Finite Group Databasebeta

← #511416

Subgroups — #511416

Order 512.

485 conjugacy classes of subgroups.

The subgroup lattice has 435 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28267C211yesmaximal, characteristic
2#28265C211yesmaximal, characteristic, complemented
3#13539C211yesmaximal, characteristic, complemented
4#13538C211yesmaximal, characteristic, complemented
5#13514C211yesmaximal, characteristic, complemented
6#13483C211yesmaximal, characteristic, complemented
7#13480C211yesmaximal, characteristic
8#13449C211yesmaximal, characteristic
9#13264C211yesmaximal, characteristic, complemented
10#13150C211yesmaximal, characteristic, complemented
11#13149C211yesmaximal, characteristic
12#13014C211yesmaximal, characteristic
13#13013C211yesmaximal, characteristic
14#12936C211yesmaximal, characteristic, complemented
15#12935C211yesmaximal, characteristic
16C22×SD32C2211yescharacteristic
17C2×C8×Q8C2211yescharacteristic
18C2×C4×Q16C2211yescharacteristic
19C2×C4×D8C2211yescharacteristic
20#2582—21no—
21#2580—21no—
22C2×#366C2211yescharacteristic
23C2×C4⋊C16C2211yescharacteristic
24C2×#358C2211yescharacteristic
25C2×#358C2211yescharacteristic
26C2×#358C2211yescharacteristic
27C2×#357C2211yescharacteristic
28C2×#357C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×#346C2211yescharacteristic
31#2068—21no—
32#2068—21no—
33#2065—21no—
34#2065—21no—
35#1970C2211yescharacteristic
36#1970C2211yescharacteristic
37#1969C2211yescharacteristic
38#1969C2211yescharacteristic
39#1942C2211yescharacteristic
40#1942C2211yescharacteristic
41#1942C2211yescharacteristic
42#1942C2211yescharacteristic
43#1846C2211yescharacteristic
44#1845C2211yescharacteristic
45#1804—21no—
46#1803—21no—
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×Q16C2311yescharacteristic
60C22×D8C2311yescharacteristic
61C2×C4×Q8—21no—
62C2×C4×Q8D411yescharacteristic
63C2×C4×D4—21no—
64C2×SD32—22no—
65C2×SD32—22no—
66C2×SD32—41no—
67C22×C16C2311yescharacteristic
68C22×C16C2311yescharacteristic
69C22×C16C2311yescharacteristic
70C22×C16C2311yescharacteristic
71C8×Q8—41no—
72C8×Q8—41no—
73C4×Q16—41no—
74C4×Q16—41no—
75C4×D8—41no—
76C4×D8—41no—
77C2×#107—21no—
78C2×#107C2311yescharacteristic
79C2×#107C2311yescharacteristic
80C2×#107C2311yescharacteristic
81C2×#107C2311yescharacteristic
82C2×#107C2311yescharacteristic
83C2×#107C2311yescharacteristic
84C2×#106—21no—
85C2×C4⋊C8D411yescharacteristic
86C2×C4⋊C8D411yescharacteristic
87C2×C4⋊C8D411yescharacteristic
88C2×#103—21no—
89C2×#103—21no—
90C2×#103—21no—
91C2×#102—21no—
92C2×#102—21no—
93C2×#102—21no—
94C2×C4×C8C2311yescharacteristic
95C2×C4×C8C2311yescharacteristic
96C2×C4×C8C2311yescharacteristic
97#399—41no—
98#398—41no—
99#398—41no—
100#393—41no—
101#391—21no—
102#391—21no—
103#391—41no—
104#390—21no—
105#390—21no—
106#389—21no—
107#389—21no—
108#389—21no—
109#389—21no—
110#389—41no—
111#385—41no—
112#385—41no—
113#384—21no—
114#382—21no—
115#366—41no—
116C4⋊C16—41no—
117#358—22no—
118#358—22no—
119#358—41no—
120#357—22no—
121#357—22no—
122#357—41no—
123#346—41no—
124#340—21no—
125#340—21no—
126#340—21no—
127#340—21no—
128C22×Q8—21no—
129C22×Q8D811yescharacteristic
130C22×D4—21no—
131C23×C4—41no—
132C2×Q16—21no—
133C2×Q16—21no—
134C2×Q16—21no—
135C2×Q16—21no—
136C2×Q16—21no—
137C2×Q16#4112yes—
138C2×D8—21no—
139C2×D8—21no—
140C2×D8—21no—
141C2×D8—21no—
142C2×D8—21no—
143C2×D8#4112yes—
144C22×C8C2×D411yescharacteristic
145C22×C8C2×D411yescharacteristic
146C22×C8C2×D411yescharacteristic
147C22×C8C2411yescharacteristic
148C4×Q8—41no—
149C4×Q8—41no—
150C4×Q8—42no—
151C4×Q8—42no—
152C4×D4—42no—
153C4×D4—42no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4D811yescharacteristic
168C2×C4⋊C4D811yescharacteristic
169C2×C4⋊C4D811yescharacteristic
170C2×C22⋊C4—41no—
171C2×C22⋊C4—41no—
172C2×C22⋊C4—41no—
173C2×C22⋊C4—41no—
174C2×C22⋊C4—41no—
175C2×C42—41no—
176C2×C42C2×D411yescharacteristic
177C2×C42C2×D411yescharacteristic
178C2×C42C2×D411yescharacteristic
179SD32—44no—
180C2×C16—21no—
181C2×C16—21no—
182C2×C16—21no—
183C2×C16—21no—
184C2×C16—21no—
185C2×C16—21no—
186C2×C16—21no—
187C2×C16—21no—
188C2×C16—21no—
189C2×C16—21no—
190C2×C16—21no—
191C2×C16—21no—
192#107—21no—
193#107—21no—
194#107—21no—
195#107—21no—
196#107—21no—
197#107—21no—
198#107—21no—
199#107—21no—
200#107—21no—
201#107—21no—
202#107—21no—
203#107—21no—
204#107—81no—
205#106—81no—
206C4⋊C8—41no—
207C4⋊C8—41no—
208C4⋊C8—41no—
209#103—42no—
210#103—81no—
211#103—81no—
212#102—42no—
213#102—81no—
214#102—81no—
215C4×C8—21no—
216C4×C8—21no—
217C4×C8—21no—
218C4×C8—21no—
219C4×C8—21no—
220C4×C8—21no—
221#95—41no—
222#95—41no—
223#95—41no—
224#95—41no—
225#95—41no—
226#95—41no—
227#95—41no—
228#95—41no—
229#95—41no—
230#95—41no—
231C24—41no—
232C2×Q8—22no—
233C2×Q8—22no—
234C2×Q8—41no—
235C2×Q8—41no—
236C2×Q8—41no—
237C2×Q8—41no—
238C2×Q8—41no—
239C2×Q8—41no—
240C2×Q8—41no—
241C2×D4—22no—
242C2×D4—41no—
243C2×D4—41no—
244C2×D4—41no—
245C2×D4—41no—
246C2×D4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—82no—
255C22×C4—161no—
256C22×C4—161no—
257C22×C4C2×D811yescharacteristic
258C22×C4C2×D811yescharacteristic
259C22×C4C2×D811yescharacteristic
260C22×C4C2×D811yescharacteristic
261C22×C4C2×D811yescharacteristic
262C22×C4C2×D811yescharacteristic
263C22×C4C22×D411yescharacteristic
264Q16—22no—
265Q16—22no—
266Q16—41no—
267Q16—41no—
268D8—22no—
269D8—22no—
270D8—41no—
271D8—41no—
272C2×C8—21no—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8C2×#4111yescharacteristic
282C2×C8C2×#4111yescharacteristic
283C2×C8H(5,𝔽2)11yescharacteristic
284C2×C8H(5,𝔽2)11yescharacteristic
285C2×C82−1+411yescharacteristic
286C2×C82−1+411yescharacteristic
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—82no—
303C4⋊C4—82no—
304C4⋊C4—161no—
305C4⋊C4—161no—
306C4⋊C4—161no—
307C4⋊C4—161no—
308C4⋊C4—161no—
309C22⋊C4—82no—
310C22⋊C4—82no—
311C22⋊C4—82no—
312C22⋊C4—161no—
313C22⋊C4—161no—
314C42—21no—
315C42—21no—
316C42—21no—
317C42—21no—
318C42—21no—
319C42—21no—
320C42—82no—
321C16—41no—
322C16—41no—
323C16—41no—
324C16—41no—
325C23—42no—
326C23—81no—
327C23—81no—
328C23—81no—
329C23—81no—
330C23—81no—
331C23—81no—
332C23C22×D811yescharacteristic
333Q8—44no—
334Q8—42no—
335Q8—42no—
336Q8—81no—
337Q8—81no—
338D4—42no—
339D4—42no—
340D4—81no—
341D4—81no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—42no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—164no—
381C2×C4#51012yes—
382C2×C4#53411yescharacteristic
383C2×C4#54811yescharacteristic
384C2×C4#54811yescharacteristic
385C2×C4#57511yescharacteristic
386C2×C4#57711yescharacteristic
387C2×C4#57711yescharacteristic
388C8—41no—
389C8—41no—
390C8—41no—
391C8#53911yescharacteristic
392C8#53911yescharacteristic
393C8#55111yescharacteristic
394C8#56311yescharacteristic
395C22—82no—
396C22—82no—
397C22—82no—
398C22—161no—
399C22—161no—
400C22—161no—
401C22—161no—
402C22#318611yescharacteristic
403C22#335811yescharacteristic
404C22#335811yescharacteristic
405C22C2×#51011yescharacteristic
406C22#341411yescharacteristic
407C22#341511yescharacteristic
408C22#341611yescharacteristic
409C4—22no—
410C4—41no—
411C4—41no—
412C4—41no—
413C4—41no—
414C4—41no—
415C4—82no—
416C4—161no—
417C4—161no—
418C4—161no—
419C4—161no—
420C4—161no—
421C4—161no—
422C4#272811yescharacteristic
423C4#320411yescharacteristic
424C4#337611yescharacteristic
425C4#337611yescharacteristic
426C2—162no—
427C2#3254411yescharacteristic
428C2#3376211yescharacteristic
429C2#3376411yescharacteristic
430C2#3393711yescharacteristic
431C2#3393811yescharacteristic
432C2#3393911yescharacteristic
433C2#3394211yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D8C23
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×C8×Q8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #28265C2semidirect
3. #13539C2semidirect
4. #13538C2semidirect
5. #13514C2semidirect
6. #13483C2semidirect
9. #13264C2semidirect
10. #13150C2semidirect
14. #12936C2semidirect

Maximal subgroups

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

subgroupindex
#282672
#282652
#135392
#135382
#135142
#134832
#134802
#134492
#132642
#131502
#131492
#130142
#130132
#129362
#129352