Finite Group Databasebeta

← #317606

Subgroups — #317606

Order 512.

598 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29288C211yesmaximal, characteristic, complemented
2#28286C211yesmaximal, characteristic
3#22299C211yesmaximal, characteristic, complemented
4#22268C211yesmaximal, characteristic
5#10658C211yesmaximal, characteristic, complemented
6#10657C211yesmaximal, characteristic, complemented
7#10537C211yesmaximal, characteristic
8#10537C211yesmaximal, characteristic
9#9564C211yesmaximal, characteristic
10#9563C211yesmaximal, characteristic
11#9562C211yesmaximal, characteristic, complemented
12#9561C211yesmaximal, characteristic, complemented
13#8787C211yesmaximal, characteristic, complemented
14#8586C211yesmaximal, characteristic, complemented
15#8585C211yesmaximal, characteristic
16C2×#492C2211yescharacteristic
17C2×#478—21no—
18C4×C4⋊Q8C2211yescharacteristic
19C4×#127C2211yescharacteristic
20C4×#125C2211yescharacteristic
21#2088—21no—
22#2087—21no—
23#2067—21no—
24#2065—21no—
25#1989—21no—
26#1988C2211yescharacteristic
27#1988C2211yescharacteristic
28#1982C2211yescharacteristic
29#1982C2211yescharacteristic
30#1957C2211yescharacteristic
31#1956C2211yescharacteristic
32#1949—21no—
33#1948C2211yescharacteristic
34#1948C2211yescharacteristic
35#1947C2211yescharacteristic
36#1947C2211yescharacteristic
37#1939—21no—
38#1939—21no—
39#1939C2211yescharacteristic
40#1939C2211yescharacteristic
41#1916—21no—
42#1916—21no—
43#1880—21no—
44#1880—21no—
45#1877—21no—
46#1877—21no—
47#1842C2211yescharacteristic
48#1842C2211yescharacteristic
49#1842C2211yescharacteristic
50#1834C2211yescharacteristic
51#1804—21no—
52C8×C4⋊C4C2211yescharacteristic
53C8×C4⋊C4C2211yescharacteristic
54#1744C2211yescharacteristic
55C2×#332C2211yescharacteristic
56C2×#332C2211yescharacteristic
57C2×#331C2211yescharacteristic
58C2×#331C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C22×SD16—21no—
68C22×SD16—21no—
69C2×C4⋊Q8C2311yescharacteristic
70C2×#127C2311yescharacteristic
71C2×#125D411yescharacteristic
72C2×C4×Q8—21no—
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75C2×C4×D4—21no—
76#492—22no—
77#492—22no—
78#478—44no—
79C2×#107—21no—
80C2×#107—21no—
81C2×#106—21no—
82C2×#106—21no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×#103—21no—
98C2×#103—21no—
99C2×#102—21no—
100C2×#102—21no—
101C2×#102—21no—
102C2×#102—21no—
103C2×#102—21no—
104C2×#102—21no—
105C2×C4×C8—21no—
106C2×C4×C8—21no—
107C2×C4×C8C2311yescharacteristic
108C2×C4×C8C2311yescharacteristic
109#391—21no—
110#390—21no—
111#390—21no—
112#390—21no—
113#389—21no—
114#389—21no—
115#389D411yescharacteristic
116#389D411yescharacteristic
117#387—21no—
118#387—21no—
119#384C2311yescharacteristic
120#384C2311yescharacteristic
121#382D411yescharacteristic
122C4×C4⋊C4—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4C2311yescharacteristic
127C4×C4⋊C4C2311yescharacteristic
128C43C2311yescharacteristic
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#336—21no—
140#336—21no—
141#336—21no—
142#336—21no—
143#336D411yescharacteristic
144#336D411yescharacteristic
145#336D411yescharacteristic
146#336D411yescharacteristic
147#332—22no—
148#332—22no—
149#331—22no—
150#331—22no—
151C22×Q8—21no—
152C22×D4—21no—
153C22×D4—21no—
154C22×D4—21no—
155C23×C4—41no—
156C2×SD16—24no—
157C2×SD16—24no—
158C2×SD16—81no—
159C2×SD16—81no—
160C2×SD16—81no—
161C2×SD16—81no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8C2×D411yescharacteristic
169C22×C8C2×D411yescharacteristic
170C22×C8C2×D411yescharacteristic
171C22×C8C2×D411yescharacteristic
172C4⋊Q8—21no—
173C4⋊Q8—21no—
174C4⋊Q8—21no—
175C4⋊Q8#4112yes—
176#127—21no—
177#127—21no—
178#127—21no—
179#127#4112yes—
180#125—22no—
181#125SD1614yescomplemented
182C4×Q8—81no—
183C4×Q8—81no—
184C4×D4—42no—
185C4×D4—42no—
186C4×D4—81no—
187C4×D4—81no—
188C4×D4—82no—
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—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4C2×D411yescharacteristic
203C2×C4⋊C4C2×D411yescharacteristic
204C2×C4⋊C4C2×D411yescharacteristic
205C2×C4⋊C4C2×D411yescharacteristic
206C2×C22⋊C4—41no—
207C2×C22⋊C4—41no—
208C2×C22⋊C4—41no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—41no—
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42#4111yescharacteristic
217C2×C42#4111yescharacteristic
218C2×C42C2411yescharacteristic
219#107—42no—
220#107—81no—
221#106—42no—
222#106—81no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—42no—
236C4⋊C8—42no—
237C4⋊C8—81no—
238C4⋊C8—81no—
239C4⋊C8—81no—
240C4⋊C8—81no—
241C4⋊C8—81no—
242C4⋊C8—81no—
243#103—81no—
244#103—81no—
245#102—42no—
246#102—42no—
247#102—81no—
248#102—81no—
249#102—81no—
250#102—81no—
251C4×C8—21no—
252C4×C8—21no—
253C4×C8—21no—
254C4×C8—21no—
255C4×C8—81no—
256C4×C8—81no—
257#95—21no—
258#95—21no—
259#95—21no—
260#95—21no—
261#95—21no—
262#95—21no—
263#95—41no—
264#95—41no—
265C24—41no—
266C2×Q8—22no—
267C2×Q8—41no—
268C2×Q8—41no—
269C2×Q8—41no—
270C2×Q8—41no—
271C2×Q8—41no—
272C2×D4—22no—
273C2×D4—22no—
274C2×D4—22no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—81no—
287C2×D4—81no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—82no—
296C22×C4—82no—
297C22×C4—82no—
298C22×C4#12111yescharacteristic
299C22×C4#12111yescharacteristic
300C22×C4#12111yescharacteristic
301C22×C4#12111yescharacteristic
302C22×C4C22×D411yescharacteristic
303C22×C4C22×D411yescharacteristic
304C22×C4C2×#4111yescharacteristic
305SD16—84no—
306SD16—84no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—41no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—24no—
340C4⋊C4—24no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—81no—
358C4⋊C4—81no—
359C4⋊C4—81no—
360C4⋊C4—81no—
361C4⋊C4—82no—
362C4⋊C4C2×SD1612yes—
363C4⋊C4C2×SD1612yes—
364C22⋊C4—82no—
365C22⋊C4—82no—
366C22⋊C4—82no—
367C42—21no—
368C42—21no—
369C42—21no—
370C42—21no—
371C42—21no—
372C42—21no—
373C42—41no—
374C42—41no—
375C42—41no—
376C42—41no—
377C42—41no—
378C42—41no—
379C42—82no—
380C42C2×SD1612yes—
381C42#13512yes—
382C42C2×#4111yescharacteristic
383C42C2×#4111yescharacteristic
384C422−1+411yescharacteristic
385C422−1+411yescharacteristic
386C23—42no—
387C23—81no—
388C23—81no—
389C23—81no—
390C23—81no—
391C23—81no—
392C23—81no—
393C23C2×#12111yescharacteristic
394Q8—42no—
395Q8—42no—
396Q8—81no—
397Q8—81no—
398D4—42no—
399D4—42no—
400D4—42no—
401D4—42no—
402D4—82no—
403D4—82no—
404D4—81no—
405D4—81no—
406D4—81no—
407D4—81no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—42no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—164no—
442C2×C4#46512yes—
443C2×C4#46512yes—
444C2×C4#52511yescharacteristic
445C2×C4#53611yescharacteristic
446C2×C4#54811yescharacteristic
447C2×C4#54811yescharacteristic
448C2×C4#55211yescharacteristic
449C2×C4#55211yescharacteristic
450C2×C4C22×SD1611yescharacteristic
451C2×C4C2×#13511yescharacteristic
452C2×C4#57611yescharacteristic
453C2×C4#57611yescharacteristic
454C2×C4#57611yescharacteristic
455C2×C4#57711yescharacteristic
456C2×C4#57711yescharacteristic
457C2×C4#57811yescharacteristic
458C8—41no—
459C8—41no—
460C8—41no—
461C8—41no—
462C8—81no—
463C8—81no—
464C8—81no—
465C8—81no—
466C8—81no—
467C8—81no—
468C22—82no—
469C22—82no—
470C22—82no—
471C22—161no—
472C22—161no—
473C22—161no—
474C22—161no—
475C22#253611yescharacteristic
476C22#304511yescharacteristic
477C22#304511yescharacteristic
478C22C2×#46511yescharacteristic
479C22#306111yescharacteristic
480C22#306811yescharacteristic
481C22#306911yescharacteristic
482C4—21no—
483C4—21no—
484C4—22no—
485C4—22no—
486C4—41no—
487C4—41no—
488C4—42no—
489C4—42no—
490C4—82no—
491C4—81no—
492C4—81no—
493C4—161no—
494C4—161no—
495C4#312211yescharacteristic
496C4#312311yescharacteristic
497C4#324111yescharacteristic
498C4#324111yescharacteristic
499C4#335911yescharacteristic
500C4#336111yescharacteristic
501C4#336211yescharacteristic
502C4#336411yescharacteristic
503C2—162no—
504C2#1940811yescharacteristic
505C2#1940911yescharacteristic
506C2#1954211yescharacteristic
507C2#1954311yescharacteristic
508C2#2818611yescharacteristic
509C2#2821411yescharacteristic
510C2#2822111yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #29288C2semidirect
3. #22299C2semidirect
5. #10658C2semidirect
6. #10657C2semidirect
11. #9562C2semidirect
12. #9561C2semidirect
13. #8787C2semidirect
14. #8586C2semidirect
181. #125SD16semidirect

Maximal subgroups

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

subgroupindex
#292882
#282862
#222992
#222682
#106582
#106572
#105372
#95642
#95632
#95622
#95612
#87872
#85862
#85852