Finite Group Databasebeta

← #121139

Subgroups — #121139

Order 512.

548 conjugacy classes of subgroups.

The subgroup lattice has 391 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#8051C211yesmaximal, characteristic
2#8024C211yesmaximal, characteristic
3#8008C211yesmaximal, characteristic
4C2×#1284C211yesmaximal, characteristic
5#7925C211yesmaximal, 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#1297—22no—
16#1294C412yescomplemented
17#1294C412yescomplemented
18#1293—22no—
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#407—21no—
39#407D412yes—
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⋊C8C2×C412yes—
54C2×C22⋊C8D412yes—
55C2×C22⋊C8Q812yes—
56#381—21no—
57#381—21no—
58#381—21no—
59#381—21no—
60#381—21no—
61#380—21no—
62#380C2×C412yes—
63#379—21no—
64C4×C22⋊C4—21no—
65C4×C22⋊C4—21no—
66C4×C22⋊C4—21no—
67C4×C22⋊C4—21no—
68C4×C22⋊C4—21no—
69C4×C22⋊C4D412yes—
70C2×#95—21no—
71C2×#95—21no—
72C2×#95C2×C411yescharacteristic
73C2×#95C2×C411yescharacteristic
74C25#4111yescharacteristic
75C23×C4—21no—
76C23×C4—21no—
77C23×C4—41no—
78C23×C4—82no—
79C23×C4C22×C411yescharacteristic
80C23×C4C2×D411yescharacteristic
81C23×C4C2×D411yescharacteristic
82C23×C4C2×D411yescharacteristic
83C23×C4C2×Q811yescharacteristic
84C23×C4#4111yescharacteristic
85C23×C4#4111yescharacteristic
86C23×C4#4111yescharacteristic
87C23×C4#4111yescharacteristic
88C23×C4#4111yescharacteristic
89C23×C4#4111yescharacteristic
90C23×C4#4111yescharacteristic
91C23×C4#4111yescharacteristic
92C23×C4#4111yescharacteristic
93C2×M16—22no—
94C2×M16—41no—
95C2×M16—41no—
96C2×M16—41no—
97C2×M16—82no—
98C2×M16—82no—
99C22×C8—22no—
100C22×C8—22no—
101C22×C8—22no—
102C22×C8—41no—
103C22×C8—41no—
104C22×C8—41no—
105C22×C8—41no—
106C22×C8—41no—
107C22×C8—82no—
108C22×C8—82no—
109#117—82no—
110C2×C4⋊C4—22no—
111C2×C22⋊C4—24no—
112C2×C22⋊C4—22no—
113C2×C22⋊C4—22no—
114C2×C22⋊C4—42no—
115C2×C22⋊C4—41no—
116C2×C22⋊C4—41no—
117C2×C22⋊C4—41no—
118C2×C22⋊C4—41no—
119C2×C22⋊C4—41no—
120C2×C22⋊C4—82no—
121C2×C42—22no—
122C2×C42—41no—
123C2×C42—41no—
124C2×C42—82no—
125C22⋊C8—22no—
126C22⋊C8—22no—
127C22⋊C8—22no—
128C22⋊C8—22no—
129C22⋊C8—22no—
130C22⋊C8—22no—
131C22⋊C8—22no—
132C22⋊C8—22no—
133C22⋊C8—41no—
134C22⋊C8—41no—
135C22⋊C8—41no—
136C22⋊C8—41no—
137C22⋊C8—41no—
138C22⋊C8—41no—
139C22⋊C8—41no—
140C22⋊C8—41no—
141#95—21no—
142#95—21no—
143#95—21no—
144#95—21no—
145#95—22no—
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—41no—
171C22×C4—41no—
172C22×C4—42no—
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—81no—
186C22×C4—81no—
187C22×C4—81no—
188C22×C4—81no—
189C22×C4#11711yescharacteristic
190C22×C4#11711yescharacteristic
191C22×C4#11711yescharacteristic
192C22×C4C4×D411yescharacteristic
193C22×C4C4×D411yescharacteristic
194C22×C4C4×Q811yescharacteristic
195C22×C4#12111yescharacteristic
196C22×C4#12111yescharacteristic
197C22×C4#12111yescharacteristic
198C22×C4C22⋊Q811yescharacteristic
199C22×C4C22⋊Q811yescharacteristic
200C22×C4C22⋊Q811yescharacteristic
201C22×C4#12311yescharacteristic
202C22×C4#12311yescharacteristic
203C22×C4#12311yescharacteristic
204C22×C4#12311yescharacteristic
205C22×C4#12311yescharacteristic
206C22×C4#12411yescharacteristic
207C22×C4#12411yescharacteristic
208C22×C4#12511yescharacteristic
209C22×C4#12511yescharacteristic
210C22×C4#12611yescharacteristic
211C22×C4#12611yescharacteristic
212C22×C4#12611yescharacteristic
213C22×C4#12611yescharacteristic
214C22×C4#12611yescharacteristic
215C22×C4#12611yescharacteristic
216C22×C4#12611yescharacteristic
217M16—82no—
218M16—82no—
219M16—82no—
220M16—82no—
221C2×C8—42no—
222C2×C8—42no—
223C2×C8—42no—
224C2×C8—42no—
225C2×C8—82no—
226C2×C8—82no—
227C2×C8—82no—
228C2×C8—82no—
229C2×C8—82no—
230C2×C8—82no—
231C2×C8—82no—
232C2×C8—82no—
233C4⋊C4—82no—
234C22⋊C4—24no—
235C22⋊C4—44no—
236C22⋊C4—42no—
237C22⋊C4—42no—
238C22⋊C4—42no—
239C22⋊C4—82no—
240C22⋊C4—82no—
241C22⋊C4—82no—
242C22⋊C4—82no—
243C42—82no—
244C42—82no—
245C42—82no—
246C23—22no—
247C23—22no—
248C23—22no—
249C23—22no—
250C23—22no—
251C23—21no—
252C23—21no—
253C23—22no—
254C23—22no—
255C23—21no—
256C23—21no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—84no—
269C23—84no—
270C23#38711yescharacteristic
271C23#38811yescharacteristic
272C23#38811yescharacteristic
273C23#39711yescharacteristic
274C23#40011yescharacteristic
275C23#40011yescharacteristic
276C23#40111yescharacteristic
277C23#44312yes—
278C23#44412yes—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
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—41no—
307C2×C4—44no—
308C2×C4—82no—
309C2×C4—82no—
310C2×C4—88no—
311C2×C4—82no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—82no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—82no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—162no—
321C2×C4—162no—
322C2×C4#38311yescharacteristic
323C2×C4#38711yescharacteristic
324C2×C4#38711yescharacteristic
325C2×C4#38811yescharacteristic
326C2×C4#39611yescharacteristic
327C2×C4#39711yescharacteristic
328C2×C4#39711yescharacteristic
329C2×C4#40011yescharacteristic
330C8—82no—
331C8—82no—
332C8—82no—
333C8—82no—
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—21no—
351C22—42no—
352C22—42no—
353C22—44no—
354C22—44no—
355C22—42no—
356C22—42no—
357C22—82no—
358C22—81no—
359C22—81no—
360C22—82no—
361C22#144311yescharacteristic
362C22#192212yes—
363C22#192912yes—
364C22#193711yescharacteristic
365C22#193811yescharacteristic
366C22#210412yes—
367C4—21no—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—84no—
376C4—162no—
377C4—161no—
378C4—161no—
379C2—22no—
380C2—21no—
381C2—21no—
382C2—22no—
383C2—22no—
384C2—41no—
385C2—41no—
386C2—41no—
387C2#1131011yescharacteristic
388C2#1131711yescharacteristic
389C2#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
16. #1294C4semidirect
17. #1294C4semidirect

Maximal subgroups

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

subgroupindex
#80512
#80242
#80082
C2×#12842
#79252
C2×#12822
#79132