Finite Group Databasebeta

← #121260

Subgroups — #121260

Order 512.

562 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 562 conjugacy classes of subgroups — too many to draw. Only its 110 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#8114C211yesmaximal, characteristic
2C2×#1297C211yesmaximal, characteristic
3#8009C211yesmaximal, characteristic
4C2×#1284C211yesmaximal, characteristic
5C2×#1284C211yesmaximal, characteristic
6#7929C211yesmaximal, characteristic
7#7929C211yesmaximal, characteristic
8C2×#407C2211yescharacteristic
9C2×#407C2211yescharacteristic
10C2×#407C2211yescharacteristic
11C2×#407C2211yescharacteristic
12C2×#381C2211yescharacteristic
13C2×C4×C22⋊C4C2211yescharacteristic
14C2×C4×C22⋊C4C2211yescharacteristic
15#1306—22no—
16#1297—22no—
17#1293—22no—
18#1284—22no—
19#1284—22no—
20#1282—22no—
21#1282C414yescomplemented
22C24×C4C2311yescharacteristic
23C22×M16—21no—
24C22×M16—21no—
25C22×M16—21no—
26C22×M16—21no—
27C22×C22⋊C4—21no—
28C22×C22⋊C4—21no—
29C22×C22⋊C4—21no—
30C22×C42—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#407—21no—
40#407—21no—
41#407—21no—
42#407—21no—
43#407—21no—
44#407—21no—
45#407—21no—
46#407—21no—
47#407D412yes—
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⋊C8C2×C412yes—
55C2×C22⋊C8D412yes—
56#381—21no—
57#381—21no—
58#381—21no—
59#381D412yes—
60C4×C22⋊C4—21no—
61C4×C22⋊C4—21no—
62C4×C22⋊C4—21no—
63C4×C22⋊C4—21no—
64C4×C22⋊C4—21no—
65C4×C22⋊C4—21no—
66C4×C22⋊C4C2×C412yes—
67C4×C22⋊C4Q812yes—
68C2×#95—21no—
69C2×#95—21no—
70C2×#95—21no—
71C2×#95—21no—
72C25#4111yescharacteristic
73C23×C4—41no—
74C23×C4—41no—
75C23×C4—41no—
76C23×C4—82no—
77C23×C4C22×C411yescharacteristic
78C23×C4C2×D411yescharacteristic
79C23×C4C2×D411yescharacteristic
80C23×C4C2×D411yescharacteristic
81C23×C4C2×Q811yescharacteristic
82C23×C4#4111yescharacteristic
83C23×C4#4111yescharacteristic
84C23×C4#4111yescharacteristic
85C23×C4#4111yescharacteristic
86C23×C4#4111yescharacteristic
87C23×C4#4111yescharacteristic
88C23×C4#4111yescharacteristic
89C23×C4#4111yescharacteristic
90C23×C4#4111yescharacteristic
91C2×M16—22no—
92C2×M16—41no—
93C2×M16—41no—
94C2×M16—41no—
95C2×M16—41no—
96C2×M16—41no—
97C2×M16—41no—
98C2×M16—41no—
99C2×M16—82no—
100C2×M16—82no—
101C2×M16—82no—
102C2×M16—82no—
103C22×C8—22no—
104C22×C8—22no—
105C22×C8—22no—
106C22×C8—41no—
107C2×C22⋊C4—22no—
108C2×C22⋊C4—22no—
109C2×C22⋊C4—22no—
110C2×C22⋊C4—44no—
111C2×C22⋊C4—44no—
112C2×C22⋊C4—41no—
113C2×C22⋊C4—41no—
114C2×C22⋊C4—41no—
115C2×C22⋊C4—82no—
116C2×C42—22no—
117C2×C42—22no—
118C2×C42—41no—
119C2×C42—41no—
120C2×C42—82no—
121C2×C42—82no—
122C22⋊C8—22no—
123C22⋊C8—22no—
124C22⋊C8—22no—
125C22⋊C8—22no—
126C22⋊C8—22no—
127C22⋊C8—22no—
128C22⋊C8—22no—
129C22⋊C8—22no—
130C22⋊C8—41no—
131C22⋊C8—41no—
132C22⋊C8—41no—
133C22⋊C8—41no—
134C22⋊C8—41no—
135C22⋊C8—41no—
136C22⋊C8—41no—
137C22⋊C8—41no—
138#95—22no—
139#95—22no—
140#95—22no—
141#95—22no—
142#95—41no—
143#95—41no—
144#95—41no—
145#95—41no—
146C24—42no—
147C24—42no—
148C24—42no—
149C24#11711yescharacteristic
150C24#12111yescharacteristic
151C24C22⋊Q811yescharacteristic
152C24#12611yescharacteristic
153C24#12611yescharacteristic
154C24#12611yescharacteristic
155C24#12611yescharacteristic
156C22×C4—42no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—44no—
163C22×C4—44no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—81no—
168C22×C4—81no—
169C22×C4—81no—
170C22×C4—81no—
171C22×C4—81no—
172C22×C4—81no—
173C22×C4—81no—
174C22×C4—81no—
175C22×C4—82no—
176C22×C4—82no—
177C22×C4—82no—
178C22×C4—82no—
179C22×C4—82no—
180C22×C4—82no—
181C22×C4—82no—
182C22×C4#11711yescharacteristic
183C22×C4#11711yescharacteristic
184C22×C4C4×D411yescharacteristic
185C22×C4C4×D411yescharacteristic
186C22×C4C4×D411yescharacteristic
187C22×C4C4×Q811yescharacteristic
188C22×C4#12111yescharacteristic
189C22×C4#12111yescharacteristic
190C22×C4C22⋊Q811yescharacteristic
191C22×C4C22⋊Q811yescharacteristic
192C22×C4#12311yescharacteristic
193C22×C4#12311yescharacteristic
194C22×C4#12311yescharacteristic
195C22×C4#12311yescharacteristic
196C22×C4#12311yescharacteristic
197C22×C4#12311yescharacteristic
198C22×C4#12411yescharacteristic
199C22×C4#12411yescharacteristic
200C22×C4#12411yescharacteristic
201C22×C4#12511yescharacteristic
202C22×C4#12511yescharacteristic
203C22×C4#12511yescharacteristic
204C22×C4#12611yescharacteristic
205C22×C4#12611yescharacteristic
206C22×C4#12611yescharacteristic
207C22×C4#12611yescharacteristic
208C22×C4#12611yescharacteristic
209C22×C4#12611yescharacteristic
210M16—82no—
211M16—82no—
212M16—82no—
213M16—82no—
214M16—82no—
215M16—82no—
216M16—82no—
217M16—82no—
218C2×C8—42no—
219C2×C8—42no—
220C2×C8—42no—
221C2×C8—42no—
222C2×C8—82no—
223C2×C8—82no—
224C2×C8—82no—
225C2×C8—82no—
226C22⋊C4—24no—
227C22⋊C4—24no—
228C22⋊C4—42no—
229C22⋊C4—42no—
230C22⋊C4—44no—
231C22⋊C4—44no—
232C22⋊C4—82no—
233C22⋊C4—82no—
234C42—82no—
235C42—82no—
236C42—82no—
237C42—82no—
238C23—22no—
239C23—22no—
240C23—22no—
241C23—22no—
242C23—22no—
243C23—22no—
244C23—22no—
245C23—22no—
246C23—22no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—88no—
260C23#38311yescharacteristic
261C23#38711yescharacteristic
262C23#38811yescharacteristic
263C23#39611yescharacteristic
264C23#39711yescharacteristic
265C23#40011yescharacteristic
266C23#40111yescharacteristic
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
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—44no—
296C2×C4—44no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4—82no—
301C2×C4—88no—
302C2×C4—88no—
303C2×C4—82no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—164no—
307C2×C4#38711yescharacteristic
308C2×C4#38711yescharacteristic
309C2×C4#38811yescharacteristic
310C2×C4#38811yescharacteristic
311C2×C4#39711yescharacteristic
312C2×C4#39711yescharacteristic
313C2×C4#40011yescharacteristic
314C2×C4#40011yescharacteristic
315C8—82no—
316C8—82no—
317C8—82no—
318C8—82no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—44no—
338C22—44no—
339C22—44no—
340C22—82no—
341C22—82no—
342C22—82no—
343C22—82no—
344C22#144311yescharacteristic
345C22#184112yes—
346C22#210212yes—
347C22#210412yes—
348C4—21no—
349C4—21no—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—84no—
357C4—84no—
358C4—162no—
359C2—22no—
360C2—22no—
361C2—22no—
362C2—41no—
363C2—41no—
364C2—41no—
365C2—41no—
366C2#1129911yescharacteristic
367C2#1131811yescharacteristic
368C2#1132111yescharacteristic
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
21. #1282C4semidirect

Maximal subgroups

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

subgroupindex
#81142
C2×#12972
#80092
C2×#12842
#79292