Finite Group Databasebeta

← #504104

Subgroups — #504104

Order 512.

583 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2227C211yesmaximal, characteristic, complemented
2C2×#2159C211yesmaximal, characteristic
3#26065C211yesmaximal, characteristic
4#23026C211yesmaximal, characteristic
5#13531C211yesmaximal, characteristic, complemented
6#13516C211yesmaximal, characteristic, complemented
7#13477C211yesmaximal, characteristic, complemented
8#13452C211yesmaximal, characteristic, complemented
9#13176C211yesmaximal, characteristic, complemented
10#13109C211yesmaximal, characteristic, complemented
11#13062C211yesmaximal, characteristic
12#12993C211yesmaximal, characteristic
13#12936C211yesmaximal, characteristic, complemented
14#12921C211yesmaximal, characteristic
15#12913C211yesmaximal, characteristic
16C22×M32C2211yescharacteristic
17C2×#500C2211yescharacteristic
18C2×#467C2211yescharacteristic
19C2×C4×Q16C2211yescharacteristic
20C2×#427C2211yescharacteristic
21C22×#107C2211yescharacteristic
22#2440—21no—
23#2362—21no—
24#2227—24no—
25#2159—24no—
26C2×#367C2211yescharacteristic
27C2×#367C2211yescharacteristic
28C2×#367C2211yescharacteristic
29C2×#366C2211yescharacteristic
30C2×#366C2211yescharacteristic
31C2×#366C2211yescharacteristic
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#1927—21no—
46#1873—21no—
47#1865—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—
67C2×#117—21no—
68C22×C4⋊C4—21no—
69C2×M32—22no—
70C2×M32—22no—
71C2×M32—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—
82#427—22no—
83#427—41no—
84C2×#107—22no—
85C2×#107—21no—
86C2×#107—21no—
87C2×#107—21no—
88C2×#107—41no—
89C2×#107C2311yescharacteristic
90C2×#107C2311yescharacteristic
91C2×#107C2311yescharacteristic
92C2×#107C2311yescharacteristic
93C2×#107C2311yescharacteristic
94C2×#107C2311yescharacteristic
95C2×#107C2311yescharacteristic
96C2×#106—21no—
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#388—41no—
110#388—41no—
111#387—41no—
112#385—41no—
113#384—21no—
114#380—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—41no—
117C4×C22⋊C4D411yescharacteristic
118#367—22no—
119#367—22no—
120#367—41no—
121#366—22no—
122#366—22no—
123#366D414yes—
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×C4C2×D411yescharacteristic
142C2×Q16—21no—
143C2×Q16—21no—
144C2×Q16—21no—
145C2×Q16—21no—
146C2×Q16—21no—
147C2×Q16C2×Q812yes—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8C2×D411yescharacteristic
153C22×C8C2×D411yescharacteristic
154C22×C8C2×D411yescharacteristic
155C22×C8C2×Q812yes—
156C22×C8#4112yes—
157C22×C8C2411yescharacteristic
158C4⋊Q8—42no—
159C4⋊Q8—81no—
160C22⋊Q8—24no—
161C22⋊Q8—42no—
162C22⋊Q8—84no—
163C4×Q8—42no—
164C4×Q8—81no—
165#117—42no—
166#117—81no—
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—21no—
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—81no—
185C2×C22⋊C4—41no—
186C2×C22⋊C4—41no—
187C2×C22⋊C4—41no—
188C2×C22⋊C4D811yescharacteristic
189C2×C22⋊C4D811yescharacteristic
190C2×C42—41no—
191C2×C42—41no—
192C2×C42—41no—
193C2×C42C2×D411yescharacteristic
194C2×C42C2×D411yescharacteristic
195M32—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—81no—
222#107C2×D412yes—
223#107C2×D412yes—
224#107C2×Q812yes—
225#106—42no—
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—161no—
268C22×C4C22⋊Q812yes—
269C22×C4C2×D811yescharacteristic
270C22×C4C2×D811yescharacteristic
271C22×C4C2×D811yescharacteristic
272C22×C4C2×D811yescharacteristic
273C22×C4C2×D811yescharacteristic
274C22×C4#13712yes—
275C22×C4C22×D411yescharacteristic
276Q16—22no—
277Q16—22no—
278Q16—41no—
279Q16—41no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—22no—
283C2×C8—22no—
284C2×C8—22no—
285C2×C8—22no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—21no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8C22⋊Q812yes—
293C2×C8C22×D411yescharacteristic
294C2×C8C22×Q811yescharacteristic
295C2×C8C2×#4111yescharacteristic
296C2×C8C2×#4111yescharacteristic
297C2×C8H(5,𝔽2)11yescharacteristic
298C2×C82−1+411yescharacteristic
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—44no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—82no—
315C4⋊C4—82no—
316C4⋊C4—82no—
317C4⋊C4—82no—
318C4⋊C4—82no—
319C4⋊C4—82no—
320C4⋊C4—82no—
321C4⋊C4—161no—
322C22⋊C4—22no—
323C22⋊C4—22no—
324C22⋊C4—44no—
325C22⋊C4—82no—
326C22⋊C4—82no—
327C42—21no—
328C42—21no—
329C42—21no—
330C42—21no—
331C42—82no—
332C42—82no—
333C42—82no—
334C16—22no—
335C16—22no—
336C16—41no—
337C16—41no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23#47412yes—
345C23C22×D811yescharacteristic
346Q8—42no—
347Q8—42no—
348Q8—81no—
349Q8—81no—
350Q8—164no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
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—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—164no—
392C2×C4#47412yes—
393C2×C4#50912yes—
394C2×C4#51012yes—
395C2×C4C2×C22⋊Q811yescharacteristic
396C2×C4D4211yescharacteristic
397C2×C4#54711yescharacteristic
398C2×C4C2×#13711yescharacteristic
399C2×C4#57611yescharacteristic
400C2×C4#57811yescharacteristic
401C8—22no—
402C8—41no—
403C8—41no—
404C8#52511yescharacteristic
405C8#53911yescharacteristic
406C8D4×Q811yescharacteristic
407C8#55411yescharacteristic
408C22—22no—
409C22—22no—
410C22—22no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22C2×#47411yescharacteristic
416C22D4×D811yescharacteristic
417C22#333411yescharacteristic
418C22C2×#50911yescharacteristic
419C22C2×#51011yescharacteristic
420C22#341611yescharacteristic
421C22#341711yescharacteristic
422C4—22no—
423C4—22no—
424C4—22no—
425C4—41no—
426C4—41no—
427C4—82no—
428C4—82no—
429C4—82no—
430C4—82no—
431C4—161no—
432C4—161no—
433C4—161no—
434C4—161no—
435C4#244911yescharacteristic
436C4#308311yescharacteristic
437C4#327011yescharacteristic
438C4#334611yescharacteristic
439C2—42no—
440C2#2633611yescharacteristic
441C2#3362911yescharacteristic
442C2#3363211yescharacteristic
443C2#3383611yescharacteristic
444C2#3383811yescharacteristic
445C2#3391011yescharacteristic
446C2#3392311yescharacteristic
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×#2227C2semidirect
5. #13531C2semidirect
6. #13516C2semidirect
7. #13477C2semidirect
8. #13452C2semidirect
9. #13176C2semidirect
10. #13109C2semidirect
13. #12936C2semidirect
79. #467Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#22272
C2×#21592
#260652
#230262
#135312
#135162
#134772
#134522
#131762
#131092
#130622
#129932
#129362
#129212
#129132