Finite Group Databasebeta

← #508552

Subgroups — #508552

Order 512.

446 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33215C211yesmaximal, characteristic
2#20821C211yesmaximal, characteristic
3#13539C211yesmaximal, characteristic, complemented
4#13539C211yesmaximal, characteristic, complemented
5#13506C211yesmaximal, characteristic, complemented
6#13505C211yesmaximal, characteristic, complemented
7#13494C211yesmaximal, characteristic
8#13494C211yesmaximal, characteristic
9#13493C211yesmaximal, characteristic
10#13492C211yesmaximal, characteristic
11#13276C211yesmaximal, characteristic, complemented
12#13275C211yesmaximal, characteristic, complemented
13#13178C211yesmaximal, characteristic, complemented
14#13178C211yesmaximal, characteristic, complemented
15#12784C211yesmaximal, characteristic
16C2×#500C2211yescharacteristic
17C2×#405C2211yescharacteristic
18#2628—21no—
19#2314—21no—
20C2×#367C2211yescharacteristic
21C2×#367C2211yescharacteristic
22C2×#366C2211yescharacteristic
23C2×#366C2211yescharacteristic
24C2×C4⋊C16C2211yescharacteristic
25C2×C4⋊C16C2211yescharacteristic
26C2×C22⋊C16C2211yescharacteristic
27C2×C22⋊C16C2211yescharacteristic
28#1993—21no—
29#1944—21no—
30#1942C2211yescharacteristic
31#1942C2211yescharacteristic
32#1846C2211yescharacteristic
33#1846C2211yescharacteristic
34#1845C2211yescharacteristic
35#1836—21no—
36#1809—21no—
37#1809—21no—
38#1808C2211yescharacteristic
39#1808C2211yescharacteristic
40#1808C2211yescharacteristic
41#1808C2211yescharacteristic
42#1776—21no—
43C4×#107C2211yescharacteristic
44C4×#107C2211yescharacteristic
45#1380C2211yescharacteristic
46#1380C2211yescharacteristic
47#1380C2211yescharacteristic
48#1380C2211yescharacteristic
49#1380C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×M16D411yescharacteristic
60C23×C8C2311yescharacteristic
61C2×C4⋊Q8—21no—
62C2×#117D411yescharacteristic
63C22×C16C2311yescharacteristic
64C22×C16C2311yescharacteristic
65C22×C16C2311yescharacteristic
66C22×C16C2311yescharacteristic
67#500—22no—
68#500—22no—
69C2×#107—21no—
70C2×#107—21no—
71C2×#107C2311yescharacteristic
72C2×#107C2311yescharacteristic
73C2×#107C2311yescharacteristic
74C2×#107C2311yescharacteristic
75C2×#107C2311yescharacteristic
76C2×#107C2311yescharacteristic
77C2×#107C2311yescharacteristic
78C2×#107C2311yescharacteristic
79C2×#106—21no—
80C2×#106—21no—
81#405—22no—
82#405—22no—
83C2×#97D411yescharacteristic
84C2×#97D411yescharacteristic
85C2×C4×C8C2311yescharacteristic
86C2×C4×C8C2311yescharacteristic
87#400—41no—
88#399—41no—
89#395—41no—
90#393—41no—
91#389—21no—
92#389—21no—
93#388—41no—
94#388—41no—
95#387—41no—
96#387—41no—
97#384—21no—
98#382—21no—
99#382—21no—
100#380—21no—
101#380—21no—
102#380—21no—
103#380—21no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4—21no—
106#367—41no—
107#367—41no—
108#366—41no—
109#366—41no—
110C4⋊C16—22no—
111C4⋊C16—22no—
112C22⋊C16—41no—
113C22⋊C16—41no—
114#340—21no—
115#340—21no—
116#340—21no—
117#340—21no—
118#340—21no—
119#340—21no—
120#340—21no—
121#340—21no—
122C22×Q8—41no—
123C23×C4C2×D411yescharacteristic
124C2×M16—22no—
125C2×M16—22no—
126C2×M16—41no—
127C22×C8—21no—
128C22×C8—21no—
129C22×C8—21no—
130C22×C8—21no—
131C22×C8—21no—
132C22×C8—21no—
133C22×C8C2×D411yescharacteristic
134C22×C8C2×D411yescharacteristic
135C22×C8C2×D411yescharacteristic
136C22×C8C2411yescharacteristic
137C4⋊Q8—42no—
138C4⋊Q8—42no—
139#117—22no—
140#117—22no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4D811yescharacteristic
155C2×C4⋊C4D811yescharacteristic
156C2×C22⋊C4—41no—
157C2×C22⋊C4—41no—
158C2×C22⋊C4—41no—
159C2×C22⋊C4—41no—
160C2×C22⋊C4D811yescharacteristic
161C2×C22⋊C4D811yescharacteristic
162C2×C42—41no—
163C2×C42—41no—
164C2×C42C2×D411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C16—21no—
167C2×C16—21no—
168C2×C16—21no—
169C2×C16—21no—
170C2×C16—21no—
171C2×C16—21no—
172C2×C16—21no—
173C2×C16—21no—
174C2×C16—21no—
175C2×C16—21no—
176C2×C16—21no—
177C2×C16—21no—
178#107—21no—
179#107—21no—
180#107—21no—
181#107—21no—
182#107—21no—
183#107—21no—
184#107—21no—
185#107—21no—
186#107—21no—
187#107—21no—
188#107—21no—
189#107—21no—
190#107—21no—
191#107—21no—
192#107—21no—
193#107—21no—
194#107—42no—
195#107—42no—
196#106—42no—
197#106—42no—
198#97—22no—
199#97—22no—
200C4×C8—21no—
201C4×C8—21no—
202C4×C8#4112yes—
203C4×C8#4112yes—
204#95—41no—
205#95—41no—
206#95—41no—
207#95—41no—
208#95—41no—
209#95—41no—
210#95—41no—
211#95—41no—
212#95—41no—
213#95—41no—
214#95—41no—
215#95—41no—
216C24C2×D811yescharacteristic
217C2×Q8—82no—
218C2×Q8—161no—
219C2×Q8—161no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4C2×D811yescharacteristic
235C22×C4C2×D811yescharacteristic
236C22×C4C2×D811yescharacteristic
237C22×C4C2×D811yescharacteristic
238C22×C4C2×D811yescharacteristic
239C22×C4C22×D411yescharacteristic
240M16—44no—
241C2×C8—21no—
242C2×C8—21no—
243C2×C8—21no—
244C2×C8—21no—
245C2×C8—21no—
246C2×C8—21no—
247C2×C8—21no—
248C2×C8—41no—
249C2×C8—41no—
250C2×C8—41no—
251C2×C8—41no—
252C2×C8#12312yes—
253C2×C8#13712yes—
254C2×C8C2×#4111yescharacteristic
255C2×C8C2×#4111yescharacteristic
256C2×C8H(5,𝔽2)11yescharacteristic
257C2×C82−1+411yescharacteristic
258C2×C82−1+411yescharacteristic
259C2×C82−1+411yescharacteristic
260C4⋊C4—22no—
261C4⋊C4—22no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—82no—
279C4⋊C4—161no—
280C4⋊C4—161no—
281C4⋊C4—161no—
282C4⋊C4—161no—
283C22⋊C4—22no—
284C22⋊C4—22no—
285C22⋊C4—161no—
286C22⋊C4—161no—
287C22⋊C4—161no—
288C22⋊C4—161no—
289C42—21no—
290C42—21no—
291C42—161no—
292C42—161no—
293C42#12312yes—
294C42#13712yes—
295C16—41no—
296C16—41no—
297C16—41no—
298C16—41no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23C22×D811yescharacteristic
307Q8—164no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4#48012yes—
350C2×C4#48012yes—
351C2×C4C2×#12311yescharacteristic
352C2×C4#53611yescharacteristic
353C2×C4#53711yescharacteristic
354C2×C4C2×#13711yescharacteristic
355C2×C4C2×#13711yescharacteristic
356C2×C4#57511yescharacteristic
357C8—22no—
358C8—22no—
359C8—41no—
360C8#54111yescharacteristic
361C8#54111yescharacteristic
362C8#54211yescharacteristic
363C8#54211yescharacteristic
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22C2×#48011yescharacteristic
372C22#322611yescharacteristic
373C22#324511yescharacteristic
374C22#341511yescharacteristic
375C22#341611yescharacteristic
376C22#341611yescharacteristic
377C22#341711yescharacteristic
378C4—22no—
379C4—22no—
380C4—41no—
381C4—41no—
382C4—41no—
383C4—161no—
384C4—161no—
385C4—161no—
386C4—161no—
387C4—161no—
388C4—161no—
389C4—161no—
390C4—161no—
391C4#262511yescharacteristic
392C4#309511yescharacteristic
393C4#323711yescharacteristic
394C4#325511yescharacteristic
395C2—81no—
396C2#2993611yescharacteristic
397C2#3365011yescharacteristic
398C2#3365111yescharacteristic
399C2#3380911yescharacteristic
400C2#3381011yescharacteristic
401C2#3382311yescharacteristic
402C2#3382411yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC24C2×D8
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#405 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #13539C2semidirect
4. #13539C2semidirect
5. #13506C2semidirect
6. #13505C2semidirect
11. #13276C2semidirect
12. #13275C2semidirect
13. #13178C2semidirect
14. #13178C2semidirect

Maximal subgroups

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

subgroupindex
#332152
#208212
#135392
#135062
#135052
#134942
#134932
#134922
#132762
#132752
#131782
#127842