Finite Group Databasebeta

← #101511

Subgroups — #101511

Order 512.

370 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 76 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#29393C211yesmaximal, characteristic
2#22011C211yesmaximal, characteristic
3#9869C211yesmaximal, characteristic, complemented
4#7639C211yesmaximal, characteristic, complemented
5#7628C211yesmaximal, characteristic
6#7043C211yesmaximal, characteristic, complemented
7#7023C211yesmaximal, characteristic, complemented
8C4×C4⋊Q8C2211yescharacteristic
9#2293C2211yescharacteristic
10#2084C411yescharacteristic
11#2071C411yescharacteristic
12#2057C411yescharacteristic
13#2023C411yescharacteristic
14#1955—21no—
15#1940—21no—
16#1925—21no—
17#1923—22no—
18#1921—21no—
19#1920—21no—
20#1864—21no—
21#1839C2211yescharacteristic
22#1818—21no—
23#1798—21no—
24#1767C2211yescharacteristic
25C4×C4⋊C8C2211yescharacteristic
26#1745C2211yescharacteristic
27#1745C2211yescharacteristic
28#1279—22no—
29#1272—21no—
30#1272—21no—
31#1271—21no—
32#1270—21no—
33C2×C4⋊Q8C2×C411yescharacteristic
34C2×C4×Q8—21no—
35C2×C4×D4D411yescharacteristic
36C2×#117—21no—
37C2×#107—21no—
38C2×#106—21no—
39C2×C4⋊C8—21no—
40C2×C4⋊C8—21no—
41C2×C4⋊C8—22no—
42C2×#102—21no—
43C2×#102—21no—
44C2×C22⋊C8—22no—
45C2×C22⋊C8—22no—
46C2×#97—21no—
47C2×#97—21no—
48C2×#97—21no—
49C2×C4×C8D412yes—
50#391—21no—
51#390C2×C411yescharacteristic
52#390D411yescharacteristic
53#389—21no—
54#389—21no—
55#389C2×C411yescharacteristic
56#389D411yescharacteristic
57#388—21no—
58#388—21no—
59#384C2×C411yescharacteristic
60C4×C4⋊C4—21no—
61C4×C4⋊C4—21no—
62C4×C4⋊C4D411yescharacteristic
63C4×C22⋊C4—21no—
64C43C2311yescharacteristic
65#340—21no—
66#340—21no—
67#340—21no—
68#340—21no—
69#336—22no—
70#336—21no—
71#336—21no—
72#336—21no—
73#336—21no—
74#336—21no—
75#336—21no—
76#336—21no—
77#336C2×C411yescharacteristic
78#336C2×C411yescharacteristic
79#336D411yescharacteristic
80#336D411yescharacteristic
81C22×Q8—21no—
82C22×D4C22⋊C411yescharacteristic
83C23×C4—21no—
84C22×C8—21no—
85C22×C8—21no—
86C22×C8—21no—
87C22×C8—22no—
88C22×C8—21no—
89C22×C8—41no—
90C22×C8—41no—
91C4⋊Q8—42no—
92C4×Q8—81no—
93C4×Q8—81no—
94C4×D4—41no—
95C4×D4—41no—
96#117—82no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—41no—
105C2×C4⋊C4C22⋊C411yescharacteristic
106C2×C4⋊C4C22⋊C411yescharacteristic
107C2×C4⋊C4C22⋊C411yescharacteristic
108C2×C22⋊C4—21no—
109C2×C22⋊C4—21no—
110C2×C22⋊C4—21no—
111C2×C22⋊C4—21no—
112C2×C22⋊C4—21no—
113C2×C42—21no—
114C2×C42—41no—
115C2×C42C22×C411yescharacteristic
116C2×C42C2×D411yescharacteristic
117C2×C42C2×D411yescharacteristic
118C2×C42C2×D411yescharacteristic
119C2×C42C2×D411yescharacteristic
120C2×C42#4111yescharacteristic
121C2×C42#4111yescharacteristic
122#107—42no—
123#106—42no—
124C4⋊C8—22no—
125C4⋊C8—41no—
126C4⋊C8—41no—
127C4⋊C8—41no—
128C4⋊C8—82no—
129#102—42no—
130#102—42no—
131C22⋊C8—82no—
132C22⋊C8—82no—
133#97—22no—
134#97—41no—
135#97—41no—
136#97—41no—
137#97—81no—
138C4×C8—42no—
139#95—21no—
140#95—21no—
141#95—21no—
142#95—41no—
143#95—41no—
144C24—21no—
145C2×Q8—22no—
146C2×Q8—41no—
147C2×Q8—81no—
148C2×Q8—81no—
149C2×D4—21no—
150C2×D4—21no—
151C2×D4—41no—
152C2×D4—41no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4C2×C22⋊C411yescharacteristic
163C22×C4C4×D411yescharacteristic
164C22×C4C4×D411yescharacteristic
165C22×C4#12011yescharacteristic
166C22×C4#12111yescharacteristic
167C22×C4#12111yescharacteristic
168C22×C4#12311yescharacteristic
169C2×C8—42no—
170C2×C8—41no—
171C2×C8—41no—
172C2×C8—41no—
173C2×C8—42no—
174C2×C8—41no—
175C2×C8—42no—
176C2×C8—42no—
177C2×C8—41no—
178C2×C8—41no—
179C2×C8—41no—
180C2×C8—41no—
181C2×C8—41no—
182C2×C8—41no—
183C2×C8—42no—
184C2×C8—41no—
185C2×C8—41no—
186C2×C8—81no—
187C2×C8—81no—
188C2×C8—81no—
189C2×C8—81no—
190C4⋊C4—22no—
191C4⋊C4—22no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—41no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—41no—
198C4⋊C4—41no—
199C4⋊C4—41no—
200C4⋊C4—41no—
201C4⋊C4—41no—
202C4⋊C4—81no—
203C4⋊C4—81no—
204C4⋊C4—81no—
205C4⋊C4—82no—
206C22⋊C4—42no—
207C22⋊C4—41no—
208C22⋊C4—41no—
209C22⋊C4—41no—
210C22⋊C4—41no—
211C22⋊C4—81no—
212C22⋊C4—81no—
213C42—21no—
214C42—21no—
215C42—21no—
216C42—21no—
217C42—21no—
218C42—21no—
219C42—21no—
220C42—21no—
221C42—41no—
222C42—41no—
223C42—41no—
224C42—42no—
225C42—81no—
226C42—81no—
227C23—41no—
228C23—41no—
229C23—41no—
230C23—41no—
231C23—41no—
232C23—41no—
233C23—41no—
234C23#38611yescharacteristic
235Q8—82no—
236Q8—82no—
237D4—42no—
238D4—81no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—42no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—82no—
267C2×C4—82no—
268C2×C4—81no—
269C2×C4—81no—
270C2×C4—81no—
271C2×C4—81no—
272C2×C4—81no—
273C2×C4#42111yescharacteristic
274C2×C4#42111yescharacteristic
275C2×C4#44311yescharacteristic
276C2×C4#44411yescharacteristic
277C2×C4#45511yescharacteristic
278C2×C4#45611yescharacteristic
279C2×C4#47111yescharacteristic
280C2×C4#47211yescharacteristic
281C8—81no—
282C8—81no—
283C8—81no—
284C8—82no—
285C8—81no—
286C8—82no—
287C8—82no—
288C22—81no—
289C22—81no—
290C22—81no—
291C22—81no—
292C22—81no—
293C22—81no—
294C22—81no—
295C22#187611yescharacteristic
296C22#188211yescharacteristic
297C22#189111yescharacteristic
298C22#189711yescharacteristic
299C22#190411yescharacteristic
300C22#190711yescharacteristic
301C22#190911yescharacteristic
302C4—21no—
303C4—21no—
304C4—21no—
305C4—21no—
306C4—21no—
307C4—21no—
308C4—21no—
309C4—21no—
310C4—41no—
311C4—41no—
312C4—41no—
313C4—81no—
314C4—82no—
315C4—81no—
316C4—81no—
317C4—161no—
318C2—161no—
319C2#905111yescharacteristic
320C2#921311yescharacteristic
321C2#927911yescharacteristic
322C2#932711yescharacteristic
323C2#937511yescharacteristic
324C2#949311yescharacteristic
325C2#952911yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
prime radicalC22×D4C22⋊C4
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. #9869C2semidirect
4. #7639C2semidirect
6. #7043C2semidirect
7. #7023C2semidirect

Maximal subgroups

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

subgroupindex
#293932
#220112
#98692
#76392
#76282
#70432
#70232