Finite Group Databasebeta

← #508547

Subgroups — #508547

Order 512.

464 conjugacy classes of subgroups.

The subgroup lattice has 405 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33208C211yesmaximal, characteristic
2#20818C211yesmaximal, characteristic
3#13539C211yesmaximal, characteristic, complemented
4#13539C211yesmaximal, characteristic, complemented
5#13507C211yesmaximal, characteristic, complemented
6#13507C211yesmaximal, characteristic, complemented
7#13494C211yesmaximal, characteristic
8#13494C211yesmaximal, characteristic
9#13493C211yesmaximal, characteristic
10#13492C211yesmaximal, characteristic
11#13277C211yesmaximal, characteristic, complemented
12#13277C211yesmaximal, characteristic, complemented
13#13178C211yesmaximal, characteristic, complemented
14#13178C211yesmaximal, characteristic, complemented
15#12783C211yesmaximal, characteristic
16C2×#499C2211yescharacteristic
17C22×C4×C8C2211yescharacteristic
18#2851—21no—
19#2283—21no—
20C2×#367C2211yescharacteristic
21C2×#367C2211yescharacteristic
22C2×#366C2211yescharacteristic
23C2×#366C2211yescharacteristic
24C2×C4⋊C16C2211yescharacteristic
25C2×C4⋊C16C2211yescharacteristic
26C2×C22⋊C16C2211yescharacteristic
27C2×C22⋊C16C2211yescharacteristic
28#1991—21no—
29#1942C2211yescharacteristic
30#1942C2211yescharacteristic
31#1941—21no—
32#1846C2211yescharacteristic
33#1846C2211yescharacteristic
34#1846C2211yescharacteristic
35#1835—21no—
36#1808C2211yescharacteristic
37#1808C2211yescharacteristic
38#1808C2211yescharacteristic
39#1808C2211yescharacteristic
40#1807—21no—
41#1807—21no—
42C4×#107C2211yescharacteristic
43C4×#107C2211yescharacteristic
44C4×#106—21no—
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
59C23×C8D411yescharacteristic
60C23×C8C2311yescharacteristic
61C2×#125—21no—
62C22×C42D411yescharacteristic
63C22×C16C2311yescharacteristic
64C22×C16C2311yescharacteristic
65C22×C16C2311yescharacteristic
66C22×C16C2311yescharacteristic
67#499—22no—
68#499—22no—
69C2×#107C2311yescharacteristic
70C2×#107C2311yescharacteristic
71C2×#107C2311yescharacteristic
72C2×#107C2311yescharacteristic
73C2×#107C2311yescharacteristic
74C2×#107C2311yescharacteristic
75C2×#107C2311yescharacteristic
76C2×#107C2311yescharacteristic
77C2×#106—21no—
78C2×#106—21no—
79C2×#106—21no—
80C2×#106—21no—
81C2×C4×C8—22no—
82C2×C4×C8—22no—
83C2×C4×C8D411yescharacteristic
84C2×C4×C8D411yescharacteristic
85C2×C4×C8C2311yescharacteristic
86C2×C4×C8C2311yescharacteristic
87#389—21no—
88#389—21no—
89#387—41no—
90#387—41no—
91#385—41no—
92#385—41no—
93#383—41no—
94#382—21no—
95#382—21no—
96#382—21no—
97#381—41no—
98#380—21no—
99#380—21no—
100#380—21no—
101#380—21no—
102C4×C4⋊C4—21no—
103C4×C4⋊C4—21no—
104C4×C22⋊C4—41no—
105#376—41no—
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—
122C23×C4D811yescharacteristic
123C23×C4D811yescharacteristic
124C23×C4C2×D411yescharacteristic
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—21no—
128C22×C8—21no—
129C22×C8—21no—
130C22×C8—21no—
131C22×C8—21no—
132C22×C8—21no—
133C22×C8—41no—
134C22×C8C2×D411yescharacteristic
135C22×C8C2×D411yescharacteristic
136C22×C8C2×D411yescharacteristic
137C22×C8C2411yescharacteristic
138#125—42no—
139#125—42no—
140C2×C4⋊C4—21no—
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—41no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—41no—
152C2×C22⋊C4—41no—
153C2×C22⋊C4—41no—
154C2×C22⋊C4—41no—
155C2×C22⋊C4—41no—
156C2×C42—22no—
157C2×C42—22no—
158C2×C42—41no—
159C2×C42—41no—
160C2×C42D811yescharacteristic
161C2×C42D811yescharacteristic
162C2×C42C2×D411yescharacteristic
163C2×C42C2×D411yescharacteristic
164C2×C16—21no—
165C2×C16—21no—
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—
176#107—21no—
177#107—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#106—42no—
193#106—42no—
194#106—42no—
195#106—42no—
196C4×C8—22no—
197C4×C8—21no—
198C4×C8—21no—
199C4×C8—22no—
200C4×C8#4112yes—
201C4×C8#4112yes—
202#95—41no—
203#95—41no—
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
217C22×C4—22no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—22no—
225C22×C4—22no—
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×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4C2×D811yescharacteristic
238C22×C4C2×D811yescharacteristic
239C22×C4C2×D811yescharacteristic
240C22×C4C2×D811yescharacteristic
241C22×C4C2×D811yescharacteristic
242C22×C4C22×D411yescharacteristic
243C2×C8—21no—
244C2×C8—21no—
245C2×C8—21no—
246C2×C8—21no—
247C2×C8—21no—
248C2×C8—21no—
249C2×C8—21no—
250C2×C8—44no—
251C2×C8—41no—
252C2×C8—41no—
253C2×C8—41no—
254C2×C8—41no—
255C2×C8#12312yes—
256C2×C8#13712yes—
257C2×C8C2×#4111yescharacteristic
258C2×C8C2×#4111yescharacteristic
259C2×C8H(5,𝔽2)11yescharacteristic
260C2×C82−1+411yescharacteristic
261C2×C82−1+411yescharacteristic
262C2×C82−1+411yescharacteristic
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—41no—
279C4⋊C4—82no—
280C4⋊C4—82no—
281C4⋊C4—82no—
282C4⋊C4—82no—
283C22⋊C4—82no—
284C22⋊C4—82no—
285C22⋊C4—82no—
286C22⋊C4—82no—
287C42—21no—
288C42—21no—
289C42—22no—
290C42—22no—
291C42—82no—
292C42—82no—
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#50812yes—
306C23C22×D811yescharacteristic
307C2×C4—21no—
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—44no—
321C2×C4—44no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
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—81no—
350C2×C4#48012yes—
351C2×C4#48012yes—
352C2×C4C2×#12311yescharacteristic
353C2×C4#53611yescharacteristic
354C2×C4#53711yescharacteristic
355C2×C4C2×#13711yescharacteristic
356C2×C4C2×#13711yescharacteristic
357C2×C4#57511yescharacteristic
358C8—22no—
359C8—22no—
360C8—41no—
361C8#54111yescharacteristic
362C8#54111yescharacteristic
363C8#54211yescharacteristic
364C8#54211yescharacteristic
365C22—22no—
366C22—22no—
367C22—22no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22C2×#48011yescharacteristic
373C22#322611yescharacteristic
374C22#324511yescharacteristic
375C22C2×#50811yescharacteristic
376C22#341411yescharacteristic
377C22#341411yescharacteristic
378C22#341411yescharacteristic
379C4—22no—
380C4—22no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—161no—
385C4—161no—
386C4—161no—
387C4—161no—
388C4—161no—
389C4—161no—
390C4—161no—
391C4—161no—
392C4#262511yescharacteristic
393C4#309511yescharacteristic
394C4#323711yescharacteristic
395C4#325511yescharacteristic
396C2—42no—
397C2#2993611yescharacteristic
398C2#3364511yescharacteristic
399C2#3364711yescharacteristic
400C2#3379811yescharacteristic
401C2#3380611yescharacteristic
402C2#3381611yescharacteristic
403C2#3382211yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC24C2×D8
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ 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. #13507C2semidirect
6. #13507C2semidirect
11. #13277C2semidirect
12. #13277C2semidirect
13. #13178C2semidirect
14. #13178C2semidirect

Maximal subgroups

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

subgroupindex
#332082
#208182
#135392
#135072
#134942
#134932
#134922
#132772
#131782
#127832