Finite Group Databasebeta

← #134419

Subgroups — #134419

Order 512.

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

normal subgroups only

This group has 444 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
1C4×#470C211yesmaximal, characteristic
2#12785C211yesmaximal, characteristic
3C2×#1340C212yesmaximal
4#10837C211yesmaximal, characteristic
5#7254C212yesmaximal
6C2×#470C2211yescharacteristic
7C2×C4×Q16C411yescharacteristic, complemented
8C2×C4×M16C2211yescharacteristic
9C4×C22⋊Q8—21no—
10#2285—21no—
11#2150—21no—
12C2×#349C2212yes—
13#1971C411yescharacteristic, complemented
14#1944C411yescharacteristic, complemented
15#1935—21no—
16#1894—21no—
17#1809C2211yescharacteristic
18C4×#106C411yescharacteristic, complemented
19C4×#103—21no—
20C2×H(3,ℤ/4)—21no—
21C2×H(3,ℤ/4)C2212yes—
22#1377—21no—
23#1340—24no—
24C22×Q16C2×C411yescharacteristic
25C22×M16D411yescharacteristic
26C22×M16C2311yescharacteristic
27C2×C22⋊Q8—21no—
28C2×C4×Q8—21no—
29C2×#117—22no—
30C22×C42D411yescharacteristic
31C2×M32—22no—
32#470—21no—
33#470—21no—
34#470D412yes—
35#470Q812yes—
36C4×Q16—41no—
37C4×Q16—41no—
38C2×#107C2×C411yescharacteristic
39C2×#106C2×C411yescharacteristic
40C2×#106C2×C411yescharacteristic
41C2×#103—21no—
42C2×#103—21no—
43C2×#103—21no—
44C4×M16—41no—
45C4×M16—41no—
46C2×#97C2×C411yescharacteristic
47C2×#97D411yescharacteristic
48C2×C4×C8C2×C411yescharacteristic
49C2×C4×C8D411yescharacteristic
50#391—21no—
51#389—21no—
52#387—41no—
53#385—41no—
54#384—41no—
55#382—41no—
56#380—21no—
57#380—41no—
58#380—41no—
59C4×C4⋊C4—22no—
60C4×C4⋊C4—22no—
61C4×C4⋊C4—21no—
62C4×C4⋊C4—41no—
63C4×C22⋊C4—41no—
64C4⋊C16—41no—
65C4⋊C16—41no—
66#349—22no—
67#349—22no—
68#340—21no—
69#340—21no—
70#340—21no—
71H(3,ℤ/4)—22no—
72H(3,ℤ/4)—22no—
73H(3,ℤ/4)—42no—
74C22×Q8—21no—
75C23×C4D811yescharacteristic
76C23×C4D811yescharacteristic
77C23×C4C2×D411yescharacteristic
78C2×Q16—21no—
79C2×Q16—41no—
80C2×Q16—41no—
81C2×Q16C4⋊C412yescomplemented
82C2×M16—21no—
83C2×M16—21no—
84C2×M16—41no—
85C2×M16—41no—
86C2×M16—41no—
87C2×M16—41no—
88C2×M16C2×D411yescharacteristic
89C2×M16C2×Q811yescharacteristic
90C2×M16#4111yescharacteristic
91C2×M16#4111yescharacteristic
92C22×C8C22⋊C411yescharacteristic
93C22×C8C22⋊C411yescharacteristic
94C22×C8C22×C411yescharacteristic
95C22×C8C2×D411yescharacteristic
96C22⋊Q8—22no—
97C22⋊Q8—22no—
98C22⋊Q8—41no—
99C22⋊Q8—41no—
100C4×Q8—41no—
101C4×Q8—41no—
102C4×Q8—41no—
103C4×Q8—41no—
104#117—42no—
105#117—42no—
106#117—42no—
107#117—42no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—41no—
112C2×C4⋊C4—42no—
113C2×C4⋊C4—42no—
114C2×C4⋊C4—42no—
115C2×C4⋊C4—41no—
116C2×C4⋊C4—41no—
117C2×C22⋊C4—41no—
118C2×C22⋊C4—42no—
119C2×C22⋊C4—41no—
120C2×C42—21no—
121C2×C42—21no—
122C2×C42—21no—
123C2×C42—21no—
124C2×C42—24no—
125C2×C42—41no—
126C2×C42—41no—
127C2×C42C22⋊C411yescharacteristic
128C2×C42C22⋊C411yescharacteristic
129C2×C42SD1611yescharacteristic
130C2×C42SD1611yescharacteristic
131M32—42no—
132M32—42no—
133C2×C16—42no—
134#107—41no—
135#106—21no—
136#106—41no—
137#106C4⋊C412yescomplemented
138#103—22no—
139#103—22no—
140#103—42no—
141#103—42no—
142#97—41no—
143#97—41no—
144C4×C8—21no—
145C4×C8—21no—
146C4×C8—21no—
147C4×C8—21no—
148#95—42no—
149#95—42no—
150#95—41no—
151#95—41no—
152#95—41no—
153#95—41no—
154#95—41no—
155C24C2×D811yescharacteristic
156C2×Q8—22no—
157C2×Q8—41no—
158C2×Q8—41no—
159C2×Q8—41no—
160C2×Q8—41no—
161C2×Q8—41no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—42no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—44no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—81no—
177C22×C4#10211yescharacteristic
178C22×C4#10211yescharacteristic
179C22×C4#10211yescharacteristic
180C22×C4#10211yescharacteristic
181C22×C4C2×C22⋊C411yescharacteristic
182C22×C4#12111yescharacteristic
183C22×C4C22⋊Q811yescharacteristic
184C22×C4C2×SD1611yescharacteristic
185C22×C4AGL(1,ℤ/8)11yescharacteristic
186C22×C4#13711yescharacteristic
187Q16—42no—
188Q16—42no—
189M16—41no—
190M16—41no—
191M16—41no—
192M16—41no—
193M16—42no—
194M16—42no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8C2×C4⋊C411yescharacteristic
206C2×C8#11711yescharacteristic
207C2×C8#12111yescharacteristic
208C2×C8C22⋊Q811yescharacteristic
209C4⋊C4—22no—
210C4⋊C4—41no—
211C4⋊C4—42no—
212C4⋊C4—42no—
213C4⋊C4—41no—
214C4⋊C4—41no—
215C4⋊C4—41no—
216C4⋊C4—41no—
217C4⋊C4—82no—
218C4⋊C4—82no—
219C4⋊C4—84no—
220C4⋊C4—84no—
221C4⋊C4—82no—
222C4⋊C4—81no—
223C4⋊C4—81no—
224C22⋊C4—42no—
225C22⋊C4—42no—
226C22⋊C4—82no—
227C22⋊C4—82no—
228C22⋊C4—82no—
229C42—21no—
230C42—21no—
231C42—21no—
232C42—21no—
233C42—21no—
234C42—21no—
235C42—44no—
236C42—44no—
237C42—82no—
238C42—81no—
239C42—81no—
240C42#9911yescharacteristic
241C42#9911yescharacteristic
242C42#10411yescharacteristic
243C42#10411yescharacteristic
244C16—82no—
245C23—21no—
246C23—21no—
247C23—41no—
248C23—41no—
249C23C2×#10211yescharacteristic
250C23#45911yescharacteristic
251C23#47411yescharacteristic
252Q8—42no—
253Q8—42no—
254Q8—81no—
255Q8—81no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—41no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4—81no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—84no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4—81no—
287C2×C4—84no—
288C2×C4—84no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4#32711yescharacteristic
292C2×C4#32711yescharacteristic
293C2×C4#32711yescharacteristic
294C2×C4#32711yescharacteristic
295C2×C4#38011yescharacteristic
296C2×C4C2×#9911yescharacteristic
297C2×C4#41011yescharacteristic
298C2×C4#41711yescharacteristic
299C2×C4C2×#10411yescharacteristic
300C2×C4#42111yescharacteristic
301C2×C4#46111yescharacteristic
302C2×C4#47611yescharacteristic
303C8—21no—
304C8—21no—
305C8—21no—
306C8—21no—
307C8—41no—
308C8—41no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—41no—
314C22—41no—
315C22—41no—
316C22—41no—
317C22—81no—
318C22C2×#32711yescharacteristic
319C22#151711yescharacteristic
320C22#180211yescharacteristic
321C4—21no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—41no—
327C4—41no—
328C4—82no—
329C4—82no—
330C4—161no—
331C4—164no—
332C4—161no—
333C4#134012yescomplemented
334C4#150911yescharacteristic
335C4#151311yescharacteristic
336C4#179911yescharacteristic
337C4#180611yescharacteristic
338C2—21no—
339C2—21no—
340C2—41no—
341C2—41no—
342C2#857111yescharacteristic
343C2C2×#134011yescharacteristic
344C2#1168511yescharacteristic
subgroupG/H
center, socleC22C2×#327
commutatorC22×C8C22×C4
Frattini, cosocleC22×M16C23
prime radicalC24C2×D8
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×M16 ◁ G
LECSG ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
7. C2×C4×Q16C4semidirect
13. #1971C4semidirect
14. #1944C4semidirect
18. C4×#106C4semidirect
81. C2×Q16C4⋊C4semidirect
137. #106C4⋊C4semidirect
333. C4#1340semidirect

Maximal subgroups

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

subgroupindex
C4×#4702
#127852
C2×#13402
#108372
#72542