Finite Group Databasebeta

← #110721

Subgroups — #110721

Order 512.

604 conjugacy classes of subgroups.

The subgroup lattice has 383 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21332C211yesmaximal, characteristic
2#20539C211yesmaximal, characteristic
3#17734C211yesmaximal, characteristic
4#17496C211yesmaximal, characteristic
5#8105C211yesmaximal, characteristic
6#8071C211yesmaximal, characteristic
7#7949C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#432C2211yescharacteristic
12C2×#432C2211yescharacteristic
13#2285C2211yescharacteristic
14C4×#117C2211yescharacteristic
15#1996—22no—
16#1996C414yes—
17#1771—22no—
18#1771—22no—
19#1750—22no—
20#1740—22no—
21#1559—22no—
22#1558C412yescomplemented
23#1556C412yescomplemented
24#1542—22no—
25#1497—22no—
26#1491C414yescomplemented
27#1481—22no—
28#1305—22no—
29#1300—22no—
30#1285—22no—
31#533—21no—
32#533—21no—
33#533—21no—
34#533—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39C2×C4⋊Q8C2×C411yescharacteristic
40C2×C4⋊Q8C2×C411yescharacteristic
41C2×#125—21no—
42C2×C22⋊Q8—21no—
43C2×C22⋊Q8C2×C412yes—
44C2×C4×Q8—21no—
45C2×C4×Q8C2×C412yes—
46C2×#117—21no—
47C2×#117—21no—
48C2×#117D412yes—
49C22×C42C2311yescharacteristic
50#432—21no—
51#432—21no—
52#432—21no—
53#432—21no—
54#432—21no—
55#432—21no—
56#432—21no—
57#432C2×C412yes—
58#432C2×C412yes—
59#432C2×C412yes—
60#432D412yes—
61#432D412yes—
62#431—21no—
63#431—21no—
64#431—21no—
65#431—21no—
66#423—21no—
67#423—21no—
68C2×C4⋊C8D412yes—
69C2×C4⋊C8D412yes—
70#407—21no—
71#407—21no—
72C2×C22⋊C8—21no—
73C2×C22⋊C8—21no—
74C4×M16—21no—
75C4×M16C2×C412yes—
76C2×#97—21no—
77C2×#97—21no—
78C2×#97—22no—
79C2×C4×C8—22no—
80#384—21no—
81#382—21no—
82#380C2×C412yes—
83#380D412yes—
84C4×C4⋊C4—21no—
85C4×C4⋊C4—21no—
86C4×C4⋊C4—21no—
87C4×C4⋊C4—21no—
88C4×C22⋊C4—21no—
89C4×C22⋊C4C2×C412yes—
90#376—21no—
91C43—21no—
92#342—28no—
93#336—24no—
94#333—24no—
95#332—24no—
96Q8⋊C8—28no—
97C22×Q8—21no—
98C22×Q8C4212yes—
99C23×C4C22×C411yescharacteristic
100C23×C4C2×D411yescharacteristic
101C23×C4C2×D411yescharacteristic
102C2×M16—22no—
103C2×M16—41no—
104C22×C8—22no—
105C22×C8—22no—
106C4⋊Q8—21no—
107C4⋊Q8—21no—
108C4⋊Q8—21no—
109C4⋊Q8—21no—
110C4⋊Q8—21no—
111C4⋊Q8—21no—
112C4⋊Q8—21no—
113C4⋊Q8—21no—
114#125—41no—
115#125—41no—
116#125—41no—
117#125—41no—
118C22⋊Q8—22no—
119C22⋊Q8—22no—
120C22⋊Q8—22no—
121C22⋊Q8—22no—
122C22⋊Q8—41no—
123C22⋊Q8—41no—
124C22⋊Q8—41no—
125C22⋊Q8—41no—
126C4×Q8—22no—
127C4×Q8—22no—
128C4×Q8—22no—
129C4×Q8—22no—
130C4×Q8—41no—
131C4×Q8—41no—
132C4×Q8—41no—
133C4×Q8—41no—
134#117—24no—
135#117—42no—
136#117—41no—
137#117—41no—
138#117—41no—
139#117—41no—
140#117—42no—
141#117—42no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4C4212yes—
152C2×C22⋊C4—22no—
153C2×C22⋊C4—22no—
154C2×C22⋊C4C22⋊C414yes—
155C2×C42—22no—
156C2×C42—22no—
157C2×C42—22no—
158C2×C42—41no—
159C2×C42—41no—
160C2×C42C22×C411yescharacteristic
161C2×C42C22×C411yescharacteristic
162C2×C42C2×D411yescharacteristic
163C2×C42C2×D411yescharacteristic
164C2×C42C2×D411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C42#4111yescharacteristic
167C2×C42#4111yescharacteristic
168C2×C42#4111yescharacteristic
169C2×C42#4111yescharacteristic
170C2×C42#4111yescharacteristic
171C2×C42#4111yescharacteristic
172C4⋊C8—22no—
173C4⋊C8—22no—
174C4⋊C8—22no—
175C4⋊C8—41no—
176C4⋊C8—41no—
177C4⋊C8—41no—
178C4⋊C8C22⋊C414yes—
179C4⋊C8C22⋊C414yes—
180C22⋊C8—22no—
181C22⋊C8—22no—
182C22⋊C8—22no—
183C22⋊C8—22no—
184C22⋊C8—22no—
185C22⋊C8—41no—
186C22⋊C8—41no—
187C22⋊C8—41no—
188#97—22no—
189#97—22no—
190#97—22no—
191#97—22no—
192#97—22no—
193#97—22no—
194#97—82no—
195C4×C8—22no—
196C4×C8—41no—
197C4×C8—82no—
198#95—22no—
199#95—22no—
200#95—22no—
201#95—41no—
202C24C2×C22⋊C411yescharacteristic
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—24no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—42no—
215C22×C4—81no—
216C22×C4—81no—
217C22×C4C2×C4211yescharacteristic
218C22×C4#11711yescharacteristic
219C22×C4C4×D411yescharacteristic
220C22×C4C4×D411yescharacteristic
221C22×C4C4×D411yescharacteristic
222C22×C4C4×D411yescharacteristic
223C22×C4C4×D411yescharacteristic
224C22×C4C4×D411yescharacteristic
225C22×C4C4×D411yescharacteristic
226C22×C4C4×D411yescharacteristic
227C22×C4#12011yescharacteristic
228C22×C4#12111yescharacteristic
229C22×C4#12111yescharacteristic
230C22×C4#12111yescharacteristic
231C22×C4#12111yescharacteristic
232C22×C4#12311yescharacteristic
233C22×C4#12411yescharacteristic
234C22×C4#12711yescharacteristic
235M16—82no—
236M16—82no—
237C2×C8—24no—
238C2×C8—24no—
239C2×C8—42no—
240C2×C8—42no—
241C2×C8—42no—
242C2×C8—42no—
243C2×C8—42no—
244C2×C8—42no—
245C4⋊C4—22no—
246C4⋊C4—22no—
247C4⋊C4—44no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—41no—
256C4⋊C4—42no—
257C4⋊C4—41no—
258C4⋊C4—82no—
259C4⋊C4—82no—
260C4⋊C4—82no—
261C22⋊C4—24no—
262C22⋊C4—42no—
263C22⋊C4—42no—
264C22⋊C4—42no—
265C22⋊C4—44no—
266C42—44no—
267C42—42no—
268C42—42no—
269C42—42no—
270C42—42no—
271C42—82no—
272C42—82no—
273C42C2×C22⋊C411yescharacteristic
274C42C2×C22⋊C411yescharacteristic
275C42#11711yescharacteristic
276C42#11711yescharacteristic
277C42C4×D411yescharacteristic
278C42C4×D411yescharacteristic
279C42C4×D411yescharacteristic
280C42C4×D411yescharacteristic
281C42#12111yescharacteristic
282C42#12111yescharacteristic
283C42#12311yescharacteristic
284C42#12311yescharacteristic
285C42#12411yescharacteristic
286C42#12411yescharacteristic
287C42#12611yescharacteristic
288C42#12611yescharacteristic
289C23—21no—
290C23—21no—
291C23—41no—
292C23—41no—
293C23C4×C22⋊C411yescharacteristic
294C23#38611yescharacteristic
295C23#39011yescharacteristic
296Q8—44no—
297Q8—82no—
298Q8—82no—
299Q8—82no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—44no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—44no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4C4×C22⋊C411yescharacteristic
329C2×C4C4×C22⋊C411yescharacteristic
330C2×C4#38611yescharacteristic
331C2×C4#38611yescharacteristic
332C2×C4#38811yescharacteristic
333C2×C4#38811yescharacteristic
334C2×C4#38811yescharacteristic
335C2×C4#38811yescharacteristic
336C2×C4#38811yescharacteristic
337C2×C4#38811yescharacteristic
338C2×C4#39011yescharacteristic
339C2×C4#39011yescharacteristic
340C2×C4#45512yes—
341C2×C4#45612yes—
342C8—44no—
343C8—44no—
344C8—82no—
345C8—82no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—81no—
355C22—81no—
356C22#143711yescharacteristic
357C22#188511yescharacteristic
358C22#188611yescharacteristic
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—21no—
369C4—84no—
370C4—84no—
371C4—82no—
372C4—82no—
373C4#188212yes—
374C4#188212yes—
375C2—21no—
376C2—21no—
377C2—41no—
378C2—41no—
379C2#1094111yescharacteristic
380C2#1095111yescharacteristic
381C2#1095711yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
socleC22#1437
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ 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
22. #1558C4semidirect
23. #1556C4semidirect
26. #1491C4semidirect

Maximal subgroups

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

subgroupindex
#213322
#205392
#177342
#174962
#81052
#80712
#79492