988C211yesmaximal, characteristic
7#7869C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C22×C4⋊C8C2211yescharacteristic
13C4×#117C2211yescharacteristic
14C4×#117C2211yescharacteristic
15#1989—22no—
16#1988C412yescomplemented
17#1987C412yescomplemented
18C8×C4⋊C4—22no—
19C4×C4⋊C8—22no—
20C8×C22⋊C4—22no—
21#1735—22no—
22#1508—22no—
23#1506C414yescomplemented
24#1503—22no—
25#1485C414yescomplemented
26#1481—22no—
27#1479—22no—
28#1297—22no—
29#1290—22no—
30#1276—22no—
31C23×C8D412yes—
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—
48C2×#117—21no—
49C2×#117—21no—
50C22×C42C2311yescharacteristic
51#432—21no—
52#432—21no—
53#432—21no—
54#432C2×C412yes—
55#431—21no—
56#431—21no—
57#431—21no—
58#431—21no—
59#431—21no—
60#431—21no—
61#431C2×C412yes—
62#431D412yes—
63#423—21no—
64#423—21no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8C2×C412yes—
70C2×C4⋊C8D412yes—
71C2×C4⋊C8D412yes—
72#407—21no—
73#407—21no—
74C2×C22⋊C8—21no—
75C4×M16—21no—
76C4×M16C2×C412yes—
77C2×C4×C8—21no—
78C2×C4×C8—24no—
79C4×C4⋊C4—21no—
80C4×C4⋊C4—21no—
81C4×C4⋊C4C2×C412yes—
82C4×C4⋊C4D412yes—
83C4×C22⋊C4—21no—
84C4×C22⋊C4—21no—
85C4×C22⋊C4—21no—
86C4×C22⋊C4D412yes—
87#376—21no—
88#376C2×C412yes—
89C43—21no—
90C43—21no—
91#340—24no—
92#340—24no—
93#336—24no—
94#328—28no—
95Q8⋊C8—28no—
96C22×Q8—21no—
97C22×Q8—21no—
98C23×C4C2×D411yescharacteristic
99C23×C4C2×D411yescharacteristic
100C23×C4#4111yescharacteristic
101C2×M16—22no—
102C2×M16—41no—
103C22×C8—22no—
104C22×C8—22no—
105C22×C8—22no—
106C22×C8—44no—
107C22×C8C22⋊C414yes—
108C4⋊Q8—21no—
109C4⋊Q8—21no—
110C4⋊Q8—21no—
111C4⋊Q8—21no—
112C4⋊Q8—21no—
113C4⋊Q8—21no—
114C4⋊Q8C4212yes—
115C4⋊Q8C4212yes—
116#125—22no—
117#125—22no—
118#125—41no—
119#125—41no—
120C22⋊Q8—22no—
121C22⋊Q8—22no—
122C22⋊Q8—22no—
123C22⋊Q8—22no—
124C22⋊Q8—41no—
125C22⋊Q8—41no—
126C22⋊Q8—41no—
127C22⋊Q8—41no—
128C4×Q8—22no—
129C4×Q8—22no—
130C4×Q8—22no—
131C4×Q8—22no—
132C4×Q8—41no—
133C4×Q8—41no—
134C4×Q8—41no—
135C4×Q8—41no—
136#117—28no—
137#117—24no—
138#117—22no—
139#117—22no—
140#117—42no—
141#117—41no—
142#117—41no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C22⋊C4—22no—
152C2×C22⋊C4—41no—
153C2×C22⋊C4—41no—
154C2×C42—22no—
155C2×C42—22no—
156C2×C42—22no—
157C2×C42—22no—
158C2×C42—41no—
159C2×C42—41no—
160C2×C42—41no—
161C2×C42C22×C411yescharacteristic
162C2×C42C22×C411yescharacteristic
163C2×C42C22×C411yescharacteristic
164C2×C42C2×D411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C42C2×D411yescharacteristic
168C2×C42#4111yescharacteristic
169C2×C42#4111yescharacteristic
170C2×C42#4111yescharacteristic
171C2×C42#4111yescharacteristic
172C2×C42#4111yescharacteristic
173C4⋊C8—22no—
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⋊C8C22⋊C414yes—
183C22⋊C8—22no—
184C22⋊C8—22no—
185C22⋊C8—22no—
186C22⋊C8—41no—
187C22⋊C8—41no—
188C22⋊C8—41no—
189#97—22no—
190#97—22no—
191C4×C8—22no—
192C4×C8—22no—
193C4×C8—22no—
194C4×C8—41no—
195C4×C8—84no—
196#95—22no—
197#95—22no—
198#95—41no—
199#95C22⋊C414yes—
200C24#12111yescharacteristic
201C2×Q8—22no—
202C2×Q8—22no—
203C2×Q8—41no—
204C2×Q8—41no—
205C2×Q8—44no—
206C2×Q8—82no—
207C22×C4—24no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—24no—
211C22×C4—42no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—81no—
215C22×C4C2×C22⋊C411yescharacteristic
216C22×C4C2×C22⋊C411yescharacteristic
217C22×C4#11711yescharacteristic
218C22×C4C4×D411yescharacteristic
219C22×C4C4×D411yescharacteristic
220C22×C4C4×D411yescharacteristic
221C22×C4C4×D411yescharacteristic
222C22×C4C4×D411yescharacteristic
223C22×C4C4×D411yescharacteristic
224C22×C4#12011yescharacteristic
225C22×C4#12111yescharacteristic
226C22×C4#12111yescharacteristic
227C22×C4#12311yescharacteristic
228C22×C4#12311yescharacteristic
229C22×C4#12411yescharacteristic
230C22×C4#12411yescharacteristic
231C22×C4#12611yescharacteristic
232C22×C4#12711yescharacteristic
233M16—82no—
234M16—82no—
235C2×C8—24no—
236C2×C8—24no—
237C2×C8—42no—
238C2×C8—44no—
239C2×C8—42no—
240C2×C8—44no—
241C2×C8—42no—
242C2×C8—42no—
243C2×C8—82no—
244C4⋊C4—28no—
245C4⋊C4—24no—
246C4⋊C4—22no—
247C4⋊C4—22no—
248C4⋊C4—22no—
249C4⋊C4—22no—
250C4⋊C4—42no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—42no—
256C4⋊C4—42no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C22⋊C4—28no—
260C22⋊C4—44no—
261C22⋊C4—42no—
262C22⋊C4—42no—
263C42—28no—
264C42—44no—
265C42—42no—
266C42—42no—
267C42—42no—
268C42—42no—
269C42—44no—
270C42—82no—
271C42—82no—
272C42C2×C4211yescharacteristic
273C42C2×C22⋊C411yescharacteristic
274C42#11711yescharacteristic
275C42#11711yescharacteristic
276C42C4×D411yescharacteristic
277C42C4×D411yescharacteristic
278C42C4×D411yescharacteristic
279C42C4×D411yescharacteristic
280C42C4×D411yescharacteristic
281C42C4×D411yescharacteristic
282C42#12111yescharacteristic
283C42#12111yescharacteristic
284C42#12111yescharacteristic
285C42#12311yescharacteristic
286C42#12411yescharacteristic
287C42#12611yescharacteristic
288C23—22no—
289C23—22no—
290C23—41no—
291C23#38611yescharacteristic
292C23#38811yescharacteristic
293C23#39011yescharacteristic
294Q8—44no—
295Q8—82no—
296Q8—82no—
297Q8—82no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—28no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—44no—
318C2×C4—44no—
319C2×C4—44no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4C4×C22⋊C411yescharacteristic
327C2×C4C4×C22⋊C411yescharacteristic
328C2×C4C4×C22⋊C411yescharacteristic
329C2×C4#38611yescharacteristic
330C2×C4#38611yescharacteristic
331C2×C4#38811yescharacteristic
332C2×C4#38811yescharacteristic
333C2×C4#38811yescharacteristic
334C2×C4#38811yescharacteristic
335C2×C4#38811yescharacteristic
336C2×C4#39011yescharacteristic
337C2×C4#39011yescharacteristic
338C2×C4#44314yes—
339C2×C4#45512yes—
340C2×C4#45612yes—
341C8—44no—
342C8—44no—
343C8—82no—
344C8—82no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—44no—
353C22#143711yescharacteristic
354C22#188212yes—
355C22#197412yes—
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—48no—
365C4—84no—
366C4—82no—
367C4—82no—
368C4#176412yes—
369C4#188212yes—
370C4#188511yescharacteristic
371C4#188611yescharacteristic
372C4#193712yes—
373C2—22no—
374C2—22no—
375C2—41no—
376C2#1092911yescharacteristic
377C2#1095111yescharacteristic
378C2#1095711yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC42C2×C42
Frattini, cosocleC22×C42C23
socleC22#1437
prime radicalC24#121
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
16. #1988C4semidirect
17. #1987C4semidirect
23. #1506C4semidirect
25. #1485C4semidirect

Maximal subgroups

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

subgroupindex
#213112
#205432
#175452
#174642
#80482
#79882
#78692