Finite Group Databasebeta

← #510901

Subgroups — #510901

Order 512.

596 conjugacy classes of subgroups.

The subgroup lattice has 497 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27929C211yesmaximal, characteristic, complemented
2#27791C211yesmaximal, characteristic, complemented
3#13526C211yesmaximal, characteristic, complemented
4#13480C211yesmaximal, characteristic, complemented
5#13448C211yesmaximal, characteristic, complemented
6#13380C211yesmaximal, characteristic, complemented
7#13363C211yesmaximal, characteristic, complemented
8#13304C211yesmaximal, characteristic
9#13282C211yesmaximal, characteristic, complemented
10#13235C211yesmaximal, characteristic, complemented
11#13176C211yesmaximal, characteristic, complemented
12#13130C211yesmaximal, characteristic, complemented
13#13120C211yesmaximal, characteristic, complemented
14#12991C211yesmaximal, characteristic, complemented
15#12919C211yesmaximal, characteristic, complemented
16C22×D16C2211yescharacteristic
17C2×#500C2211yescharacteristic
18C2×#480—21no—
19C2×#466C2211yescharacteristic
20C2×#466C2211yescharacteristic
21C22×#107C2211yescharacteristic
22#2501—21no—
23#2491—21no—
24C2×#366C2211yescharacteristic
25C2×#366C2211yescharacteristic
26C2×#366C2211yescharacteristic
27C2×C4⋊C16C2211yescharacteristic
28C2×#357C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×#357C2211yescharacteristic
31C2×#357C2211yescharacteristic
32C2×#357C2211yescharacteristic
33C2×#357C2211yescharacteristic
34C2×C22⋊C16C2211yescharacteristic
35C2×C22⋊C16C2211yescharacteristic
36C2×C4×C16C2211yescharacteristic
37#2093—21no—
38#2093—21no—
39#2033—21no—
40#2023—21no—
41#1970C2211yescharacteristic
42#1970C2211yescharacteristic
43#1970C2211yescharacteristic
44#1970C2211yescharacteristic
45#1925—21no—
46#1845C2211yescharacteristic
47#1845C2211yescharacteristic
48#1808C2211yescharacteristic
49C4×#107C2211yescharacteristic
50C8×C22⋊C4C2211yescharacteristic
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×#123—41no—
66C2×#121—21no—
67C2×#121—21no—
68C22×C4⋊C4—21no—
69C2×D16—22no—
70C2×D16—41no—
71C2×D16—41no—
72C22×C16C2311yescharacteristic
73C22×C16C2311yescharacteristic
74C22×C16C2311yescharacteristic
75C22×C16C2311yescharacteristic
76#500—22no—
77#500—41no—
78#480—44no—
79#466—22no—
80#466—41no—
81#466—41no—
82#466—41no—
83C2×#107—21no—
84C2×#107—21no—
85C2×#107—21no—
86C2×#107—21no—
87C2×#107—41no—
88C2×#107—41no—
89C2×#107C2311yescharacteristic
90C2×#107C2311yescharacteristic
91C2×#107C2311yescharacteristic
92C2×#107C2311yescharacteristic
93C2×#107C2311yescharacteristic
94C2×#107C2311yescharacteristic
95C2×#102—21no—
96C2×#102—21no—
97C2×#102—21no—
98C2×#102—21no—
99C2×#102—21no—
100C2×#102—21no—
101C2×C22⋊C8D411yescharacteristic
102C2×C4×C8C2311yescharacteristic
103C2×C4×C8C2311yescharacteristic
104#399—41no—
105#399—41no—
106#394—41no—
107#393—41no—
108#390—21no—
109#390—21no—
110#390—21no—
111#390—21no—
112#388—41no—
113#384—21no—
114#384—21no—
115#380—21no—
116C4×C4⋊C4—21no—
117C4×C22⋊C4D411yescharacteristic
118#366—41no—
119#366—41no—
120#366—41no—
121C4⋊C16—22no—
122#357—22no—
123#357—41no—
124#357—41no—
125#357—41no—
126#357—41no—
127#357—41no—
128C22⋊C16—22no—
129C22⋊C16—41no—
130C4×C16—41no—
131#340—21no—
132#340—21no—
133#336D411yescharacteristic
134#336D411yescharacteristic
135C22×Q8—41no—
136C22×D4—21no—
137C22×D4—21no—
138C22×D4—41no—
139C22×D4—41no—
140C23×C4—41no—
141C23×C4C2×D411yescharacteristic
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—21no—
153C2×D8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8C2×D411yescharacteristic
161C22×C8C2×D411yescharacteristic
162C22×C8C2×D411yescharacteristic
163C22×C8C2411yescharacteristic
164C4⋊Q8—42no—
165C4⋊Q8—81no—
166#124—84no—
167#123—84no—
168#123—84no—
169#121—42no—
170#121—42no—
171#121—42no—
172#121—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—42no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—41no—
186C2×C4⋊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⋊C4—41no—
195C2×C22⋊C4—41no—
196C2×C22⋊C4—41no—
197C2×C22⋊C4—41no—
198C2×C22⋊C4D811yescharacteristic
199C2×C22⋊C4D811yescharacteristic
200C2×C42—41no—
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D411yescharacteristic
203D16—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×C16—21no—
214C2×C16—21no—
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—21no—
226#107—21no—
227#107—42no—
228#107—42no—
229#107—42no—
230#107—42no—
231#107#4112yes—
232#102—42no—
233#102—42no—
234#102—42no—
235#102—42no—
236#102—42no—
237#102—81no—
238C22⋊C8—22no—
239C4×C8—21no—
240C4×C8—21no—
241C4×C8—21no—
242C4×C8—21no—
243#95—41no—
244#95—41no—
245#95D811yescharacteristic
246#95D811yescharacteristic
247C24—41no—
248C24—41no—
249C24C2×D811yescharacteristic
250C2×Q8—82no—
251C2×Q8—82no—
252C2×Q8—161no—
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—41no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—82no—
266C2×D4—82no—
267C2×D4—82no—
268C2×D4—82no—
269C2×D4—82no—
270C2×D4—161no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
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×C4C2×D811yescharacteristic
287C22×C4C2×D811yescharacteristic
288C22×C4C2×D811yescharacteristic
289C22×C4C2×D811yescharacteristic
290C22×C4C2×D811yescharacteristic
291C22×C4C22×D411yescharacteristic
292D8—41no—
293D8—41no—
294D8—41no—
295D8—41no—
296D8—41no—
297D8—41no—
298D8—41no—
299D8—41no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8AGL(1,ℤ/8)12yes—
313C2×C8C2×#4111yescharacteristic
314C2×C8C2×#4111yescharacteristic
315C2×C8H(5,𝔽2)11yescharacteristic
316C2×C8H(5,𝔽2)11yescharacteristic
317C2×C8H(5,𝔽2)11yescharacteristic
318C2×C82−1+411yescharacteristic
319C4⋊C4—22no—
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—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—
337C22⋊C4—22no—
338C22⋊C4—82no—
339C22⋊C4—82no—
340C22⋊C4—82no—
341C22⋊C4—82no—
342C22⋊C4—82no—
343C22⋊C4—82no—
344C22⋊C4—82no—
345C22⋊C4—82no—
346C22⋊C4—82no—
347C22⋊C4—82no—
348C22⋊C4—161no—
349C22⋊C4D1614yescomplemented
350C42—21no—
351C42—21no—
352C42—21no—
353C42—21no—
354C42—82no—
355C16—22no—
356C16—41no—
357C16—41no—
358C16—41no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—81no—
366C23—81no—
367C23—81no—
368C23—81no—
369C23—81no—
370C23—81no—
371C23—81no—
372C23—81no—
373C23—81no—
374C23—81no—
375C23—81no—
376C23—81no—
377C23—81no—
378C23—81no—
379C23C2×D1612yes—
380C23C22×D811yescharacteristic
381Q8—164no—
382D4—81no—
383D4—81no—
384D4—81no—
385D4—81no—
386D4—81no—
387D4—81no—
388D4—81no—
389D4—81no—
390D4—164no—
391D4—164no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—42no—
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—81no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—164no—
427C2×C4C2×D1612yes—
428C2×C4C2×D1612yes—
429C2×C4#50912yes—
430C2×C4#53511yescharacteristic
431C2×C4#54611yescharacteristic
432C2×C4#54711yescharacteristic
433C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
434C2×C4#57711yescharacteristic
435C2×C4#57811yescharacteristic
436C8—22no—
437C8—41no—
438C8—41no—
439C8#53811yescharacteristic
440C8#54211yescharacteristic
441C8#55911yescharacteristic
442C8#56211yescharacteristic
443C22—22no—
444C22—22no—
445C22—22no—
446C22—41no—
447C22—41no—
448C22—41no—
449C22—41no—
450C22—161no—
451C22—161no—
452C22—161no—
453C22—161no—
454C22—161no—
455C22—161no—
456C22—161no—
457C22—161no—
458C22—161no—
459C22—161no—
460C22—161no—
461C22—161no—
462C22—161no—
463C22—161no—
464C22#320511yescharacteristic
465C22#329411yescharacteristic
466C22#333411yescharacteristic
467C22C22×D1611yescharacteristic
468C22C2×#50911yescharacteristic
469C22#341611yescharacteristic
470C22#341711yescharacteristic
471C4—22no—
472C4—22no—
473C4—22no—
474C4—41no—
475C4—41no—
476C4—82no—
477C4—161no—
478C4—161no—
479C4—161no—
480C4—161no—
481C4—161no—
482C4#271011yescharacteristic
483C4#322511yescharacteristic
484C4#330811yescharacteristic
485C4#335411yescharacteristic
486C2—42no—
487C2—321no—
488C2—321no—
489C2#3204611yescharacteristic
490C2#3377111yescharacteristic
491C2#3379111yescharacteristic
492C2#3386111yescharacteristic
493C2#3387511yescharacteristic
494C2#3390111yescharacteristic
495C2#3392111yescharacteristic
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. #27929C2semidirect
2. #27791C2semidirect
3. #13526C2semidirect
4. #13480C2semidirect
5. #13448C2semidirect
6. #13380C2semidirect
7. #13363C2semidirect
9. #13282C2semidirect
10. #13235C2semidirect
11. #13176C2semidirect
12. #13130C2semidirect
13. #13120C2semidirect
14. #12991C2semidirect
15. #12919C2semidirect
349. C22⋊C4D16semidirect

Maximal subgroups

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

subgroupindex
#279292
#277912
#135262
#134802
#134482
#133802
#133632
#133042
#132822
#132352
#131762
#131302
#131202
#129912
#129192