Finite Group Databasebeta

← #504108

Subgroups — #504108

Order 512.

611 conjugacy classes of subgroups.

The subgroup lattice has 467 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2226C211yesmaximal, characteristic, complemented
2C2×#2159C211yesmaximal, characteristic, complemented
3#26065C211yesmaximal, characteristic, complemented
4#23027C211yesmaximal, characteristic
5#13526C211yesmaximal, characteristic, complemented
6#13515C211yesmaximal, characteristic, complemented
7#13474C211yesmaximal, characteristic, complemented
8#13451C211yesmaximal, characteristic, complemented
9#13177C211yesmaximal, characteristic, complemented
10#13110C211yesmaximal, characteristic, complemented
11#13061C211yesmaximal, characteristic
12#12991C211yesmaximal, characteristic
13#12937C211yesmaximal, characteristic, complemented
14#12921C211yesmaximal, characteristic, complemented
15#12913C211yesmaximal, characteristic, complemented
16C22×M32C2211yescharacteristic
17C2×#500C2211yescharacteristic
18C2×#466C2211yescharacteristic
19C2×C4×D8C2211yescharacteristic
20C2×#427C2211yescharacteristic
21C22×#107C2211yescharacteristic
22#2440—21no—
23#2363—21no—
24#2226—24no—
25#2159—24no—
26C2×#367C2211yescharacteristic
27C2×#367C2211yescharacteristic
28C2×#367C2211yescharacteristic
29C2×#366C2211yescharacteristic
30C2×#366C2211yescharacteristic
31C2×#366C2211yescharacteristic
32C2×C4⋊C16C2211yescharacteristic
33C2×C4⋊C16C2211yescharacteristic
34C2×#357C2211yescharacteristic
35C2×#357C2211yescharacteristic
36C2×#357C2211yescharacteristic
37C2×#357C2211yescharacteristic
38C2×C22⋊C16C2211yescharacteristic
39#2083—21no—
40#2057—21no—
41#1970C2211yescharacteristic
42#1942C2211yescharacteristic
43#1942C2211yescharacteristic
44#1942C2211yescharacteristic
45#1926—21no—
46#1875—21no—
47#1864—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×D8C2311yescharacteristic
62C23×C8C2311yescharacteristic
63C2×C4⋊Q8—21no—
64C2×C22⋊Q8—41no—
65C2×#121—21no—
66C2×C4×D4—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#466—22no—
79#466Q814yescomplemented
80C4×D8—22no—
81C4×D8—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×#102—21no—
98C2×#102—21no—
99C2×#102—21no—
100C2×C22⋊C8D411yescharacteristic
101C2×C4×C8C2311yescharacteristic
102C2×C4×C8C2311yescharacteristic
103#398—41no—
104#395—41no—
105#390—21no—
106#390—41no—
107#389—21no—
108#389—21no—
109#389—21no—
110#388—41no—
111#386—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#367D414yescomplemented
121#366—22no—
122#366—22no—
123#366—41no—
124C4⋊C16—22no—
125C4⋊C16—22no—
126#357—22no—
127#357—22no—
128#357—22no—
129#357—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—41no—
139C22×D4—21no—
140C22×D4—41no—
141C23×C4—41no—
142C23×C4—41no—
143C23×C4C2×D411yescharacteristic
144C2×D8—21no—
145C2×D8—21no—
146C2×D8—21no—
147C2×D8—21no—
148C2×D8—21no—
149C2×D8C2×Q812yes—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8C2×D411yescharacteristic
155C22×C8C2×D411yescharacteristic
156C22×C8C2×D411yescharacteristic
157C22×C8C2×Q812yes—
158C22×C8#4112yes—
159C22×C8C2411yescharacteristic
160C4⋊Q8—42no—
161C4⋊Q8—81no—
162C22⋊Q8—84no—
163#121—24no—
164#121—42no—
165C4×D4—42no—
166C4×D4—81no—
167#117—42no—
168#117—81no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—42no—
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⋊C4—41no—
189C2×C22⋊C4—41no—
190C2×C22⋊C4—41no—
191C2×C22⋊C4D811yescharacteristic
192C2×C22⋊C4D811yescharacteristic
193C2×C42—41no—
194C2×C42—41no—
195C2×C42C2×D411yescharacteristic
196C2×C42C2×D411yescharacteristic
197M32—28no—
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×C16—21no—
207C2×C16—21no—
208C2×C16C2×D412yes—
209C2×C16#4112yes—
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—21no—
220#107—21no—
221#107—42no—
222#107—42no—
223#107—81no—
224#107C2×D412yes—
225#107C2×D412yes—
226#107C2×Q812yes—
227#106—42no—
228#102—24no—
229#102—42no—
230#102—81no—
231C22⋊C8—22no—
232C4×C8—21no—
233C4×C8—21no—
234C4×C8—21no—
235C4×C8—21no—
236#95—41no—
237#95—41no—
238#95—41no—
239#95—41no—
240#95—41no—
241#95D811yescharacteristic
242#95D811yescharacteristic
243C24—41no—
244C24C2×D811yescharacteristic
245C2×Q8—82no—
246C2×Q8—82no—
247C2×Q8—161no—
248C2×D4—22no—
249C2×D4—44no—
250C2×D4—41no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—41no—
255C2×D4—161no—
256C2×D4—161no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—82no—
269C22×C4—82no—
270C22×C4—82no—
271C22×C4—82no—
272C22×C4—161no—
273C22×C4—161no—
274C22×C4C22⋊Q812yes—
275C22×C4C2×D811yescharacteristic
276C22×C4C2×D811yescharacteristic
277C22×C4C2×D811yescharacteristic
278C22×C4C2×D811yescharacteristic
279C22×C4C2×D811yescharacteristic
280C22×C4#13712yes—
281C22×C4C22×D411yescharacteristic
282D8—22no—
283D8—22no—
284D8—41no—
285D8—41no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—22no—
289C2×C8—22no—
290C2×C8—22no—
291C2×C8—22no—
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8C22⋊Q812yes—
299C2×C8C22×D411yescharacteristic
300C2×C8C22×Q811yescharacteristic
301C2×C8C2×#4111yescharacteristic
302C2×C8C2×#4111yescharacteristic
303C2×C8H(5,𝔽2)11yescharacteristic
304C2×C82−1+411yescharacteristic
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—41no—
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—41no—
318C4⋊C4—41no—
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—22no—
328C22⋊C4—44no—
329C22⋊C4—82no—
330C22⋊C4—82no—
331C22⋊C4—82no—
332C22⋊C4—82no—
333C22⋊C4—161no—
334C42—21no—
335C42—21no—
336C42—21no—
337C42—21no—
338C42—82no—
339C42—161no—
340C16—22no—
341C16—22no—
342C16—41no—
343C16—41no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—42no—
351C23—81no—
352C23—81no—
353C23—81no—
354C23—81no—
355C23—81no—
356C23—81no—
357C23#47412yes—
358C23C22×D811yescharacteristic
359Q8—164no—
360D4—42no—
361D4—42no—
362D4—81no—
363D4—81no—
364D4—164no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
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—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—164no—
403C2×C4—164no—
404C2×C4#47412yes—
405C2×C4#50912yes—
406C2×C4#51012yes—
407C2×C4C2×C22⋊Q811yescharacteristic
408C2×C4D4211yescharacteristic
409C2×C4#54711yescharacteristic
410C2×C4C2×#13711yescharacteristic
411C2×C4#57611yescharacteristic
412C2×C4#57811yescharacteristic
413C8—22no—
414C8—41no—
415C8—41no—
416C8#52511yescharacteristic
417C8#53911yescharacteristic
418C8D4×Q811yescharacteristic
419C8#55411yescharacteristic
420C22—22no—
421C22—22no—
422C22—22no—
423C22—41no—
424C22—41no—
425C22—41no—
426C22—41no—
427C22—82no—
428C22—82no—
429C22—82no—
430C22—161no—
431C22—161no—
432C22—161no—
433C22—161no—
434C22C2×#47411yescharacteristic
435C22D4×D811yescharacteristic
436C22#333411yescharacteristic
437C22C2×#50911yescharacteristic
438C22C2×#51011yescharacteristic
439C22#341511yescharacteristic
440C22#341611yescharacteristic
441C4—22no—
442C4—22no—
443C4—22no—
444C4—41no—
445C4—41no—
446C4—82no—
447C4—82no—
448C4—82no—
449C4—161no—
450C4—161no—
451C4—161no—
452C4—161no—
453C4#244911yescharacteristic
454C4#308311yescharacteristic
455C4#327011yescharacteristic
456C4#334611yescharacteristic
457C2—42no—
458C2—162no—
459C2#2633611yescharacteristic
460C2#3362911yescharacteristic
461C2#3363111yescharacteristic
462C2#3383211yescharacteristic
463C2#3384011yescharacteristic
464C2#3391211yescharacteristic
465C2#3392411yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC2×#466C22
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×#2226C2semidirect
2. C2×#2159C2semidirect
3. #26065C2semidirect
5. #13526C2semidirect
6. #13515C2semidirect
7. #13474C2semidirect
8. #13451C2semidirect
9. #13177C2semidirect
10. #13110C2semidirect
13. #12937C2semidirect
14. #12921C2semidirect
15. #12913C2semidirect
79. #466Q8semidirect
120. #367D4semidirect

Maximal subgroups

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

subgroupindex
C2×#22262
C2×#21592
#260652
#230272
#135262
#135152
#134742
#134512
#131772
#131102
#130612
#129912
#129372
#129212
#129132