Finite Group Databasebeta

← #137828

Subgroups — #137828

Order 512.

334 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 92 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#33488C211yesmaximal, characteristic
2C2×#1585C211yesmaximal, characteristic
3#11642C211yesmaximal, characteristic, complemented
4#11631C211yesmaximal, characteristic
5#11495C211yesmaximal, characteristic, complemented
6#7195C211yesmaximal, characteristic, complemented
7#7146C211yesmaximal, characteristic, complemented
8C2×#446C2211yescharacteristic
9C2×C4×SD16C411yescharacteristic
10C2×C8×D4C2211yescharacteristic
11#2169—21no—
12#2168—22no—
13D4×C16—21no—
14#2163C2211yescharacteristic
15#2163C2211yescharacteristic
16#2162C2211yescharacteristic
17#2162C2211yescharacteristic
18#2112—21no—
19C2×C22⋊C16—21no—
20#1585—21no—
21#1585—21no—
22#1585—21no—
23#1585—21no—
24C2×#335C411yescharacteristic
25C2×Q8⋊C8C411yescharacteristic
26C2×D4⋊C8C411yescharacteristic
27C2×#322C2211yescharacteristic
28#1338—22no—
29D4⋊C16—21no—
30D4⋊C16—21no—
31#1322—21no—
32#1322—21no—
33C22×SD16C811yescharacteristic
34C23×C8—21no—
35C2×C4×Q8C2×C411yescharacteristic
36C2×C4×D4C2×C411yescharacteristic
37C2×M32—22no—
38C22×C16—22no—
39#446—21no—
40#446—21no—
41#446—21no—
42#446—21no—
43C4×SD16—21no—
44C4×SD16—21no—
45C4×SD16—21no—
46C4×SD16—21no—
47C8×D4—21no—
48C8×D4—21no—
49C8×D4D411yescharacteristic
50C8×D4D411yescharacteristic
51C8×D4D411yescharacteristic
52C8×D4D411yescharacteristic
53C2×#106C811yescharacteristic
54C2×C4⋊C8—21no—
55C2×C4⋊C8C2×C411yescharacteristic
56C2×C4⋊C8C2×C411yescharacteristic
57C2×#103C811yescharacteristic
58C2×#102C811yescharacteristic
59C2×C22⋊C8—21no—
60C2×#97—21no—
61C2×C4×C8C2×C411yescharacteristic
62C2×C4×C8C2×C411yescharacteristic
63C2×C4×C8C2311yescharacteristic
64C4⋊C16—21no—
65C4⋊C16—21no—
66C4⋊C16—21no—
67C4⋊C16—21no—
68C22⋊C16—22no—
69C22⋊C16—22no—
70C22⋊C16—21no—
71C22⋊C16—21no—
72C22⋊C16—21no—
73C22⋊C16D411yescharacteristic
74C22⋊C16D411yescharacteristic
75#346—21no—
76#346D412yes—
77C4×C16—21no—
78C4×C16—21no—
79#335—21no—
80#335—21no—
81Q8⋊C8—21no—
82Q8⋊C8—21no—
83D4⋊C8—21no—
84D4⋊C8—21no—
85#322—21no—
86#322—21no—
87C22×Q8C2×C811yescharacteristic
88C22×D4C2×C811yescharacteristic
89C23×C4—21no—
90C2×SD16—21no—
91C2×SD16—21no—
92C2×SD16—42no—
93C22×C8—21no—
94C22×C8—21no—
95C22×C8—21no—
96C22×C8—21no—
97C22×C8—21no—
98C22×C8—21no—
99C22×C8—21no—
100C22×C8—82no—
101C22×C8C2×C811yescharacteristic
102C22×C8C2×C811yescharacteristic
103C22×C8C2×D411yescharacteristic
104C22×C8#4111yescharacteristic
105C4×Q8—21no—
106C4×Q8—21no—
107C4×Q8—21no—
108C4×Q8—21no—
109C4×D4—21no—
110C4×D4—21no—
111C4×D4C22⋊C411yescharacteristic
112C4×D4C22⋊C411yescharacteristic
113C4×D4M1611yescharacteristic
114C4×D4M1611yescharacteristic
115C2×C4⋊C4—21no—
116C2×C4⋊C4C2×C811yescharacteristic
117C2×C4⋊C4C2×C811yescharacteristic
118C2×C22⋊C4—21no—
119C2×C42—21no—
120C2×C42C22×C411yescharacteristic
121M32—82no—
122C2×C16—21no—
123C2×C16—21no—
124C2×C16—22no—
125C2×C16—41no—
126C2×C16—41no—
127C2×C16—82no—
128#106—21no—
129#106—21no—
130C4⋊C8—21no—
131C4⋊C8—21no—
132C4⋊C8—21no—
133C4⋊C8—41no—
134C4⋊C8—41no—
135C4⋊C8C22⋊C411yescharacteristic
136C4⋊C8C22⋊C411yescharacteristic
137#103—21no—
138#103—21no—
139#102—21no—
140#102—21no—
141C22⋊C8—21no—
142C22⋊C8—21no—
143C22⋊C8—21no—
144C22⋊C8—21no—
145#97—41no—
146#97—41no—
147C4×C8—21no—
148C4×C8—21no—
149C4×C8—21no—
150C4×C8—21no—
151C4×C8C2×D411yescharacteristic
152C4×C8C2×D411yescharacteristic
153C4×C8C2×D411yescharacteristic
154C4×C8#4111yescharacteristic
155C24—21no—
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—41no—
159C2×Q8—41no—
160C2×D4—22no—
161C2×D4—21no—
162C2×D4—41no—
163C2×D4C22⋊C811yescharacteristic
164C2×D4C22⋊C811yescharacteristic
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—81no—
173C22×C4—81no—
174C22×C4C4×D411yescharacteristic
175C22×C4C22×C811yescharacteristic
176C22×C4#13111yescharacteristic
177SD16—42no—
178SD16—42no—
179C2×C8—22no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—41no—
185C2×C8—41no—
186C2×C8—42no—
187C2×C8—42no—
188C2×C8—41no—
189C2×C8—41no—
190C2×C8—41no—
191C2×C8—41no—
192C2×C8—41no—
193C2×C8—81no—
194C2×C8—81no—
195C2×C8—81no—
196C2×C8—81no—
197C2×C8#12011yescharacteristic
198C2×C8#12111yescharacteristic
199C2×C8#12111yescharacteristic
200C2×C8#12311yescharacteristic
201C4⋊C4—21no—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—41no—
205C4⋊C4—41no—
206C4⋊C4C22⋊C811yescharacteristic
207C4⋊C4C22⋊C811yescharacteristic
208C22⋊C4—21no—
209C22⋊C4—21no—
210C22⋊C4—21no—
211C22⋊C4—21no—
212C42—41no—
213C42—41no—
214C42C2×C22⋊C411yescharacteristic
215C42C4×D411yescharacteristic
216C42C2×M1611yescharacteristic
217C42#13111yescharacteristic
218C16—41no—
219C16—42no—
220C16—41no—
221C16—81no—
222C16—81no—
223C23—21no—
224C23—21no—
225C23—42no—
226C23—41no—
227C23—41no—
228C23—81no—
229C23C8×D411yescharacteristic
230Q8—41no—
231Q8—41no—
232Q8—81no—
233D4—21no—
234D4—21no—
235D4—21no—
236D4—42no—
237D4#34912yes—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—81no—
255C2×C4—81no—
256C2×C4—81no—
257C2×C4—81no—
258C2×C4#38611yescharacteristic
259C2×C4C2×C22⋊C811yescharacteristic
260C2×C4#40811yescharacteristic
261C2×C4C8×D411yescharacteristic
262C2×C4#43511yescharacteristic
263C2×C4#43511yescharacteristic
264C8—21no—
265C8—21no—
266C8—21no—
267C8—21no—
268C8—42no—
269C8—41no—
270C8—41no—
271C8—41no—
272C8—41no—
273C8—81no—
274C22—21no—
275C22—21no—
276C22—22no—
277C22—41no—
278C22—41no—
279C22—41no—
280C22—41no—
281C22—41no—
282C22—82no—
283C22—82no—
284C22#186911yescharacteristic
285C22#217011yescharacteristic
286C22#217111yescharacteristic
287C4—21no—
288C4—21no—
289C4—21no—
290C4—21no—
291C4—41no—
292C4—41no—
293C4—81no—
294C4—81no—
295C4#190211yescharacteristic
296C4#190411yescharacteristic
297C4C2×#34911yescharacteristic
298C4#211511yescharacteristic
299C2—21no—
300C2—21no—
301C2—42no—
302C2—81no—
303C2#908711yescharacteristic
304C2#1298011yescharacteristic
305C2#1298511yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC22×C4C22×C8
Frattini, cosocleC2×C4×C8C23
socleC22#1869
prime radicalC22×D4C2×C8
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11642C2semidirect
5. #11495C2semidirect
6. #7195C2semidirect
7. #7146C2semidirect

Maximal subgroups

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

subgroupindex
#334882
C2×#15852
#116422
#116312
#114952
#71952
#71462