Finite Group Databasebeta

← #505490

Subgroups — #505490

Order 512.

612 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27839C211yesmaximal, characteristic
2C22⋊C4×Q16C211yesmaximal, characteristic
3#13494C211yesmaximal, characteristic
4#13494C211yesmaximal, characteristic
5#13481C211yesmaximal, characteristic, complemented
6#13477C211yesmaximal, characteristic, complemented
7#13148C211yesmaximal, characteristic, complemented
8#13132C211yesmaximal, characteristic, complemented
9#13120C211yesmaximal, characteristic
10#12938C211yesmaximal, characteristic, complemented
11#12938C211yesmaximal, characteristic, complemented
12#12812C211yesmaximal, characteristic, complemented
13#12732C211yesmaximal, characteristic
14#12732C211yesmaximal, characteristic
15C4×#358C211yesmaximal, characteristic, complemented
16C23×Q16C2211yescharacteristic
17C2×C4×Q16C2211yescharacteristic
18C2×C4×Q16C2211yescharacteristic
19#2496—21no—
20Q8×C22⋊C4—21no—
21C2×#367C2211yescharacteristic
22C2×#367C2211yescharacteristic
23C2×C4⋊C16C2211yescharacteristic
24C2×#358C2211yescharacteristic
25C2×#358C2211yescharacteristic
26C2×#358C2211yescharacteristic
27C2×#358C2211yescharacteristic
28C2×#358C2211yescharacteristic
29C2×#358C2211yescharacteristic
30C2×#358C2211yescharacteristic
31C2×#358C2211yescharacteristic
32C2×C22⋊C16C2211yescharacteristic
33C2×C22⋊C16C2211yescharacteristic
34C2×C4×C16C2211yescharacteristic
35#2099—21no—
36#2075—21no—
37#1969C2211yescharacteristic
38#1969C2211yescharacteristic
39#1942C2211yescharacteristic
40#1942C2211yescharacteristic
41#1918—21no—
42#1878—21no—
43#1878—21no—
44#1845C2211yescharacteristic
45#1808C2211yescharacteristic
46#1808C2211yescharacteristic
47#1808C2211yescharacteristic
48#1787—21no—
49C4×#107C2211yescharacteristic
50C8×C22⋊C4C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C23×Q8—21no—
60C22×Q16—22no—
61C22×Q16—22no—
62C22×Q16—22no—
63C22×Q16—22no—
64C22×Q16—21no—
65C22×Q16—41no—
66C22×Q16—41no—
67C22×Q16C2311yescharacteristic
68C22×Q16C2311yescharacteristic
69C23×C8C2311yescharacteristic
70C2×#126—41no—
71C2×C4×Q8—21no—
72C2×C4×Q8—21no—
73C22×C16C2311yescharacteristic
74C22×C16C2311yescharacteristic
75C22×C16C2311yescharacteristic
76C22×C16C2311yescharacteristic
77C4×Q16—22no—
78C4×Q16—22no—
79C4×Q16—41no—
80C4×Q16—41no—
81C2×#107—21no—
82C2×#107C2311yescharacteristic
83C2×#107C2311yescharacteristic
84C2×#107C2311yescharacteristic
85C2×#107C2311yescharacteristic
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#103—21no—
89C2×#103—21no—
90C2×#103—21no—
91C2×#103—21no—
92C2×C22⋊C8D411yescharacteristic
93C2×C4×C8C2311yescharacteristic
94C2×C4×C8C2311yescharacteristic
95#400—41no—
96#397—41no—
97#391—21no—
98#391—21no—
99#391—41no—
100#391—41no—
101#389—21no—
102#389—21no—
103#387—41no—
104#384—21no—
105#380—21no—
106#380—21no—
107#380—21no—
108#380—41no—
109C4×C4⋊C4—21no—
110C4×C22⋊C4—41no—
111C4×C22⋊C4D411yescharacteristic
112#367—41no—
113#367—41no—
114C4⋊C16—41no—
115#358—22no—
116#358—22no—
117#358—22no—
118#358—22no—
119#358—22no—
120#358—22no—
121#358—41no—
122#358—41no—
123C22⋊C16—41no—
124C22⋊C16—41no—
125C4×C16—41no—
126#340—21no—
127#340—21no—
128#340—21no—
129#340—21no—
130#340—21no—
131#340—21no—
132#336D411yescharacteristic
133#336D411yescharacteristic
134C22×Q8—24no—
135C22×Q8—21no—
136C22×Q8—21no—
137C22×Q8—42no—
138C22×Q8—42no—
139C22×Q8—42no—
140C22×Q8—41no—
141C22×Q8—81no—
142C22×Q8—81no—
143C23×C4—41no—
144C23×C4C2×D411yescharacteristic
145C2×Q16—24no—
146C2×Q16—21no—
147C2×Q16—21no—
148C2×Q16—21no—
149C2×Q16—21no—
150C2×Q16—21no—
151C2×Q16—21no—
152C2×Q16—21no—
153C2×Q16—21no—
154C2×Q16—42no—
155C2×Q16—42no—
156C2×Q16—44no—
157C2×Q16—44no—
158C2×Q16—44no—
159C2×Q16—44no—
160C2×Q16#4112yes—
161C2×Q16#4112yes—
162C2×Q16#4112yes—
163C2×Q16#4112yes—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8C2×D411yescharacteristic
171C22×C8C2×D411yescharacteristic
172C22×C8C2×D411yescharacteristic
173C22×C8C2411yescharacteristic
174#126—84no—
175C4×Q8—42no—
176C4×Q8—42no—
177C4×Q8—81no—
178C4×Q8—81no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4—41no—
191C2×C22⋊C4—41no—
192C2×C22⋊C4—41no—
193C2×C22⋊C4—41no—
194C2×C22⋊C4D811yescharacteristic
195C2×C22⋊C4D811yescharacteristic
196C2×C42—41no—
197C2×C42—41no—
198C2×C42—41no—
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C16—21no—
202C2×C16—21no—
203C2×C16—21no—
204C2×C16—21no—
205C2×C16—21no—
206C2×C16—21no—
207C2×C16—21no—
208C2×C16—21no—
209C2×C16—21no—
210C2×C16—21no—
211C2×C16—21no—
212C2×C16—21no—
213#107—21no—
214#107—21no—
215#107—21no—
216#107—21no—
217#107—21no—
218#107—21no—
219#107—21no—
220#107—21no—
221#107—21no—
222#107—21no—
223#107—21no—
224#107—21no—
225#107—42no—
226#103—42no—
227#103—42no—
228#103—81no—
229#103—81no—
230C22⋊C8—41no—
231C4×C8—21no—
232C4×C8—21no—
233C4×C8—21no—
234C4×C8—21no—
235#95—41no—
236#95—41no—
237#95—41no—
238#95—41no—
239#95—41no—
240#95—41no—
241#95—41no—
242#95—41no—
243#95—41no—
244#95—41no—
245#95D811yescharacteristic
246#95D811yescharacteristic
247C24C2×D811yescharacteristic
248C2×Q8—22no—
249C2×Q8—22no—
250C2×Q8—22no—
251C2×Q8—22no—
252C2×Q8—44no—
253C2×Q8—41no—
254C2×Q8—41no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×Q8—48no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×Q8—82no—
263C2×Q8—82no—
264C2×Q8—84no—
265C2×Q8—84no—
266C2×Q8—84no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—41no—
274C22×C4—44no—
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×C4C2×D811yescharacteristic
285C22×C4C2×D811yescharacteristic
286C22×C4C2×D811yescharacteristic
287C22×C4C2×D811yescharacteristic
288C22×C4C2×D811yescharacteristic
289C22×C4C22×D411yescharacteristic
290Q16—22no—
291Q16—22no—
292Q16—22no—
293Q16—22no—
294Q16—22no—
295Q16—48no—
296Q16—41no—
297Q16—41no—
298Q16#12614yescomplemented
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8C2×#4111yescharacteristic
313C2×C8C2×#4111yescharacteristic
314C2×C8C2×#4111yescharacteristic
315C2×C8C2×#4111yescharacteristic
316C2×C8H(5,𝔽2)11yescharacteristic
317C2×C8H(5,𝔽2)11yescharacteristic
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—82no—
331C4⋊C4—82no—
332C4⋊C4—82no—
333C4⋊C4—82no—
334C4⋊C4—82no—
335C4⋊C4—82no—
336C22⋊C4—41no—
337C22⋊C4—82no—
338C22⋊C4—82no—
339C22⋊C4—82no—
340C22⋊C4SD3214yes—
341C42—21no—
342C42—21no—
343C42—21no—
344C42—21no—
345C42—82no—
346C42—82no—
347C42—82no—
348C16—41no—
349C16—41no—
350C16—41no—
351C16—41no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23C2×SD3212yes—
359C23C22×D811yescharacteristic
360Q8—24no—
361Q8—42no—
362Q8—42no—
363Q8—42no—
364Q8—42no—
365Q8—42no—
366Q8—88no—
367Q8—81no—
368Q8—81no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—88no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—81no—
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×C4C2×SD3212yes—
412C2×C4C2×SD3212yes—
413C2×C4#52511yescharacteristic
414C2×C4#54611yescharacteristic
415C2×C4#54611yescharacteristic
416C2×C4#57511yescharacteristic
417C2×C4#57711yescharacteristic
418C2×C4#57711yescharacteristic
419C8—41no—
420C8—41no—
421C8—41no—
422C8C2×#12611yescharacteristic
423C8#53811yescharacteristic
424C8#55111yescharacteristic
425C8#55111yescharacteristic
426C22—22no—
427C22—22no—
428C22—22no—
429C22—41no—
430C22—41no—
431C22—41no—
432C22—41no—
433C22#310011yescharacteristic
434C22#329411yescharacteristic
435C22#329411yescharacteristic
436C22C22×SD3211yescharacteristic
437C22#341411yescharacteristic
438C22#341711yescharacteristic
439C22#341711yescharacteristic
440C4—22no—
441C4—22no—
442C4—41no—
443C4—41no—
444C4—41no—
445C4—44no—
446C4—82no—
447C4—161no—
448C4—161no—
449C4—161no—
450C4—161no—
451C4—161no—
452C4—161no—
453C4#248011yescharacteristic
454C4#312311yescharacteristic
455C4#331111yescharacteristic
456C4#332311yescharacteristic
457C2—42no—
458C2#2751811yescharacteristic
459C2#3366311yescharacteristic
460C2#3368711yescharacteristic
461C2#3386511yescharacteristic
462C2#3386511yescharacteristic
463C2#3388011yescharacteristic
464C2#3389411yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC24C2×D8
LCSG ▷ C22×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
5. #13481C2semidirect
6. #13477C2semidirect
7. #13148C2semidirect
8. #13132C2semidirect
10. #12938C2semidirect
11. #12938C2semidirect
12. #12812C2semidirect
15. C4×#358C2semidirect
298. Q16#126semidirect

Maximal subgroups

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

subgroupindex
#278392
C22⋊C4×Q162
#134942
#134812
#134772
#131482
#131322
#131202
#129382
#128122
#127322
C4×#3582