Finite Group Databasebeta

← #511334

Subgroups — #511334

Order 512.

413 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28013C211yesmaximal, characteristic
2#27993C211yesmaximal, characteristic, complemented
3#13539C211yesmaximal, characteristic, complemented
4#13515C211yesmaximal, characteristic, complemented
5#13482C211yesmaximal, characteristic
6#13476C211yesmaximal, characteristic
7#13476C211yesmaximal, characteristic
8#13474C211yesmaximal, characteristic
9#13178C211yesmaximal, characteristic, complemented
10#13178C211yesmaximal, characteristic, complemented
11#13166C211yesmaximal, characteristic, complemented
12#13159C211yesmaximal, characteristic, complemented
13#13147C211yesmaximal, characteristic
14#13130C211yesmaximal, characteristic
15#13109C211yesmaximal, characteristic, complemented
16#2513—21no—
17#2509—21no—
18C2×#367C2211yescharacteristic
19C2×#367C2211yescharacteristic
20C2×#366C2211yescharacteristic
21C2×#366C2211yescharacteristic
22C2×C4⋊C16C2211yescharacteristic
23C2×C4⋊C16C2211yescharacteristic
24C2×C4⋊C16C2211yescharacteristic
25C2×#357C2211yescharacteristic
26C2×#357C2211yescharacteristic
27C2×#357C2211yescharacteristic
28C2×#357C2211yescharacteristic
29C2×C4×C16C2211yescharacteristic
30#2101—21no—
31#2088—21no—
32#2073—21no—
33#2065—21no—
34#1970C2211yescharacteristic
35#1970C2211yescharacteristic
36#1970C2211yescharacteristic
37#1942C2211yescharacteristic
38#1942C2211yescharacteristic
39#1942C2211yescharacteristic
40#1942C2211yescharacteristic
41#1942C2211yescharacteristic
42#1926—21no—
43#1919—21no—
44#1846C2211yescharacteristic
45#1846C2211yescharacteristic
46C4×#107C2211yescharacteristic
47C4×#107C2211yescharacteristic
48C8×C4⋊C4C2211yescharacteristic
49#1380C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×D8C2311yescharacteristic
60C22×C16C2311yescharacteristic
61C22×C16C2311yescharacteristic
62C22×C16C2311yescharacteristic
63C22×C16C2311yescharacteristic
64C2×#107C2311yescharacteristic
65C2×#107C2311yescharacteristic
66C2×#107C2311yescharacteristic
67C2×#107C2311yescharacteristic
68C2×#107C2311yescharacteristic
69C2×#107C2311yescharacteristic
70C2×#107C2311yescharacteristic
71C2×#106—21no—
72C2×#106—21no—
73C2×C4⋊C8D411yescharacteristic
74C2×#102—21no—
75C2×#102—21no—
76C2×#102—21no—
77C2×C4×C8C2311yescharacteristic
78C2×C4×C8C2311yescharacteristic
79C2×C4×C8C2311yescharacteristic
80#400—41no—
81#400—41no—
82#397—41no—
83#397—41no—
84#390—21no—
85#390—21no—
86#390—21no—
87#389—21no—
88#389—21no—
89#389—21no—
90#389—21no—
91#389—21no—
92#389—41no—
93#388—41no—
94#388—41no—
95#387—41no—
96#382—21no—
97#382—21no—
98C4×C4⋊C4—21no—
99C4×C4⋊C4—21no—
100C4×C4⋊C4D411yescharacteristic
101#367—22no—
102#367—41no—
103#366—22no—
104#366—41no—
105C4⋊C16—22no—
106C4⋊C16—22no—
107C4⋊C16—41no—
108#357—22no—
109#357—22no—
110#357—41no—
111#357—41no—
112C4×C16—41no—
113#340—21no—
114#340—21no—
115#340—21no—
116#340—21no—
117#340—21no—
118#340—21no—
119#340—21no—
120#336D411yescharacteristic
121#336D411yescharacteristic
122C22×D4—21no—
123C2×D8—21no—
124C2×D8—21no—
125C2×D8—21no—
126C2×D8—21no—
127C2×D8—21no—
128C2×D8—21no—
129C22×C8C2×D411yescharacteristic
130C22×C8C2×D411yescharacteristic
131C22×C8C2×D411yescharacteristic
132C22×C8C2411yescharacteristic
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—41no—
141C2×C4⋊C4—41no—
142C2×C4⋊C4—41no—
143C2×C4⋊C4—41no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4D811yescharacteristic
146C2×C4⋊C4D811yescharacteristic
147C2×C22⋊C4—41no—
148C2×C22⋊C4—41no—
149C2×C22⋊C4—41no—
150C2×C22⋊C4—41no—
151C2×C22⋊C4—41no—
152C2×C22⋊C4—41no—
153C2×C42—41no—
154C2×C42—41no—
155C2×C42C2×D411yescharacteristic
156C2×C42C2×D411yescharacteristic
157C2×C42C2×D411yescharacteristic
158C2×C16—21no—
159C2×C16—21no—
160C2×C16—21no—
161C2×C16—21no—
162C2×C16—21no—
163C2×C16—21no—
164C2×C16—21no—
165C2×C16—21no—
166C2×C16—21no—
167C2×C16—21no—
168C2×C16—21no—
169C2×C16#4112yes—
170#107—21no—
171#107—21no—
172#107—21no—
173#107—21no—
174#107—21no—
175#107—21no—
176#107—21no—
177#107—21no—
178#107—21no—
179#107—21no—
180#107—21no—
181#107—21no—
182#107—21no—
183#107#4112yes—
184#106—42no—
185#106—81no—
186C4⋊C8—41no—
187#102—42no—
188#102—81no—
189#102—81no—
190C4×C8—21no—
191C4×C8—21no—
192C4×C8—21no—
193C4×C8—21no—
194C4×C8—21no—
195C4×C8—21no—
196#95—41no—
197#95—41no—
198#95—41no—
199#95—41no—
200#95—41no—
201#95—41no—
202#95—41no—
203#95—41no—
204#95—41no—
205#95—41no—
206#95—41no—
207#95D811yescharacteristic
208#95D811yescharacteristic
209C24—41no—
210C2×D4—41no—
211C2×D4—41no—
212C2×D4—41no—
213C2×D4—41no—
214C2×D4—41no—
215C2×D4—41no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4C2×D811yescharacteristic
224C22×C4C2×D811yescharacteristic
225C22×C4C2×D811yescharacteristic
226C22×C4C2×D811yescharacteristic
227C22×C4C2×D811yescharacteristic
228C22×C4C2×D811yescharacteristic
229C22×C4C22×D411yescharacteristic
230D8—41no—
231D8—41no—
232D8—41no—
233D8—41no—
234C2×C8—21no—
235C2×C8—21no—
236C2×C8—21no—
237C2×C8—21no—
238C2×C8—21no—
239C2×C8—21no—
240C2×C8—21no—
241C2×C8—21no—
242C2×C8—21no—
243C2×C8C2×#4111yescharacteristic
244C2×C8C2×#4111yescharacteristic
245C2×C8H(5,𝔽2)11yescharacteristic
246C2×C82−1+411yescharacteristic
247C2×C82−1+411yescharacteristic
248C2×C82−1+411yescharacteristic
249C4⋊C4—22no—
250C4⋊C4—22no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—82no—
266C4⋊C4—82no—
267C4⋊C4—161no—
268C4⋊C4—161no—
269C4⋊C4—161no—
270C22⋊C4—82no—
271C22⋊C4—82no—
272C22⋊C4—161no—
273C22⋊C4—161no—
274C22⋊C4—161no—
275C22⋊C4—161no—
276C42—21no—
277C42—21no—
278C42—21no—
279C42—21no—
280C42—21no—
281C42—21no—
282C42—161no—
283C42—161no—
284C16—22no—
285C16—41no—
286C16—41no—
287C16—41no—
288C23—81no—
289C23—81no—
290C23—81no—
291C23—81no—
292C23—81no—
293C23—81no—
294C23—81no—
295C23C22×D811yescharacteristic
296D4—81no—
297D4—81no—
298D4—81no—
299D4—81no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
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—42no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—81no—
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#50812yes—
339C2×C4#53711yescharacteristic
340C2×C4#54711yescharacteristic
341C2×C4#54811yescharacteristic
342C2×C4#57511yescharacteristic
343C2×C4#57711yescharacteristic
344C2×C4#57811yescharacteristic
345C8—41no—
346C8—41no—
347C8—41no—
348C8#54111yescharacteristic
349C8#54111yescharacteristic
350C8#55311yescharacteristic
351C8#56311yescharacteristic
352C22—161no—
353C22—161no—
354C22—161no—
355C22—161no—
356C22—161no—
357C22—161no—
358C22—161no—
359C22#324511yescharacteristic
360C22#333411yescharacteristic
361C22#335811yescharacteristic
362C22C2×#50811yescharacteristic
363C22#341411yescharacteristic
364C22#341511yescharacteristic
365C22#341611yescharacteristic
366C4—22no—
367C4—41no—
368C4—41no—
369C4—41no—
370C4—41no—
371C4—41no—
372C4—82no—
373C4—161no—
374C4—161no—
375C4—161no—
376C4—161no—
377C4—161no—
378C4—161no—
379C4#272511yescharacteristic
380C4#325811yescharacteristic
381C4#335111yescharacteristic
382C4#337711yescharacteristic
383C2—321no—
384C2#3247211yescharacteristic
385C2#3381611yescharacteristic
386C2#3382311yescharacteristic
387C2#3390711yescharacteristic
388C2#3392211yescharacteristic
389C2#3392811yescharacteristic
390C2#3393811yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D8C23
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ C8×C4⋊C4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #27993C2semidirect
3. #13539C2semidirect
4. #13515C2semidirect
9. #13178C2semidirect
10. #13178C2semidirect
11. #13166C2semidirect
12. #13159C2semidirect
15. #13109C2semidirect

Maximal subgroups

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

subgroupindex
#280132
#279932
#135392
#135152
#134822
#134762
#134742
#131782
#131662
#131592
#131472
#131302
#131092