Finite Group Databasebeta

← #504651

Subgroups — #504651

Order 512.

609 conjugacy classes of subgroups.

The subgroup lattice has 483 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2236C211yesmaximal, characteristic, complemented
2C2×#2224C211yesmaximal, characteristic, complemented
3#27792C211yesmaximal, characteristic, complemented
4#23025C211yesmaximal, characteristic
5#13477C211yesmaximal, characteristic, complemented
6#13451C211yesmaximal, characteristic, complemented
7#13381C211yesmaximal, characteristic
8#13305C211yesmaximal, characteristic
9#13238C211yesmaximal, characteristic, complemented
10#13133C211yesmaximal, characteristic, complemented
11#13130C211yesmaximal, characteristic, complemented
12#13121C211yesmaximal, characteristic, complemented
13#13014C211yesmaximal, characteristic, complemented
14#12919C211yesmaximal, characteristic, complemented
15#12674C211yesmaximal, characteristic, complemented
16C22×SD32C2211yescharacteristic
17C2×#500C2211yescharacteristic
18C2×#467C2211yescharacteristic
19C2×#466C2211yescharacteristic
20C2×C4×D8C2211yescharacteristic
21C22×#107C2211yescharacteristic
22#2492—21no—
23#2361—21no—
24#2236—24no—
25#2224—24no—
26C2×#367C2211yescharacteristic
27C2×#366C2211yescharacteristic
28C2×#366C2211yescharacteristic
29C2×#366C2211yescharacteristic
30C2×#366C2211yescharacteristic
31C2×C4⋊C16C2211yescharacteristic
32C2×#358C2211yescharacteristic
33C2×#358C2211yescharacteristic
34C2×#358C2211yescharacteristic
35C2×#357C2211yescharacteristic
36C2×#357C2211yescharacteristic
37C2×#357C2211yescharacteristic
38C2×C22⋊C16C2211yescharacteristic
39C2×C22⋊C16C2211yescharacteristic
40C2×#346C2211yescharacteristic
41#2036—21no—
42#2024—21no—
43#1970C2211yescharacteristic
44#1969C2211yescharacteristic
45#1969C2211yescharacteristic
46#1942C2211yescharacteristic
47#1928—21no—
48#1925—21no—
49#1874—21no—
50#1862—21no—
51#1808C2211yescharacteristic
52C4×#107C2211yescharacteristic
53C4×#107C2211yescharacteristic
54C8×C22⋊C4C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1374C2211yescharacteristic
60#1374C2211yescharacteristic
61C22×Q16C2311yescharacteristic
62C22×D8C2311yescharacteristic
63C23×C8C2311yescharacteristic
64C2×C4⋊Q8—21no—
65C2×C22⋊Q8—21no—
66C2×#121—21no—
67C2×C4×D4—21no—
68C2×C4×D4—41no—
69C22×C4⋊C4—21no—
70C2×SD32—22no—
71C2×SD32—22no—
72C2×SD32—41no—
73C22×C16C2311yescharacteristic
74C22×C16C2311yescharacteristic
75C22×C16C2311yescharacteristic
76C22×C16C2311yescharacteristic
77#500—22no—
78#500—41no—
79#467—22no—
80#467—41no—
81#466—22no—
82#466—22no—
83C4×D8—22no—
84C4×D8—41no—
85C2×#107—22no—
86C2×#107—21no—
87C2×#107—21no—
88C2×#107—41no—
89C2×#107C2311yescharacteristic
90C2×#107C2311yescharacteristic
91C2×#107C2311yescharacteristic
92C2×#107C2311yescharacteristic
93C2×#107C2311yescharacteristic
94C2×#107C2311yescharacteristic
95C2×#103—21no—
96C2×#103—21no—
97C2×#103—21no—
98C2×#102—21no—
99C2×#102—21no—
100C2×#102—21no—
101C2×C22⋊C8D411yescharacteristic
102C2×C4×C8C2311yescharacteristic
103C2×C4×C8C2311yescharacteristic
104#400—41no—
105#393—41no—
106#391—21no—
107#391—21no—
108#391—41no—
109#390—21no—
110#389—21no—
111#388—41no—
112#388—41no—
113#386—41no—
114#385—41no—
115#380—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C22⋊C4D411yescharacteristic
119#367—22no—
120#366—22no—
121#366—22no—
122#366—41no—
123#366D414yescomplemented
124C4⋊C16—22no—
125#358—22no—
126#358—22no—
127#358—22no—
128#357—22no—
129#357—22no—
130#357—22no—
131C22⋊C16—22no—
132C22⋊C16—22no—
133#346—22no—
134#340—21no—
135#340—21no—
136#340—21no—
137#340—21no—
138#336D411yescharacteristic
139#336D411yescharacteristic
140C22×Q8—21no—
141C22×Q8—41no—
142C22×D4—21no—
143C22×D4—41no—
144C23×C4—41no—
145C23×C4—41no—
146C23×C4C2×D411yescharacteristic
147C2×Q16—21no—
148C2×Q16—21no—
149C2×Q16—21no—
150C2×Q16—21no—
151C2×Q16—21no—
152C2×Q16—21no—
153C2×D8—21no—
154C2×D8—21no—
155C2×D8—21no—
156C2×D8—21no—
157C2×D8—21no—
158C2×D8#4112yes—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8C2×D411yescharacteristic
166C22×C8C2×D411yescharacteristic
167C22×C8C2×D411yescharacteristic
168C22×C8C2411yescharacteristic
169C4⋊Q8—42no—
170C4⋊Q8—81no—
171C22⋊Q8—42no—
172C22⋊Q8—81no—
173#121—24no—
174#121—42no—
175C4×D4—42no—
176C4×D4—81no—
177C4×D4—84no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—81no—
190C2×C22⋊C4—41no—
191C2×C22⋊C4—41no—
192C2×C22⋊C4—41no—
193C2×C22⋊C4—41no—
194C2×C22⋊C4—41no—
195C2×C22⋊C4—41no—
196C2×C22⋊C4D811yescharacteristic
197C2×C22⋊C4D811yescharacteristic
198C2×C42—41no—
199C2×C42—41no—
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202SD32—44no—
203C2×C16—21no—
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—
215#107—21no—
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—42no—
225#107—42no—
226#107C2×D412yes—
227#107C2×D412yes—
228#107#4112yes—
229#103—42no—
230#103—42no—
231#103—81no—
232#102—24no—
233#102—42no—
234#102—81no—
235C22⋊C8—41no—
236C4×C8—21no—
237C4×C8—21no—
238C4×C8—21no—
239C4×C8—21no—
240#95—41no—
241#95—41no—
242#95—41no—
243#95—41no—
244#95—41no—
245#95—41no—
246#95—41no—
247#95—41no—
248#95D811yescharacteristic
249#95D811yescharacteristic
250C24—41no—
251C24C2×D811yescharacteristic
252C2×Q8—41no—
253C2×Q8—41no—
254C2×Q8—41no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×Q8—41no—
258C2×Q8—82no—
259C2×Q8—82no—
260C2×Q8—161no—
261C2×D4—22no—
262C2×D4—44no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—82no—
269C2×D4—82no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—82no—
284C22×C4—82no—
285C22×C4—82no—
286C22×C4—82no—
287C22×C4—161no—
288C22×C4—161no—
289C22×C4C2×D811yescharacteristic
290C22×C4C2×D811yescharacteristic
291C22×C4C2×D811yescharacteristic
292C22×C4C2×D811yescharacteristic
293C22×C4C2×D811yescharacteristic
294C22×C4C22×D411yescharacteristic
295Q16—41no—
296Q16—41no—
297Q16—41no—
298Q16—41no—
299D8—22no—
300D8—22no—
301D8—41no—
302D8—41no—
303C2×C8—21no—
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—41no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8C22×D411yescharacteristic
317C2×C8C2×#4111yescharacteristic
318C2×C8C2×#4111yescharacteristic
319C2×C8C2×#4111yescharacteristic
320C2×C8H(5,𝔽2)11yescharacteristic
321C2×C82−1+411yescharacteristic
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—82no—
335C4⋊C4—82no—
336C4⋊C4—82no—
337C4⋊C4—82no—
338C22⋊C4—22no—
339C22⋊C4—22no—
340C22⋊C4—44no—
341C22⋊C4—82no—
342C22⋊C4—82no—
343C22⋊C4—82no—
344C22⋊C4—161no—
345C22⋊C4—161no—
346C42—21no—
347C42—21no—
348C42—21no—
349C42—21no—
350C42—82no—
351C42—161no—
352C16—22no—
353C16—22no—
354C16—41no—
355C16—41no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—42no—
363C23—81no—
364C23—81no—
365C23—81no—
366C23—81no—
367C23—81no—
368C23—81no—
369C23#51012yes—
370C23C22×D811yescharacteristic
371Q8—81no—
372Q8—81no—
373Q8—81no—
374Q8—81no—
375Q8—164no—
376D4—42no—
377D4—42no—
378D4—88no—
379D4—81no—
380D4—81no—
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—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
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—81no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—164no—
421C2×C4—164no—
422C2×C4#50912yes—
423C2×C4#52511yescharacteristic
424C2×C4D4211yescharacteristic
425C2×C4#54711yescharacteristic
426C2×C4#57511yescharacteristic
427C2×C4#57611yescharacteristic
428C2×C4#57811yescharacteristic
429C8—41no—
430C8—41no—
431C8—41no—
432C8#53211yescharacteristic
433C8#54211yescharacteristic
434C8#54811yescharacteristic
435C8#55111yescharacteristic
436C22—22no—
437C22—22no—
438C22—22no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22—82no—
444C22—82no—
445C22—82no—
446C22—161no—
447C22—161no—
448C22—161no—
449C22—161no—
450C22#310011yescharacteristic
451C22D4×D811yescharacteristic
452C22#333411yescharacteristic
453C22C2×#50911yescharacteristic
454C22C2×#51011yescharacteristic
455C22#341511yescharacteristic
456C22#341711yescharacteristic
457C4—22no—
458C4—41no—
459C4—41no—
460C4—41no—
461C4—41no—
462C4—82no—
463C4—82no—
464C4—82no—
465C4—161no—
466C4—161no—
467C4—161no—
468C4—161no—
469C4#246111yescharacteristic
470C4#312411yescharacteristic
471C4#327711yescharacteristic
472C4#334711yescharacteristic
473C2—42no—
474C2—162no—
475C2#2678411yescharacteristic
476C2#3367011yescharacteristic
477C2#3368411yescharacteristic
478C2#3383011yescharacteristic
479C2#3383611yescharacteristic
480C2#3391311yescharacteristic
481C2#3392011yescharacteristic
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×#2236C2semidirect
2. C2×#2224C2semidirect
3. #27792C2semidirect
5. #13477C2semidirect
6. #13451C2semidirect
9. #13238C2semidirect
10. #13133C2semidirect
11. #13130C2semidirect
12. #13121C2semidirect
13. #13014C2semidirect
14. #12919C2semidirect
15. #12674C2semidirect
123. #366D4semidirect

Maximal subgroups

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

subgroupindex
C2×#22362
C2×#22242
#277922
#230252
#134772
#134512
#133812
#133052
#132382
#131332
#131302
#131212
#130142
#129192
#126742