Finite Group Databasebeta

← #119288

Subgroups — #119288

Order 512.

650 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21324C211yesmaximal, characteristic
2C2×#1556C211yesmaximal, characteristic
3#14663C211yesmaximal, characteristic
4#8018C211yesmaximal, characteristic
5#8018C211yesmaximal, characteristic
6C2×#1288C211yesmaximal, characteristic
7C2×#1288C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9C2×#432C2211yescharacteristic
10C2×#432C2211yescharacteristic
11C2×C4×M16C2211yescharacteristic
12#2285C2211yescharacteristic
13C2×#383C2211yescharacteristic
14C2×#383C2211yescharacteristic
15#1993—22no—
16#1990C414yescomplemented
17#1736—22no—
18#1556—22no—
19#1556—22no—
20C2×#332—22no—
21#1446—22no—
22#1445—22no—
23#1445—22no—
24#1294—22no—
25#1294—22no—
26#1288—22no—
27#1288—22no—
28C23×Q8—21no—
29C22×M16—21no—
30C2×C4⋊Q8—21no—
31C2×C4⋊Q8—21no—
32C2×C4⋊Q8—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⋊Q8C2×C412yes—
41C2×C4⋊Q8D412yes—
42C2×#117—21no—
43C22×C4⋊C4—21no—
44C22×C4⋊C4D412yes—
45C22×C42C2311yescharacteristic
46#432—21no—
47#432—21no—
48#432—21no—
49#432—21no—
50#432—21no—
51#432—21no—
52#432D412yes—
53#432D412yes—
54C2×C4⋊C8—21no—
55C2×C4⋊C8—21no—
56C2×C22⋊C8—21no—
57C2×C22⋊C8—21no—
58C4×M16—21no—
59C4×M16—21no—
60C4×M16—21no—
61C4×M16—21no—
62C2×#97—21no—
63C2×#97—21no—
64C2×#97—21no—
65C2×C4×C8C2×C412yes—
66#384C2×C412yes—
67#383—21no—
68#383—21no—
69#383—21no—
70#383—21no—
71#383—21no—
72#383—21no—
73#383—21no—
74#383—21no—
75#382—21no—
76#380—21no—
77#380—21no—
78C4×C4⋊C4—21no—
79C4×C4⋊C4—21no—
80C2×#95—21no—
81C2×#95—21no—
82C2×#95—21no—
83C2×#95—21no—
84C2×#95—28no—
85C2×#95D412yes—
86C2×#95D412yes—
87#340—28no—
88#332—28no—
89C22×Q8—22no—
90C22×Q8—22no—
91C22×Q8—41no—
92C22×Q8—41no—
93C22×Q8—41no—
94C22×Q8—41no—
95C22×Q8—84no—
96C23×C4—22no—
97C23×C4—22no—
98C23×C4—22no—
99C23×C4C2×D411yescharacteristic
100C23×C4C2×D411yescharacteristic
101C23×C4C2×D411yescharacteristic
102C2×M16—24no—
103C2×M16—42no—
104C2×M16—41no—
105C2×M16—41no—
106C22×C8—22no—
107C22×C8—41no—
108C22×C8—41no—
109C4⋊Q8—22no—
110C4⋊Q8—22no—
111C4⋊Q8—22no—
112C4⋊Q8—22no—
113C4⋊Q8—22no—
114C4⋊Q8—22no—
115C4⋊Q8—22no—
116C4⋊Q8—22no—
117C4⋊Q8—41no—
118C4⋊Q8—41no—
119C4⋊Q8—41no—
120C4⋊Q8—41no—
121C4⋊Q8—41no—
122C4⋊Q8—41no—
123C4⋊Q8—41no—
124C4⋊Q8—41no—
125#117—44no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4—41no—
138C2×C4⋊C4—41no—
139C2×C4⋊C4—41no—
140C2×C4⋊C4C22⋊C414yes—
141C2×C22⋊C4—41no—
142C2×C42—41no—
143C2×C42C22×C411yescharacteristic
144C2×C42C2×D411yescharacteristic
145C2×C42C2×D411yescharacteristic
146C2×C42C2×D411yescharacteristic
147C2×C42#4111yescharacteristic
148C2×C42#4111yescharacteristic
149C2×C42#4111yescharacteristic
150C2×C42#4111yescharacteristic
151C2×C42#4111yescharacteristic
152C2×C42#4111yescharacteristic
153C2×C42#4111yescharacteristic
154C2×C42#4111yescharacteristic
155C4⋊C8—22no—
156C4⋊C8—22no—
157C4⋊C8—41no—
158C4⋊C8—41no—
159C22⋊C8—22no—
160C22⋊C8—22no—
161C22⋊C8—41no—
162C22⋊C8—41no—
163#97—22no—
164#97—22no—
165#97—41no—
166#97—41no—
167#97—41no—
168#97—41no—
169C4×C8—22no—
170C4×C8—22no—
171#95—22no—
172#95—22no—
173#95—22no—
174#95—22no—
175#95—41no—
176#95—41no—
177#95—41no—
178#95—41no—
179#95—41no—
180#95—41no—
181#95—41no—
182#95—41no—
183#95—41no—
184#95—41no—
185#95—88no—
186C24#12011yescharacteristic
187C2×Q8—42no—
188C2×Q8—42no—
189C2×Q8—42no—
190C2×Q8—42no—
191C2×Q8—84no—
192C2×Q8—84no—
193C2×Q8—84no—
194C2×Q8—84no—
195C2×Q8—84no—
196C2×Q8—84no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—24no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—48no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—44no—
206C22×C4—44no—
207C22×C4—44no—
208C22×C4—44no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4C2×C22⋊C411yescharacteristic
217C22×C4C4×D411yescharacteristic
218C22×C4C4×D411yescharacteristic
219C22×C4#12111yescharacteristic
220C22×C4#12111yescharacteristic
221C22×C4#12111yescharacteristic
222C22×C4#12111yescharacteristic
223C22×C4#12111yescharacteristic
224C22×C4#12111yescharacteristic
225C22×C4#12111yescharacteristic
226C22×C4#12111yescharacteristic
227C22×C4#12311yescharacteristic
228C22×C4#12311yescharacteristic
229C22×C4#12311yescharacteristic
230C22×C4#12311yescharacteristic
231C22×C4#12311yescharacteristic
232C22×C4#12411yescharacteristic
233C22×C4#12411yescharacteristic
234M16—44no—
235M16—44no—
236C2×C8—24no—
237C2×C8—42no—
238C2×C8—42no—
239C2×C8—42no—
240C2×C8—42no—
241C2×C8—82no—
242C2×C8—82no—
243C4⋊C4—24no—
244C4⋊C4—24no—
245C4⋊C4—44no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—82no—
253C4⋊C4—82no—
254C22⋊C4—82no—
255C42—82no—
256C42#11711yescharacteristic
257C42#11711yescharacteristic
258C42C4×D411yescharacteristic
259C42C4×D411yescharacteristic
260C42#12011yescharacteristic
261C42#12311yescharacteristic
262C42#12311yescharacteristic
263C42#12311yescharacteristic
264C42#12311yescharacteristic
265C42#12311yescharacteristic
266C42#12411yescharacteristic
267C42#12411yescharacteristic
268C42#12611yescharacteristic
269C42#12611yescharacteristic
270C42#12611yescharacteristic
271C42#12611yescharacteristic
272C23—22no—
273C23—22no—
274C23#38611yescharacteristic
275C23#39411yescharacteristic
276C23#39411yescharacteristic
277C23#45412yes—
278C23#45612yes—
279Q8—84no—
280Q8—84no—
281Q8—84no—
282Q8—84no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—48no—
304C2×C4—44no—
305C2×C4—84no—
306C2×C4—84no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4—82no—
310C2×C4—82no—
311C2×C4—84no—
312C2×C4—84no—
313C2×C4—84no—
314C2×C4—84no—
315C2×C4—82no—
316C2×C4—82no—
317C2×C4—82no—
318C2×C4—82no—
319C2×C4—82no—
320C2×C4#38111yescharacteristic
321C2×C4#38611yescharacteristic
322C2×C4#38811yescharacteristic
323C2×C4#38811yescharacteristic
324C2×C4#38811yescharacteristic
325C2×C4#38811yescharacteristic
326C2×C4#39411yescharacteristic
327C2×C4#39411yescharacteristic
328C2×C4#39711yescharacteristic
329C2×C4#39711yescharacteristic
330C2×C4#39711yescharacteristic
331C2×C4#39711yescharacteristic
332C8—44no—
333C8—82no—
334C8—82no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—24no—
340C22—24no—
341C22#144011yescharacteristic
342C22#188611yescharacteristic
343C22#188811yescharacteristic
344C22#190412yes—
345C22#190512yes—
346C22#203812yes—
347C22#203812yes—
348C4—21no—
349C4—21no—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—84no—
361C4—162no—
362C4—162no—
363C4—162no—
364C2—22no—
365C2—22no—
366C2#942712yes—
367C2#943012yes—
368C2#1116211yescharacteristic
369C2#1118311yescharacteristic
370C2#1118411yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
16. #1990C4semidirect

Maximal subgroups

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

subgroupindex
#213242
C2×#15562
#146632
#80182
C2×#12882