Finite Group Databasebeta

← #504116

Subgroups — #504116

Order 512.

561 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2229C211yesmaximal, characteristic, complemented
2C2×#2159C211yesmaximal, characteristic
3#26067C211yesmaximal, characteristic
4#23026C211yesmaximal, characteristic
5#13529C211yesmaximal, characteristic, complemented
6#13515C211yesmaximal, characteristic, complemented
7#13475C211yesmaximal, characteristic, complemented
8#13453C211yesmaximal, characteristic, complemented
9#13178C211yesmaximal, characteristic, complemented
10#13110C211yesmaximal, characteristic, complemented
11#13062C211yesmaximal, characteristic
12#12993C211yesmaximal, characteristic
13#12938C211yesmaximal, characteristic, complemented
14#12921C211yesmaximal, characteristic
15#12913C211yesmaximal, characteristic
16C22×M32C2211yescharacteristic
17C2×#499C2211yescharacteristic
18C2×#467C2211yescharacteristic
19C2×C4×Q16C2211yescharacteristic
20C2×#427C2211yescharacteristic
21C22×#107C2211yescharacteristic
22#2442—21no—
23#2362—21no—
24#2229—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#2085—21no—
40#2059—21no—
41#1969C2211yescharacteristic
42#1942C2211yescharacteristic
43#1942C2211yescharacteristic
44#1942C2211yescharacteristic
45#1927—21no—
46#1873—21no—
47#1865—21no—
48#1853—21no—
49#1846C2211yescharacteristic
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×#125—21no—
64C2×C22⋊Q8—21no—
65C2×C4×Q8—21no—
66C2×#117—21no—
67C22×C4⋊C4—21no—
68C2×M32—22no—
69C2×M32—22no—
70C2×M32—22no—
71C22×C16C2311yescharacteristic
72C22×C16C2311yescharacteristic
73C22×C16C2311yescharacteristic
74C22×C16C2311yescharacteristic
75#499—22no—
76#499—41no—
77#467—22no—
78#467Q814yes—
79C4×Q16—22no—
80C4×Q16—41no—
81#427—22no—
82#427—41no—
83C2×#107—22no—
84C2×#107—21no—
85C2×#107—41no—
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#107C2311yescharacteristic
89C2×#107C2311yescharacteristic
90C2×#107C2311yescharacteristic
91C2×#107C2311yescharacteristic
92C2×#107C2311yescharacteristic
93C2×#106—21no—
94C2×#106—21no—
95C2×#106—21no—
96C2×#103—21no—
97C2×#103—21no—
98C2×#103—21no—
99C2×C22⋊C8D411yescharacteristic
100C2×C4×C8C2311yescharacteristic
101C2×C4×C8C2311yescharacteristic
102#400—41no—
103#398—41no—
104#391—21no—
105#389—21no—
106#389—21no—
107#389—21no—
108#388—41no—
109#388—41no—
110#387—41no—
111#385—41no—
112#382—21no—
113#380—21no—
114#380—41no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—41no—
117C4×C22⋊C4D411yescharacteristic
118#367—22no—
119#367—41no—
120#367D414yes—
121#366—22no—
122#366—22no—
123#366—22no—
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—
139C23×C4—41no—
140C23×C4C2×D411yescharacteristic
141C2×Q16—21no—
142C2×Q16—21no—
143C2×Q16—21no—
144C2×Q16—21no—
145C2×Q16—21no—
146C2×Q16C2×Q812yes—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8C2×D411yescharacteristic
152C22×C8C2×D411yescharacteristic
153C22×C8C2×D411yescharacteristic
154C22×C8C2×Q812yes—
155C22×C8#4112yes—
156C22×C8C2411yescharacteristic
157#125—42no—
158#125—81no—
159C22⋊Q8—24no—
160C22⋊Q8—42no—
161C4×Q8—42no—
162C4×Q8—81no—
163#117—42no—
164#117—81no—
165C2×C4⋊C4—21no—
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—42no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—41no—
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—81no—
182C2×C22⋊C4—41no—
183C2×C22⋊C4—41no—
184C2×C22⋊C4—41no—
185C2×C22⋊C4D811yescharacteristic
186C2×C22⋊C4D811yescharacteristic
187C2×C42—41no—
188C2×C42—41no—
189C2×C42—41no—
190C2×C42C2×D411yescharacteristic
191C2×C42C2×D411yescharacteristic
192M32—28no—
193C2×C16—21no—
194C2×C16—21no—
195C2×C16—21no—
196C2×C16—21no—
197C2×C16—21no—
198C2×C16—21no—
199C2×C16—21no—
200C2×C16—21no—
201C2×C16—21no—
202C2×C16—21no—
203C2×C16C2×D412yes—
204C2×C16#4112yes—
205#107—21no—
206#107—21no—
207#107—21no—
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—42no—
217#107C2×D412yes—
218#107C2×D412yes—
219#107C2×Q812yes—
220#106—42no—
221#106—42no—
222#106—81no—
223#103—24no—
224#103—42no—
225#103—81no—
226C22⋊C8—22no—
227C4×C8—21no—
228C4×C8—21no—
229C4×C8—21no—
230C4×C8—21no—
231#95—41no—
232#95—41no—
233#95—41no—
234#95—41no—
235#95—41no—
236#95—41no—
237#95—41no—
238#95—41no—
239#95—41no—
240#95D811yescharacteristic
241#95D811yescharacteristic
242C24C2×D811yescharacteristic
243C2×Q8—22no—
244C2×Q8—41no—
245C2×Q8—41no—
246C2×Q8—41no—
247C2×Q8—41no—
248C2×Q8—41no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—82no—
262C22×C4—82no—
263C22×C4—161no—
264C22×C4C22⋊Q812yes—
265C22×C4C2×D811yescharacteristic
266C22×C4C2×D811yescharacteristic
267C22×C4C2×D811yescharacteristic
268C22×C4C2×D811yescharacteristic
269C22×C4C2×D811yescharacteristic
270C22×C4#13712yes—
271C22×C4C22×D411yescharacteristic
272Q16—22no—
273Q16—22no—
274Q16—41no—
275Q16—41no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—22no—
279C2×C8—22no—
280C2×C8—22no—
281C2×C8—22no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8C22⋊Q812yes—
289C2×C8C22×D411yescharacteristic
290C2×C8C22×Q811yescharacteristic
291C2×C8C2×#4111yescharacteristic
292C2×C8C2×#4111yescharacteristic
293C2×C8H(5,𝔽2)11yescharacteristic
294C2×C82−1+411yescharacteristic
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—44no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—82no—
311C4⋊C4—82no—
312C4⋊C4—82no—
313C4⋊C4—82no—
314C4⋊C4—82no—
315C4⋊C4—161no—
316C4⋊C4—161no—
317C22⋊C4—41no—
318C22⋊C4—41no—
319C22⋊C4—44no—
320C22⋊C4—82no—
321C22⋊C4—82no—
322C42—21no—
323C42—21no—
324C42—21no—
325C42—21no—
326C42—82no—
327C42—82no—
328C42—82no—
329C16—22no—
330C16—22no—
331C16—41no—
332C16—41no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23#47412yes—
340C23C22×D811yescharacteristic
341Q8—42no—
342Q8—42no—
343Q8—81no—
344Q8—81no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
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—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—164no—
388C2×C4#47412yes—
389C2×C4C2×C22⋊Q811yescharacteristic
390C2×C4D4211yescharacteristic
391C2×C4#54711yescharacteristic
392C2×C4C2×#13711yescharacteristic
393C2×C4#57611yescharacteristic
394C2×C4#57811yescharacteristic
395C8—22no—
396C8—41no—
397C8—41no—
398C8#52511yescharacteristic
399C8#53911yescharacteristic
400C8D4×Q811yescharacteristic
401C8#55411yescharacteristic
402C22—22no—
403C22—22no—
404C22—22no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22C2×#47411yescharacteristic
410C22D4×D811yescharacteristic
411C22#333411yescharacteristic
412C22#341411yescharacteristic
413C22#341411yescharacteristic
414C22#341611yescharacteristic
415C22#341711yescharacteristic
416C4—22no—
417C4—41no—
418C4—41no—
419C4—41no—
420C4—41no—
421C4—82no—
422C4—82no—
423C4—82no—
424C4—82no—
425C4—161no—
426C4—161no—
427C4—161no—
428C4—161no—
429C4#244911yescharacteristic
430C4#308311yescharacteristic
431C4#327011yescharacteristic
432C4#334611yescharacteristic
433C2—42no—
434C2#2633611yescharacteristic
435C2#3363011yescharacteristic
436C2#3363211yescharacteristic
437C2#3383511yescharacteristic
438C2#3384111yescharacteristic
439C2#3391611yescharacteristic
440C2#3391811yescharacteristic
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×#2229C2semidirect
5. #13529C2semidirect
6. #13515C2semidirect
7. #13475C2semidirect
8. #13453C2semidirect
9. #13178C2semidirect
10. #13110C2semidirect
13. #12938C2semidirect

Maximal subgroups

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

subgroupindex
C2×#22292
C2×#21592
#260672
#230262
#135292
#135152
#134752
#134532
#131782
#131102
#130622
#129932
#129382
#129212
#129132