Finite Group Databasebeta

← #339367

Subgroups — #339367

Order 512.

541 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28799C211yesmaximal, characteristic, complemented
2#27728C211yesmaximal, characteristic
3#26300C211yesmaximal, characteristic
4#26040C211yesmaximal, characteristic
5#10679C211yesmaximal, characteristic, complemented
6#10621C211yesmaximal, characteristic
7#10593C211yesmaximal, characteristic, complemented
8#10578C211yesmaximal, characteristic, complemented
9#10412C211yesmaximal, characteristic
10#10349C211yesmaximal, characteristic, complemented
11#9615C211yesmaximal, characteristic, complemented
12#9576C211yesmaximal, characteristic, complemented
13#9545C211yesmaximal, characteristic
14#9535C211yesmaximal, characteristic, complemented
15#8890C211yesmaximal, characteristic
16C2×#474—21no—
17#2571C2211yescharacteristic
18#2563C2211yescharacteristic
19#2551C2211yescharacteristic
20D4×C4⋊C4C2211yescharacteristic
21#2101—21no—
22#2086—21no—
23#2085—21no—
24#2077—21no—
25#2077—21no—
26#2073—21no—
27#2058—21no—
28#2057—21no—
29#1993—21no—
30#1990C2211yescharacteristic
31#1990C2211yescharacteristic
32#1987C2211yescharacteristic
33#1987C2211yescharacteristic
34#1956C2211yescharacteristic
35#1954C2211yescharacteristic
36#1948C2211yescharacteristic
37#1948C2211yescharacteristic
38#1947C2211yescharacteristic
39#1947C2211yescharacteristic
40#1943—21no—
41#1939C2211yescharacteristic
42#1934—21no—
43#1862—21no—
44#1853—21no—
45#1845C2211yescharacteristic
46#1835—21no—
47#1834C2211yescharacteristic
48#1834C2211yescharacteristic
49#1823—21no—
50#1802C2211yescharacteristic
51#1802C2211yescharacteristic
52C4×#106C2211yescharacteristic
53#1770C2211yescharacteristic
54C8×C4⋊C4C2211yescharacteristic
55C4×#102C2211yescharacteristic
56C2×#335C2211yescharacteristic
57C2×#333C2211yescharacteristic
58C2×D4⋊C8C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1279C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1277C2211yescharacteristic
66#1273C2211yescharacteristic
67C2×C4⋊Q8—21no—
68C2×C4⋊Q8—21no—
69C2×C4⋊Q8C2311yescharacteristic
70C2×C4×D4—21no—
71C2×C4×D4C2311yescharacteristic
72C22×C4⋊C4—21no—
73#474—44no—
74C2×#107—21no—
75C2×#107—21no—
76C2×#107—21no—
77C2×#107—21no—
78C2×#106—21no—
79C2×#106—21no—
80C2×#106D411yescharacteristic
81C2×#106D411yescharacteristic
82C2×C4⋊C8—21no—
83C2×C4⋊C8C2311yescharacteristic
84C2×C4⋊C8C2311yescharacteristic
85C2×C4⋊C8C2311yescharacteristic
86C2×C4⋊C8C2311yescharacteristic
87C2×#103—41no—
88C2×#102—21no—
89C2×#102—21no—
90C2×#102D411yescharacteristic
91C2×#102D411yescharacteristic
92C2×C22⋊C8—21no—
93C2×C22⋊C8—21no—
94C2×#97—21no—
95C2×C4×C8—21no—
96C2×C4×C8C2311yescharacteristic
97C2×C4×C8C2311yescharacteristic
98C2×C4×C8C2311yescharacteristic
99#400—41no—
100#395—41no—
101#393—41no—
102#390—21no—
103#389—21no—
104#389—21no—
105#389—21no—
106#389—21no—
107#389—21no—
108#385—41no—
109#384C2311yescharacteristic
110#384C2311yescharacteristic
111#384C2311yescharacteristic
112#382—21no—
113#382—21no—
114#382—21no—
115#380—21no—
116#380—21no—
117#380—21no—
118C4×C4⋊C4C2311yescharacteristic
119C4×C4⋊C4C2311yescharacteristic
120C4×C4⋊C4C2311yescharacteristic
121#340—21no—
122#340—21no—
123#340—21no—
124#340—21no—
125#340—21no—
126#340—21no—
127#340—21no—
128#340—21no—
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—41no—
138#340D411yescharacteristic
139#340D411yescharacteristic
140#340D411yescharacteristic
141#340D411yescharacteristic
142#336—21no—
143#336—21no—
144#336—21no—
145#335—22no—
146#333—22no—
147D4⋊C8—22no—
148C22×Q8—21no—
149C22×Q8—41no—
150C22×D4C2×D411yescharacteristic
151C23×C4—21no—
152C23×C4—41no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8C2×D411yescharacteristic
159C22×C8C2×D411yescharacteristic
160C22×C8C2×D411yescharacteristic
161C22×C8C2×D411yescharacteristic
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164C4⋊Q8—21no—
165C4⋊Q8—21no—
166C4⋊Q8—42no—
167C4⋊Q8—42no—
168C4⋊Q8—81no—
169C4⋊Q8—81no—
170C4×D4—21no—
171C4×D4—21no—
172C4×D4—21no—
173C4×D4—21no—
174C4×D4—42no—
175C4×D4—81no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
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—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—81no—
193C2×C4⋊C4C2×D411yescharacteristic
194C2×C4⋊C4C2×D411yescharacteristic
195C2×C4⋊C4C2×D411yescharacteristic
196C2×C4⋊C4C2×D411yescharacteristic
197C2×C4⋊C4C2×D411yescharacteristic
198C2×C4⋊C4C2×D411yescharacteristic
199C2×C4⋊C4C2×D411yescharacteristic
200C2×C22⋊C4—21no—
201C2×C22⋊C4—41no—
202C2×C22⋊C4—41no—
203C2×C22⋊C4—41no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42C2411yescharacteristic
208#107—42no—
209#107—42no—
210#107—42no—
211#107—81no—
212#106—22no—
213#106—41no—
214#106—81no—
215#106—81no—
216C4⋊C8—21no—
217C4⋊C8—21no—
218C4⋊C8—21no—
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8—21no—
222C4⋊C8—21no—
223C4⋊C8—42no—
224C4⋊C8#4112yes—
225#103—82no—
226#102—22no—
227#102—22no—
228#102—42no—
229#102—42no—
230C22⋊C8—42no—
231C22⋊C8—81no—
232#97—81no—
233C4×C8—21no—
234C4×C8—21no—
235C4×C8—21no—
236C4×C8—21no—
237C4×C8—21no—
238C4×C8—21no—
239C4×C8—42no—
240#95—21no—
241#95—21no—
242#95—21no—
243#95—41no—
244#95—41no—
245#95—41no—
246#95—41no—
247#95—41no—
248#95—41no—
249#95—41no—
250C24—21no—
251C2×Q8—41no—
252C2×Q8—41no—
253C2×Q8—82no—
254C2×Q8—81no—
255C2×Q8—81no—
256C2×Q8—82no—
257C2×Q8—161no—
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—21no—
261C2×D4—21no—
262C2×D4—21no—
263C2×D4AGL(1,ℤ/8)12yes—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—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×C4C22×D411yescharacteristic
285C22×C4C22×D411yescharacteristic
286C22×C4H(5,𝔽2)11yescharacteristic
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—21no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—22no—
293C2×C8—21no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8#13712yes—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—82no—
344C4⋊C4—82no—
345C4⋊C4—81no—
346C4⋊C4—81no—
347C4⋊C4—81no—
348C4⋊C4—82no—
349C4⋊C4—161no—
350C4⋊C4#13512yes—
351C22⋊C4—41no—
352C22⋊C4—41no—
353C22⋊C4—82no—
354C22⋊C4—161no—
355C22⋊C4—161no—
356C42—41no—
357C42—41no—
358C42—41no—
359C42—41no—
360C42—41no—
361C42—41no—
362C42C2×#4111yescharacteristic
363C422−1+411yescharacteristic
364C422−1+411yescharacteristic
365C422−1+411yescharacteristic
366C23—22no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23—41no—
371C23—41no—
372C23—41no—
373C23D4211yescharacteristic
374Q8—82no—
375Q8—161no—
376Q8—164no—
377D4—22no—
378D4—22no—
379D4—41no—
380D4—41no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—22no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
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#53611yescharacteristic
422C2×C4#53611yescharacteristic
423C2×C4#54211yescharacteristic
424C2×C4#54711yescharacteristic
425C2×C4#54811yescharacteristic
426C2×C4#56311yescharacteristic
427C2×C4C2×#13511yescharacteristic
428C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
429C2×C4C2×#13711yescharacteristic
430C2×C4#57511yescharacteristic
431C2×C4#57611yescharacteristic
432C2×C4#57711yescharacteristic
433C2×C4#57811yescharacteristic
434C2×C4#57811yescharacteristic
435C8—22no—
436C8—41no—
437C8—42no—
438C8—41no—
439C8—41no—
440C8—81no—
441C8—81no—
442C8—81no—
443C8—81no—
444C22—42no—
445C22—42no—
446C22—42no—
447C22—81no—
448C22—81no—
449C22—81no—
450C22—81no—
451C22#267411yescharacteristic
452C22#326511yescharacteristic
453C22#326911yescharacteristic
454C22#327611yescharacteristic
455C22#328011yescharacteristic
456C22#328811yescharacteristic
457C22#329011yescharacteristic
458C4—21no—
459C4—21no—
460C4—42no—
461C4—81no—
462C4—81no—
463C4—81no—
464C4—81no—
465C4—81no—
466C4—81no—
467C4—161no—
468C4—161no—
469C4—161no—
470C4—161no—
471C4#323311yescharacteristic
472C4#323811yescharacteristic
473C4#324211yescharacteristic
474C4#324311yescharacteristic
475C4#334311yescharacteristic
476C4#334411yescharacteristic
477C4#336411yescharacteristic
478C4#337011yescharacteristic
479C2—82no—
480C2#1977311yescharacteristic
481C2#1990511yescharacteristic
482C2#2008311yescharacteristic
483C2#2022911yescharacteristic
484C2#3080111yescharacteristic
485C2#3080911yescharacteristic
486C2#3083911yescharacteristic
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. #28799C2semidirect
5. #10679C2semidirect
7. #10593C2semidirect
8. #10578C2semidirect
10. #10349C2semidirect
11. #9615C2semidirect
12. #9576C2semidirect
14. #9535C2semidirect

Maximal subgroups

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

subgroupindex
#287992
#277282
#263002
#260402
#106792
#106212
#105932
#105782
#104122
#103492
#96152
#95762
#95452
#95352
#88902