Finite Group Databasebeta

← #504700

Subgroups — #504700

Order 512.

584 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2241C211yesmaximal, characteristic, complemented
2C2×#2175C211yesmaximal, characteristic
3#26484C211yesmaximal, characteristic
4#26255C211yesmaximal, characteristic, complemented
5#13475C211yesmaximal, characteristic, complemented
6#13451C211yesmaximal, characteristic, complemented
7#13449C211yesmaximal, characteristic
8#13327C211yesmaximal, characteristic
9#13238C211yesmaximal, characteristic
10#13150C211yesmaximal, characteristic, complemented
11#13150C211yesmaximal, characteristic, complemented
12#13147C211yesmaximal, characteristic
13#13014C211yesmaximal, characteristic
14#12936C211yesmaximal, characteristic, complemented
15#12663C211yesmaximal, characteristic, complemented
16C22×SD32C2211yescharacteristic
17C22×SD32C2211yescharacteristic
18C2×#500C2211yescharacteristic
19C2×#494C2211yescharacteristic
20C2×C4×Q16C2211yescharacteristic
21C2×C4×D8C2211yescharacteristic
22#2452—21no—
23#2446—21no—
24#2241—24no—
25#2175—24no—
26C2×#366C2211yescharacteristic
27C2×#366C2211yescharacteristic
28C2×C4⋊C16C2211yescharacteristic
29C2×#358C2211yescharacteristic
30C2×#358C2211yescharacteristic
31C2×#358C2211yescharacteristic
32C2×#358C2211yescharacteristic
33C2×#358C2211yescharacteristic
34C2×#358C2211yescharacteristic
35C2×#357C2211yescharacteristic
36C2×#357C2211yescharacteristic
37C2×#346C2211yescharacteristic
38C2×#346C2211yescharacteristic
39C2×C4×C16C2211yescharacteristic
40#2067—21no—
41#2056—21no—
42#2043—21no—
43#2023—21no—
44#1970C2211yescharacteristic
45#1970C2211yescharacteristic
46#1969C2211yescharacteristic
47#1969C2211yescharacteristic
48#1969C2211yescharacteristic
49#1942C2211yescharacteristic
50#1942C2211yescharacteristic
51#1942C2211yescharacteristic
52#1878—21no—
53#1862—21no—
54C8×C4⋊C4C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1374C2211yescharacteristic
60#1374C2211yescharacteristic
61C22×Q16—21no—
62C22×Q16C2311yescharacteristic
63C22×Q16C2311yescharacteristic
64C22×D8C2311yescharacteristic
65C2×C4⋊Q8—21no—
66C2×C4⋊Q8—21no—
67C2×C22⋊Q8—41no—
68C2×C4×Q8—21no—
69C2×C4×D4—21no—
70C2×SD32—22no—
71C2×SD32—22no—
72C2×SD32—22no—
73C2×SD32—22no—
74C2×SD32—22no—
75C2×SD32—41no—
76C22×C16C2311yescharacteristic
77C22×C16C2311yescharacteristic
78C22×C16C2311yescharacteristic
79C22×C16C2311yescharacteristic
80#500—22no—
81#500—41no—
82#494—22no—
83#494—41no—
84C4×Q16—22no—
85C4×Q16—41no—
86C4×D8—22no—
87C4×D8—41no—
88C2×#107—21no—
89C2×#107C2311yescharacteristic
90C2×#107C2311yescharacteristic
91C2×#107C2311yescharacteristic
92C2×#107C2311yescharacteristic
93C2×#107C2311yescharacteristic
94C2×C4⋊C8D411yescharacteristic
95C2×#103—21no—
96C2×#103—21no—
97C2×#103—21no—
98C2×#103—21no—
99C2×#102—21no—
100C2×#102—21no—
101C2×#102—21no—
102C2×C4×C8C2311yescharacteristic
103C2×C4×C8C2311yescharacteristic
104C2×C4×C8C2311yescharacteristic
105#400—41no—
106#398—41no—
107#396—41no—
108#393—41no—
109#391—21no—
110#391—21no—
111#391—21no—
112#390—21no—
113#390—21no—
114#389—21no—
115#389—21no—
116#389—21no—
117#387—41no—
118#387—41no—
119#385—41no—
120C4×C4⋊C4D411yescharacteristic
121#366—22no—
122#366—41no—
123C4⋊C16—22no—
124#358—22no—
125#358—22no—
126#358—22no—
127#358—22no—
128#358—22no—
129#358D414yes—
130#357—22no—
131#357—22no—
132#346—22no—
133#346—22no—
134C4×C16—22no—
135#340—21no—
136#340—21no—
137#340—21no—
138#336D411yescharacteristic
139#336D411yescharacteristic
140C22×Q8—21no—
141C22×Q8—21no—
142C22×Q8—41no—
143C22×D4—21no—
144C23×C4—41no—
145C2×Q16—21no—
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—21no—
155C2×Q16—21no—
156C2×Q16—42no—
157C2×Q16—42no—
158C2×Q16—81no—
159C2×Q16C2×D412yes—
160C2×D8—21no—
161C2×D8—21no—
162C2×D8—21no—
163C2×D8—21no—
164C2×D8—21no—
165C2×D8#4112yes—
166C22×C8C2×D411yescharacteristic
167C22×C8C2×D411yescharacteristic
168C22×C8C2×D411yescharacteristic
169C22×C8C2411yescharacteristic
170C4⋊Q8—42no—
171C4⋊Q8—42no—
172C4⋊Q8—81no—
173C4⋊Q8—81no—
174C22⋊Q8—84no—
175C4×Q8—42no—
176C4×Q8—81no—
177C4×D4—42no—
178C4×D4—81no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—41no—
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×C4⋊C4D811yescharacteristic
192C2×C4⋊C4D811yescharacteristic
193C2×C22⋊C4—41no—
194C2×C22⋊C4—41no—
195C2×C22⋊C4—41no—
196C2×C22⋊C4—41no—
197C2×C22⋊C4—41no—
198C2×C42—41no—
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202SD32—28no—
203SD32—44no—
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—
213C2×C16C2×D412yes—
214C2×C16#4112yes—
215C2×C16#4112yes—
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#107C2×D412yes—
227C4⋊C8—22no—
228#103—42no—
229#103—81no—
230#103—81no—
231#103—81no—
232#102—42no—
233#102—42no—
234#102—81no—
235C4×C8—21no—
236C4×C8—21no—
237C4×C8—21no—
238C4×C8—21no—
239C4×C8—21no—
240C4×C8—21no—
241#95—41no—
242#95—41no—
243#95—41no—
244#95—41no—
245#95—41no—
246#95—41no—
247#95—41no—
248#95—41no—
249#95—41no—
250#95D811yescharacteristic
251#95D811yescharacteristic
252C24—41no—
253C2×Q8—22no—
254C2×Q8—41no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—82no—
266C2×Q8—82no—
267C2×Q8—161no—
268C2×D4—22no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—82no—
282C22×C4—82no—
283C22×C4—161no—
284C22×C4C2×D811yescharacteristic
285C22×C4C2×D811yescharacteristic
286C22×C4C2×D811yescharacteristic
287C22×C4C2×D811yescharacteristic
288C22×C4C2×D811yescharacteristic
289C22×C4C2×D811yescharacteristic
290C22×C4C22×D411yescharacteristic
291Q16—22no—
292Q16—22no—
293Q16—41no—
294Q16—41no—
295Q16—41no—
296Q16—41no—
297Q16—41no—
298Q16—41no—
299Q16—84no—
300D8—22no—
301D8—22no—
302D8—41no—
303D8—41no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8#13512yes—
313C2×C8C22×D411yescharacteristic
314C2×C8C2×#4111yescharacteristic
315C2×C8C2×#4111yescharacteristic
316C2×C8C2×#4111yescharacteristic
317C2×C8H(5,𝔽2)11yescharacteristic
318C2×C82−1+411yescharacteristic
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—22no—
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—41no—
331C4⋊C4—82no—
332C4⋊C4—82no—
333C4⋊C4—82no—
334C4⋊C4—82no—
335C4⋊C4—82no—
336C4⋊C4—82no—
337C4⋊C4—82no—
338C22⋊C4—82no—
339C22⋊C4—82no—
340C22⋊C4—82no—
341C22⋊C4—82no—
342C22⋊C4—161no—
343C42—21no—
344C42—21no—
345C42—21no—
346C42—21no—
347C42—21no—
348C42—21no—
349C42—82no—
350C16—22no—
351C16—22no—
352C16—22no—
353C16—41no—
354C23—42no—
355C23—81no—
356C23—81no—
357C23—81no—
358C23—81no—
359C23—81no—
360C23—81no—
361C23C22×D811yescharacteristic
362Q8—42no—
363Q8—42no—
364Q8—81no—
365Q8—81no—
366Q8—81no—
367Q8—81no—
368Q8—81no—
369Q8—81no—
370Q8—164no—
371D4—42no—
372D4—42no—
373D4—81no—
374D4—81no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—42no—
392C2×C4—42no—
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×C4—81no—
412C2×C4—164no—
413C2×C4#50912yes—
414C2×C4#51012yes—
415C2×C4#53211yescharacteristic
416C2×C4D4211yescharacteristic
417C2×C4#54811yescharacteristic
418C2×C4C2×#13511yescharacteristic
419C2×C4#57611yescharacteristic
420C2×C4#57711yescharacteristic
421C8—22no—
422C8—41no—
423C8—41no—
424C8#52511yescharacteristic
425C8#54211yescharacteristic
426C8#54711yescharacteristic
427C8#55111yescharacteristic
428C22—82no—
429C22—82no—
430C22—82no—
431C22—161no—
432C22—161no—
433C22—161no—
434C22—161no—
435C22#316911yescharacteristic
436C22D4×D811yescharacteristic
437C22#335811yescharacteristic
438C22C2×#50911yescharacteristic
439C22C2×#51011yescharacteristic
440C22#341611yescharacteristic
441C22#341711yescharacteristic
442C4—22no—
443C4—22no—
444C4—41no—
445C4—41no—
446C4—41no—
447C4—41no—
448C4—82no—
449C4—82no—
450C4—161no—
451C4—161no—
452C4—161no—
453C4—161no—
454C4—161no—
455C4#246111yescharacteristic
456C4#317911yescharacteristic
457C4#328211yescharacteristic
458C4#336111yescharacteristic
459C2—162no—
460C2#2681811yescharacteristic
461C2#3374511yescharacteristic
462C2#3375111yescharacteristic
463C2#3383111yescharacteristic
464C2#3383611yescharacteristic
465C2#3393411yescharacteristic
466C2#3393911yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D8C23
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ C8×C4⋊C4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#2241C2semidirect
4. #26255C2semidirect
5. #13475C2semidirect
6. #13451C2semidirect
10. #13150C2semidirect
11. #13150C2semidirect
14. #12936C2semidirect
15. #12663C2semidirect

Maximal subgroups

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

subgroupindex
C2×#22412
C2×#21752
#264842
#262552
#134752
#134512
#134492
#133272
#132382
#131502
#131472
#130142
#129362
#126632