Finite Group Databasebeta

← #134522

Subgroups — #134522

Order 512.

444 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 94 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#21536C211yesmaximal, characteristic
2#12786C211yesmaximal, characteristic, complemented
3C2×#1339C212yesmaximal
4#10866C211yesmaximal, characteristic, complemented
5#7254C212yesmaximal, complemented
6C2×#469C2211yescharacteristic
7C2×C4×D8C411yescharacteristic, complemented
8C2×#427—21no—
9C2×#405C2211yescharacteristic
10#2324—21no—
11#2150—21no—
12C2×#349C2212yes—
13#1972C411yescharacteristic, complemented
14#1944C411yescharacteristic, complemented
15#1934—21no—
16#1893—21no—
17#1809C2211yescharacteristic
18C4×#106C411yescharacteristic, complemented
19#1762—21no—
20#1740—21no—
21C2×H(3,ℤ/4)C2212yes—
22#1377—21no—
23#1339—24no—
24C22×D8C2×C411yescharacteristic
25C22×M16C2311yescharacteristic
26C23×C8D411yescharacteristic
27C2×#121—21no—
28C2×C4×D4—21no—
29C2×#117—22no—
30C2×#117D411yescharacteristic
31C2×M32—22no—
32#469—21no—
33#469—21no—
34#469D412yes—
35#469Q812yes—
36C4×D8—41no—
37C4×D8—41no—
38#427—24no—
39#427—42no—
40C2×#107—41no—
41C2×#107C2×C411yescharacteristic
42C2×#106—41no—
43C2×#106C2×C411yescharacteristic
44C2×#106C2×C411yescharacteristic
45C2×#102—21no—
46C2×#102—21no—
47C2×#102—21no—
48#405—41no—
49#405—41no—
50C2×#97C2×C411yescharacteristic
51C2×#97D411yescharacteristic
52C2×C4×C8C2×C411yescharacteristic
53C2×C4×C8D411yescharacteristic
54#390—21no—
55#390—41no—
56#389—21no—
57#388—41no—
58#386—41no—
59#385—41no—
60#380—21no—
61C4×C4⋊C4—21no—
62C4⋊C16—41no—
63C4⋊C16—41no—
64#349—22no—
65#349—22no—
66#342—41no—
67#342—41no—
68#340—21no—
69#340—21no—
70#340—21no—
71H(3,ℤ/4)—22no—
72H(3,ℤ/4)—22no—
73C22×D4—21no—
74C22×D4—41no—
75C23×C4—41no—
76C23×C4C2×D411yescharacteristic
77C2×D8—21no—
78C2×D8—41no—
79C2×D8—41no—
80C2×D8C4⋊C412yes—
81C2×M16—41no—
82C2×M16—41no—
83C2×M16C2×D411yescharacteristic
84C2×M16C2×Q811yescharacteristic
85C2×M16#4111yescharacteristic
86C2×M16#4111yescharacteristic
87C22×C8—21no—
88C22×C8—21no—
89C22×C8—21no—
90C22×C8—21no—
91C22×C8—22no—
92C22×C8C22⋊C411yescharacteristic
93C22×C8C22⋊C411yescharacteristic
94C22×C8C22×C411yescharacteristic
95C22×C8C2×D411yescharacteristic
96#121—22no—
97#121—22no—
98#121—41no—
99#121—41no—
100C4×D4—42no—
101C4×D4—42no—
102#117—24no—
103#117—41no—
104#117—41no—
105#117—42no—
106#117—42no—
107#117—42no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—42no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4SD1611yescharacteristic
114C2×C4⋊C4SD1611yescharacteristic
115C2×C22⋊C4—41no—
116C2×C22⋊C4—42no—
117C2×C22⋊C4—41no—
118C2×C22⋊C4—41no—
119C2×C22⋊C4—41no—
120C2×C22⋊C4—41no—
121C2×C22⋊C4D811yescharacteristic
122C2×C22⋊C4D811yescharacteristic
123C2×C42—42no—
124C2×C42—41no—
125C2×C42C22⋊C411yescharacteristic
126C2×C42C22⋊C411yescharacteristic
127M32—42no—
128M32—42no—
129C2×C16—42no—
130#107—44no—
131#107—41no—
132#106—21no—
133#106—44no—
134#106—41no—
135#106C4⋊C412yes—
136#102—22no—
137#102—22no—
138#102—41no—
139#102—41no—
140#102—42no—
141#97—41no—
142#97—41no—
143C4×C8—21no—
144C4×C8—21no—
145C4×C8—21no—
146C4×C8—21no—
147#95—41no—
148#95—41no—
149#95—41no—
150C24—41no—
151C24C2×D811yescharacteristic
152C2×D4—22no—
153C2×D4—42no—
154C2×D4—42no—
155C2×D4—41no—
156C2×D4—41no—
157C2×D4—41no—
158C2×D4—41no—
159C2×D4—41no—
160C2×D4—161no—
161C2×D4—161no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—82no—
170C22×C4—82no—
171C22×C4—161no—
172C22×C4—161no—
173C22×C4#10211yescharacteristic
174C22×C4#10211yescharacteristic
175C22×C4#10211yescharacteristic
176C22×C4#10211yescharacteristic
177C22×C4C2×C22⋊C411yescharacteristic
178C22×C4#12111yescharacteristic
179C22×C4C22⋊Q811yescharacteristic
180C22×C4C2×SD1611yescharacteristic
181C22×C4AGL(1,ℤ/8)11yescharacteristic
182C22×C4#13711yescharacteristic
183D8—42no—
184D8—42no—
185M16—41no—
186M16—41no—
187M16—41no—
188M16—41no—
189C2×C8—22no—
190C2×C8—22no—
191C2×C8—22no—
192C2×C8—22no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8#9911yescharacteristic
202C2×C8#9911yescharacteristic
203C2×C8#10412yes—
204C2×C8C2×C4⋊C411yescharacteristic
205C2×C8#11711yescharacteristic
206C2×C8#12111yescharacteristic
207C2×C8C22⋊Q811yescharacteristic
208C4⋊C4—22no—
209C4⋊C4—41no—
210C4⋊C4—41no—
211C4⋊C4—44no—
212C4⋊C4—41no—
213C4⋊C4—41no—
214C4⋊C4—41no—
215C4⋊C4—41no—
216C4⋊C4—41no—
217C4⋊C4—82no—
218C4⋊C4—161no—
219C22⋊C4—41no—
220C22⋊C4—41no—
221C22⋊C4—42no—
222C22⋊C4—42no—
223C22⋊C4—44no—
224C22⋊C4—82no—
225C22⋊C4—82no—
226C22⋊C4—161no—
227C22⋊C4—161no—
228C22⋊C4—161no—
229C42—21no—
230C42—21no—
231C42—21no—
232C42—21no—
233C42—44no—
234C42—82no—
235C42—161no—
236C16—82no—
237C23—21no—
238C23—21no—
239C23—41no—
240C23—41no—
241C23—42no—
242C23—81no—
243C23—81no—
244C23—81no—
245C23—81no—
246C23—81no—
247C23—81no—
248C23C2×#10211yescharacteristic
249C23#45911yescharacteristic
250C23#47411yescharacteristic
251D4—42no—
252D4—42no—
253D4—81no—
254D4—81no—
255D4—162no—
256D4—162no—
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—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—42no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4—84no—
279C2×C4—81no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—164no—
286C2×C4—162no—
287C2×C4—162no—
288C2×C4#38011yescharacteristic
289C2×C4C2×#9911yescharacteristic
290C2×C4#41011yescharacteristic
291C2×C4#41711yescharacteristic
292C2×C4C2×#10411yescharacteristic
293C2×C4#42111yescharacteristic
294C2×C4#46111yescharacteristic
295C2×C4#47611yescharacteristic
296C8—21no—
297C8—21no—
298C8—21no—
299C8—21no—
300C8—21no—
301C8—21no—
302C8—22no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—41no—
308C22—41no—
309C22—41no—
310C22—41no—
311C22—81no—
312C22—82no—
313C22—82no—
314C22—82no—
315C22—161no—
316C22—161no—
317C22—161no—
318C22—161no—
319C22#150211yescharacteristic
320C22#151911yescharacteristic
321C22#180211yescharacteristic
322C4—21no—
323C4—21no—
324C4—41no—
325C4—41no—
326C4—41no—
327C4—41no—
328C4—82no—
329C4—161no—
330C4—164no—
331C4—161no—
332C4#150411yescharacteristic
333C4#151211yescharacteristic
334C4#179911yescharacteristic
335C4#180611yescharacteristic
336C2—21no—
337C2—21no—
338C2—41no—
339C2—41no—
340C2—162no—
341C2#857711yescharacteristic
342C2#1186711yescharacteristic
343C2#1186811yescharacteristic
subgroupG/H
center, socleC22#1519
commutatorC22×C8C22×C4
Frattini, cosocleC22×M16C23
prime radicalC2×#469C22
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#405 ◁ G
LECSG ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12786C2semidirect
4. #10866C2semidirect
5. #7254C2semidirect
7. C2×C4×D8C4semidirect
13. #1972C4semidirect
14. #1944C4semidirect
18. C4×#106C4semidirect

Maximal subgroups

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

subgroupindex
#215362
#127862
C2×#13392
#108662
#72542