Finite Group Databasebeta

← #507232

Subgroups — #507232

Order 512.

597 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27791C211yesmaximal, characteristic, complemented
2#25034C211yesmaximal, characteristic, complemented
3#13492C211yesmaximal, characteristic, complemented
4#13480C211yesmaximal, characteristic, complemented
5#13363C211yesmaximal, characteristic, complemented
6#13304C211yesmaximal, characteristic
7#13282C211yesmaximal, characteristic, complemented
8#13235C211yesmaximal, characteristic, complemented
9#13147C211yesmaximal, characteristic, complemented
10#13109C211yesmaximal, characteristic, complemented
11#13059C211yesmaximal, characteristic, complemented
12#13059C211yesmaximal, characteristic, complemented
13#13012C211yesmaximal, characteristic, complemented
14C4×#357C211yesmaximal, characteristic, complemented
15C16×C22⋊C4C211yesmaximal, characteristic, complemented
16C22×D16C2211yescharacteristic
17C23×C16C2211yescharacteristic
18C2×#500C2211yescharacteristic
19C2×#466C2211yescharacteristic
20C2×#466C2211yescharacteristic
21C2×C4×D8C2211yescharacteristic
22#2491—21no—
23#2422—21no—
24C2×#366C2211yescharacteristic
25C2×#357C2211yescharacteristic
26C2×#357C2211yescharacteristic
27C2×#357C2211yescharacteristic
28C2×#357C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×#357C2211yescharacteristic
31C2×C22⋊C16C2211yescharacteristic
32C2×C4×C16C2211yescharacteristic
33C2×C4×C16C2211yescharacteristic
34#2071—21no—
35#2033—21no—
36#2033—21no—
37#2023—21no—
38#1970C2211yescharacteristic
39#1970C2211yescharacteristic
40#1970C2211yescharacteristic
41#1942C2211yescharacteristic
42#1925—21no—
43#1862—21no—
44#1845C2211yescharacteristic
45#1808C2211yescharacteristic
46#1808C2211yescharacteristic
47C4×#107C2211yescharacteristic
48C8×C22⋊C4C2211yescharacteristic
49#1380C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×D8C2311yescharacteristic
60C22×D8C2311yescharacteristic
61C23×C8C2311yescharacteristic
62C2×C4⋊Q8—21no—
63C2×#124—41no—
64C2×#123—41no—
65C2×#121—21no—
66C2×#121—21no—
67C2×C4×D4—21no—
68C2×D16—22no—
69C2×D16—22no—
70C2×D16—41no—
71C22×C16—22no—
72C22×C16—41no—
73C22×C16—41no—
74C22×C16C2311yescharacteristic
75C22×C16C2311yescharacteristic
76C22×C16C2311yescharacteristic
77C22×C16C2311yescharacteristic
78#500—22no—
79#500—41no—
80#466—22no—
81#466—22no—
82#466—41no—
83#466—41no—
84C4×D8—22no—
85C4×D8—41no—
86C2×#107—21no—
87C2×#107—21no—
88C2×#107C2311yescharacteristic
89C2×#107C2311yescharacteristic
90C2×#107C2311yescharacteristic
91C2×#107C2311yescharacteristic
92C2×#107C2311yescharacteristic
93C2×#107C2311yescharacteristic
94C2×#102—21no—
95C2×#102—21no—
96C2×#102—21no—
97C2×#102—21no—
98C2×#102—21no—
99C2×#102—21no—
100C2×C22⋊C8D411yescharacteristic
101C2×C4×C8C2311yescharacteristic
102C2×C4×C8C2311yescharacteristic
103#397—41no—
104#394—41no—
105#394—41no—
106#393—41no—
107#390—21no—
108#390—21no—
109#390—21no—
110#389—21no—
111#388—41no—
112#385—41no—
113#384—21no—
114#380—21no—
115#380—21no—
116C4×C4⋊C4—21no—
117C4×C22⋊C4D411yescharacteristic
118#366—22no—
119#357—22no—
120#357—22no—
121#357—22no—
122#357—41no—
123#357—41no—
124#357—41no—
125C22⋊C16—22no—
126C4×C16—41no—
127C4×C16—41no—
128#340—21no—
129#340—21no—
130#340—21no—
131#340—21no—
132#336D411yescharacteristic
133#336D411yescharacteristic
134C22×Q8—41no—
135C22×D4—21no—
136C22×D4—21no—
137C22×D4—41no—
138C22×D4—41no—
139C23×C4—41no—
140C23×C4C2×D411yescharacteristic
141C2×D8—21no—
142C2×D8—21no—
143C2×D8—21no—
144C2×D8—21no—
145C2×D8—21no—
146C2×D8—21no—
147C2×D8—21no—
148C2×D8—21no—
149C2×D8—21no—
150C2×D8—21no—
151C2×D8—21no—
152C2×D8#4112yes—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8C2×D411yescharacteristic
160C22×C8C2×D411yescharacteristic
161C22×C8C2×D411yescharacteristic
162C22×C8C2411yescharacteristic
163C4⋊Q8—42no—
164C4⋊Q8—81no—
165#124—84no—
166#123—84no—
167#121—42no—
168#121—42no—
169#121—81no—
170#121—81no—
171C4×D4—42no—
172C4×D4—81no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4—41no—
183C2×C22⋊C4—41no—
184C2×C22⋊C4—41no—
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⋊C4—41no—
192C2×C22⋊C4—41no—
193C2×C22⋊C4—41no—
194C2×C22⋊C4D811yescharacteristic
195C2×C22⋊C4D811yescharacteristic
196C2×C42—41no—
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D411yescharacteristic
199D16—44no—
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×C16—21no—
209C2×C16—21no—
210C2×C16—44no—
211C2×C16#4112yes—
212C2×C16#4112yes—
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—21no—
222#107—21no—
223#107—21no—
224#107—42no—
225#107—81no—
226#107#4112yes—
227#102—42no—
228#102—42no—
229#102—42no—
230#102—42no—
231#102—42no—
232#102—81no—
233C22⋊C8—22no—
234C4×C8—21no—
235C4×C8—21no—
236C4×C8—21no—
237C4×C8—21no—
238#95—41no—
239#95—41no—
240#95—41no—
241#95—41no—
242#95D811yescharacteristic
243#95D811yescharacteristic
244C24—41no—
245C24—41no—
246C24C2×D811yescharacteristic
247C2×Q8—82no—
248C2×Q8—82no—
249C2×Q8—161no—
250C2×D4—22no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—41no—
261C2×D4—41no—
262C2×D4—82no—
263C2×D4—82no—
264C2×D4—82no—
265C2×D4—82no—
266C2×D4—161no—
267C2×D4—161no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—82no—
281C22×C4—82no—
282C22×C4—161no—
283C22×C4C2×D811yescharacteristic
284C22×C4C2×D811yescharacteristic
285C22×C4C2×D811yescharacteristic
286C22×C4C2×D811yescharacteristic
287C22×C4C2×D811yescharacteristic
288C22×C4C22×D411yescharacteristic
289D8—22no—
290D8—22no—
291D8—41no—
292D8—41no—
293D8—41no—
294D8—41no—
295D8—41no—
296D8—41no—
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8#13512yes—
310C2×C8C2×#4111yescharacteristic
311C2×C8C2×#4111yescharacteristic
312C2×C8C2×#4111yescharacteristic
313C2×C8C2×#4111yescharacteristic
314C2×C8H(5,𝔽2)11yescharacteristic
315C2×C8H(5,𝔽2)11yescharacteristic
316C4⋊C4—22no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—82no—
329C4⋊C4—82no—
330C4⋊C4—82no—
331C4⋊C4—161no—
332C22⋊C4—22no—
333C22⋊C4—82no—
334C22⋊C4—82no—
335C22⋊C4—82no—
336C22⋊C4—82no—
337C22⋊C4—82no—
338C22⋊C4—82no—
339C22⋊C4—82no—
340C22⋊C4—82no—
341C22⋊C4—82no—
342C22⋊C4—82no—
343C22⋊C4—161no—
344C22⋊C4D1614yescomplemented
345C42—21no—
346C42—21no—
347C42—21no—
348C42—21no—
349C42—82no—
350C16—22no—
351C16—22no—
352C16—41no—
353C16—41no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—42no—
361C23—81no—
362C23—81no—
363C23—81no—
364C23—81no—
365C23—81no—
366C23—81no—
367C23—81no—
368C23—81no—
369C23—81no—
370C23—81no—
371C23—81no—
372C23—81no—
373C23—81no—
374C23C2×D1612yes—
375C23C22×D811yescharacteristic
376Q8—164no—
377D4—42no—
378D4—42no—
379D4—81no—
380D4—81no—
381D4—81no—
382D4—81no—
383D4—81no—
384D4—81no—
385D4—164no—
386D4—164no—
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—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—42no—
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—81no—
421C2×C4—164no—
422C2×C4C2×D1612yes—
423C2×C4C2×D1612yes—
424C2×C4#50812yes—
425C2×C4#53211yescharacteristic
426C2×C4#54611yescharacteristic
427C2×C4#54611yescharacteristic
428C2×C4C2×#13511yescharacteristic
429C2×C4#57711yescharacteristic
430C2×C4#57711yescharacteristic
431C8—22no—
432C8—41no—
433C8—41no—
434C8#52911yescharacteristic
435C8#53811yescharacteristic
436C8#55111yescharacteristic
437C8#55911yescharacteristic
438C22—22no—
439C22—22no—
440C22—22no—
441C22—41no—
442C22—41no—
443C22—41no—
444C22—41no—
445C22—82no—
446C22—82no—
447C22—82no—
448C22—161no—
449C22—161no—
450C22—161no—
451C22—161no—
452C22—161no—
453C22—161no—
454C22—161no—
455C22—161no—
456C22—161no—
457C22—161no—
458C22—161no—
459C22#316911yescharacteristic
460C22#329411yescharacteristic
461C22#329411yescharacteristic
462C22C22×D1611yescharacteristic
463C22C2×#50811yescharacteristic
464C22#341611yescharacteristic
465C22#341611yescharacteristic
466C4—22no—
467C4—22no—
468C4—22no—
469C4—41no—
470C4—41no—
471C4—82no—
472C4—161no—
473C4—161no—
474C4—161no—
475C4—161no—
476C4—161no—
477C4#255711yescharacteristic
478C4#317311yescharacteristic
479C4#330111yescharacteristic
480C4#330311yescharacteristic
481C2—42no—
482C2—162no—
483C2—321no—
484C2#2897511yescharacteristic
485C2#3373311yescharacteristic
486C2#3374211yescharacteristic
487C2#3386111yescharacteristic
488C2#3386111yescharacteristic
489C2#3386811yescharacteristic
490C2#3387011yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radical#13304C2
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. #27791C2semidirect
2. #25034C2semidirect
3. #13492C2semidirect
4. #13480C2semidirect
5. #13363C2semidirect
7. #13282C2semidirect
8. #13235C2semidirect
9. #13147C2semidirect
10. #13109C2semidirect
11. #13059C2semidirect
12. #13059C2semidirect
13. #13012C2semidirect
14. C4×#357C2semidirect
15. C16×C22⋊C4C2semidirect
344. C22⋊C4D16semidirect

Maximal subgroups

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

subgroupindex
#277912
#250342
#134922
#134802
#133632
#133042
#132822
#132352
#131472
#131092
#130592
#130122
C4×#3572
C16×C22⋊C42