Finite Group Databasebeta

← #111119

Subgroups — #111119

Order 512.

660 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#21820C211yesmaximal, characteristic
2#20735C211yesmaximal, characteristic
3C2×#1491C211yesmaximal, characteristic
4#17319C211yesmaximal, characteristic
5#7986C211yesmaximal, characteristic
6#7972C211yesmaximal, characteristic
7#7928C211yesmaximal, characteristic
8C22×C4×Q8C2211yescharacteristic
9C2×#432C2211yescharacteristic
10C2×#432C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×C4×M16C2211yescharacteristic
14#2283C2211yescharacteristic
15#1985C414yescomplemented
16#1984—22no—
17#1823C412yes—
18#1817—22no—
19#1805—22no—
20#1787C412yes—
21C4×#103—22no—
22#1491—22no—
23#1491C414yescomplemented
24C2×Q8⋊C8—22no—
25#1452—22no—
26#1451—22no—
27#1451—22no—
28#1290—22no—
29#1288—22no—
30#1282—22no—
31C23×Q8C2×C411yescharacteristic
32C22×M16—21no—
33C2×C4×Q8—21no—
34C2×C4×Q8—21no—
35C2×C4×Q8—21no—
36C2×C4×Q8—21no—
37C2×C4×Q8—21no—
38C2×C4×Q8—21no—
39C2×C4×Q8—21no—
40C2×C4×Q8—21no—
41C2×C4×Q8—21no—
42C2×C4×Q8—21no—
43C2×C4×Q8C2×C412yes—
44C2×C4×Q8C2×C412yes—
45C22×C4⋊C4—21no—
46C22×C4⋊C4C2×C411yescharacteristic
47C22×C42—21no—
48C22×C42C2311yescharacteristic
49#432—21no—
50#432—21no—
51#432—21no—
52#432—21no—
53#432—21no—
54#432—21no—
55#432—21no—
56#432C2×C412yes—
57#431—21no—
58#431—21no—
59#431—21no—
60#431—21no—
61#431—21no—
62#431—21no—
63#431—21no—
64#431—21no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8C2×C412yes—
68C2×C4⋊C8D412yes—
69C2×#103—24no—
70C2×C22⋊C8—21no—
71C2×C22⋊C8—21no—
72C2×C22⋊C8C2×C412yes—
73C2×C22⋊C8D412yes—
74C4×M16—21no—
75C4×M16—21no—
76C4×M16C2×C412yes—
77C4×M16D412yes—
78C2×#97—21no—
79C2×#97—21no—
80C2×#97C2×C412yes—
81C2×C4×C8D412yes—
82C2×C4×C8D412yes—
83C2×C4×C8D412yes—
84#383—21no—
85#382—21no—
86#381—21no—
87#380C2×C412yes—
88C4×C4⋊C4—21no—
89C4×C22⋊C4—21no—
90#376—21no—
91#340—24no—
92#336—28no—
93Q8⋊C8—28no—
94#322—28no—
95C22×Q8—21no—
96C22×Q8—21no—
97C22×Q8—21no—
98C22×Q8—21no—
99C22×Q8—21no—
100C22×Q8—44no—
101C22×Q8C4212yes—
102C23×C4—21no—
103C23×C4—21no—
104C23×C4C22×C411yescharacteristic
105C23×C4C2×D411yescharacteristic
106C23×C4#4111yescharacteristic
107C2×M16—22no—
108C2×M16—22no—
109C2×M16—44no—
110C22×C8—22no—
111C22×C8—22no—
112C22×C8—41no—
113C22×C8C22⋊C414yes—
114C22×C8C22⋊C414yes—
115C4×Q8—22no—
116C4×Q8—22no—
117C4×Q8—22no—
118C4×Q8—22no—
119C4×Q8—22no—
120C4×Q8—22no—
121C4×Q8—22no—
122C4×Q8—22no—
123C4×Q8—41no—
124C4×Q8—41no—
125C4×Q8—41no—
126C4×Q8—41no—
127C4×Q8—41no—
128C4×Q8—41no—
129C4×Q8—41no—
130C4×Q8—41no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—41no—
140C2×C4⋊C4—41no—
141C2×C4⋊C4—41no—
142C2×C4⋊C4—41no—
143C2×C4⋊C4C4212yes—
144C2×C22⋊C4—22no—
145C2×C42—22no—
146C2×C42—22no—
147C2×C42—41no—
148C2×C42—41no—
149C2×C42—41no—
150C2×C42—41no—
151C2×C42—41no—
152C2×C42C22×C411yescharacteristic
153C2×C42C22×C411yescharacteristic
154C2×C42C2×D411yescharacteristic
155C2×C42C2×D411yescharacteristic
156C2×C42C2×D411yescharacteristic
157C2×C42C2×D411yescharacteristic
158C2×C42C2×D411yescharacteristic
159C2×C42#4111yescharacteristic
160C2×C42#4111yescharacteristic
161C2×C42#4111yescharacteristic
162C2×C42#4111yescharacteristic
163C2×C42#4111yescharacteristic
164C4⋊C8—22no—
165C4⋊C8—22no—
166C4⋊C8—22no—
167C4⋊C8—22no—
168C4⋊C8—22no—
169C4⋊C8—41no—
170C4⋊C8—41no—
171C4⋊C8—41no—
172#103—48no—
173C22⋊C8—22no—
174C22⋊C8—22no—
175C22⋊C8—22no—
176C22⋊C8—22no—
177C22⋊C8—22no—
178C22⋊C8—41no—
179C22⋊C8—41no—
180C22⋊C8—41no—
181#97—22no—
182#97—22no—
183#97—22no—
184#97—22no—
185#97—22no—
186#97—41no—
187C4×C8—22no—
188C4×C8—22no—
189C4×C8—22no—
190C4×C8—22no—
191C4×C8—22no—
192C4×C8C22⋊C414yes—
193#95—22no—
194#95—41no—
195#95—41no—
196#95—41no—
197C24C4×D411yescharacteristic
198C2×Q8—24no—
199C2×Q8—24no—
200C2×Q8—24no—
201C2×Q8—22no—
202C2×Q8—22no—
203C2×Q8—44no—
204C2×Q8—42no—
205C2×Q8—42no—
206C2×Q8—42no—
207C2×Q8—44no—
208C2×Q8—44no—
209C2×Q8—41no—
210C2×Q8—41no—
211C22×C4—24no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—44no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—44no—
222C22×C4—42no—
223C22×C4C2×C4211yescharacteristic
224C22×C4C2×C22⋊C411yescharacteristic
225C22×C4C2×C22⋊C411yescharacteristic
226C22×C4#11711yescharacteristic
227C22×C4#11711yescharacteristic
228C22×C4C4×D411yescharacteristic
229C22×C4C4×D411yescharacteristic
230C22×C4C4×D411yescharacteristic
231C22×C4C4×D411yescharacteristic
232C22×C4C4×D411yescharacteristic
233C22×C4#12111yescharacteristic
234C22×C4#12111yescharacteristic
235C22×C4#12111yescharacteristic
236C22×C4#12311yescharacteristic
237C22×C4#12411yescharacteristic
238C22×C4#12411yescharacteristic
239C22×C4#12611yescharacteristic
240C22×C4#12711yescharacteristic
241M16—44no—
242M16—44no—
243C2×C8—24no—
244C2×C8—24no—
245C2×C8—24no—
246C2×C8—44no—
247C2×C8—44no—
248C2×C8—44no—
249C2×C8—44no—
250C2×C8—42no—
251C2×C8—42no—
252C2×C8—82no—
253C4⋊C4—22no—
254C4⋊C4—22no—
255C4⋊C4—44no—
256C4⋊C4—41no—
257C4⋊C4—42no—
258C4⋊C4—41no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C22⋊C4—44no—
263C42—42no—
264C42—42no—
265C42—42no—
266C42—42no—
267C42—82no—
268C42C2×C22⋊C411yescharacteristic
269C42#11711yescharacteristic
270C42C4×D411yescharacteristic
271C42C4×D411yescharacteristic
272C42C4×D411yescharacteristic
273C42C4×D411yescharacteristic
274C42C4×D411yescharacteristic
275C42C4×D411yescharacteristic
276C42#12011yescharacteristic
277C42#12111yescharacteristic
278C42#12111yescharacteristic
279C42#12111yescharacteristic
280C42#12311yescharacteristic
281C42#12311yescharacteristic
282C42#12411yescharacteristic
283C42#12611yescharacteristic
284C23—41no—
285C23—41no—
286C23C4×C22⋊C411yescharacteristic
287C23#38811yescharacteristic
288C23#39011yescharacteristic
289C23#44414yes—
290Q8—24no—
291Q8—24no—
292Q8—42no—
293Q8—42no—
294Q8—44no—
295Q8—44no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—24no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—42no—
316C2×C4—44no—
317C2×C4—44no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—44no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4C4×C22⋊C411yescharacteristic
327C2×C4C4×C22⋊C411yescharacteristic
328C2×C4#38611yescharacteristic
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
338C8—44no—
339C8—44no—
340C8—44no—
341C8—82no—
342C8—82no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—44no—
350C22#143711yescharacteristic
351C22#176512yes—
352C22#193812yes—
353C22#197512yes—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—44no—
365C4—82no—
366C4—162no—
367C4#189112yes—
368C4#190912yes—
369C2—22no—
370C2—22no—
371C2—22no—
372C2#1093011yescharacteristic
373C2#1093711yescharacteristic
374C2#1094711yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
socleC22#1437
prime radicalC24C4×D4
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
15. #1985C4semidirect
23. #1491C4semidirect

Maximal subgroups

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

subgroupindex
#218202
#207352
C2×#14912
#173192
#79862
#79722
#79282