Finite Group Databasebeta

← #509064

Subgroups — #509064

Order 512.

415 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33272C211yesmaximal, characteristic
2#33261C211yesmaximal, characteristic, complemented
3#13537C211yesmaximal, characteristic, complemented
4#13527C211yesmaximal, characteristic
5#13516C211yesmaximal, characteristic, complemented
6#13516C211yesmaximal, characteristic, complemented
7#13507C211yesmaximal, characteristic, complemented
8#13482C211yesmaximal, characteristic
9#13476C211yesmaximal, characteristic
10#13474C211yesmaximal, characteristic
11#13167C211yesmaximal, characteristic, complemented
12#13166C211yesmaximal, characteristic, complemented
13#13149C211yesmaximal, characteristic
14#13147C211yesmaximal, characteristic
15#12654C211yesmaximal, characteristic, complemented
16#2647—21no—
17#2644—21no—
18C2×#367C2211yescharacteristic
19C2×#367C2211yescharacteristic
20C2×#367C2211yescharacteristic
21C2×#366C2211yescharacteristic
22C2×C4⋊C16C2211yescharacteristic
23C2×C4⋊C16C2211yescharacteristic
24C2×#357C2211yescharacteristic
25C2×#357C2211yescharacteristic
26C2×#357C2211yescharacteristic
27C2×#357C2211yescharacteristic
28C2×#346C2211yescharacteristic
29C2×C4×C16C2211yescharacteristic
30#1970C2211yescharacteristic
31#1970C2211yescharacteristic
32#1970C2211yescharacteristic
33#1944—21no—
34#1943—21no—
35#1942C2211yescharacteristic
36#1942C2211yescharacteristic
37#1942C2211yescharacteristic
38#1942C2211yescharacteristic
39#1942C2211yescharacteristic
40#1942C2211yescharacteristic
41#1942C2211yescharacteristic
42#1936—21no—
43#1936—21no—
44#1936—21no—
45#1846C2211yescharacteristic
46#1845C2211yescharacteristic
47#1836—21no—
48#1726C2211yescharacteristic
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×#107—21no—
65C2×#107C2311yescharacteristic
66C2×#107C2311yescharacteristic
67C2×#107C2311yescharacteristic
68C2×#107C2311yescharacteristic
69C2×#107C2311yescharacteristic
70C2×#107C2311yescharacteristic
71C2×#107C2311yescharacteristic
72C2×#106—21no—
73C2×#102—21no—
74C2×#102—21no—
75C2×#102—21no—
76C2×#97D411yescharacteristic
77C2×#97D411yescharacteristic
78C2×#97D411yescharacteristic
79C2×C4×C8C2311yescharacteristic
80C2×C4×C8C2311yescharacteristic
81C2×C4×C8C2311yescharacteristic
82#400—41no—
83#400—41no—
84#398—41no—
85#398—41no—
86#397—41no—
87#397—41no—
88#397—41no—
89#396—41no—
90#390—21no—
91#390—21no—
92#390—21no—
93#389—21no—
94#389—21no—
95#389—21no—
96#389—21no—
97#389—21no—
98#389—21no—
99#389—21no—
100#384—21no—
101#382—21no—
102#376D411yescharacteristic
103#367—22no—
104#367—22no—
105#367—41no—
106#366—41no—
107C4⋊C16—22no—
108C4⋊C16—22no—
109#357—22no—
110#357—22no—
111#357—41no—
112#357—41no—
113#346—41no—
114C4×C16—41no—
115#340—21no—
116#340—21no—
117#340—21no—
118#340—21no—
119#340—21no—
120#340—21no—
121#340—21no—
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⋊C4—41no—
146C2×C4⋊C4—41no—
147C2×C4⋊C4—41no—
148C2×C4⋊C4—41no—
149C2×C22⋊C4—41no—
150C2×C22⋊C4—41no—
151C2×C22⋊C4—41no—
152C2×C22⋊C4—41no—
153C2×C22⋊C4—41no—
154C2×C22⋊C4—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#4112yes—
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—21no—
184#107—42no—
185#106—42no—
186#102—42no—
187#102—42no—
188#102—42no—
189#97—22no—
190#97—22no—
191#97—41no—
192C4×C8—21no—
193C4×C8—21no—
194C4×C8—21no—
195C4×C8—21no—
196C4×C8—21no—
197C4×C8—21no—
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
209#95D811yescharacteristic
210#95D811yescharacteristic
211C24—41no—
212C2×D4—41no—
213C2×D4—41no—
214C2×D4—41no—
215C2×D4—41no—
216C2×D4—41no—
217C2×D4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4C2×D811yescharacteristic
226C22×C4C2×D811yescharacteristic
227C22×C4C2×D811yescharacteristic
228C22×C4C2×D811yescharacteristic
229C22×C4C2×D811yescharacteristic
230C22×C4C2×D811yescharacteristic
231C22×C4C22×D411yescharacteristic
232D8—41no—
233D8—41no—
234D8—41no—
235D8—41no—
236C2×C8—21no—
237C2×C8—21no—
238C2×C8—21no—
239C2×C8—21no—
240C2×C8—21no—
241C2×C8—21no—
242C2×C8—21no—
243C2×C8#13512yes—
244C2×C8#13712yes—
245C2×C8C2×#4111yescharacteristic
246C2×C8C2×#4111yescharacteristic
247C2×C8H(5,𝔽2)11yescharacteristic
248C2×C82−1+411yescharacteristic
249C2×C82−1+411yescharacteristic
250C2×C82−1+411yescharacteristic
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—161no—
266C4⋊C4—161no—
267C4⋊C4—161no—
268C4⋊C4—161no—
269C4⋊C4—161no—
270C4⋊C4—161no—
271C4⋊C4—161no—
272C4⋊C4—161no—
273C4⋊C4—161no—
274C22⋊C4—161no—
275C22⋊C4—161no—
276C22⋊C4—161no—
277C22⋊C4—161no—
278C22⋊C4—161no—
279C22⋊C4—161no—
280C42—21no—
281C42—21no—
282C42—21no—
283C42—21no—
284C42—21no—
285C42—21no—
286C16—22no—
287C16—22no—
288C16—41no—
289C16—41no—
290C23—81no—
291C23—81no—
292C23—81no—
293C23—81no—
294C23—81no—
295C23—81no—
296C23—81no—
297C23C22×D811yescharacteristic
298D4—81no—
299D4—81no—
300D4—81no—
301D4—81no—
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—21no—
318C2×C4—21no—
319C2×C4—21no—
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—81no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4#53211yescharacteristic
342C2×C4#53611yescharacteristic
343C2×C4#53711yescharacteristic
344C2×C4C2×#13511yescharacteristic
345C2×C4C2×#13711yescharacteristic
346C2×C4#57511yescharacteristic
347C8—22no—
348C8—22no—
349C8—41no—
350C8#53911yescharacteristic
351C8#53911yescharacteristic
352C8#54111yescharacteristic
353C8#54111yescharacteristic
354C22—161no—
355C22—161no—
356C22—161no—
357C22—161no—
358C22—161no—
359C22—161no—
360C22—161no—
361C22#316911yescharacteristic
362C22#322611yescharacteristic
363C22#324511yescharacteristic
364C22#341511yescharacteristic
365C22#341611yescharacteristic
366C22#341611yescharacteristic
367C22#341611yescharacteristic
368C4—41no—
369C4—41no—
370C4—41no—
371C4—41no—
372C4—41no—
373C4—41no—
374C4—161no—
375C4—161no—
376C4—161no—
377C4—161no—
378C4—161no—
379C4—161no—
380C4—161no—
381C4#266411yescharacteristic
382C4#317211yescharacteristic
383C4#323211yescharacteristic
384C4#324811yescharacteristic
385C2—321no—
386C2#3037611yescharacteristic
387C2#3374311yescharacteristic
388C2#3374411yescharacteristic
389C2#3380711yescharacteristic
390C2#3381011yescharacteristic
391C2#3382311yescharacteristic
392C2#3382911yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D8C23
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #376 ◁ #1726 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #33261C2semidirect
3. #13537C2semidirect
5. #13516C2semidirect
6. #13516C2semidirect
7. #13507C2semidirect
11. #13167C2semidirect
12. #13166C2semidirect
15. #12654C2semidirect

Maximal subgroups

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

subgroupindex
#332722
#332612
#135372
#135272
#135162
#135072
#134822
#134762
#134742
#131672
#131662
#131492
#131472
#126542