Finite Group Databasebeta

← #101459

Subgroups — #101459

Order 512.

481 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 481 conjugacy classes of subgroups — too many to draw. Only its 96 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#28160C211yesmaximal, characteristic
2#22264C211yesmaximal, characteristic
3#9874C211yesmaximal, characteristic, complemented
4#8588C211yesmaximal, characteristic
5#7652C211yesmaximal, characteristic, complemented
6#7040C211yesmaximal, characteristic, complemented
7#7018C211yesmaximal, characteristic, complemented
8C2×#460—21no—
9C2×#436—21no—
10C4×C4⋊Q8C2211yescharacteristic
11C4×#127C2211yescharacteristic
12#2071C412yes—
13#2013C412yes—
14#1954—21no—
15#1877—22no—
16#1870—22no—
17#1870—22no—
18#1842C2211yescharacteristic
19#1831C2211yescharacteristic
20#1831C2211yescharacteristic
21#1831C2211yescharacteristic
22#1803—21no—
23#1744C2211yescharacteristic
24C2×#331—21no—
25#1296—21no—
26#1279—21no—
27#1279—21no—
28#1271—22no—
29C22×SD16—22no—
30C22×M16—22no—
31C2×C4⋊Q8C2×C411yescharacteristic
32C2×#127D411yescharacteristic
33C2×C4×Q8—21no—
34C2×C4×D4—21no—
35C2×C4×D4—21no—
36C2×C4×D4D412yes—
37#460—42no—
38#460—42no—
39#436—42no—
40#436—42no—
41C2×C4⋊C8—21no—
42C2×C4⋊C8—21no—
43C2×C4⋊C8—21no—
44C2×C4⋊C8—21no—
45C2×C4⋊C8—21no—
46C2×C4⋊C8—21no—
47C2×#103—22no—
48C2×#102—22no—
49C2×C22⋊C8—22no—
50C2×C22⋊C8—22no—
51C2×C22⋊C8—22no—
52C2×C4×C8D411yescharacteristic
53C2×C4×C8D411yescharacteristic
54#391—21no—
55#390—21no—
56#390—21no—
57#390C2×C412yes—
58#389—21no—
59#389—21no—
60#384C2×C411yescharacteristic
61#384D411yescharacteristic
62C4×C4⋊C4—21no—
63C4×C4⋊C4—21no—
64C43C2311yescharacteristic
65#340—22no—
66#336—21no—
67#336—21no—
68#336—21no—
69#336—21no—
70#336—21no—
71#336—21no—
72#336C2×C412yes—
73#336D412yes—
74#331—42no—
75C22×Q8—21no—
76C22×D4—21no—
77C22×D4—21no—
78C22×D4C22⋊C412yes—
79C23×C4—22no—
80C2×SD16—44no—
81C2×SD16—44no—
82C2×SD16—82no—
83C2×M16—44no—
84C2×M16—82no—
85C2×M16—82no—
86C22×C8—21no—
87C22×C8—22no—
88C22×C8—22no—
89C22×C8—21no—
90C22×C8—41no—
91C22×C8—41no—
92C4⋊Q8—42no—
93#127—21no—
94#127—21no—
95#127—21no—
96#127—21no—
97C4×Q8—41no—
98C4×Q8—41no—
99C4×Q8—41no—
100C4×Q8—41no—
101C4×D4—41no—
102C4×D4—41no—
103C4×D4—41no—
104C4×D4—41no—
105C4×D4—82no—
106C4×D4—82no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4C22⋊C412yes—
116C2×C22⋊C4—22no—
117C2×C22⋊C4—22no—
118C2×C22⋊C4—22no—
119C2×C42—41no—
120C2×C42C22×C411yescharacteristic
121C2×C42C2×D411yescharacteristic
122C2×C42C2×D411yescharacteristic
123C2×C42C2×D411yescharacteristic
124C2×C42C2×D411yescharacteristic
125C2×C42#4111yescharacteristic
126C2×C42#4111yescharacteristic
127C4⋊C8—41no—
128C4⋊C8—41no—
129C4⋊C8—41no—
130C4⋊C8—41no—
131C4⋊C8—41no—
132C4⋊C8—41no—
133C4⋊C8—41no—
134C4⋊C8—41no—
135C4⋊C8—41no—
136C4⋊C8—41no—
137C4⋊C8—81no—
138#103—82no—
139#102—44no—
140C22⋊C8—44no—
141C22⋊C8—82no—
142C22⋊C8—82no—
143C4×C8—21no—
144C4×C8—21no—
145C4×C8—21no—
146C4×C8SD1612yescomplemented
147#95—41no—
148#95—41no—
149C24—22no—
150C2×Q8—22no—
151C2×Q8—41no—
152C2×Q8—41no—
153C2×Q8—41no—
154C2×Q8—41no—
155C2×Q8—41no—
156C2×D4—22no—
157C2×D4—22no—
158C2×D4—42no—
159C2×D4—42no—
160C2×D4—41no—
161C2×D4—41no—
162C2×D4—41no—
163C2×D4—41no—
164C2×D4—41no—
165C2×D4—41no—
166C2×D4—41no—
167C2×D4—41no—
168C2×D4—81no—
169C2×D4—81no—
170C22×C4—22no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—41no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—41no—
179C22×C4C2×C22⋊C411yescharacteristic
180C22×C4C4×D411yescharacteristic
181C22×C4C4×D411yescharacteristic
182C22×C4#12011yescharacteristic
183C22×C4#12111yescharacteristic
184C22×C4#12111yescharacteristic
185C22×C4#12311yescharacteristic
186SD16—88no—
187M16—88no—
188C2×C8—22no—
189C2×C8—41no—
190C2×C8—41no—
191C2×C8—41no—
192C2×C8—41no—
193C2×C8—42no—
194C2×C8—42no—
195C2×C8—42no—
196C2×C8—42no—
197C2×C8—42no—
198C2×C8—42no—
199C2×C8—41no—
200C2×C8—81no—
201C2×C8—81no—
202C2×C8—81no—
203C2×C8—81no—
204C2×C8—81no—
205C2×C8—81no—
206C4⋊C4—22no—
207C4⋊C4—22no—
208C4⋊C4—42no—
209C4⋊C4—41no—
210C4⋊C4—41no—
211C4⋊C4—42no—
212C4⋊C4—41no—
213C4⋊C4—41no—
214C4⋊C4—41no—
215C4⋊C4—41no—
216C4⋊C4—81no—
217C4⋊C4—81no—
218C4⋊C4—81no—
219C4⋊C4—81no—
220C22⋊C4—42no—
221C22⋊C4—42no—
222C22⋊C4—82no—
223C22⋊C4—82no—
224C42—21no—
225C42—21no—
226C42—41no—
227C42—41no—
228C42—41no—
229C42—41no—
230C42—81no—
231C42—81no—
232C42C2×SD1611yescharacteristic
233C42#13511yescharacteristic
234C42AGL(1,ℤ/8)11yescharacteristic
235C42AGL(1,ℤ/8)11yescharacteristic
236C42AGL(1,ℤ/8)11yescharacteristic
237C42AGL(1,ℤ/8)11yescharacteristic
238C42#13711yescharacteristic
239C42#13711yescharacteristic
240C23—42no—
241C23—42no—
242C23—42no—
243C23—42no—
244C23—42no—
245C23—42no—
246C23—42no—
247C23#38611yescharacteristic
248Q8—42no—
249Q8—42no—
250Q8—81no—
251Q8—81no—
252D4—44no—
253D4—82no—
254D4—82no—
255D4—82no—
256D4—81no—
257D4—81no—
258D4—81no—
259D4—81no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—82no—
279C2×C4—82no—
280C2×C4—82no—
281C2×C4—82no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4—81no—
287C2×C4#41711yescharacteristic
288C2×C4#41811yescharacteristic
289C2×C4C4×SD1611yescharacteristic
290C2×C4#44011yescharacteristic
291C2×C4#45011yescharacteristic
292C2×C4#45211yescharacteristic
293C2×C4#46111yescharacteristic
294C2×C4#46311yescharacteristic
295C2×C4#46811yescharacteristic
296C2×C4#46811yescharacteristic
297C2×C4#46911yescharacteristic
298C2×C4#46911yescharacteristic
299C2×C4#47111yescharacteristic
300C2×C4#47311yescharacteristic
301C8—42no—
302C8—81no—
303C8—82no—
304C8—82no—
305C8—161no—
306C8—161no—
307C22—82no—
308C22—82no—
309C22—82no—
310C22—82no—
311C22—82no—
312C22—82no—
313C22—82no—
314C22#187711yescharacteristic
315C22#188111yescharacteristic
316C22#189111yescharacteristic
317C22#189511yescharacteristic
318C22#190411yescharacteristic
319C22#190811yescharacteristic
320C22#191111yescharacteristic
321C4—21no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—21no—
327C4—21no—
328C4—21no—
329C4—41no—
330C4—41no—
331C4—82no—
332C4—82no—
333C4—161no—
334C4#169111yescharacteristic
335C4#169311yescharacteristic
336C4#169911yescharacteristic
337C4#170111yescharacteristic
338C2—162no—
339C2#904711yescharacteristic
340C2#922511yescharacteristic
341C2#929511yescharacteristic
342C2#930911yescharacteristic
343C2#937311yescharacteristic
344C2#948911yescharacteristic
345C2#952711yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
prime radicalC2×#127D4
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #9874C2semidirect
5. #7652C2semidirect
6. #7040C2semidirect
7. #7018C2semidirect
146. C4×C8SD16semidirect

Maximal subgroups

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

subgroupindex
#281602
#222642
#98742
#85882
#76522
#70402
#70182