Finite Group Databasebeta

← #308419

Subgroups — #308419

Order 512.

603 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26466C211yesmaximal, characteristic, complemented
2#26020C211yesmaximal, characteristic
3#23066C211yesmaximal, characteristic
4#22708C211yesmaximal, characteristic
5#10681C211yesmaximal, characteristic, complemented
6#10655C211yesmaximal, characteristic, complemented
7#10598C211yesmaximal, characteristic, complemented
8#10574C211yesmaximal, characteristic, complemented
9#9622C211yesmaximal, characteristic, complemented
10#9540C211yesmaximal, characteristic, complemented
11#9219C211yesmaximal, characteristic
12#8984C211yesmaximal, characteristic
13#8923C211yesmaximal, characteristic, complemented
14#8889C211yesmaximal, characteristic
15#8845C211yesmaximal, characteristic
16C2×#477—21no—
17C2×#474C2211yescharacteristic
18C2×C8×D4C2211yescharacteristic
19#2551C2211yescharacteristic
20#2365C2211yescharacteristic
21#2354C2211yescharacteristic
22#2354C2211yescharacteristic
23#2068—21no—
24#2067—21no—
25#2024—21no—
26#2023—21no—
27#1990C2211yescharacteristic
28#1990C2211yescharacteristic
29#1988—21no—
30#1981C2211yescharacteristic
31#1956C2211yescharacteristic
32#1948C2211yescharacteristic
33#1948C2211yescharacteristic
34#1948C2211yescharacteristic
35#1947C2211yescharacteristic
36#1947C2211yescharacteristic
37#1939C2211yescharacteristic
38#1939C2211yescharacteristic
39#1936—21no—
40#1926—21no—
41#1921—21no—
42#1918—21no—
43#1863—21no—
44#1854—21no—
45#1854—21no—
46#1844C2211yescharacteristic
47#1834C2211yescharacteristic
48#1834C2211yescharacteristic
49#1807—21no—
50#1802C2211yescharacteristic
51#1776—21no—
52C4×#107C2211yescharacteristic
53#1770C2211yescharacteristic
54C4×#102C2211yescharacteristic
55#1735—21no—
56C2×#334C2211yescharacteristic
57C2×#332C2211yescharacteristic
58C2×D4⋊C8C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1279C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1277C2211yescharacteristic
66#1273C2211yescharacteristic
67C23×C8—21no—
68C2×C4⋊Q8—21no—
69C2×C4⋊Q8—21no—
70C2×C4⋊Q8C2311yescharacteristic
71C2×C4×Q8—21no—
72C2×C4×D4C2311yescharacteristic
73C2×#117—21no—
74C2×#117—21no—
75#477—44no—
76#474—22no—
77#474Q814yescomplemented
78C8×D4—22no—
79C8×D4—22no—
80C2×#107—21no—
81C2×#107—21no—
82C2×#107—21no—
83C2×#107D411yescharacteristic
84C2×#107D411yescharacteristic
85C2×#106—21no—
86C2×#106—21no—
87C2×#106—21no—
88C2×#106—41no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×#103—21no—
96C2×#103—21no—
97C2×#103—21no—
98C2×#103—21no—
99C2×#102—21no—
100C2×#102—21no—
101C2×#102D411yescharacteristic
102C2×#102D411yescharacteristic
103C2×C22⋊C8—21no—
104C2×#97—21no—
105C2×C4×C8C2311yescharacteristic
106C2×C4×C8C2311yescharacteristic
107C2×C4×C8C2311yescharacteristic
108#395—41no—
109#391—21no—
110#390—21no—
111#390—21no—
112#389—21no—
113#389—21no—
114#389—21no—
115#388—41no—
116#388—41no—
117#387—41no—
118#384C2311yescharacteristic
119#384C2311yescharacteristic
120#384C2311yescharacteristic
121#380—21no—
122#380—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4C2311yescharacteristic
127C4×C4⋊C4C2311yescharacteristic
128C4×C4⋊C4C2311yescharacteristic
129#340—21no—
130#340—21no—
131#340—21no—
132#340—21no—
133#340—21no—
134#340—21no—
135#340—21no—
136#340—21no—
137#340—21no—
138#340—21no—
139#340—21no—
140#340—21no—
141#340—41no—
142#340D411yescharacteristic
143#340D411yescharacteristic
144#340D411yescharacteristic
145#340D411yescharacteristic
146#336—21no—
147#336—21no—
148#336—21no—
149#334—22no—
150#332—22no—
151D4⋊C8—22no—
152C22×Q8—21no—
153C22×Q8—41no—
154C22×D4C2×D411yescharacteristic
155C23×C4—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—24no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—81no—
163C22×C8—81no—
164C22×C8C2×D411yescharacteristic
165C22×C8C2×D411yescharacteristic
166C22×C8C2×D411yescharacteristic
167C22×C8C2×D411yescharacteristic
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170C4⋊Q8—21no—
171C4⋊Q8—42no—
172C4⋊Q8—42no—
173C4⋊Q8—81no—
174C4⋊Q8—81no—
175C4⋊Q8C2×Q812yes—
176C4×Q8—42no—
177C4×Q8—81no—
178C4×D4—21no—
179C4×D4—21no—
180C4×D4C2×Q812yes—
181C4×D4#4112yes—
182#117—42no—
183#117—42no—
184#117—81no—
185#117—81no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4C2×D411yescharacteristic
201C2×C4⋊C4C2×D411yescharacteristic
202C2×C4⋊C4C2×D411yescharacteristic
203C2×C4⋊C4C2×D411yescharacteristic
204C2×C4⋊C4C2×D411yescharacteristic
205C2×C4⋊C4C2×D411yescharacteristic
206C2×C4⋊C4C2×D411yescharacteristic
207C2×C22⋊C4—21no—
208C2×C22⋊C4—41no—
209C2×C22⋊C4—41no—
210C2×C22⋊C4—41no—
211C2×C22⋊C4—41no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—41no—
216C2×C42—41no—
217C2×C42—41no—
218C2×C42C2411yescharacteristic
219#107—24no—
220#107—22no—
221#107—41no—
222#107—42no—
223#107—42no—
224#106—42no—
225#106—81no—
226#106—81no—
227#106—82no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—42no—
235C4⋊C8—81no—
236C4⋊C8C2×Q812yes—
237C4⋊C8#4112yes—
238#103—42no—
239#103—42no—
240#103—42no—
241#103—81no—
242#102—24no—
243#102—22no—
244#102—41no—
245#102—81no—
246C22⋊C8—24no—
247#97—42no—
248C4×C8—21no—
249C4×C8—21no—
250C4×C8—21no—
251C4×C8—21no—
252C4×C8—21no—
253C4×C8—21no—
254#95—21no—
255#95—21no—
256#95—21no—
257#95—41no—
258#95—41no—
259#95—41no—
260#95—41no—
261#95—41no—
262C24—21no—
263C2×Q8—22no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×Q8—41no—
268C2×Q8—41no—
269C2×Q8—82no—
270C2×Q8—82no—
271C2×Q8—161no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4—21no—
277C2×D4C22⋊Q812yes—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4C22×D411yescharacteristic
296C22×C4C22×D411yescharacteristic
297C22×C4H(5,𝔽2)11yescharacteristic
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—22no—
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—21no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—84no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—82no—
354C4⋊C4—81no—
355C4⋊C4—81no—
356C4⋊C4—81no—
357C4⋊C4—82no—
358C4⋊C4—82no—
359C4⋊C4C22⋊Q812yes—
360C4⋊C4#13512yes—
361C4⋊C4AGL(1,ℤ/8)12yes—
362C22⋊C4—22no—
363C22⋊C4—22no—
364C22⋊C4—82no—
365C22⋊C4—82no—
366C22⋊C4—161no—
367C22⋊C4—161no—
368C42—41no—
369C42—41no—
370C42—41no—
371C42—41no—
372C42—41no—
373C42—41no—
374C42—82no—
375C42—82no—
376C42—82no—
377C42C22×Q811yescharacteristic
378C42C2×#4111yescharacteristic
379C42C2×#4111yescharacteristic
380C422−1+411yescharacteristic
381C23—22no—
382C23—41no—
383C23—41no—
384C23—41no—
385C23—41no—
386C23—41no—
387C23—41no—
388C23D4211yescharacteristic
389Q8—42no—
390Q8—42no—
391Q8—81no—
392Q8—81no—
393Q8—164no—
394D4—22no—
395D4—22no—
396D4—41no—
397D4—41no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4C2×C22⋊Q811yescharacteristic
438C2×C4#52511yescharacteristic
439C2×C4#53911yescharacteristic
440C2×C4#54711yescharacteristic
441C2×C4D4×Q811yescharacteristic
442C2×C4#55411yescharacteristic
443C2×C4C2×#13511yescharacteristic
444C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
445C2×C4#57511yescharacteristic
446C2×C4#57511yescharacteristic
447C2×C4#57611yescharacteristic
448C2×C4#57711yescharacteristic
449C2×C4#57711yescharacteristic
450C2×C4#57811yescharacteristic
451C8—42no—
452C8—42no—
453C8—41no—
454C8—41no—
455C8—41no—
456C8—41no—
457C8—42no—
458C8—81no—
459C8—81no—
460C22—42no—
461C22—42no—
462C22—42no—
463C22—81no—
464C22—81no—
465C22—81no—
466C22—81no—
467C22#244911yescharacteristic
468C22#326411yescharacteristic
469C22#326711yescharacteristic
470C22#327511yescharacteristic
471C22#328511yescharacteristic
472C22#328911yescharacteristic
473C22#329111yescharacteristic
474C4—21no—
475C4—21no—
476C4—42no—
477C4—42no—
478C4—42no—
479C4—81no—
480C4—81no—
481C4—82no—
482C4—81no—
483C4—81no—
484C4—161no—
485C4—161no—
486C4—161no—
487C4#307611yescharacteristic
488C4#308211yescharacteristic
489C4#311511yescharacteristic
490C4#311811yescharacteristic
491C4#334011yescharacteristic
492C4#334911yescharacteristic
493C4#338411yescharacteristic
494C4#338911yescharacteristic
495C2—82no—
496C2#1983511yescharacteristic
497C2#1993911yescharacteristic
498C2#2002611yescharacteristic
499C2#2024111yescharacteristic
500C2#2633911yescharacteristic
501C2#2638811yescharacteristic
502C2#2640111yescharacteristic
subgroupG/H
center, socleC23D42
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #26466C2semidirect
5. #10681C2semidirect
6. #10655C2semidirect
7. #10598C2semidirect
8. #10574C2semidirect
9. #9622C2semidirect
10. #9540C2semidirect
13. #8923C2semidirect
77. #474Q8semidirect

Maximal subgroups

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

subgroupindex
#264662
#260202
#230662
#227082
#106812
#106552
#105982
#105742
#96222
#95402
#92192
#89842
#89232
#88892
#88452