Finite Group Databasebeta

← #510671

Subgroups — #510671

Order 512.

466 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28013C211yesmaximal, characteristic
2#26485C211yesmaximal, characteristic, complemented
3#13539C211yesmaximal, characteristic, complemented
4#13514C211yesmaximal, characteristic, complemented
5#13474C211yesmaximal, characteristic
6#13449C211yesmaximal, characteristic
7#13449C211yesmaximal, characteristic
8#13363C211yesmaximal, characteristic
9#13276C211yesmaximal, characteristic, complemented
10#13160C211yesmaximal, characteristic, complemented
11#13150C211yesmaximal, characteristic, complemented
12#13149C211yesmaximal, characteristic
13#13133C211yesmaximal, characteristic, complemented
14#13013C211yesmaximal, characteristic
15#12814C211yesmaximal, characteristic, complemented
16C22×SD32C2211yescharacteristic
17C2×#500C2211yescharacteristic
18C2×C4×Q16C2211yescharacteristic
19#2513—21no—
20#2451—21no—
21C2×#366C2211yescharacteristic
22C2×C4⋊C16C2211yescharacteristic
23C2×C4⋊C16C2211yescharacteristic
24C2×C4⋊C16C2211yescharacteristic
25C2×#358C2211yescharacteristic
26C2×#358C2211yescharacteristic
27C2×#358C2211yescharacteristic
28C2×#357C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×#357C2211yescharacteristic
31C2×#346C2211yescharacteristic
32#2073—21no—
33#2068—21no—
34#2065—21no—
35#2024—21no—
36#1970C2211yescharacteristic
37#1970C2211yescharacteristic
38#1970C2211yescharacteristic
39#1969C2211yescharacteristic
40#1969C2211yescharacteristic
41#1942C2211yescharacteristic
42#1942C2211yescharacteristic
43#1942C2211yescharacteristic
44#1926—21no—
45#1863—21no—
46#1846C2211yescharacteristic
47C4×#107C2211yescharacteristic
48C8×C4⋊C4C2211yescharacteristic
49#1380C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×Q16C2311yescharacteristic
60C22×D8C2311yescharacteristic
61C2×C4⋊Q8—21no—
62C2×C4×Q8—21no—
63C2×SD32—22no—
64C2×SD32—41no—
65C2×SD32—41no—
66C22×C16C2311yescharacteristic
67C22×C16C2311yescharacteristic
68C22×C16C2311yescharacteristic
69C22×C16C2311yescharacteristic
70#500—22no—
71#500—41no—
72C4×Q16—22no—
73C4×Q16—41no—
74C2×#107—21no—
75C2×#107C2311yescharacteristic
76C2×#107C2311yescharacteristic
77C2×#107C2311yescharacteristic
78C2×#107C2311yescharacteristic
79C2×#107C2311yescharacteristic
80C2×#107C2311yescharacteristic
81C2×#106—21no—
82C2×C4⋊C8D411yescharacteristic
83C2×#103—21no—
84C2×#103—21no—
85C2×#103—21no—
86C2×#102—21no—
87C2×#102—21no—
88C2×#102—21no—
89C2×C4×C8C2311yescharacteristic
90C2×C4×C8C2311yescharacteristic
91C2×C4×C8C2311yescharacteristic
92#398—41no—
93#397—41no—
94#397—41no—
95#395—41no—
96#391—21no—
97#391—21no—
98#391—41no—
99#390—21no—
100#390—21no—
101#390—21no—
102#389—21no—
103#389—21no—
104#389—21no—
105#388—41no—
106#388—41no—
107#387—41no—
108#382—21no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4D411yescharacteristic
111#366—41no—
112C4⋊C16—41no—
113C4⋊C16—41no—
114C4⋊C16—41no—
115#358—22no—
116#358—22no—
117#358—41no—
118#357—22no—
119#357—41no—
120#357—41no—
121#346—41no—
122#340—21no—
123#340—21no—
124#340—21no—
125#340—21no—
126#336D411yescharacteristic
127#336D411yescharacteristic
128C22×Q8—21no—
129C22×Q8—41no—
130C22×D4—21no—
131C2×Q16—21no—
132C2×Q16—21no—
133C2×Q16—21no—
134C2×Q16—21no—
135C2×Q16—21no—
136C2×Q16#4112yes—
137C2×D8—21no—
138C2×D8—21no—
139C2×D8—21no—
140C2×D8—21no—
141C2×D8—21no—
142C2×D8—21no—
143C22×C8C2×D411yescharacteristic
144C22×C8C2×D411yescharacteristic
145C22×C8C2×D411yescharacteristic
146C22×C8C2411yescharacteristic
147C4⋊Q8—42no—
148C4⋊Q8—81no—
149C4×Q8—42no—
150C4×Q8—81no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4D811yescharacteristic
163C2×C4⋊C4D811yescharacteristic
164C2×C22⋊C4—41no—
165C2×C22⋊C4—41no—
166C2×C22⋊C4—41no—
167C2×C22⋊C4—41no—
168C2×C22⋊C4—41no—
169C2×C22⋊C4—41no—
170C2×C42—41no—
171C2×C42—41no—
172C2×C42C2×D411yescharacteristic
173C2×C42C2×D411yescharacteristic
174C2×C42C2×D411yescharacteristic
175SD32—44no—
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—
186C2×C16—21no—
187C2×C16—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—21no—
198#107—21no—
199#107—42no—
200#107#4112yes—
201#106—81no—
202C4⋊C8—41no—
203#103—42no—
204#103—42no—
205#103—81no—
206#102—42no—
207#102—81no—
208#102—81no—
209C4×C8—21no—
210C4×C8—21no—
211C4×C8—21no—
212C4×C8—21no—
213C4×C8—21no—
214C4×C8—21no—
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#95D811yescharacteristic
226#95D811yescharacteristic
227C24—41no—
228C2×Q8—22no—
229C2×Q8—41no—
230C2×Q8—41no—
231C2×Q8—41no—
232C2×Q8—41no—
233C2×Q8—41no—
234C2×Q8—82no—
235C2×Q8—82no—
236C2×Q8—161no—
237C2×D4—41no—
238C2×D4—41no—
239C2×D4—41no—
240C2×D4—41no—
241C2×D4—41no—
242C2×D4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4C2×D811yescharacteristic
251C22×C4C2×D811yescharacteristic
252C22×C4C2×D811yescharacteristic
253C22×C4C2×D811yescharacteristic
254C22×C4C2×D811yescharacteristic
255C22×C4C2×D811yescharacteristic
256C22×C4C22×D411yescharacteristic
257Q16—22no—
258Q16—22no—
259Q16—41no—
260Q16—41no—
261D8—41no—
262D8—41no—
263D8—41no—
264D8—41no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—21no—
272C2×C8—21no—
273C2×C8—21no—
274C2×C8C2×#4111yescharacteristic
275C2×C8C2×#4111yescharacteristic
276C2×C8H(5,𝔽2)11yescharacteristic
277C2×C8H(5,𝔽2)11yescharacteristic
278C2×C82−1+411yescharacteristic
279C2×C82−1+411yescharacteristic
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—82no—
295C4⋊C4—82no—
296C4⋊C4—161no—
297C4⋊C4—161no—
298C4⋊C4—161no—
299C22⋊C4—82no—
300C22⋊C4—82no—
301C22⋊C4—161no—
302C22⋊C4—161no—
303C22⋊C4—161no—
304C22⋊C4—161no—
305C42—21no—
306C42—21no—
307C42—21no—
308C42—21no—
309C42—21no—
310C42—21no—
311C42—82no—
312C42—161no—
313C16—41no—
314C16—41no—
315C16—41no—
316C16—41no—
317C23—81no—
318C23—81no—
319C23—81no—
320C23—81no—
321C23—81no—
322C23—81no—
323C23—81no—
324C23C22×D811yescharacteristic
325Q8—42no—
326Q8—42no—
327Q8—81no—
328Q8—81no—
329Q8—164no—
330D4—81no—
331D4—81no—
332D4—81no—
333D4—81no—
334C2×C4—21no—
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—42no—
352C2×C4—42no—
353C2×C4—81no—
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#51012yes—
373C2×C4#53711yescharacteristic
374C2×C4#54611yescharacteristic
375C2×C4#54811yescharacteristic
376C2×C4#57511yescharacteristic
377C2×C4#57711yescharacteristic
378C2×C4#57711yescharacteristic
379C8—41no—
380C8—41no—
381C8—41no—
382C8#53811yescharacteristic
383C8#54211yescharacteristic
384C8#55211yescharacteristic
385C8#56311yescharacteristic
386C22—161no—
387C22—161no—
388C22—161no—
389C22—161no—
390C22—161no—
391C22—161no—
392C22—161no—
393C22#324511yescharacteristic
394C22#329411yescharacteristic
395C22#335811yescharacteristic
396C22C2×#51011yescharacteristic
397C22#341411yescharacteristic
398C22#341511yescharacteristic
399C22#341611yescharacteristic
400C4—22no—
401C4—41no—
402C4—41no—
403C4—41no—
404C4—41no—
405C4—41no—
406C4—82no—
407C4—82no—
408C4—161no—
409C4—161no—
410C4—161no—
411C4—161no—
412C4—161no—
413C4#270211yescharacteristic
414C4#326311yescharacteristic
415C4#332611yescharacteristic
416C4#337611yescharacteristic
417C2—321no—
418C2#3180211yescharacteristic
419C2#3381811yescharacteristic
420C2#3382311yescharacteristic
421C2#3388611yescharacteristic
422C2#3388711yescharacteristic
423C2#3393811yescharacteristic
424C2#3394211yescharacteristic
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
2. #26485C2semidirect
3. #13539C2semidirect
4. #13514C2semidirect
9. #13276C2semidirect
10. #13160C2semidirect
11. #13150C2semidirect
13. #13133C2semidirect
15. #12814C2semidirect

Maximal subgroups

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

subgroupindex
#280132
#264852
#135392
#135142
#134742
#134492
#133632
#132762
#131602
#131502
#131492
#131332
#130132
#128142