Finite Group Databasebeta

← #505555

Subgroups — #505555

Order 512.

517 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28012C211yesmaximal, characteristic, complemented
2#22682C211yesmaximal, characteristic
3#13539C211yesmaximal, characteristic, complemented
4#13539C211yesmaximal, characteristic, complemented
5#13483C211yesmaximal, characteristic, complemented
6#13477C211yesmaximal, characteristic, complemented
7#13150C211yesmaximal, characteristic, complemented
8#13110C211yesmaximal, characteristic, complemented
9#13105C211yesmaximal, characteristic, complemented
10#13014C211yesmaximal, characteristic
11#13014C211yesmaximal, characteristic
12#13014C211yesmaximal, characteristic
13#12937C211yesmaximal, characteristic
14#12813C211yesmaximal, characteristic
15#12637C211yesmaximal, characteristic
16C22×SD32C2211yescharacteristic
17C2×C4×D8C2211yescharacteristic
18C2×C4×D8C2211yescharacteristic
19C2×C4×D8C2211yescharacteristic
20#2514—21no—
21#2349—21no—
22C2×#367C2211yescharacteristic
23C2×C4⋊C16C2211yescharacteristic
24C2×#358C2211yescharacteristic
25C2×#358C2211yescharacteristic
26C2×#358C2211yescharacteristic
27C2×#357C2211yescharacteristic
28C2×#357C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×#346C2211yescharacteristic
31#2074—21no—
32#2066—21no—
33#1969C2211yescharacteristic
34#1969C2211yescharacteristic
35#1969C2211yescharacteristic
36#1942C2211yescharacteristic
37#1942C2211yescharacteristic
38#1942C2211yescharacteristic
39#1927—21no—
40#1864—21no—
41#1864—21no—
42#1846C2211yescharacteristic
43#1846C2211yescharacteristic
44#1804—21no—
45C4×#107C2211yescharacteristic
46C8×C4⋊C4C2211yescharacteristic
47#1380C2211yescharacteristic
48#1380C2211yescharacteristic
49#1380C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1374C2211yescharacteristic
56#1374C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×Q16C2311yescharacteristic
60C22×D8C2311yescharacteristic
61C2×C4×D4—21no—
62C2×C4×D4—21no—
63C2×C4×D4—21no—
64C2×SD32—22no—
65C2×SD32—22no—
66C2×SD32—22no—
67C22×C16C2311yescharacteristic
68C22×C16C2311yescharacteristic
69C22×C16C2311yescharacteristic
70C22×C16C2311yescharacteristic
71C4×D8—22no—
72C4×D8—22no—
73C4×D8—22no—
74C4×D8—22no—
75C4×D8—41no—
76C4×D8—41no—
77C2×#107C2311yescharacteristic
78C2×#107C2311yescharacteristic
79C2×#107C2311yescharacteristic
80C2×#107C2311yescharacteristic
81C2×#107C2311yescharacteristic
82C2×#107C2311yescharacteristic
83C2×#106—21no—
84C2×#106—21no—
85C2×C4⋊C8D411yescharacteristic
86C2×#103—21no—
87C2×#103—21no—
88C2×#103—21no—
89C2×#102—21no—
90C2×#102—21no—
91C2×#102—21no—
92C2×C4×C8C2311yescharacteristic
93C2×C4×C8C2311yescharacteristic
94C2×C4×C8C2311yescharacteristic
95#401—41no—
96#400—41no—
97#391—21no—
98#391—21no—
99#391—21no—
100#389—21no—
101#389—21no—
102#389—21no—
103#387—41no—
104#385—41no—
105#385—41no—
106#383—41no—
107#382—21no—
108#382—21no—
109#380—41no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4D411yescharacteristic
112C2×#95—41no—
113#367—22no—
114C4⋊C16—41no—
115#358—22no—
116#358—41no—
117#358—41no—
118#357—22no—
119#357—22no—
120#357—22no—
121#346—41no—
122#340—21no—
123#340—21no—
124#340—21no—
125#340—21no—
126#336D411yescharacteristic
127#336D411yescharacteristic
128C22×Q8—21no—
129C22×D4—21no—
130C23×C4—41no—
131C23×C4—41no—
132C23×C4—41no—
133C2×Q16—21no—
134C2×Q16—21no—
135C2×Q16—21no—
136C2×Q16—21no—
137C2×Q16—21no—
138C2×Q16—21no—
139C2×D8—21no—
140C2×D8—21no—
141C2×D8—21no—
142C2×D8#4112yes—
143C2×D8#4112yes—
144C2×D8#4112yes—
145C22×C8C2×D411yescharacteristic
146C22×C8C2×D411yescharacteristic
147C22×C8C2×D411yescharacteristic
148C22×C8C2411yescharacteristic
149C4×D4—24no—
150C4×D4—42no—
151C4×D4—42no—
152C4×D4—42no—
153C4×D4—81no—
154C4×D4—81no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4D811yescharacteristic
165C2×C4⋊C4D811yescharacteristic
166C2×C22⋊C4—41no—
167C2×C22⋊C4—41no—
168C2×C22⋊C4—41no—
169C2×C42—41no—
170C2×C42C2×D411yescharacteristic
171C2×C42C2×D411yescharacteristic
172C2×C42C2×D411yescharacteristic
173SD32—44no—
174C2×C16—21no—
175C2×C16—21no—
176C2×C16—21no—
177C2×C16—21no—
178C2×C16—21no—
179C2×C16—21no—
180C2×C16—21no—
181C2×C16—21no—
182C2×C16—21no—
183C2×C16—21no—
184C2×C16—21no—
185C2×C16—21no—
186#107—21no—
187#107—21no—
188#107—21no—
189#107—21no—
190#107—21no—
191#107—21no—
192#107—21no—
193#107—21no—
194#107—21no—
195#107—21no—
196#107—21no—
197#107#4112yes—
198#106—81no—
199#106—81no—
200C4⋊C8—41no—
201#103—42no—
202#103—81no—
203#103—81no—
204#102—24no—
205#102—42no—
206#102—42no—
207C4×C8—21no—
208C4×C8—21no—
209C4×C8—21no—
210C4×C8—21no—
211C4×C8—21no—
212C4×C8—21no—
213#95—41no—
214#95—41no—
215#95—41no—
216#95—41no—
217#95—41no—
218#95—41no—
219#95—41no—
220#95—41no—
221#95—41no—
222#95—41no—
223#95—41no—
224#95—41no—
225#95—41no—
226#95—41no—
227#95D811yescharacteristic
228#95D811yescharacteristic
229C24—41no—
230C2×Q8—41no—
231C2×Q8—41no—
232C2×Q8—41no—
233C2×Q8—41no—
234C2×Q8—41no—
235C2×Q8—41no—
236C2×D4—22no—
237C2×D4—22no—
238C2×D4—22no—
239C2×D4—41no—
240C2×D4—41no—
241C2×D4—41no—
242C22×C4—44no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—82no—
251C22×C4—82no—
252C22×C4—82no—
253C22×C4—82no—
254C22×C4—161no—
255C22×C4—161no—
256C22×C4—161no—
257C22×C4—161no—
258C22×C4C2×D811yescharacteristic
259C22×C4C2×D811yescharacteristic
260C22×C4C2×D811yescharacteristic
261C22×C4C2×D811yescharacteristic
262C22×C4C2×D811yescharacteristic
263C22×C4C2×D811yescharacteristic
264C22×C4C22×D411yescharacteristic
265Q16—41no—
266Q16—41no—
267Q16—41no—
268Q16—41no—
269D8—22no—
270D8—22no—
271D8—22no—
272D8#12614yes—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8C2×#4111yescharacteristic
283C2×C8C2×#4111yescharacteristic
284C2×C8C2×#4111yescharacteristic
285C2×C8C2×#4111yescharacteristic
286C2×C8H(5,𝔽2)11yescharacteristic
287C2×C82−1+411yescharacteristic
288C4⋊C4—22no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—82no—
303C4⋊C4—161no—
304C4⋊C4—161no—
305C22⋊C4—44no—
306C22⋊C4—82no—
307C22⋊C4—82no—
308C42—21no—
309C42—21no—
310C42—21no—
311C42—21no—
312C42—21no—
313C42—21no—
314C42—82no—
315C16—41no—
316C16—41no—
317C16—41no—
318C16—41no—
319C23—42no—
320C23—42no—
321C23—42no—
322C23—81no—
323C23—81no—
324C23—81no—
325C23—81no—
326C23C22×D811yescharacteristic
327Q8—81no—
328Q8—81no—
329Q8—81no—
330Q8—81no—
331D4—24no—
332D4—42no—
333D4—42no—
334D4—42no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—42no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
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—164no—
375C2×C4—164no—
376C2×C4—164no—
377C2×C4#52511yescharacteristic
378C2×C4#54611yescharacteristic
379C2×C4#54811yescharacteristic
380C2×C4#57511yescharacteristic
381C2×C4#57711yescharacteristic
382C2×C4#57711yescharacteristic
383C8—41no—
384C8—41no—
385C8—41no—
386C8C2×#12611yescharacteristic
387C8#53911yescharacteristic
388C8#55211yescharacteristic
389C8#55311yescharacteristic
390C22—44no—
391C22—82no—
392C22—82no—
393C22—82no—
394C22—82no—
395C22—82no—
396C22—82no—
397C22#310011yescharacteristic
398C22#329411yescharacteristic
399C22#335811yescharacteristic
400C22#341411yescharacteristic
401C22#341411yescharacteristic
402C22#341611yescharacteristic
403C22#341611yescharacteristic
404C4—41no—
405C4—41no—
406C4—41no—
407C4—41no—
408C4—41no—
409C4—41no—
410C4—82no—
411C4—161no—
412C4—161no—
413C4—161no—
414C4—161no—
415C4—161no—
416C4—161no—
417C4#248111yescharacteristic
418C4#312511yescharacteristic
419C4#330711yescharacteristic
420C4#337611yescharacteristic
421C2—84no—
422C2#2755011yescharacteristic
423C2#3367111yescharacteristic
424C2#3368311yescharacteristic
425C2#3387811yescharacteristic
426C2#3389011yescharacteristic
427C2#3394311yescharacteristic
428C2#3394311yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D8C23
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ C8×C4⋊C4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #28012C2semidirect
3. #13539C2semidirect
4. #13539C2semidirect
5. #13483C2semidirect
6. #13477C2semidirect
7. #13150C2semidirect
8. #13110C2semidirect
9. #13105C2semidirect

Maximal subgroups

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

subgroupindex
#280122
#226822
#135392
#134832
#134772
#131502
#131102
#131052
#130142
#129372
#128132
#126372