Finite Group Databasebeta

← #121289

Subgroups — #121289

Order 512.

548 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#8057C211yesmaximal, characteristic
2#8051C211yesmaximal, characteristic
3#8024C211yesmaximal, characteristic
4C2×#1284C211yesmaximal, characteristic
5#7927C211yesmaximal, characteristic
6C2×#1282C211yesmaximal, characteristic
7#7913C211yesmaximal, characteristic
8C2×#407C2211yescharacteristic
9C2×#407C2211yescharacteristic
10C22×C22⋊C8C2211yescharacteristic
11C22×C22⋊C8C2211yescharacteristic
12#2280C2211yescharacteristic
13C2×#381C2211yescharacteristic
14C2×C4×C22⋊C4C2211yescharacteristic
15#1298—22no—
16#1297—22no—
17#1294C412yescomplemented
18#1294C412yescomplemented
19#1284—22no—
20#1282—22no—
21#1282—22no—
22#1280—22no—
23C24×C4C2311yescharacteristic
24C22×M16—21no—
25C22×M16—21no—
26C23×C8—21no—
27C23×C8—21no—
28C2×#117—21no—
29C22×C22⋊C4—21no—
30C22×C22⋊C4—21no—
31C22×C42—21no—
32#407—21no—
33#407—21no—
34#407—21no—
35#407—21no—
36#407—21no—
37#407—21no—
38#407D412yes—
39#407Q812yes—
40C2×C22⋊C8—21no—
41C2×C22⋊C8—21no—
42C2×C22⋊C8—21no—
43C2×C22⋊C8—21no—
44C2×C22⋊C8—21no—
45C2×C22⋊C8—21no—
46C2×C22⋊C8—21no—
47C2×C22⋊C8—21no—
48C2×C22⋊C8—21no—
49C2×C22⋊C8—21no—
50C2×C22⋊C8—21no—
51C2×C22⋊C8—21no—
52C2×C22⋊C8—21no—
53C2×C22⋊C8—21no—
54C2×C22⋊C8C2×C412yes—
55C2×C22⋊C8D412yes—
56#381—21no—
57#381—21no—
58#381—21no—
59#381—21no—
60#381—21no—
61#380—21no—
62#380C2×C411yescharacteristic
63#380C2×C411yescharacteristic
64#379—21no—
65C4×C22⋊C4—21no—
66C4×C22⋊C4—21no—
67C4×C22⋊C4—21no—
68C4×C22⋊C4—21no—
69C4×C22⋊C4—21no—
70C4×C22⋊C4D412yes—
71C2×#95—21no—
72C2×#95—21no—
73C2×#95C2×C412yes—
74C25#4111yescharacteristic
75C23×C4—22no—
76C23×C4—41no—
77C23×C4—82no—
78C23×C4C22×C411yescharacteristic
79C23×C4C2×D411yescharacteristic
80C23×C4C2×D411yescharacteristic
81C23×C4C2×D411yescharacteristic
82C23×C4C2×Q811yescharacteristic
83C23×C4#4111yescharacteristic
84C23×C4#4111yescharacteristic
85C23×C4#4111yescharacteristic
86C23×C4#4111yescharacteristic
87C23×C4#4111yescharacteristic
88C23×C4#4111yescharacteristic
89C23×C4#4111yescharacteristic
90C23×C4#4111yescharacteristic
91C23×C4#4111yescharacteristic
92C2×M16—22no—
93C2×M16—22no—
94C2×M16—41no—
95C2×M16—41no—
96C2×M16—82no—
97C2×M16—82no—
98C22×C8—22no—
99C22×C8—22no—
100C22×C8—41no—
101C22×C8—41no—
102C22×C8—41no—
103C22×C8—41no—
104C22×C8—41no—
105C22×C8—41no—
106C22×C8—82no—
107C22×C8—82no—
108#117—82no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C22⋊C4—24no—
112C2×C22⋊C4—22no—
113C2×C22⋊C4—21no—
114C2×C22⋊C4—21no—
115C2×C22⋊C4—42no—
116C2×C22⋊C4—41no—
117C2×C22⋊C4—41no—
118C2×C22⋊C4—41no—
119C2×C22⋊C4—41no—
120C2×C22⋊C4—41no—
121C2×C22⋊C4—82no—
122C2×C42—22no—
123C2×C42—41no—
124C2×C42—41no—
125C2×C42—82no—
126C22⋊C8—22no—
127C22⋊C8—22no—
128C22⋊C8—22no—
129C22⋊C8—22no—
130C22⋊C8—22no—
131C22⋊C8—22no—
132C22⋊C8—22no—
133C22⋊C8—22no—
134C22⋊C8—41no—
135C22⋊C8—41no—
136C22⋊C8—41no—
137C22⋊C8—41no—
138C22⋊C8—41no—
139C22⋊C8—41no—
140C22⋊C8—41no—
141C22⋊C8—41no—
142#95—22no—
143#95—22no—
144#95—21no—
145#95—21no—
146#95—22no—
147#95—41no—
148#95—41no—
149#95—41no—
150#95—41no—
151C24—22no—
152C24—22no—
153C24—42no—
154C24—81no—
155C24C4×D411yescharacteristic
156C24#12311yescharacteristic
157C24#12411yescharacteristic
158C24#12511yescharacteristic
159C24#12611yescharacteristic
160C24#12611yescharacteristic
161C24#12611yescharacteristic
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—44no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—81no—
176C22×C4—81no—
177C22×C4—81no—
178C22×C4—81no—
179C22×C4—81no—
180C22×C4—81no—
181C22×C4—81no—
182C22×C4—81no—
183C22×C4—82no—
184C22×C4—82no—
185C22×C4—82no—
186C22×C4—82no—
187C22×C4#11711yescharacteristic
188C22×C4#11711yescharacteristic
189C22×C4#11711yescharacteristic
190C22×C4C4×D411yescharacteristic
191C22×C4C4×D411yescharacteristic
192C22×C4C4×Q811yescharacteristic
193C22×C4#12111yescharacteristic
194C22×C4#12111yescharacteristic
195C22×C4#12111yescharacteristic
196C22×C4C22⋊Q811yescharacteristic
197C22×C4C22⋊Q811yescharacteristic
198C22×C4C22⋊Q811yescharacteristic
199C22×C4#12311yescharacteristic
200C22×C4#12311yescharacteristic
201C22×C4#12311yescharacteristic
202C22×C4#12311yescharacteristic
203C22×C4#12311yescharacteristic
204C22×C4#12411yescharacteristic
205C22×C4#12411yescharacteristic
206C22×C4#12511yescharacteristic
207C22×C4#12511yescharacteristic
208C22×C4#12611yescharacteristic
209C22×C4#12611yescharacteristic
210C22×C4#12611yescharacteristic
211C22×C4#12611yescharacteristic
212C22×C4#12611yescharacteristic
213C22×C4#12611yescharacteristic
214C22×C4#12611yescharacteristic
215M16—82no—
216M16—82no—
217M16—82no—
218M16—82no—
219C2×C8—42no—
220C2×C8—42no—
221C2×C8—42no—
222C2×C8—42no—
223C2×C8—82no—
224C2×C8—82no—
225C2×C8—82no—
226C2×C8—82no—
227C2×C8—82no—
228C2×C8—82no—
229C2×C8—82no—
230C2×C8—82no—
231C4⋊C4—81no—
232C4⋊C4—81no—
233C22⋊C4—24no—
234C22⋊C4—44no—
235C22⋊C4—42no—
236C22⋊C4—42no—
237C22⋊C4—42no—
238C22⋊C4—82no—
239C22⋊C4—82no—
240C22⋊C4—82no—
241C22⋊C4—82no—
242C42—82no—
243C42—82no—
244C42—82no—
245C23—22no—
246C23—22no—
247C23—22no—
248C23—22no—
249C23—22no—
250C23—21no—
251C23—21no—
252C23—22no—
253C23—22no—
254C23—21no—
255C23—21no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23—84no—
268C23—84no—
269C23#38711yescharacteristic
270C23#38811yescharacteristic
271C23#38811yescharacteristic
272C23#39711yescharacteristic
273C23#40011yescharacteristic
274C23#40011yescharacteristic
275C23#40111yescharacteristic
276C23#44312yes—
277C23#44412yes—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
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—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—44no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4—88no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—82no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—82no—
316C2×C4—82no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4—162no—
320C2×C4—162no—
321C2×C4#38311yescharacteristic
322C2×C4#38711yescharacteristic
323C2×C4#38711yescharacteristic
324C2×C4#38811yescharacteristic
325C2×C4#39611yescharacteristic
326C2×C4#39711yescharacteristic
327C2×C4#39711yescharacteristic
328C2×C4#40011yescharacteristic
329C8—82no—
330C8—82no—
331C8—82no—
332C8—82no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—42no—
351C22—42no—
352C22—44no—
353C22—44no—
354C22—42no—
355C22—42no—
356C22—82no—
357C22—81no—
358C22—81no—
359C22—82no—
360C22#144311yescharacteristic
361C22#192212yes—
362C22#192912yes—
363C22#193711yescharacteristic
364C22#193811yescharacteristic
365C22#210412yes—
366C4—21no—
367C4—21no—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—84no—
375C4—161no—
376C4—161no—
377C4—162no—
378C2—22no—
379C2—21no—
380C2—21no—
381C2—22no—
382C2—22no—
383C2—41no—
384C2—41no—
385C2—41no—
386C2#1131311yescharacteristic
387C2#1131711yescharacteristic
388C2#1131811yescharacteristic
subgroupG/H
center, socleC22#1443
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25#41
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
17. #1294C4semidirect
18. #1294C4semidirect

Maximal subgroups

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

subgroupindex
#80572
#80512
#80242
C2×#12842
#79272
C2×#12822
#79132