Finite Group Databasebeta

← #498366

Subgroups — #498366

Order 512.

652 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2219C211yesmaximal, characteristic
2C2×#2176C211yesmaximal, characteristic, complemented
3C2×#2159C211yesmaximal, characteristic
4#23026C211yesmaximal, characteristic
5#23024C211yesmaximal, characteristic
6#13236C211yesmaximal, characteristic, complemented
7#13167C211yesmaximal, characteristic, complemented
8#13133C211yesmaximal, characteristic, complemented
9#13132C211yesmaximal, characteristic, complemented
10#12994C211yesmaximal, characteristic
11#12993C211yesmaximal, characteristic
12#12938C211yesmaximal, characteristic, complemented
13#12936C211yesmaximal, characteristic, complemented
14C4×#367C211yesmaximal, characteristic, complemented
15#12612C211yesmaximal, characteristic
16C22×Q32C2211yescharacteristic
17C22×M32C2211yescharacteristic
18C2×#467C2211yescharacteristic
19C2×#467C2211yescharacteristic
20C2×C4×Q16C2211yescharacteristic
21C2×C4×Q16C2211yescharacteristic
22C2×#427C2211yescharacteristic
23C22×#107C2211yescharacteristic
24#2362—21no—
25#2360—21no—
26#2219C418yescomplemented
27#2176—24no—
28#2159—24no—
29C2×#367C2211yescharacteristic
30C2×#367C2211yescharacteristic
31C2×#367C2211yescharacteristic
32C2×#366C2211yescharacteristic
33C2×#366C2211yescharacteristic
34C2×#358C2211yescharacteristic
35C2×#358C2211yescharacteristic
36C2×#358C2211yescharacteristic
37C2×#358C2211yescharacteristic
38C2×#358C2211yescharacteristic
39C2×#358C2211yescharacteristic
40C2×C22⋊C16C2211yescharacteristic
41C2×#346C2211yescharacteristic
42C2×C4×C16C2211yescharacteristic
43#1969C2211yescharacteristic
44#1969C2211yescharacteristic
45#1942C2211yescharacteristic
46#1942C2211yescharacteristic
47#1928—21no—
48#1927—21no—
49#1873—21no—
50#1872—21no—
51#1865—21no—
52#1863—21no—
53C4×#107C2211yescharacteristic
54C4×#107C2211yescharacteristic
55C8×C22⋊C4C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1380C2211yescharacteristic
60#1374C2211yescharacteristic
61#1374C2211yescharacteristic
62C22×Q16C2311yescharacteristic
63C22×Q16C2311yescharacteristic
64C23×C8C2311yescharacteristic
65C2×C22⋊Q8—21no—
66C2×C22⋊Q8—21no—
67C2×C4×Q8—21no—
68C2×C4×Q8—21no—
69C2×#117—21no—
70C22×C4⋊C4—21no—
71C2×Q32—22no—
72C2×Q32—22no—
73C2×Q32C2×C414yescomplemented
74C2×M32—22no—
75C2×M32—22no—
76C2×M32C2×C414yes—
77C22×C16C2311yescharacteristic
78C22×C16C2311yescharacteristic
79C22×C16C2311yescharacteristic
80C22×C16C2311yescharacteristic
81#467—22no—
82#467—22no—
83#467C2×C414yes—
84#467C2×C414yes—
85C4×Q16—22no—
86C4×Q16—22no—
87C4×Q16—41no—
88C4×Q16—41no—
89#427—22no—
90#427—41no—
91C2×#107—22no—
92C2×#107—21no—
93C2×#107—41no—
94C2×#107C2311yescharacteristic
95C2×#107C2311yescharacteristic
96C2×#107C2311yescharacteristic
97C2×#107C2311yescharacteristic
98C2×#107C2311yescharacteristic
99C2×#107C2311yescharacteristic
100C2×#106—21no—
101C2×#103—21no—
102C2×#103—21no—
103C2×#103—21no—
104C2×#103—21no—
105C2×#103—21no—
106C2×#103—21no—
107C2×C22⋊C8D411yescharacteristic
108C2×C4×C8C2311yescharacteristic
109C2×C4×C8C2311yescharacteristic
110#391—21no—
111#391—21no—
112#389—21no—
113#389—21no—
114#389—41no—
115#388—41no—
116#388—41no—
117#387—41no—
118#387—41no—
119#385—41no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—41no—
123C4×C22⋊C4—41no—
124C4×C22⋊C4D411yescharacteristic
125#367—22no—
126#367C2×C414yescomplemented
127#367D414yes—
128#366—22no—
129#366—22no—
130#358—22no—
131#358—22no—
132#358—41no—
133#358—41no—
134#358C2×C414yescomplemented
135#358C2×C414yescomplemented
136C22⋊C16—22no—
137#346—22no—
138C4×C16—22no—
139#340—21no—
140#340—21no—
141#340—21no—
142#340—21no—
143#336D411yescharacteristic
144#336D411yescharacteristic
145C22×Q8—21no—
146C22×Q8—21no—
147C23×C4—41no—
148C23×C4C2×D411yescharacteristic
149C2×Q16—21no—
150C2×Q16—21no—
151C2×Q16—21no—
152C2×Q16—21no—
153C2×Q16—21no—
154C2×Q16—21no—
155C2×Q16—21no—
156C2×Q16—21no—
157C2×Q16—21no—
158C2×Q16—21no—
159C2×Q16C22×C412yes—
160C2×Q16C22×C412yes—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8C22×C412yes—
166C22×C8C2×D411yescharacteristic
167C22×C8C2×D411yescharacteristic
168C22×C8C2×D411yescharacteristic
169C22×C8#4112yes—
170C22×C8C2411yescharacteristic
171C22⋊Q8—24no—
172C22⋊Q8—24no—
173C22⋊Q8—42no—
174C22⋊Q8—42no—
175C4×Q8—42no—
176C4×Q8—42no—
177C4×Q8—81no—
178C4×Q8—81no—
179#117—42no—
180#117—81no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—81no—
196C2×C22⋊C4—41no—
197C2×C22⋊C4—41no—
198C2×C22⋊C4—41no—
199C2×C22⋊C4D811yescharacteristic
200C2×C22⋊C4D811yescharacteristic
201C2×C42—41no—
202C2×C42—41no—
203C2×C42—41no—
204C2×C42—41no—
205C2×C42—41no—
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208Q32—28no—
209M32—28no—
210C2×C16—21no—
211C2×C16—21no—
212C2×C16—21no—
213C2×C16—21no—
214C2×C16—21no—
215C2×C16—21no—
216C2×C16—21no—
217C2×C16—21no—
218C2×C16C22×C412yes—
219C2×C16C22×C412yes—
220C2×C16C2×D412yes—
221C2×C16#4112yes—
222#107—21no—
223#107—21no—
224#107—21no—
225#107—21no—
226#107—21no—
227#107—21no—
228#107—21no—
229#107—21no—
230#107—42no—
231#107C22×C412yes—
232#107C22×C412yes—
233#107C2×D412yes—
234#107C2×D412yes—
235#106—42no—
236#103—24no—
237#103—24no—
238#103—42no—
239#103—42no—
240#103—81no—
241#103—81no—
242C22⋊C8—22no—
243C4×C8—21no—
244C4×C8—21no—
245C4×C8—21no—
246C4×C8—21no—
247#95—41no—
248#95—41no—
249#95—41no—
250#95—41no—
251#95—41no—
252#95—41no—
253#95—41no—
254#95—41no—
255#95D811yescharacteristic
256#95D811yescharacteristic
257C24C2×D811yescharacteristic
258C2×Q8—22no—
259C2×Q8—22no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×Q8—41no—
268C2×Q8—41no—
269C2×Q8—41no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—82no—
283C22×C4—82no—
284C22×C4—161no—
285C22×C4C4×D412yes—
286C22×C4C2×D811yescharacteristic
287C22×C4C2×D811yescharacteristic
288C22×C4C2×D811yescharacteristic
289C22×C4C2×D811yescharacteristic
290C22×C4C2×D811yescharacteristic
291C22×C4#13512yes—
292C22×C4C22×D411yescharacteristic
293Q16—22no—
294Q16—22no—
295Q16—22no—
296Q16—22no—
297Q16—41no—
298Q16—41no—
299Q16—41no—
300Q16—41no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—22no—
306C2×C8—22no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8C4×D412yes—
314C2×C8C23×C411yescharacteristic
315C2×C8C22×D411yescharacteristic
316C2×C8C2×#4111yescharacteristic
317C2×C8C2×#4111yescharacteristic
318C2×C8H(5,𝔽2)11yescharacteristic
319C2×C82−1+411yescharacteristic
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—44no—
329C4⋊C4—44no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—82no—
335C4⋊C4—82no—
336C4⋊C4—82no—
337C4⋊C4—82no—
338C4⋊C4—161no—
339C22⋊C4—22no—
340C22⋊C4—41no—
341C22⋊C4—44no—
342C22⋊C4—44no—
343C22⋊C4—82no—
344C42—21no—
345C42—21no—
346C42—21no—
347C42—21no—
348C42—82no—
349C42—82no—
350C42—82no—
351C42—82no—
352C42—161no—
353C16—22no—
354C16—41no—
355C16—41no—
356C16C4×D414yes—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23C4×D812yes—
364C23C22×D811yescharacteristic
365Q8—42no—
366Q8—42no—
367Q8—42no—
368Q8—42no—
369Q8—81no—
370Q8—81no—
371Q8—81no—
372Q8—81no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—164no—
415C2×C4C4×D812yes—
416C2×C4#51012yes—
417C2×C4C2×C4×D411yescharacteristic
418C2×C4D4211yescharacteristic
419C2×C4#54711yescharacteristic
420C2×C4C2×#13511yescharacteristic
421C2×C4#57611yescharacteristic
422C2×C4#57811yescharacteristic
423C8—22no—
424C8—41no—
425C8—41no—
426C8C2×C4×D411yescharacteristic
427C8#52011yescharacteristic
428C8#54811yescharacteristic
429C8#55311yescharacteristic
430C22—22no—
431C22—22no—
432C22—22no—
433C22—41no—
434C22—41no—
435C22—41no—
436C22—41no—
437C22C2×C4×D811yescharacteristic
438C22D4×D811yescharacteristic
439C22#333411yescharacteristic
440C22C2×#51011yescharacteristic
441C22#341411yescharacteristic
442C22#341611yescharacteristic
443C22#341711yescharacteristic
444C4—22no—
445C4—22no—
446C4—41no—
447C4—41no—
448C4—41no—
449C4—82no—
450C4—82no—
451C4—82no—
452C4—82no—
453C4—82no—
454C4—82no—
455C4—161no—
456C4—161no—
457C4#236111yescharacteristic
458C4#295111yescharacteristic
459C4#328511yescharacteristic
460C4#333911yescharacteristic
461C2—42no—
462C2#2302511yescharacteristic
463C2C4×#51011yescharacteristic
464C2#3345511yescharacteristic
465C2#3384111yescharacteristic
466C2#3384211yescharacteristic
467C2#3391111yescharacteristic
468C2#3391611yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC24C2×D8
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ C8×C22⋊C4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2176C2semidirect
6. #13236C2semidirect
7. #13167C2semidirect
8. #13133C2semidirect
9. #13132C2semidirect
12. #12938C2semidirect
13. #12936C2semidirect
14. C4×#367C2semidirect
26. #2219C4semidirect
73. C2×Q32C2×C4semidirect
126. #367C2×C4semidirect
134. #358C2×C4semidirect
135. #358C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#22192
C2×#21762
C2×#21592
#230262
#230242
#132362
#131672
#131332
#131322
#129942
#129932
#129382
#129362
C4×#3672
#126122