Finite Group Databasebeta

← #511285

Subgroups — #511285

Order 512.

516 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28527C211yesmaximal, characteristic
2#28015C211yesmaximal, characteristic, complemented
3#13531C211yesmaximal, characteristic, complemented
4#13481C211yesmaximal, characteristic, complemented
5#13480C211yesmaximal, characteristic
6#13452C211yesmaximal, characteristic, complemented
7#13449C211yesmaximal, characteristic
8#13364C211yesmaximal, characteristic, complemented
9#13237C211yesmaximal, characteristic
10#13177C211yesmaximal, characteristic, complemented
11#13131C211yesmaximal, characteristic
12#13104C211yesmaximal, characteristic, complemented
13#13014C211yesmaximal, characteristic
14#12937C211yesmaximal, characteristic
15#12888C211yesmaximal, characteristic, complemented
16C22×SD32C2211yescharacteristic
17C2×#500C2211yescharacteristic
18C2×#474—21no—
19C2×C4×D8C2211yescharacteristic
20#2542—21no—
21#2511—21no—
22C2×#367C2211yescharacteristic
23C2×#367C2211yescharacteristic
24C2×#367C2211yescharacteristic
25C2×C4⋊C16C2211yescharacteristic
26C2×#358C2211yescharacteristic
27C2×#358C2211yescharacteristic
28C2×#358C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×#357C2211yescharacteristic
31C2×#357C2211yescharacteristic
32C2×C4×C16C2211yescharacteristic
33#2093—21no—
34#2093—21no—
35#2072—21no—
36#2068—21no—
37#1970C2211yescharacteristic
38#1970C2211yescharacteristic
39#1969C2211yescharacteristic
40#1969C2211yescharacteristic
41#1969C2211yescharacteristic
42#1942C2211yescharacteristic
43#1928—21no—
44#1845C2211yescharacteristic
45#1845C2211yescharacteristic
46#1845C2211yescharacteristic
47C4×#107C2211yescharacteristic
48C8×C4⋊C4C2211yescharacteristic
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×C22⋊Q8—41no—
63C2×C22⋊Q8—41no—
64C2×C4×D4—21no—
65C2×SD32—22no—
66C2×SD32—22no—
67C2×SD32—41no—
68C22×C16C2311yescharacteristic
69C22×C16C2311yescharacteristic
70C22×C16C2311yescharacteristic
71C22×C16C2311yescharacteristic
72#500—41no—
73#500—41no—
74#474—44no—
75C4×D8—41no—
76C4×D8—41no—
77C2×#107—21no—
78C2×#107—21no—
79C2×#107—21no—
80C2×#107—21no—
81C2×#107C2311yescharacteristic
82C2×#107C2311yescharacteristic
83C2×#107C2311yescharacteristic
84C2×#107C2311yescharacteristic
85C2×#107C2311yescharacteristic
86C2×#107C2311yescharacteristic
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#400—41no—
99#399—41no—
100#399—41no—
101#391—21no—
102#391—21no—
103#391—21no—
104#390—21no—
105#390—21no—
106#389—21no—
107#387—41no—
108#387—41no—
109#384—21no—
110#384—21no—
111#384—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4D411yescharacteristic
114#367—41no—
115#367—41no—
116#367—41no—
117C4⋊C16—22no—
118#358—41no—
119#358—41no—
120#358—41no—
121#357—22no—
122#357—22no—
123#357—41no—
124C4×C16—22no—
125#340—21no—
126#340—21no—
127#336D411yescharacteristic
128#336D411yescharacteristic
129C22×Q8—21no—
130C22×Q8—41no—
131C22×D4—21no—
132C23×C4—41no—
133C2×Q16—21no—
134C2×Q16—21no—
135C2×Q16—21no—
136C2×Q16—21no—
137C2×Q16—21no—
138C2×Q16—21no—
139C2×D8—21no—
140C2×D8—21no—
141C2×D8—21no—
142C2×D8—21no—
143C2×D8—21no—
144C2×D8#4112yes—
145C22×C8C2×D411yescharacteristic
146C22×C8C2×D411yescharacteristic
147C22×C8C2×D411yescharacteristic
148C22×C8C2411yescharacteristic
149C4⋊Q8—42no—
150C4⋊Q8—42no—
151C22⋊Q8—84no—
152C22⋊Q8—84no—
153C4×D4—42no—
154C4×D4—42no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
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⋊C4—41no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4D811yescharacteristic
170C2×C4⋊C4D811yescharacteristic
171C2×C22⋊C4—41no—
172C2×C22⋊C4—41no—
173C2×C22⋊C4—41no—
174C2×C22⋊C4—41no—
175C2×C22⋊C4—41no—
176C2×C42—41no—
177C2×C42C2×D411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C42C2×D411yescharacteristic
180SD32—44no—
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—
192C2×C16#4112yes—
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—21no—
205#107—42no—
206#107—42no—
207#107—42no—
208#107—81no—
209C4⋊C8—22no—
210#103—42no—
211#103—81no—
212#103—81no—
213#102—42no—
214#102—42no—
215#102—42no—
216C4×C8—21no—
217C4×C8—21no—
218C4×C8—21no—
219C4×C8—21no—
220C4×C8—21no—
221C4×C8—21no—
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#95D811yescharacteristic
231#95D811yescharacteristic
232C24—41no—
233C2×Q8—41no—
234C2×Q8—41no—
235C2×Q8—41no—
236C2×Q8—41no—
237C2×Q8—41no—
238C2×Q8—41no—
239C2×Q8—82no—
240C2×Q8—82no—
241C2×Q8—82no—
242C2×D4—22no—
243C2×D4—41no—
244C2×D4—41no—
245C2×D4—41no—
246C2×D4—41no—
247C2×D4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—82no—
256C22×C4—82no—
257C22×C4—82no—
258C22×C4C2×D811yescharacteristic
259C22×C4C2×D811yescharacteristic
260C22×C4C2×D811yescharacteristic
261C22×C4C2×D811yescharacteristic
262C22×C4C2×D811yescharacteristic
263C22×C4C2×D811yescharacteristic
264C22×C4C22×D411yescharacteristic
265Q16—41no—
266Q16—41no—
267Q16—41no—
268Q16—41no—
269D8—22no—
270D8—22no—
271D8—41no—
272D8—41no—
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×C8AGL(1,ℤ/8)12yes—
282C2×C8C2×#4111yescharacteristic
283C2×C8C2×#4111yescharacteristic
284C2×C8H(5,𝔽2)11yescharacteristic
285C2×C8H(5,𝔽2)11yescharacteristic
286C2×C82−1+411yescharacteristic
287C2×C82−1+411yescharacteristic
288C4⋊C4—22no—
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—82no—
302C4⋊C4—82no—
303C4⋊C4—82no—
304C4⋊C4—82no—
305C4⋊C4—82no—
306C4⋊C4—82no—
307C4⋊C4—161no—
308C4⋊C4—161no—
309C4⋊C4SD3214yescomplemented
310C22⋊C4—82no—
311C22⋊C4—82no—
312C22⋊C4—82no—
313C22⋊C4—82no—
314C22⋊C4—82no—
315C42—21no—
316C42—21no—
317C42—21no—
318C42—21no—
319C42—21no—
320C42—21no—
321C42—161no—
322C16—22no—
323C16—41no—
324C16—41no—
325C16—41no—
326C23—42no—
327C23—81no—
328C23—81no—
329C23—81no—
330C23—81no—
331C23—81no—
332C23—81no—
333C23C22×D811yescharacteristic
334Q8—81no—
335Q8—81no—
336Q8—81no—
337Q8—81no—
338Q8—164no—
339D4—42no—
340D4—42no—
341D4—81no—
342D4—81no—
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—81no—
358C2×C4—81no—
359C2×C4—81no—
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—164no—
379C2×C4C2×SD3212yes—
380C2×C4C2×SD3212yes—
381C2×C4C2×SD3212yes—
382C2×C4#50912yes—
383C2×C4#53511yescharacteristic
384C2×C4#54711yescharacteristic
385C2×C4#54811yescharacteristic
386C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
387C2×C4#57711yescharacteristic
388C2×C4#57811yescharacteristic
389C8—22no—
390C8—41no—
391C8—41no—
392C8#53911yescharacteristic
393C8#54211yescharacteristic
394C8#55111yescharacteristic
395C8#56311yescharacteristic
396C22—82no—
397C22—82no—
398C22—82no—
399C22—161no—
400C22—161no—
401C22—161no—
402C22—161no—
403C22#320511yescharacteristic
404C22#333411yescharacteristic
405C22#335811yescharacteristic
406C22C22×SD3211yescharacteristic
407C22C2×#50911yescharacteristic
408C22#341611yescharacteristic
409C22#341711yescharacteristic
410C4—22no—
411C4—22no—
412C4—22no—
413C4—22no—
414C4—41no—
415C4—41no—
416C4—161no—
417C4—161no—
418C4—161no—
419C4—161no—
420C4—161no—
421C4—161no—
422C4—161no—
423C4#272311yescharacteristic
424C4#322511yescharacteristic
425C4#334511yescharacteristic
426C4#336711yescharacteristic
427C2—162no—
428C2#3242611yescharacteristic
429C2#3377411yescharacteristic
430C2#3379111yescharacteristic
431C2#3390211yescharacteristic
432C2#3391011yescharacteristic
433C2#3392611yescharacteristic
434C2#3394011yescharacteristic
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. #28015C2semidirect
3. #13531C2semidirect
4. #13481C2semidirect
6. #13452C2semidirect
8. #13364C2semidirect
10. #13177C2semidirect
12. #13104C2semidirect
15. #12888C2semidirect
309. C4⋊C4SD32semidirect

Maximal subgroups

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

subgroupindex
#285272
#280152
#135312
#134812
#134802
#134522
#134492
#133642
#132372
#131772
#131312
#131042
#130142
#129372
#128882