Finite Group Databasebeta

← #510289

Subgroups — #510289

Order 512.

461 conjugacy classes of subgroups.

The subgroup lattice has 407 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28832C211yesmaximal, characteristic, complemented
2#26484C211yesmaximal, characteristic
3#13539C211yesmaximal, characteristic, complemented
4#13537C211yesmaximal, characteristic, complemented
5#13516C211yesmaximal, characteristic, complemented
6#13482C211yesmaximal, characteristic
7#13482C211yesmaximal, characteristic
8#13452C211yesmaximal, characteristic, complemented
9#13283C211yesmaximal, characteristic, complemented
10#13149C211yesmaximal, characteristic
11#13131C211yesmaximal, characteristic
12#13111C211yesmaximal, characteristic, complemented
13#13105C211yesmaximal, characteristic, complemented
14#12937C211yesmaximal, characteristic
15#12811C211yesmaximal, characteristic
16C2×#500C2211yescharacteristic
17C2×C4×D8C2211yescharacteristic
18#2570—21no—
19#2452—21no—
20C2×#367C2211yescharacteristic
21C2×#367C2211yescharacteristic
22C2×C4⋊C16C2211yescharacteristic
23C2×#357C2211yescharacteristic
24C2×#357C2211yescharacteristic
25C2×#357C2211yescharacteristic
26C2×#357C2211yescharacteristic
27C2×#346C2211yescharacteristic
28#2085—21no—
29#2084—21no—
30#2067—21no—
31#2023—21no—
32#1970C2211yescharacteristic
33#1970C2211yescharacteristic
34#1949—21no—
35#1942C2211yescharacteristic
36#1942C2211yescharacteristic
37#1942C2211yescharacteristic
38#1942C2211yescharacteristic
39#1862—21no—
40#1846C2211yescharacteristic
41#1846C2211yescharacteristic
42#1845C2211yescharacteristic
43C4×#107C2211yescharacteristic
44C8×C4⋊C4C2211yescharacteristic
45#1380C2211yescharacteristic
46#1380C2211yescharacteristic
47#1380C2211yescharacteristic
48#1380C2211yescharacteristic
49#1380C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1374C2211yescharacteristic
56#1374C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×D8C2311yescharacteristic
60C2×C4⋊Q8—21no—
61C2×C22⋊Q8—41no—
62C2×C4×D4—21no—
63C22×C16C2311yescharacteristic
64C22×C16C2311yescharacteristic
65C22×C16C2311yescharacteristic
66C22×C16C2311yescharacteristic
67#500—22no—
68#500—41no—
69C4×D8—22no—
70C4×D8—41no—
71C2×#107—21no—
72C2×#107—21no—
73C2×#107C2311yescharacteristic
74C2×#107C2311yescharacteristic
75C2×#107C2311yescharacteristic
76C2×#107C2311yescharacteristic
77C2×#107C2311yescharacteristic
78C2×#107C2311yescharacteristic
79C2×#107C2311yescharacteristic
80C2×#106—21no—
81C2×#106—21no—
82C2×C4⋊C8D411yescharacteristic
83C2×#102—21no—
84C2×#102—21no—
85C2×#102—21no—
86C2×C4×C8C2311yescharacteristic
87C2×C4×C8C2311yescharacteristic
88C2×C4×C8C2311yescharacteristic
89#400—41no—
90#398—41no—
91#396—41no—
92#393—41no—
93#390—21no—
94#390—21no—
95#389—21no—
96#389—21no—
97#389—21no—
98#389—21no—
99#387—41no—
100#387—41no—
101#385—41no—
102#384—21no—
103#382—21no—
104#382—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4D411yescharacteristic
107#367—22no—
108#367—41no—
109C4⋊C16—41no—
110#357—22no—
111#357—22no—
112#357—22no—
113#357—41no—
114#346—41no—
115#340—21no—
116#340—21no—
117#340—21no—
118#340—21no—
119#340—21no—
120#336D411yescharacteristic
121#336D411yescharacteristic
122C22×Q8—41no—
123C22×D4—21no—
124C23×C4—41no—
125C2×D8—21no—
126C2×D8—21no—
127C2×D8—21no—
128C2×D8—21no—
129C2×D8—21no—
130C2×D8#4112yes—
131C22×C8C2×D411yescharacteristic
132C22×C8C2×D411yescharacteristic
133C22×C8C2×D411yescharacteristic
134C22×C8C2411yescharacteristic
135C4⋊Q8—42no—
136C4⋊Q8—81no—
137C22⋊Q8—84no—
138C4×D4—42no—
139C4×D4—81no—
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—41no—
148C2×C4⋊C4—41no—
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⋊C4—41no—
161C2×C42—41no—
162C2×C42C2×D411yescharacteristic
163C2×C42C2×D411yescharacteristic
164C2×C42C2×D411yescharacteristic
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—
176C2×C16—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—42no—
191#107—81no—
192#107#4112yes—
193#106—81no—
194#106—81no—
195C4⋊C8—22no—
196#102—42no—
197#102—42no—
198#102—81no—
199C4×C8—21no—
200C4×C8—21no—
201C4×C8—21no—
202C4×C8—21no—
203C4×C8—21no—
204C4×C8—21no—
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#95D811yescharacteristic
215#95D811yescharacteristic
216C24—41no—
217C2×Q8—82no—
218C2×Q8—82no—
219C2×Q8—161no—
220C2×D4—22no—
221C2×D4—41no—
222C2×D4—41no—
223C2×D4—41no—
224C2×D4—41no—
225C2×D4—41no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—82no—
234C22×C4—82no—
235C22×C4—161no—
236C22×C4C2×D811yescharacteristic
237C22×C4C2×D811yescharacteristic
238C22×C4C2×D811yescharacteristic
239C22×C4C2×D811yescharacteristic
240C22×C4C2×D811yescharacteristic
241C22×C4C2×D811yescharacteristic
242C22×C4C22×D411yescharacteristic
243D8—22no—
244D8—22no—
245D8—41no—
246D8—41no—
247C2×C8—21no—
248C2×C8—21no—
249C2×C8—21no—
250C2×C8—21no—
251C2×C8—21no—
252C2×C8—21no—
253C2×C8—21no—
254C2×C8—21no—
255C2×C8#13712yes—
256C2×C8C2×#4111yescharacteristic
257C2×C8C2×#4111yescharacteristic
258C2×C8H(5,𝔽2)11yescharacteristic
259C2×C82−1+411yescharacteristic
260C2×C82−1+411yescharacteristic
261C2×C82−1+411yescharacteristic
262C4⋊C4—22no—
263C4⋊C4—22no—
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—82no—
280C4⋊C4—82no—
281C4⋊C4—82no—
282C4⋊C4—82no—
283C4⋊C4—161no—
284C4⋊C4—161no—
285C22⋊C4—82no—
286C22⋊C4—82no—
287C22⋊C4—82no—
288C22⋊C4—82no—
289C22⋊C4—161no—
290C42—21no—
291C42—21no—
292C42—21no—
293C42—21no—
294C42—21no—
295C42—21no—
296C42—161no—
297C16—41no—
298C16—41no—
299C16—41no—
300C16—41no—
301C23—42no—
302C23—81no—
303C23—81no—
304C23—81no—
305C23—81no—
306C23—81no—
307C23—81no—
308C23C22×D811yescharacteristic
309Q8—164no—
310D4—42no—
311D4—42no—
312D4—81no—
313D4—81no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—42no—
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—81no—
351C2×C4—81no—
352C2×C4—164no—
353C2×C4#50912yes—
354C2×C4#53611yescharacteristic
355C2×C4#54611yescharacteristic
356C2×C4#54811yescharacteristic
357C2×C4C2×#13711yescharacteristic
358C2×C4#57711yescharacteristic
359C2×C4#57711yescharacteristic
360C8—22no—
361C8—41no—
362C8—41no—
363C8#53911yescharacteristic
364C8#54111yescharacteristic
365C8#55211yescharacteristic
366C8#56311yescharacteristic
367C22—82no—
368C22—82no—
369C22—82no—
370C22—161no—
371C22—161no—
372C22—161no—
373C22—161no—
374C22#322611yescharacteristic
375C22#329411yescharacteristic
376C22#335811yescharacteristic
377C22C2×#50911yescharacteristic
378C22#341411yescharacteristic
379C22#341611yescharacteristic
380C22#341611yescharacteristic
381C4—22no—
382C4—41no—
383C4—41no—
384C4—41no—
385C4—41no—
386C4—41no—
387C4—82no—
388C4—161no—
389C4—161no—
390C4—161no—
391C4—161no—
392C4—161no—
393C4—161no—
394C4#269011yescharacteristic
395C4#324411yescharacteristic
396C4#331611yescharacteristic
397C4#337211yescharacteristic
398C2—162no—
399C2#3145411yescharacteristic
400C2#3380111yescharacteristic
401C2#3381011yescharacteristic
402C2#3387711yescharacteristic
403C2#3389311yescharacteristic
404C2#3394011yescharacteristic
405C2#3394311yescharacteristic
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
1. #28832C2semidirect
3. #13539C2semidirect
4. #13537C2semidirect
5. #13516C2semidirect
8. #13452C2semidirect
9. #13283C2semidirect
12. #13111C2semidirect
13. #13105C2semidirect

Maximal subgroups

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

subgroupindex
#288322
#264842
#135392
#135372
#135162
#134822
#134522
#132832
#131492
#131312
#131112
#131052
#129372
#128112