Finite Group Databasebeta

← #6499025

Subgroups — #6499025

Order 512.

1493 conjugacy classes of subgroups.

The subgroup lattice has 455 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35288C212yesmaximal, complemented
2#33291C211yesmaximal, characteristic, complemented
3#33241C211yesmaximal, characteristic, complemented
4#33048C212yesmaximal, complemented
5#30204C212yesmaximal, complemented
6#29876C211yesmaximal, characteristic, complemented
7#29806C211yesmaximal, characteristic
8#29096C218yesmaximal, complemented
9#29000C218yesmaximal, complemented
10#22269C211yesmaximal, characteristic, complemented
11#22247C211yesmaximal, characteristic, complemented
12#21646C212yesmaximal, complemented
13#20472C211yesmaximal, characteristic, complemented
14C2×#500C2211yescharacteristic
15C2×#493C2211yescharacteristic
16C2×#486C2212yes—
17C2×#431C2211yescharacteristic
18C2×C4×M16C2211yescharacteristic
19#2859C2212yes—
20#2858C2212yes—
21#2857C2212yes—
22#2856C2212yes—
23#2644—22no—
24#2624—22no—
25#2513C2214yes—
26#2490C2214yes—
27#2454C2214yes—
28#2448—22no—
29#2447—22no—
30#2427—22no—
31#2426—22no—
32#2420C2214yes—
33#2324—22no—
34#2319—22no—
35C4×C4⋊Q8C2212yes—
36C4×#127C2212yes—
37C4×#117C2211yescharacteristic
38#2086C2214yes—
39#2075C2214yes—
40#2073C2218yes—
41#2054C2214yes—
42#2025C2218yes—
43#1999C2214yes—
44#1991C2218yes—
45#1990C2212yes—
46#1990C2212yes—
47#1970C2218yes—
48#1957C2212yes—
49#1957C2212yes—
50#1936C2218yes—
51#1936C2218yes—
52#1918C2218yes—
53#1895C2218yes—
54#1879C2218yes—
55#1834C2212yes—
56#1834C2211yescharacteristic
57#1834C2211yescharacteristic
58#1771C2214yes—
59#1762C2218yes—
60#1753C2214yes—
61#1744C2211yescharacteristic
62#1744C2211yescharacteristic
63C4×#97C2211yescharacteristic
64C42×C8C2211yescharacteristic
65C22×D8C2314yes—
66C22×M16C2312yes—
67C2×C4⋊Q8C2312yes—
68C2×#127C2312yes—
69C2×#125C2314yes—
70C2×#124—22no—
71C2×#124—22no—
72C2×C22⋊Q8—22no—
73C2×#121—22no—
74C2×#121—22no—
75C2×#121C2314yes—
76C2×C4×Q8—22no—
77C2×C4×D4—22no—
78C2×C4×D4—22no—
79C2×C4×D4C2314yes—
80C2×#117D412yes—
81C22×C42C2311yescharacteristic
82#500—24no—
83#493—24no—
84#486—28no—
85#431—24no—
86C2×#107C2314yes—
87C2×C4⋊C8C2312yes—
88C2×#102C2318yes—
89C2×#102C2318yes—
90C2×#102C2318yes—
91C2×C22⋊C8C2312yes—
92C4×M16—24no—
93C2×#97C2312yes—
94C2×#97C2314yes—
95C2×C4×C8C2311yescharacteristic
96C2×C4×C8C2311yescharacteristic
97C2×C4×C8C2314yes—
98C2×C4×C8C2311yescharacteristic
99C2×C4×C8C2311yescharacteristic
100#399—24no—
101#397—24no—
102#397—24no—
103#397—24no—
104#397—24no—
105#396—24no—
106#394—24no—
107#394—24no—
108#391—22no—
109#391—22no—
110#390—24no—
111#390—22no—
112#390—22no—
113#390—22no—
114#390—22no—
115#390—22no—
116#390—22no—
117#390C2314yes—
118#390C2314yes—
119#390C2314yes—
120#390C2314yes—
121#389—22no—
122#389—22no—
123#389—22no—
124#389—22no—
125#389—22no—
126#389—22no—
127#389C2314yes—
128#389C2314yes—
129#388—24no—
130#388—24no—
131#388—24no—
132#386—24no—
133#385—24no—
134#384C2312yes—
135#384C2312yes—
136#382C2314yes—
137#381—24no—
138#380—22no—
139#380—24no—
140#380C2314yes—
141C4×C4⋊C4—22no—
142C4×C4⋊C4D411yescharacteristic
143C4×C4⋊C4D411yescharacteristic
144C4×C4⋊C4C2314yes—
145C4×C4⋊C4C2312yes—
146C4×C22⋊C4D411yescharacteristic
147C4×C22⋊C4D411yescharacteristic
148C4×C22⋊C4C2312yes—
149#376D412yes—
150C43C2311yescharacteristic
151C43C2311yescharacteristic
152#340C2314yes—
153#340C2318yes—
154#340C2318yes—
155#340C2314yes—
156#340C2318yes—
157#336C2314yes—
158#336C2314yes—
159C22×Q8—22no—
160C22×D4—24no—
161C22×D4—22no—
162C22×D4—22no—
163C22×D4C2414yes—
164C23×C4—24no—
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2411yescharacteristic
168C2×D8—28no—
169C2×D8—48no—
170C2×M16—24no—
171C2×M16—44no—
172C22×C8C2412yes—
173C22×C8C2412yes—
174C22×C8C2412yes—
175C22×C8C2412yes—
176C4⋊Q8—24no—
177C4⋊Q8—24no—
178#127—24no—
179#127—24no—
180#125—48no—
181#124—84no—
182#124—84no—
183C22⋊Q8—84no—
184#121—48no—
185#121—48no—
186#121—84no—
187C4×Q8—84no—
188C4×D4—48no—
189C4×D4—48no—
190C4×D4—84no—
191#117—44no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4C2×D412yes—
202C2×C4⋊C4C2×D412yes—
203C2×C4⋊C4C2414yes—
204C2×C4⋊C4C2414yes—
205C2×C4⋊C4C2414yes—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4C2×D412yes—
218C2×C22⋊C4C2×D412yes—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—24no—
222C2×C42C2×D412yes—
223C2×C42C2×D412yes—
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42C2412yes—
231C2×C42C2412yes—
232C2×C42C2411yescharacteristic
233C2×C42C2411yescharacteristic
234#107—28no—
235C4⋊C8—24no—
236#102—216no—
237#102—48no—
238#102—48no—
239C22⋊C8—24no—
240#97—24no—
241#97—44no—
242C4×C8—22no—
243C4×C8—22no—
244C4×C8—44no—
245C4×C8#4114yes—
246C4×C8#4114yes—
247#95—24no—
248#95—24no—
249#95—24no—
250#95—24no—
251#95—24no—
252#95—24no—
253#95C2×D412yes—
254#95C2×D412yes—
255#95C2×D412yes—
256#95C2×D412yes—
257C24—24no—
258C24C22×D411yescharacteristic
259C2×Q8—44no—
260C2×Q8—44no—
261C2×Q8—44no—
262C2×D4—24no—
263C2×D4—24no—
264C2×D4—28no—
265C2×D4—28no—
266C2×D4—48no—
267C2×D4—44no—
268C2×D4—44no—
269C2×D4—44no—
270C2×D4—44no—
271C2×D4—44no—
272C2×D4—44no—
273C2×D4—84no—
274C2×D4—84no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—24no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—48no—
287C22×C4—84no—
288C22×C4—84no—
289C22×C4C22×D411yescharacteristic
290C22×C4C22×D412yes—
291C22×C4C22×D411yescharacteristic
292C22×C4C22×D412yes—
293C22×C4C22×D412yes—
294C22×C4C22×D411yescharacteristic
295C22×C4C22×D412yes—
296C22×C4C22×D411yescharacteristic
297C22×C4C22×D411yescharacteristic
298C22×C4C2511yescharacteristic
299D8—88no—
300M16—48no—
301C2×C8—24no—
302C2×C8—24no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—24no—
306C2×C8—24no—
307C2×C8C2×#4114yes—
308C2×C8C2×#4114yes—
309C4⋊C4—24no—
310C4⋊C4—24no—
311C4⋊C4—28no—
312C4⋊C4—28no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—44no—
316C4⋊C4—44no—
317C4⋊C4—44no—
318C4⋊C4—44no—
319C4⋊C4—44no—
320C4⋊C4—84no—
321C4⋊C4—84no—
322C4⋊C4—84no—
323C4⋊C4—84no—
324C22⋊C4—42no—
325C22⋊C4—42no—
326C22⋊C4—48no—
327C22⋊C4—48no—
328C22⋊C4—84no—
329C22⋊C4—84no—
330C22⋊C4—84no—
331C22⋊C4—84no—
332C22⋊C4—84no—
333C22⋊C4—84no—
334C22⋊C4—84no—
335C22⋊C4—84no—
336C22⋊C4—84no—
337C42—22no—
338C42—22no—
339C42—24no—
340C42—24no—
341C42—22no—
342C42—22no—
343C42—22no—
344C42—22no—
345C42—21no—
346C42—21no—
347C42—24no—
348C42—24no—
349C42—84no—
350C42C2×#4112yes—
351C42C2×#4112yes—
352C23—21no—
353C23—22no—
354C23—22no—
355C23—21no—
356C23—21no—
357C23—44no—
358C23—44no—
359C23—44no—
360C23—44no—
361C23—44no—
362C23—44no—
363C23—44no—
364C23C23×D411yescharacteristic
365Q8—88no—
366D4—48no—
367D4—48no—
368D4—88no—
369D4—88no—
370D4—168no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—168no—
404C2×C4#53212yes—
405C2×C4#53212yes—
406C2×C4#57514yes—
407C2×C4#57612yes—
408C2×C4#57612yes—
409C2×C4C22×#4111yescharacteristic
410C2×C4C22×#4111yescharacteristic
411C2×C4#58514yes—
412C8—24no—
413C8—24no—
414C8—42no—
415C8—42no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22—42no—
420C22—42no—
421C22—84no—
422C22—84no—
423C22—84no—
424C22—84no—
425C22—84no—
426C22—84no—
427C22—84no—
428C22C2×#53211yescharacteristic
429C22#350012yes—
430C22C2×#57512yes—
431C22C2×#57611yescharacteristic
432C22C2×#57611yescharacteristic
433C4—41no—
434C4—41no—
435C4—41no—
436C4—42no—
437C4—42no—
438C4—42no—
439C4—42no—
440C4—84no—
441C4—84no—
442C4—84no—
443C4#317614yes—
444C4#318314yes—
445C4#345512yes—
446C4#351012yes—
447C2—81no—
448C2—164no—
449C2#3533111yescharacteristic
450C2C2×#317611yescharacteristic
451C2C2×#318311yescharacteristic
452C2#6254612yes—
453C2#6254612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#29806C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35288C2semidirect
2. #33291C2semidirect
3. #33241C2semidirect
4. #33048C2semidirect
5. #30204C2semidirect
6. #29876C2semidirect
8. #29096C2semidirect
9. #29000C2semidirect
10. #22269C2semidirect
11. #22247C2semidirect
12. #21646C2semidirect
13. #20472C2semidirect

Maximal subgroups

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

subgroupindex
#352882
#332912
#332412
#330482
#302042
#298762
#298062
#290962
#290002
#222692
#222472
#216462
#204722