Finite Group Databasebeta

← #323815

Subgroups — #323815

Order 512.

720 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30304C211yesmaximal, characteristic, complemented
2#29218C211yesmaximal, characteristic, complemented
3#29177C211yesmaximal, characteristic, complemented
4#28514C211yesmaximal, characteristic
5#28508C211yesmaximal, characteristic
6#22268C211yesmaximal, characteristic
7#10680C211yesmaximal, characteristic, complemented
8#10678C211yesmaximal, characteristic
9#10674C211yesmaximal, characteristic, complemented
10#10672C211yesmaximal, characteristic
11#9842C211yesmaximal, characteristic
12#9829C211yesmaximal, characteristic
13#8821C211yesmaximal, characteristic, complemented
14#8820C211yesmaximal, characteristic, complemented
15#8265C211yesmaximal, characteristic, complemented
16C2×#492C2211yescharacteristic
17C2×#478—22no—
18C2×#474—22no—
19#2868C2211yescharacteristic
20#2859C2211yescharacteristic
21C4×C4⋊Q8C2211yescharacteristic
22C4×#127C2211yescharacteristic
23#2100—22no—
24#2093—22no—
25#2092—22no—
26#2088—22no—
27#2083—22no—
28#2057—22no—
29#1990C2211yescharacteristic
30#1990C2211yescharacteristic
31#1990C2211yescharacteristic
32#1990C2211yescharacteristic
33#1987—22no—
34#1957C2211yescharacteristic
35#1957C2211yescharacteristic
36#1957C2211yescharacteristic
37#1957C2211yescharacteristic
38#1957C2211yescharacteristic
39#1956C2211yescharacteristic
40#1919—22no—
41#1880—22no—
42#1877—22no—
43#1845C2211yescharacteristic
44#1845C2211yescharacteristic
45#1845C2211yescharacteristic
46#1845C2211yescharacteristic
47#1845C2211yescharacteristic
48#1844C2211yescharacteristic
49#1844C2211yescharacteristic
50#1834C2211yescharacteristic
51#1834C2211yescharacteristic
52#1834C2211yescharacteristic
53#1834C2211yescharacteristic
54#1834C2211yescharacteristic
55#1834C2211yescharacteristic
56C4×C4⋊C8C2211yescharacteristic
57C4×C4⋊C8C2211yescharacteristic
58#1744C2211yescharacteristic
59C2×#322C2211yescharacteristic
60C2×#322C2211yescharacteristic
61#1273C2211yescharacteristic
62#1273C2211yescharacteristic
63C22×SD16—22no—
64C2×C4⋊Q8—21no—
65C2×C4⋊Q8—21no—
66C2×C4⋊Q8—21no—
67C2×C4⋊Q8C2311yescharacteristic
68C2×#127C2311yescharacteristic
69C2×#125—21no—
70C2×C4×Q8—21no—
71C2×C4×D4—21no—
72C2×C4×D4—21no—
73C2×C4×D4—21no—
74#492—22no—
75#492Q814yescomplemented
76#478—48no—
77#474—48no—
78C2×#107—22no—
79C2×#107—22no—
80C2×#107—22no—
81C2×#107—22no—
82C2×#107—22no—
83C2×#106—22no—
84C2×#106—22no—
85C2×C4⋊C8D412yes—
86C2×C4⋊C8D412yes—
87C2×#103—22no—
88C2×#102—22no—
89C2×#102—22no—
90C2×#102—22no—
91C2×#102—22no—
92C2×#102—22no—
93C2×C4×C8C2311yescharacteristic
94C2×C4×C8C2311yescharacteristic
95C2×C4×C8C2311yescharacteristic
96C2×C4×C8C2311yescharacteristic
97C2×C4×C8C2311yescharacteristic
98C2×C4×C8C2311yescharacteristic
99#391—21no—
100#390—21no—
101#390—21no—
102#390—21no—
103#389—21no—
104#389—21no—
105#389—21no—
106#389—21no—
107#389—21no—
108#389—21no—
109#384—21no—
110#384—21no—
111#384—21no—
112#384C2311yescharacteristic
113#384C2311yescharacteristic
114#384C2311yescharacteristic
115#384C2311yescharacteristic
116#384C2311yescharacteristic
117#384C2311yescharacteristic
118#382—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—21no—
121C43C2311yescharacteristic
122#340—22no—
123#340—22no—
124#340—22no—
125#340—22no—
126#340—22no—
127#340—22no—
128#340—22no—
129#340—22no—
130#340—22no—
131#340—22no—
132#336D412yes—
133#336D412yes—
134#336D412yes—
135#336D412yes—
136#322—22no—
137#322—22no—
138C22×Q8—21no—
139C22×Q8—41no—
140C22×D4—21no—
141C22×D4—21no—
142C22×D4—21no—
143C23×C4—41no—
144C2×SD16—28no—
145C2×SD16—82no—
146C2×SD16—82no—
147C22×C8C2×D412yes—
148C22×C8C2×D412yes—
149C22×C8C2×D412yes—
150C22×C8C2×D412yes—
151C22×C8C2×D412yes—
152C22×C8C2×D412yes—
153C4⋊Q8—21no—
154C4⋊Q8—21no—
155C4⋊Q8—21no—
156C4⋊Q8—44no—
157C4⋊Q8—44no—
158C4⋊Q8—82no—
159C4⋊Q8C2×Q812yes—
160#127—21no—
161#127—21no—
162#127—21no—
163#127C2×Q812yes—
164#125—44no—
165C4×Q8—82no—
166C4×D4—44no—
167C4×D4—44no—
168C4×D4—82no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
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—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—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—41no—
195C2×C22⋊C4—41no—
196C2×C22⋊C4—41no—
197C2×C22⋊C4—41no—
198C2×C42—41no—
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2411yescharacteristic
206#107—44no—
207#107—44no—
208#107—44no—
209#107—82no—
210#107—82no—
211#106—44no—
212#106—44no—
213C4⋊C8—24no—
214C4⋊C8—24no—
215#103—82no—
216#102—44no—
217#102—44no—
218#102—44no—
219#102—44no—
220#102—82no—
221C4×C8—21no—
222C4×C8—21no—
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×C8C2×Q812yes—
232C4×C8#4112yes—
233#95—41no—
234#95—41no—
235#95—41no—
236#95—41no—
237C24—41no—
238C2×Q8—22no—
239C2×Q8—41no—
240C2×Q8—42no—
241C2×Q8—42no—
242C2×Q8—82no—
243C2×Q8—82no—
244C2×Q8—82no—
245C2×D4—22no—
246C2×D4—22no—
247C2×D4—22no—
248C2×D4—41no—
249C2×D4—41no—
250C2×D4—41no—
251C2×D4—42no—
252C2×D4—42no—
253C2×D4—42no—
254C2×D4—42no—
255C2×D4—42no—
256C2×D4—42no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—82no—
265C22×C4—82no—
266C22×C4—82no—
267C22×C4#12711yescharacteristic
268C22×C4#12711yescharacteristic
269C22×C4#12711yescharacteristic
270C22×C4#12711yescharacteristic
271C22×C4C22×D411yescharacteristic
272C22×C4C22×D411yescharacteristic
273C22×C4C22×D411yescharacteristic
274SD16—88no—
275C2×C8—22no—
276C2×C8—22no—
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—22no—
280C2×C8—22no—
281C2×C8—22no—
282C2×C8—22no—
283C2×C8—22no—
284C2×C8—22no—
285C2×C8—22no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—22no—
289C2×C8—22no—
290C2×C8—22no—
291C2×C8C22⋊Q814yes—
292C2×C8AGL(1,ℤ/8)14yes—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—82no—
316C4⋊C4—82no—
317C4⋊C4—82no—
318C4⋊C4—82no—
319C4⋊C4—82no—
320C4⋊C4—82no—
321C22⋊C4—82no—
322C22⋊C4—82no—
323C22⋊C4—82no—
324C42—22no—
325C42—22no—
326C42—22no—
327C42—22no—
328C42—22no—
329C42—82no—
330C42AGL(1,ℤ/8)14yes—
331C42C22×Q811yescharacteristic
332C42C2×#4111yescharacteristic
333C422−1+411yescharacteristic
334C422−1+411yescharacteristic
335C23—42no—
336C23—81no—
337C23—81no—
338C23—81no—
339C23—81no—
340C23—81no—
341C23—81no—
342C23C2×#12711yescharacteristic
343Q8—44no—
344Q8—82no—
345Q8—164no—
346D4—44no—
347D4—44no—
348D4—44no—
349D4—82no—
350D4—82no—
351D4—82no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—42no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—164no—
384C2×C4#49612yes—
385C2×C4#49612yes—
386C2×C4C2×C22⋊Q811yescharacteristic
387C2×C4#53611yescharacteristic
388C2×C4#54711yescharacteristic
389C2×C4#54711yescharacteristic
390C2×C4D4×Q811yescharacteristic
391C2×C4D4×Q811yescharacteristic
392C2×C4C2×AGL(1,ℤ/8)12yes—
393C2×C4C2×AGL(1,ℤ/8)12yes—
394C2×C4#57612yes—
395C2×C4#57712yes—
396C2×C4#57812yes—
397C2×C4#57812yes—
398C8—24no—
399C8—24no—
400C8—42no—
401C8—42no—
402C8—42no—
403C8—42no—
404C22—82no—
405C22—82no—
406C22—82no—
407C22—161no—
408C22—161no—
409C22—161no—
410C22—161no—
411C22#254211yescharacteristic
412C22C2×#49611yescharacteristic
413C22#315111yescharacteristic
414C22#315411yescharacteristic
415C22#315511yescharacteristic
416C22#315611yescharacteristic
417C22#315611yescharacteristic
418C4—22no—
419C4—22no—
420C4—41no—
421C4—41no—
422C4—82no—
423C4—161no—
424C4—161no—
425C4—161no—
426C4—161no—
427C4—161no—
428C4—161no—
429C4#308112yes—
430C4#324112yes—
431C4#334312yes—
432C4#335312yes—
433C4#338612yes—
434C4#338612yes—
435C2—162no—
436C2#1967611yescharacteristic
437C2#1970011yescharacteristic
438C2#1972511yescharacteristic
439C2#1974411yescharacteristic
440C2#2852511yescharacteristic
441C2#2854011yescharacteristic
442C2#2854511yescharacteristic
subgroupG/H
center, socleC23C2×#127
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. #30304C2semidirect
2. #29218C2semidirect
3. #29177C2semidirect
7. #10680C2semidirect
9. #10674C2semidirect
13. #8821C2semidirect
14. #8820C2semidirect
15. #8265C2semidirect
75. #492Q8semidirect

Maximal subgroups

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

subgroupindex
#303042
#292182
#291772
#285142
#285082
#222682
#106802
#106782
#106742
#106722
#98422
#98292
#88212
#88202
#82652