Finite Group Databasebeta

← #504089

Subgroups — #504089

Order 512.

600 conjugacy classes of subgroups.

The subgroup lattice has 451 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2225C211yesmaximal, characteristic, complemented
2C22×#367C211yesmaximal, characteristic
3#26065C211yesmaximal, characteristic
4#23024C211yesmaximal, characteristic
5#13531C211yesmaximal, characteristic, complemented
6#13516C211yesmaximal, characteristic, complemented
7#13477C211yesmaximal, characteristic, complemented
8#13452C211yesmaximal, characteristic, complemented
9#13177C211yesmaximal, characteristic, complemented
10#13110C211yesmaximal, characteristic, complemented
11#13060C211yesmaximal, characteristic
12#12993C211yesmaximal, characteristic
13#12938C211yesmaximal, characteristic, complemented
14#12920C211yesmaximal, characteristic
15#12912C211yesmaximal, characteristic
16C23×C16C2211yescharacteristic
17C2×#500C2211yescharacteristic
18C2×#467C2211yescharacteristic
19C2×C4×Q16C2211yescharacteristic
20C22×#107C2211yescharacteristic
21C22×#107C2211yescharacteristic
22#2440—21no—
23#2360—21no—
24#2225—24no—
25C2×#367—24no—
26C2×#367C2211yescharacteristic
27C2×#367C2211yescharacteristic
28C2×#367C2211yescharacteristic
29C2×#367C2211yescharacteristic
30C2×#367C2211yescharacteristic
31C2×#367C2211yescharacteristic
32C2×C4⋊C16C2211yescharacteristic
33C2×C4⋊C16C2211yescharacteristic
34C2×#358C2211yescharacteristic
35C2×#358C2211yescharacteristic
36C2×#358C2211yescharacteristic
37C2×#358C2211yescharacteristic
38C2×C22⋊C16C2211yescharacteristic
39#2083—21no—
40#2057—21no—
41#1969C2211yescharacteristic
42#1942C2211yescharacteristic
43#1942C2211yescharacteristic
44#1942C2211yescharacteristic
45#1928—21no—
46#1872—21no—
47#1863—21no—
48#1852—21no—
49#1845C2211yescharacteristic
50#1808C2211yescharacteristic
51C4×#107C2211yescharacteristic
52C8×C22⋊C4C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1374C2211yescharacteristic
60#1374C2211yescharacteristic
61C22×Q16C2311yescharacteristic
62C23×C8C2311yescharacteristic
63C2×C4⋊Q8—21no—
64C2×C22⋊Q8—21no—
65C2×C22⋊Q8—41no—
66C2×C4×Q8—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C16—22no—
70C22×C16—22no—
71C22×C16—22no—
72C22×C16C2311yescharacteristic
73C22×C16C2311yescharacteristic
74C22×C16C2311yescharacteristic
75C22×C16C2311yescharacteristic
76#500—22no—
77#500—41no—
78#467—22no—
79#467Q814yescomplemented
80C4×Q16—22no—
81C4×Q16—41no—
82C2×#107—22no—
83C2×#107—22no—
84C2×#107—21no—
85C2×#107—21no—
86C2×#107—21no—
87C2×#107—21no—
88C2×#107—41no—
89C2×#107—41no—
90C2×#107C2311yescharacteristic
91C2×#107C2311yescharacteristic
92C2×#107C2311yescharacteristic
93C2×#107C2311yescharacteristic
94C2×#107C2311yescharacteristic
95C2×#107C2311yescharacteristic
96C2×#107C2311yescharacteristic
97C2×#103—21no—
98C2×#103—21no—
99C2×#103—21no—
100C2×C22⋊C8D411yescharacteristic
101C2×C4×C8C2311yescharacteristic
102C2×C4×C8C2311yescharacteristic
103#398—41no—
104#395—41no—
105#391—21no—
106#389—21no—
107#389—21no—
108#389—21no—
109#389—41no—
110#387—41no—
111#385—41no—
112#385—41no—
113#384—21no—
114#380—21no—
115C4×C4⋊C4—21no—
116C4×C22⋊C4—41no—
117C4×C22⋊C4D411yescharacteristic
118#367—22no—
119#367—22no—
120#367—22no—
121#367—22no—
122#367—41no—
123#367D414yes—
124C4⋊C16—22no—
125C4⋊C16—22no—
126#358—22no—
127#358—22no—
128#358—22no—
129#358—41no—
130C22⋊C16—22no—
131#340—21no—
132#340—21no—
133#340—21no—
134#340—21no—
135#340—21no—
136#336D411yescharacteristic
137#336D411yescharacteristic
138C22×Q8—21no—
139C22×Q8—41no—
140C23×C4—41no—
141C23×C4—41no—
142C23×C4C2×D411yescharacteristic
143C2×Q16—21no—
144C2×Q16—21no—
145C2×Q16—21no—
146C2×Q16—21no—
147C2×Q16—21no—
148C2×Q16C2×Q812yes—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8C2×D411yescharacteristic
154C22×C8C2×D411yescharacteristic
155C22×C8C2×D411yescharacteristic
156C22×C8C2×Q812yes—
157C22×C8#4112yes—
158C22×C8C2411yescharacteristic
159C4⋊Q8—42no—
160C4⋊Q8—81no—
161C22⋊Q8—24no—
162C22⋊Q8—42no—
163C22⋊Q8—84no—
164C4×Q8—42no—
165C4×Q8—81no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—81no—
186C2×C4⋊C4—81no—
187C2×C22⋊C4—41no—
188C2×C22⋊C4—41no—
189C2×C22⋊C4D811yescharacteristic
190C2×C22⋊C4D811yescharacteristic
191C2×C42—41no—
192C2×C42—41no—
193C2×C42C2×D411yescharacteristic
194C2×C42C2×D411yescharacteristic
195C2×C16—28no—
196C2×C16—21no—
197C2×C16—21no—
198C2×C16—21no—
199C2×C16—21no—
200C2×C16—21no—
201C2×C16—21no—
202C2×C16—21no—
203C2×C16—21no—
204C2×C16—21no—
205C2×C16—21no—
206C2×C16C2×D412yes—
207C2×C16#4112yes—
208#107—21no—
209#107—21no—
210#107—21no—
211#107—21no—
212#107—21no—
213#107—21no—
214#107—21no—
215#107—21no—
216#107—21no—
217#107—21no—
218#107—21no—
219#107—42no—
220#107—42no—
221#107—42no—
222#107—81no—
223#107C2×D412yes—
224#107C2×D412yes—
225#107C2×Q812yes—
226#103—24no—
227#103—42no—
228#103—81no—
229C22⋊C8—22no—
230C4×C8—21no—
231C4×C8—21no—
232C4×C8—21no—
233C4×C8—21no—
234#95—41no—
235#95—41no—
236#95—41no—
237#95—41no—
238#95—41no—
239#95—41no—
240#95—41no—
241#95D811yescharacteristic
242#95D811yescharacteristic
243C24C2×D811yescharacteristic
244C2×Q8—22no—
245C2×Q8—41no—
246C2×Q8—41no—
247C2×Q8—41no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×Q8—82no—
251C2×Q8—82no—
252C2×Q8—161no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—82no—
266C22×C4—82no—
267C22×C4—82no—
268C22×C4—82no—
269C22×C4—161no—
270C22×C4—161no—
271C22×C4C22⋊Q812yes—
272C22×C4C2×D811yescharacteristic
273C22×C4C2×D811yescharacteristic
274C22×C4C2×D811yescharacteristic
275C22×C4C2×D811yescharacteristic
276C22×C4C2×D811yescharacteristic
277C22×C4#13712yes—
278C22×C4C22×D411yescharacteristic
279Q16—22no—
280Q16—22no—
281Q16—41no—
282Q16—41no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—22no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—22no—
289C2×C8—21no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—21no—
295C2×C8C22⋊Q812yes—
296C2×C8C22×D411yescharacteristic
297C2×C8C22×Q811yescharacteristic
298C2×C8C2×#4111yescharacteristic
299C2×C8C2×#4111yescharacteristic
300C2×C8H(5,𝔽2)11yescharacteristic
301C2×C82−1+411yescharacteristic
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—22no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—44no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—82no—
318C4⋊C4—82no—
319C4⋊C4—82no—
320C4⋊C4—82no—
321C4⋊C4—82no—
322C4⋊C4—82no—
323C4⋊C4—82no—
324C4⋊C4—161no—
325C4⋊C4—161no—
326C22⋊C4—22no—
327C22⋊C4—44no—
328C22⋊C4—82no—
329C22⋊C4SD3214yes—
330C42—21no—
331C42—21no—
332C42—21no—
333C42—21no—
334C42—82no—
335C42—161no—
336C16—22no—
337C16—22no—
338C16—41no—
339C16—41no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23#47412yes—
345C23C2×SD3212yes—
346C23#51012yes—
347C23C22×D811yescharacteristic
348Q8—42no—
349Q8—42no—
350Q8—81no—
351Q8—81no—
352Q8—164no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
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—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—164no—
394C2×C4—164no—
395C2×C4#47412yes—
396C2×C4C2×SD3212yes—
397C2×C4C2×SD3212yes—
398C2×C4C2×C22⋊Q811yescharacteristic
399C2×C4D4211yescharacteristic
400C2×C4#54711yescharacteristic
401C2×C4C2×#13711yescharacteristic
402C2×C4#57611yescharacteristic
403C2×C4#57811yescharacteristic
404C8—22no—
405C8—41no—
406C8—41no—
407C8#52511yescharacteristic
408C8#53911yescharacteristic
409C8D4×Q811yescharacteristic
410C8#55411yescharacteristic
411C22—22no—
412C22—22no—
413C22—22no—
414C22—22no—
415C22—22no—
416C22—22no—
417C22#222514yescomplemented
418C22C2×#47411yescharacteristic
419C22D4×D811yescharacteristic
420C22#333411yescharacteristic
421C22C22×SD3211yescharacteristic
422C22C2×#51011yescharacteristic
423C22#341611yescharacteristic
424C22#341611yescharacteristic
425C4—22no—
426C4—22no—
427C4—22no—
428C4—41no—
429C4—41no—
430C4—82no—
431C4—82no—
432C4—82no—
433C4—82no—
434C4—161no—
435C4—161no—
436C4—161no—
437C4—161no—
438C4#244911yescharacteristic
439C4#308311yescharacteristic
440C4#327011yescharacteristic
441C4#334611yescharacteristic
442C2—24no—
443C2#2633611yescharacteristic
444C2C2×#222511yescharacteristic
445C2#3363411yescharacteristic
446C2#3383211yescharacteristic
447C2D4×SD3211yescharacteristic
448C2#3390211yescharacteristic
449C2#3391111yescharacteristic
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
1. C2×#2225C2semidirect
5. #13531C2semidirect
6. #13516C2semidirect
7. #13477C2semidirect
8. #13452C2semidirect
9. #13177C2semidirect
10. #13110C2semidirect
13. #12938C2semidirect
79. #467Q8semidirect
417. C22#2225semidirect

Maximal subgroups

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

subgroupindex
C2×#22252
C22×#3672
#260652
#230242
#135312
#135162
#134772
#134522
#131772
#131102
#130602
#129932
#129382
#129202
#129122