Finite Group Databasebeta

← #110492

Subgroups — #110492

Order 512.

588 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21826C211yesmaximal, characteristic
2#20548C211yesmaximal, characteristic
3#17749C211yesmaximal, characteristic
4#17479C211yesmaximal, characteristic
5#8116C211yesmaximal, characteristic
6#7962C211yesmaximal, characteristic
7#7858C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#423C2211yescharacteristic
13#2283C2211yescharacteristic
14#2283C2211yescharacteristic
15#1996—22no—
16#1996C414yes—
17#1820—22no—
18#1770—21no—
19C8×C4⋊C4—21no—
20#1768—22no—
21#1753—22no—
22#1559—22no—
23#1550—22no—
24#1548C412yescomplemented
25#1546C412yescomplemented
26#1497—22no—
27#1493—21no—
28#1493—21no—
29#1479C414yescomplemented
30#1306—22no—
31#1286—22no—
32#1274—22no—
33C22×M16—21no—
34#533—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533—21no—
42C2×C4⋊Q8C2×C411yescharacteristic
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×#125—21no—
45C2×C22⋊Q8—21no—
46C2×C22⋊Q8C2×C412yes—
47C2×C4×Q8—21no—
48C2×C4×Q8C2×C412yes—
49C2×#117—21no—
50C22×C42C2311yescharacteristic
51#432—21no—
52#432—21no—
53#432—21no—
54#432—21no—
55#431—21no—
56#431—21no—
57#431—21no—
58#431—21no—
59#431C2×C412yes—
60#431C2×C412yes—
61#431C2×C412yes—
62#431D412yes—
63#423—21no—
64#423—21no—
65#423—21no—
66#423—21no—
67#423C2×C412yes—
68#423D412yes—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8D412yes—
72#407—21no—
73#407—21no—
74C2×C22⋊C8—21no—
75C4×M16—21no—
76C4×M16—21no—
77C2×#97—22no—
78C2×C4×C8—22no—
79C2×C4×C8D411yescharacteristic
80C2×C4×C8D411yescharacteristic
81#383—21no—
82#383—21no—
83#382—21no—
84#382—21no—
85#381—21no—
86#381D412yes—
87#380—21no—
88#380C2×C412yes—
89C4×C4⋊C4—21no—
90C4×C4⋊C4—21no—
91C4×C22⋊C4C2×C412yes—
92C4×C22⋊C4D412yes—
93#376—21no—
94#376—21no—
95#342—28no—
96#336—22no—
97#336—22no—
98#333—24no—
99#332—24no—
100Q8⋊C8—24no—
101Q8⋊C8—24no—
102C22×Q8—21no—
103C22×Q8C4212yes—
104C23×C4C22×C411yescharacteristic
105C23×C4C2×D411yescharacteristic
106C23×C4C2×D411yescharacteristic
107C2×M16—22no—
108C2×M16—22no—
109C2×M16—41no—
110C2×M16—41no—
111C2×M16—82no—
112C22×C8—22no—
113C22×C8—21no—
114C22×C8—21no—
115C4⋊Q8—21no—
116C4⋊Q8—21no—
117C4⋊Q8—21no—
118C4⋊Q8—21no—
119C4⋊Q8—21no—
120C4⋊Q8—21no—
121C4⋊Q8—21no—
122C4⋊Q8—21no—
123#125—41no—
124#125—41no—
125#125—41no—
126#125—41no—
127C22⋊Q8—22no—
128C22⋊Q8—22no—
129C22⋊Q8—22no—
130C22⋊Q8—22no—
131C22⋊Q8—41no—
132C22⋊Q8—41no—
133C22⋊Q8—41no—
134C22⋊Q8—41no—
135C4×Q8—22no—
136C4×Q8—22no—
137C4×Q8—22no—
138C4×Q8—22no—
139C4×Q8—41no—
140C4×Q8—41no—
141C4×Q8—41no—
142C4×Q8—41no—
143#117—41no—
144#117—41no—
145#117—41no—
146#117—41no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4C4212yes—
155C2×C22⋊C4—22no—
156C2×C22⋊C4—22no—
157C2×C22⋊C4C22⋊C414yes—
158C2×C42—22no—
159C2×C42—22no—
160C2×C42—22no—
161C2×C42C22×C411yescharacteristic
162C2×C42C22×C411yescharacteristic
163C2×C42C2×D411yescharacteristic
164C2×C42C2×D411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C42#4111yescharacteristic
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—22no—
180C4⋊C8—41no—
181C4⋊C8—41no—
182C4⋊C8C22⋊C414yes—
183C22⋊C8—22no—
184C22⋊C8—22no—
185C22⋊C8—22no—
186C22⋊C8—22no—
187C22⋊C8—41no—
188C22⋊C8—41no—
189#97—41no—
190#97—41no—
191#97—82no—
192C4×C8—22no—
193C4×C8—22no—
194C4×C8—21no—
195C4×C8—21no—
196C4×C8—82no—
197C4×C8C22⋊C412yes—
198C4×C8C22⋊C412yes—
199#95—22no—
200#95—22no—
201#95—22no—
202#95—22no—
203#95—22no—
204#95—41no—
205#95—41no—
206#95—41no—
207C24C2×C22⋊C411yescharacteristic
208C2×Q8—22no—
209C2×Q8—22no—
210C2×Q8—44no—
211C2×Q8—41no—
212C2×Q8—41no—
213C2×Q8—81no—
214C2×Q8—81no—
215C22×C4—24no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—41no—
219C22×C4—42no—
220C22×C4—41no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4C2×C4211yescharacteristic
224C22×C4#11711yescharacteristic
225C22×C4C4×D411yescharacteristic
226C22×C4C4×D411yescharacteristic
227C22×C4C4×D411yescharacteristic
228C22×C4C4×D411yescharacteristic
229C22×C4C4×D411yescharacteristic
230C22×C4C4×D411yescharacteristic
231C22×C4C4×D411yescharacteristic
232C22×C4C4×D411yescharacteristic
233C22×C4#12011yescharacteristic
234C22×C4#12111yescharacteristic
235C22×C4#12111yescharacteristic
236C22×C4#12111yescharacteristic
237C22×C4#12111yescharacteristic
238C22×C4#12311yescharacteristic
239C22×C4#12411yescharacteristic
240C22×C4#12711yescharacteristic
241M16—82no—
242M16—82no—
243M16—82no—
244M16—82no—
245C2×C8—22no—
246C2×C8—22no—
247C2×C8—24no—
248C2×C8—41no—
249C2×C8—42no—
250C2×C8—41no—
251C2×C8—42no—
252C2×C8—41no—
253C2×C8—41no—
254C2×C8—42no—
255C2×C8—42no—
256C4⋊C4—22no—
257C4⋊C4—22no—
258C4⋊C4—44no—
259C4⋊C4—42no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—81no—
270C4⋊C4—81no—
271C22⋊C4—42no—
272C22⋊C4—42no—
273C22⋊C4—44no—
274C22⋊C4—44no—
275C42—42no—
276C42—42no—
277C42—44no—
278C42—42no—
279C42—42no—
280C42C2×C22⋊C411yescharacteristic
281C42C2×C22⋊C411yescharacteristic
282C42#11711yescharacteristic
283C42#11711yescharacteristic
284C42C4×D411yescharacteristic
285C42C4×D411yescharacteristic
286C42C4×D411yescharacteristic
287C42C4×D411yescharacteristic
288C42#12111yescharacteristic
289C42#12111yescharacteristic
290C42#12311yescharacteristic
291C42#12311yescharacteristic
292C42#12411yescharacteristic
293C42#12411yescharacteristic
294C42#12611yescharacteristic
295C42#12611yescharacteristic
296C23—22no—
297C23—41no—
298C23—41no—
299C23C4×C22⋊C411yescharacteristic
300C23#38611yescharacteristic
301C23#39011yescharacteristic
302Q8—44no—
303Q8—82no—
304Q8—82no—
305Q8—81no—
306Q8—81no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—44no—
329C2×C4—44no—
330C2×C4—44no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4—82no—
338C2×C4C4×C22⋊C411yescharacteristic
339C2×C4C4×C22⋊C411yescharacteristic
340C2×C4#38611yescharacteristic
341C2×C4#38611yescharacteristic
342C2×C4#38811yescharacteristic
343C2×C4#38811yescharacteristic
344C2×C4#38811yescharacteristic
345C2×C4#38811yescharacteristic
346C2×C4#38811yescharacteristic
347C2×C4#38811yescharacteristic
348C2×C4#39011yescharacteristic
349C2×C4#39011yescharacteristic
350C8—42no—
351C8—42no—
352C8—44no—
353C8—82no—
354C8—82no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—82no—
364C22#143711yescharacteristic
365C22#190912yes—
366C4—21no—
367C4—21no—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—84no—
377C4—82no—
378C4—82no—
379C4—162no—
380C4#175712yes—
381C4#190411yescharacteristic
382C4#190511yescharacteristic
383C2—22no—
384C2—41no—
385C2—41no—
386C2#1094711yescharacteristic
387C2#1095411yescharacteristic
388C2#1096111yescharacteristic
subgroupG/H
center, socleC22#1437
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ 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
24. #1548C4semidirect
25. #1546C4semidirect
29. #1479C4semidirect

Maximal subgroups

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

subgroupindex
#218262
#205482
#177492
#174792
#81162
#79622
#78582