Finite Group Databasebeta

← #138283

Subgroups — #138283

Order 512.

501 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 501 conjugacy classes of subgroups — too many to draw. Only its 102 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#33129C211yesmaximal, characteristic
2C2×C4×#108C211yesmaximal, characteristic, complemented
3#12844C211yesmaximal, characteristic, complemented
4C2×#1349C212yesmaximal
5#7259C212yesmaximal, complemented
6C2×#466C2211yescharacteristic
7C22×#108—21no—
8C2×C4×M16—21no—
9C22×C4×C8C2211yescharacteristic
10#2854—21no—
11C2×C22⋊C16C2212yes—
12#1970C411yescharacteristic
13#1970C411yescharacteristic
14#1942C411yescharacteristic
15#1942C411yescharacteristic
16#1934—21no—
17#1934—21no—
18#1893—21no—
19#1808C2211yescharacteristic
20C4×#108—24no—
21C2×#341C2212yes—
22#1374—21no—
23#1374—21no—
24#1349—24no—
25C22×D8C2×C411yescharacteristic
26C22×M16—22no—
27C23×C8D411yescharacteristic
28C23×C8C2311yescharacteristic
29C2×#123—41no—
30C2×#121—21no—
31C22×C42D411yescharacteristic
32C22×C16—22no—
33#466—22no—
34#466—22no—
35C2×#108—24no—
36C2×#108—24no—
37C2×#108—24no—
38C2×#107C2×C411yescharacteristic
39C2×#107C2×C411yescharacteristic
40C2×#107C2×C411yescharacteristic
41C2×#102—21no—
42C2×#102—21no—
43C2×#102—21no—
44C4×M16—24no—
45C4×M16—42no—
46C2×#97—41no—
47C2×C4×C8—22no—
48C2×C4×C8—22no—
49C2×C4×C8—41no—
50C2×C4×C8C2×C411yescharacteristic
51C2×C4×C8C2×C411yescharacteristic
52C2×C4×C8D411yescharacteristic
53C2×C4×C8D411yescharacteristic
54#390—21no—
55#390—21no—
56#389—21no—
57#389—21no—
58#388—41no—
59#388—41no—
60#386—41no—
61#380—21no—
62C22⋊C16—22no—
63C22⋊C16—22no—
64#341—22no—
65#341D414yes—
66#340—21no—
67#340—21no—
68#340—21no—
69C22×D4—21no—
70C22×D4—41no—
71C23×C4SD1611yescharacteristic
72C23×C4SD1611yescharacteristic
73C23×C4C2×D411yescharacteristic
74C2×D8—22no—
75C2×D8—41no—
76C2×D8—41no—
77C2×M16—24no—
78C2×M16—24no—
79C2×M16—24no—
80C2×M16—42no—
81C2×M16—42no—
82C2×M16—42no—
83C22×C8—21no—
84C22×C8—21no—
85C22×C8—22no—
86C22×C8—22no—
87C22×C8—41no—
88C22×C8—41no—
89C22×C8—42no—
90C22×C8C22⋊C411yescharacteristic
91C22×C8C22⋊C411yescharacteristic
92C22×C8C22×C411yescharacteristic
93C22×C8C2×D411yescharacteristic
94C22×C8C2×D412yes—
95C22×C8#4112yes—
96#123—84no—
97#121—41no—
98#121—41no—
99#121—41no—
100#121—41no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—41no—
105C2×C4⋊C4—41no—
106C2×C22⋊C4—41no—
107C2×C22⋊C4—41no—
108C2×C22⋊C4—41no—
109C2×C22⋊C4—41no—
110C2×C22⋊C4—41no—
111C2×C22⋊C4—41no—
112C2×C42—21no—
113C2×C42—21no—
114C2×C42—21no—
115C2×C42—21no—
116C2×C42C22⋊C411yescharacteristic
117C2×C42C22⋊C411yescharacteristic
118C2×C42D811yescharacteristic
119C2×C42D811yescharacteristic
120C2×C16—42no—
121C2×C16—42no—
122C2×C16—42no—
123#108—216no—
124#107—22no—
125#107—41no—
126#107—41no—
127#102—42no—
128#102—42no—
129#102—42no—
130#97—44no—
131C4×C8—22no—
132C4×C8—22no—
133C4×C8—22no—
134C4×C8—22no—
135C4×C8—44no—
136#95—41no—
137#95—41no—
138#95—41no—
139C24—41no—
140C24C2×SD1611yescharacteristic
141C2×D4—41no—
142C2×D4—41no—
143C2×D4—41no—
144C2×D4—41no—
145C2×D4—41no—
146C2×D4—41no—
147C2×D4—82no—
148C2×D4—82no—
149C2×D4—81no—
150C2×D4—81no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4#10211yescharacteristic
167C22×C4#10211yescharacteristic
168C22×C4#10211yescharacteristic
169C22×C4#10211yescharacteristic
170C22×C4C2×C22⋊C411yescharacteristic
171C22×C4#12011yescharacteristic
172C22×C4#12311yescharacteristic
173C22×C4C2×D811yescharacteristic
174C22×C4AGL(1,ℤ/8)11yescharacteristic
175C22×C4#13711yescharacteristic
176D8—82no—
177M16—28no—
178M16—44no—
179M16—44no—
180M16—44no—
181C2×C8—24no—
182C2×C8—24no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—42no—
192C2×C8—42no—
193C2×C8—44no—
194C2×C8—44no—
195C2×C8—44no—
196C2×C8#10412yes—
197C2×C8#10412yes—
198C2×C8C4×D412yes—
199C2×C8#12112yes—
200C4⋊C4—41no—
201C4⋊C4—41no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4—41no—
205C4⋊C4—41no—
206C4⋊C4—82no—
207C4⋊C4—82no—
208C22⋊C4—82no—
209C22⋊C4—82no—
210C22⋊C4—82no—
211C22⋊C4—82no—
212C22⋊C4—82no—
213C22⋊C4—81no—
214C22⋊C4—81no—
215C42—21no—
216C42—21no—
217C42—21no—
218C42—21no—
219C42—21no—
220C42—21no—
221C42—21no—
222C42—21no—
223C16—82no—
224C23—21no—
225C23—21no—
226C23—21no—
227C23—21no—
228C23—21no—
229C23—21no—
230C23—81no—
231C23—81no—
232C23—81no—
233C23—81no—
234C23—81no—
235C23—81no—
236C23—81no—
237C23C2×#10211yescharacteristic
238C23#44811yescharacteristic
239C23#48111yescharacteristic
240D4—81no—
241D4—81no—
242D4—81no—
243D4—81no—
244D4—162no—
245D4—162no—
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—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—81no—
271C2×C4—81no—
272C2×C4—81no—
273C2×C4—81no—
274C2×C4—81no—
275C2×C4—81no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4#35912yes—
280C2×C4#36112yes—
281C2×C4#38611yescharacteristic
282C2×C4#41011yescharacteristic
283C2×C4#41011yescharacteristic
284C2×C4#41711yescharacteristic
285C2×C4C2×#10411yescharacteristic
286C2×C4C2×#10411yescharacteristic
287C2×C4#44711yescharacteristic
288C2×C4#48011yescharacteristic
289C8—22no—
290C8—22no—
291C8—22no—
292C8—22no—
293C8—48no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—41no—
301C22—41no—
302C22—41no—
303C22—41no—
304C22—161no—
305C22—161no—
306C22—161no—
307C22—161no—
308C22—161no—
309C22—161no—
310C22—161no—
311C22#150211yescharacteristic
312C22#150211yescharacteristic
313C22#187411yescharacteristic
314C22C2×#35911yescharacteristic
315C22#214311yescharacteristic
316C22C2×#36111yescharacteristic
317C22#214811yescharacteristic
318C4—21no—
319C4—21no—
320C4—22no—
321C4—22no—
322C4—41no—
323C4—41no—
324C4—161no—
325C4—161no—
326C4—161no—
327C4#150911yescharacteristic
328C4#150911yescharacteristic
329C4#189611yescharacteristic
330C4#189911yescharacteristic
331C2—41no—
332C2—41no—
333C2—321no—
334C2#914811yescharacteristic
335C2#1174711yescharacteristic
336C2#1174911yescharacteristic
337C2#1175011yescharacteristic
338C2#1175111yescharacteristic
339C2#1299011yescharacteristic
340C2#1300411yescharacteristic
subgroupG/H
center, socleC23C2×#102
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radicalC2×#466C22
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×C4×#108C2semidirect
3. #12844C2semidirect
5. #7259C2semidirect

Maximal subgroups

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

subgroupindex
#331292
C2×C4×#1082
#128442
C2×#13492
#72592