Finite Group Databasebeta

← #511534

Subgroups — #511534

Order 512.

489 conjugacy classes of subgroups.

The subgroup lattice has 437 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28268C211yesmaximal, characteristic, complemented
2#28267C211yesmaximal, characteristic
3#13529C211yesmaximal, characteristic, complemented
4#13528C211yesmaximal, characteristic
5#13507C211yesmaximal, characteristic, complemented
6#13507C211yesmaximal, characteristic, complemented
7#13477C211yesmaximal, characteristic, complemented
8#13476C211yesmaximal, characteristic
9#13264C211yesmaximal, characteristic, complemented
10#13238C211yesmaximal, characteristic
11#13168C211yesmaximal, characteristic, complemented
12#13014C211yesmaximal, characteristic
13#13013C211yesmaximal, characteristic
14#12938C211yesmaximal, characteristic, complemented
15#12937C211yesmaximal, characteristic
16C22×SD32C2211yescharacteristic
17C2×C8×Q8C2211yescharacteristic
18C2×C4×Q16C2211yescharacteristic
19C2×C4×D8C2211yescharacteristic
20#2581—21no—
21#2580—21no—
22C2×#367C2211yescharacteristic
23C2×#367C2211yescharacteristic
24C2×#367C2211yescharacteristic
25C2×#366C2211yescharacteristic
26C2×#366C2211yescharacteristic
27C2×C4⋊C16C2211yescharacteristic
28C2×#358C2211yescharacteristic
29C2×#358C2211yescharacteristic
30C2×#358C2211yescharacteristic
31C2×#357C2211yescharacteristic
32C2×#357C2211yescharacteristic
33C2×#357C2211yescharacteristic
34C2×#346C2211yescharacteristic
35#2066—21no—
36#2066—21no—
37#2065—21no—
38#2065—21no—
39#1970C2211yescharacteristic
40#1970C2211yescharacteristic
41#1969C2211yescharacteristic
42#1969C2211yescharacteristic
43#1942C2211yescharacteristic
44#1942C2211yescharacteristic
45#1942C2211yescharacteristic
46#1942C2211yescharacteristic
47#1846C2211yescharacteristic
48#1846C2211yescharacteristic
49#1805—21no—
50#1804—21no—
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—41no—
65C2×SD32—41no—
66C2×SD32Q814yescomplemented
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×#107C2311yescharacteristic
78C2×#107C2311yescharacteristic
79C2×#107C2311yescharacteristic
80C2×#107C2311yescharacteristic
81C2×#107C2311yescharacteristic
82C2×#107C2311yescharacteristic
83C2×#106—21no—
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#400—41no—
98#399—41no—
99#395—41no—
100#393—41no—
101#391—21no—
102#391—21no—
103#390—21no—
104#390—21no—
105#389—21no—
106#389—21no—
107#389—21no—
108#389—21no—
109#387—41no—
110#387—41no—
111#385—41no—
112#385—41no—
113#382—21no—
114#382—21no—
115#367—41no—
116#367—41no—
117#367—41no—
118#366—22no—
119#366—41no—
120C4⋊C16—41no—
121#358—22no—
122#358—41no—
123#358—41no—
124#357—22no—
125#357—41no—
126#357—41no—
127#346—41no—
128#340—21no—
129#340—21no—
130#340—21no—
131#340—21no—
132C22×Q8—21no—
133C22×Q8D811yescharacteristic
134C22×D4—21no—
135C23×C4—41no—
136C2×Q16—21no—
137C2×Q16—21no—
138C2×Q16—21no—
139C2×Q16—21no—
140C2×Q16—21no—
141C2×Q16C2×Q812yes—
142C2×D8—21no—
143C2×D8—21no—
144C2×D8—21no—
145C2×D8—21no—
146C2×D8—21no—
147C2×D8C2×Q812yes—
148C22×C8C2×D411yescharacteristic
149C22×C8C2×D411yescharacteristic
150C22×C8C2×D411yescharacteristic
151C22×C8C2411yescharacteristic
152C4×Q8—41no—
153C4×Q8—41no—
154C4×Q8—42no—
155C4×Q8—42no—
156C4×D4—42no—
157C4×D4—42no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4D811yescharacteristic
170C2×C4⋊C4D811yescharacteristic
171C2×C4⋊C4D811yescharacteristic
172C2×C22⋊C4—41no—
173C2×C22⋊C4—41no—
174C2×C22⋊C4—41no—
175C2×C22⋊C4—41no—
176C2×C22⋊C4—41no—
177C2×C42—41no—
178C2×C42C2×D411yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C42C2×D411yescharacteristic
181SD32—44no—
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—
192C2×C16C2×Q812yes—
193C2×C16#4112yes—
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—21no—
205#107—21no—
206#106—81no—
207#106—81no—
208C4⋊C8—41no—
209C4⋊C8—41no—
210C4⋊C8—41no—
211#103—42no—
212#103—81no—
213#103—81no—
214#102—42no—
215#102—81no—
216#102—81no—
217C4×C8—21no—
218C4×C8—21no—
219C4×C8—21no—
220C4×C8—21no—
221C4×C8—21no—
222C4×C8—21no—
223#95—41no—
224#95—41no—
225#95—41no—
226#95—41no—
227#95—41no—
228#95—41no—
229#95—41no—
230#95—41no—
231#95—41no—
232#95—41no—
233#95—41no—
234#95—41no—
235C24—41no—
236C2×Q8—22no—
237C2×Q8—41no—
238C2×Q8—41no—
239C2×Q8—41no—
240C2×Q8—41no—
241C2×Q8—41no—
242C2×Q8—41no—
243C2×Q8—41no—
244C2×Q8—41no—
245C2×D4—22no—
246C2×D4—41no—
247C2×D4—41no—
248C2×D4—41no—
249C2×D4—41no—
250C2×D4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—82no—
259C22×C4—161no—
260C22×C4—161no—
261C22×C4C2×D811yescharacteristic
262C22×C4C2×D811yescharacteristic
263C22×C4C2×D811yescharacteristic
264C22×C4C2×D811yescharacteristic
265C22×C4C2×D811yescharacteristic
266C22×C4C2×D811yescharacteristic
267C22×C4C22×D411yescharacteristic
268Q16—22no—
269Q16—22no—
270Q16—41no—
271Q16—41no—
272D8—22no—
273D8—22no—
274D8—41no—
275D8—41no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8C22×Q811yescharacteristic
286C2×C8C2×#4111yescharacteristic
287C2×C8H(5,𝔽2)11yescharacteristic
288C2×C8H(5,𝔽2)11yescharacteristic
289C2×C8H(5,𝔽2)11yescharacteristic
290C2×C82−1+411yescharacteristic
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—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—82no—
307C4⋊C4—82no—
308C4⋊C4—82no—
309C4⋊C4—161no—
310C4⋊C4—161no—
311C22⋊C4—82no—
312C22⋊C4—82no—
313C22⋊C4—82no—
314C22⋊C4—161no—
315C22⋊C4—161no—
316C42—21no—
317C42—21no—
318C42—21no—
319C42—21no—
320C42—21no—
321C42—21no—
322C42—82no—
323C16—22no—
324C16—22no—
325C16—41no—
326C16—41no—
327C23—42no—
328C23—81no—
329C23—81no—
330C23—81no—
331C23—81no—
332C23—81no—
333C23—81no—
334C23C22×D811yescharacteristic
335Q8—44no—
336Q8—42no—
337Q8—42no—
338Q8—81no—
339Q8—81no—
340D4—42no—
341D4—42no—
342D4—81no—
343D4—81no—
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—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—42no—
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—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—164no—
384C2×C4#53411yescharacteristic
385C2×C4D4×Q811yescharacteristic
386C2×C4#55011yescharacteristic
387C2×C4#57511yescharacteristic
388C2×C4#57611yescharacteristic
389C2×C4#57811yescharacteristic
390C8—41no—
391C8—41no—
392C8—41no—
393C8#53911yescharacteristic
394C8#54311yescharacteristic
395C8#55411yescharacteristic
396C8#56211yescharacteristic
397C22—82no—
398C22—82no—
399C22—82no—
400C22—161no—
401C22—161no—
402C22—161no—
403C22—161no—
404C22#318611yescharacteristic
405C22Q8×D811yescharacteristic
406C22#339111yescharacteristic
407C22#341411yescharacteristic
408C22#341411yescharacteristic
409C22#341611yescharacteristic
410C22#341611yescharacteristic
411C4—41no—
412C4—41no—
413C4—41no—
414C4—41no—
415C4—41no—
416C4—41no—
417C4—82no—
418C4—161no—
419C4—161no—
420C4—161no—
421C4—161no—
422C4—161no—
423C4—161no—
424C4#273211yescharacteristic
425C4#320311yescharacteristic
426C4#338911yescharacteristic
427C4#340211yescharacteristic
428C2—162no—
429C2#3264811yescharacteristic
430C2#3376311yescharacteristic
431C2#3376611yescharacteristic
432C2#3395511yescharacteristic
433C2#3395511yescharacteristic
434C2#3396811yescharacteristic
435C2#3396811yescharacteristic
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
1. #28268C2semidirect
3. #13529C2semidirect
5. #13507C2semidirect
6. #13507C2semidirect
7. #13477C2semidirect
9. #13264C2semidirect
11. #13168C2semidirect
14. #12938C2semidirect
66. C2×SD32Q8semidirect

Maximal subgroups

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

subgroupindex
#282682
#282672
#135292
#135282
#135072
#134772
#134762
#132642
#132382
#131682
#130142
#130132
#129382
#129372