Finite Group Databasebeta

← #138587

Subgroups — #138587

Order 512.

367 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 367 conjugacy classes of subgroups — too many to draw. Only its 100 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
1D4×M32C211yesmaximal, characteristic
2#19105C211yesmaximal, characteristic
3#13287C211yesmaximal, characteristic, complemented
4#11936C211yesmaximal, characteristic, complemented
5#8642C211yesmaximal, characteristic
6#7278C211yesmaximal, characteristic, complemented
7#7158C211yesmaximal, characteristic, complemented
8C22×M32—21no—
9#3160C2211yescharacteristic
10#3101—21no—
11C2×C8×D4C2211yescharacteristic
12#2169—21no—
13#2168—22no—
14D4×C16—21no—
15#2150C2211yescharacteristic
16#2112—21no—
17C4×M32C2211yescharacteristic
18#1844C2211yescharacteristic
19#1802—21no—
20#1675C411yescharacteristic, complemented
21#1675C411yescharacteristic, complemented
22#1673C411yescharacteristic, complemented
23#1673C411yescharacteristic, complemented
24#1671—21no—
25#1671—21no—
26#1564C2211yescharacteristic
27C2×#334C2211yescharacteristic
28#1542—21no—
29#1386—21no—
30#1386—21no—
31#1363—22no—
32#1296—21no—
33C23×C8—21no—
34#533—21no—
35C2×C4×D4D411yescharacteristic
36C2×M32—22no—
37C2×M32—21no—
38C2×M32—42no—
39C2×M32—42no—
40C2×M32D411yescharacteristic
41C2×M32D411yescharacteristic
42C22×C16—22no—
43#500C2×C411yescharacteristic
44#500C2×C411yescharacteristic
45#499—21no—
46#499—21no—
47#498—21no—
48#483—41no—
49#476—21no—
50#476—21no—
51#461—21no—
52#461—21no—
53#450—41no—
54C4×SD16—41no—
55C4×SD16—41no—
56C8×D4—21no—
57C8×D4—21no—
58C8×D4C2×C411yescharacteristic
59C8×D4C2×C411yescharacteristic
60C8×D4D411yescharacteristic
61C8×D4D411yescharacteristic
62#431—21no—
63#427—21no—
64#427—21no—
65C2×#106—21no—
66C2×#106—21no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8D411yescharacteristic
69C2×#102—41no—
70C2×C22⋊C8—21no—
71C2×C4×C8C2311yescharacteristic
72#384—21no—
73C4⋊C16—21no—
74C4⋊C16D412yes—
75C22⋊C16—22no—
76C22⋊C16—22no—
77#346—21no—
78C4×C16—21no—
79#340—41no—
80#335C2×C411yescharacteristic
81#335C2×C411yescharacteristic
82#334—21no—
83#334—21no—
84#334—21no—
85#333—21no—
86#333—21no—
87#332—21no—
88#332—21no—
89C22×D4D811yescharacteristic
90C23×C4—21no—
91C2×SD16—41no—
92C2×SD16—41no—
93C22×C8—21no—
94C22×C8—21no—
95C22×C8—21no—
96C22×C8—21no—
97C22×C8—21no—
98C22×C8—41no—
99C22×C8—41no—
100C22×C8—41no—
101C22×C8—41no—
102C22×C8—41no—
103C22×C8C2×D411yescharacteristic
104C22×C8#4111yescharacteristic
105C4⋊Q8—21no—
106C4⋊Q8—21no—
107#125—41no—
108C22⋊Q8—41no—
109C22⋊Q8—41no—
110C4×Q8—41no—
111C4×Q8—41no—
112C4×D4—21no—
113C4×D4—21no—
114C4×D4C22⋊C411yescharacteristic
115C4×D4C22⋊C411yescharacteristic
116C4×D4SD1611yescharacteristic
117C4×D4SD1611yescharacteristic
118#117—41no—
119C2×C4⋊C4—41no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4D811yescharacteristic
123C2×C22⋊C4—21no—
124C2×C42C2×D411yescharacteristic
125M32—21no—
126M32—21no—
127M32—21no—
128M32—44no—
129M32—41no—
130M32—41no—
131M32SD1612yescomplemented
132C2×C16—21no—
133C2×C16—21no—
134C2×C16—22no—
135C2×C16—44no—
136#107—21no—
137#107—21no—
138#107—21no—
139#107—21no—
140#106—41no—
141#106—41no—
142#106—41no—
143#106—41no—
144#106—41no—
145#106—41no—
146#106—81no—
147C4⋊C8—21no—
148C4⋊C8—21no—
149C4⋊C8—21no—
150C4⋊C8—41no—
151C4⋊C8—41no—
152C4⋊C8C22⋊C411yescharacteristic
153C4⋊C8C22⋊C411yescharacteristic
154#103—41no—
155#103—41no—
156#102—41no—
157#102—41no—
158#102—82no—
159C22⋊C8—21no—
160C22⋊C8—21no—
161C22⋊C8—21no—
162C22⋊C8—21no—
163C22⋊C8—41no—
164C4×C8—41no—
165C4×C8C22×C411yescharacteristic
166C4×C8C2×D411yescharacteristic
167C4×C8C2×D411yescharacteristic
168C4×C8#4111yescharacteristic
169C24—21no—
170C2×Q8—41no—
171C2×Q8—41no—
172C2×D4—22no—
173C2×D4—21no—
174C2×D4—41no—
175C2×D4#10211yescharacteristic
176C2×D4#10211yescharacteristic
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—81no—
187C22×C4#12111yescharacteristic
188C22×C4C2×D811yescharacteristic
189C22×C4#13511yescharacteristic
190SD16—82no—
191SD16—82no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—41no—
197C2×C8—41no—
198C2×C8—41no—
199C2×C8—41no—
200C2×C8—41no—
201C2×C8—41no—
202C2×C8—41no—
203C2×C8—41no—
204C2×C8—41no—
205C2×C8—41no—
206C2×C8—41no—
207C2×C8—41no—
208C2×C8—81no—
209C2×C8—81no—
210C2×C8—81no—
211C2×C8C4×D411yescharacteristic
212C2×C8C4×D411yescharacteristic
213C2×C8#12011yescharacteristic
214C2×C8#12311yescharacteristic
215C2×C8C2×SD1611yescharacteristic
216C2×C8#13511yescharacteristic
217C2×C8AGL(1,ℤ/8)11yescharacteristic
218C2×C8#13711yescharacteristic
219C4⋊C4—21no—
220C4⋊C4—41no—
221C4⋊C4—41no—
222C4⋊C4—41no—
223C4⋊C4—41no—
224C4⋊C4—81no—
225C4⋊C4—81no—
226C4⋊C4—81no—
227C4⋊C4—81no—
228C4⋊C4—81no—
229C4⋊C4—81no—
230C4⋊C4—161no—
231C4⋊C4#10211yescharacteristic
232C4⋊C4#10211yescharacteristic
233C22⋊C4—21no—
234C22⋊C4—21no—
235C22⋊C4—21no—
236C22⋊C4—21no—
237C22⋊C4—81no—
238C42—81no—
239C42C2×C22⋊C411yescharacteristic
240C42#12111yescharacteristic
241C42C2×SD1611yescharacteristic
242C42#13511yescharacteristic
243C16—22no—
244C16—42no—
245C16—41no—
246C23—21no—
247C23—21no—
248C23—42no—
249C23—41no—
250C23—41no—
251C23—81no—
252C23#46611yescharacteristic
253Q8—81no—
254Q8—81no—
255Q8—161no—
256D4—21no—
257D4—21no—
258D4—21no—
259D4—42no—
260D4#36012yes—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4—161no—
279C2×C4—161no—
280C2×C4—161no—
281C2×C4#38611yescharacteristic
282C2×C4C2×#10211yescharacteristic
283C2×C4#41611yescharacteristic
284C2×C4#46111yescharacteristic
285C2×C4#46311yescharacteristic
286C2×C4#46511yescharacteristic
287C2×C4#46511yescharacteristic
288C2×C4#46611yescharacteristic
289C2×C4#47111yescharacteristic
290C2×C4#47311yescharacteristic
291C8—21no—
292C8—21no—
293C8—41no—
294C8—41no—
295C8—81no—
296C8—81no—
297C8—161no—
298C8C4×SD1611yescharacteristic
299C8#44011yescharacteristic
300C8#45011yescharacteristic
301C8#45211yescharacteristic
302C22—21no—
303C22—21no—
304C22—22no—
305C22—41no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22—41no—
310C22—82no—
311C22—82no—
312C22#167311yescharacteristic
313C22#168111yescharacteristic
314C22#189311yescharacteristic
315C4—21no—
316C4—21no—
317C4—21no—
318C4—41no—
319C4—41no—
320C4—161no—
321C4—161no—
322C4—321no—
323C4#167111yescharacteristic
324C4#167911yescharacteristic
325C4#187711yescharacteristic
326C4#190811yescharacteristic
327C4C2×#36011yescharacteristic
328C4#214511yescharacteristic
329C2—21no—
330C2—21no—
331C2—42no—
332C2—81no—
333C2#921711yescharacteristic
334C2#1306511yescharacteristic
335C2#1307011yescharacteristic
subgroupG/H
center, socleC22#1893
commutatorC4×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC22×D4D8
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C8×D4 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #13287C2semidirect
4. #11936C2semidirect
6. #7278C2semidirect
7. #7158C2semidirect
20. #1675C4semidirect
21. #1675C4semidirect
22. #1673C4semidirect
23. #1673C4semidirect
131. M32SD16semidirect

Maximal subgroups

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

subgroupindex
D4×M322
#191052
#132872
#119362
#86422
#72782
#71582