Finite Group Databasebeta

← #507807

Subgroups — #507807

Order 512.

495 conjugacy classes of subgroups.

The subgroup lattice has 437 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33272C211yesmaximal, characteristic, complemented
2#21282C211yesmaximal, characteristic, complemented
3#13514C211yesmaximal, characteristic, complemented
4#13476C211yesmaximal, characteristic, complemented
5#13476C211yesmaximal, characteristic, complemented
6#13448C211yesmaximal, characteristic
7#13235C211yesmaximal, characteristic
8#13166C211yesmaximal, characteristic, complemented
9#13149C211yesmaximal, characteristic, complemented
10#13149C211yesmaximal, characteristic, complemented
11#13131C211yesmaximal, characteristic, complemented
12#13109C211yesmaximal, characteristic, complemented
13#13012C211yesmaximal, characteristic
14#12937C211yesmaximal, characteristic, complemented
15#12654C211yesmaximal, characteristic, complemented
16C22×D16C2211yescharacteristic
17C2×C4×D8C2211yescharacteristic
18#2644—21no—
19#2319—21no—
20C2×#367C2211yescharacteristic
21C2×#367C2211yescharacteristic
22C2×#366C2211yescharacteristic
23C2×C4⋊C16C2211yescharacteristic
24C2×C4⋊C16C2211yescharacteristic
25C2×#357C2211yescharacteristic
26C2×#357C2211yescharacteristic
27C2×#357C2211yescharacteristic
28C2×#357C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×#357C2211yescharacteristic
31C2×#346C2211yescharacteristic
32C2×C4×C16C2211yescharacteristic
33#1970C2211yescharacteristic
34#1970C2211yescharacteristic
35#1970C2211yescharacteristic
36#1970C2211yescharacteristic
37#1970C2211yescharacteristic
38#1942C2211yescharacteristic
39#1942C2211yescharacteristic
40#1942C2211yescharacteristic
41#1942C2211yescharacteristic
42#1942C2211yescharacteristic
43#1936—21no—
44#1936—21no—
45#1936—21no—
46#1936—21no—
47#1936—21no—
48C4×#107C2211yescharacteristic
49#1762—21no—
50#1726C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×D8C2311yescharacteristic
60C22×D8C2311yescharacteristic
61C2×C4×D4—21no—
62C2×D16—22no—
63C2×D16—22no—
64C2×D16—41no—
65C22×C16C2311yescharacteristic
66C22×C16C2311yescharacteristic
67C22×C16C2311yescharacteristic
68C22×C16C2311yescharacteristic
69C4×D8—22no—
70C4×D8—22no—
71C2×#107C2311yescharacteristic
72C2×#107C2311yescharacteristic
73C2×#107C2311yescharacteristic
74C2×#107C2311yescharacteristic
75C2×#107C2311yescharacteristic
76C2×#107C2311yescharacteristic
77C2×#102—21no—
78C2×#102—21no—
79C2×#102—21no—
80C2×#102—21no—
81C2×#102—21no—
82C2×#102—21no—
83C2×#97D411yescharacteristic
84C2×#97D411yescharacteristic
85C2×#97D411yescharacteristic
86C2×C4×C8C2311yescharacteristic
87C2×C4×C8C2311yescharacteristic
88C2×C4×C8C2311yescharacteristic
89#397—41no—
90#397—41no—
91#397—41no—
92#396—41no—
93#390—21no—
94#390—21no—
95#390—21no—
96#390—21no—
97#390—21no—
98#389—21no—
99#389—21no—
100#389—21no—
101#389—21no—
102#389—21no—
103#388—41no—
104#388—41no—
105#381—41no—
106#380—41no—
107C4×C4⋊C4—21no—
108#376D411yescharacteristic
109#367—22no—
110#367—41no—
111#366—22no—
112C4⋊C16—22no—
113C4⋊C16—22no—
114#357—22no—
115#357—22no—
116#357—22no—
117#357—22no—
118#357—41no—
119#357—41no—
120#346—41no—
121C4×C16—41no—
122#340—21no—
123#340—21no—
124#340—21no—
125#340—21no—
126#340—21no—
127#340—21no—
128C22×D4—21no—
129C22×D4—21no—
130C23×C4—41no—
131C2×D8—21no—
132C2×D8—21no—
133C2×D8—21no—
134C2×D8—21no—
135C2×D8—21no—
136C2×D8—21no—
137C2×D8—21no—
138C2×D8—21no—
139C2×D8—21no—
140C2×D8—21no—
141C2×D8—21no—
142C2×D8#4112yes—
143C22×C8C2×D411yescharacteristic
144C22×C8C2×D411yescharacteristic
145C22×C8C2×D411yescharacteristic
146C22×C8C2411yescharacteristic
147C4×D4—24no—
148C4×D4—42no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—41no—
160C2×C22⋊C4—41no—
161C2×C22⋊C4—41no—
162C2×C22⋊C4—41no—
163C2×C22⋊C4—41no—
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×C22⋊C4—41no—
171C2×C42—41no—
172C2×C42C2×D411yescharacteristic
173C2×C42C2×D411yescharacteristic
174C2×C42C2×D411yescharacteristic
175D16—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#4112yes—
187C2×C16#4112yes—
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#4112yes—
200#102—24no—
201#102—42no—
202#102—42no—
203#102—42no—
204#102—42no—
205#102—42no—
206#97—22no—
207#97—41no—
208#97—41no—
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#95D811yescharacteristic
222#95D811yescharacteristic
223#95D811yescharacteristic
224#95D811yescharacteristic
225C24—41no—
226C24—41no—
227C2×D4—22no—
228C2×D4—41no—
229C2×D4—41no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4—41no—
233C2×D4—41no—
234C2×D4—41no—
235C2×D4—41no—
236C2×D4—41no—
237C2×D4—41no—
238C2×D4—41no—
239C22×C4—44no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—161no—
247C22×C4—161no—
248C22×C4C2×D811yescharacteristic
249C22×C4C2×D811yescharacteristic
250C22×C4C2×D811yescharacteristic
251C22×C4C2×D811yescharacteristic
252C22×C4C2×D811yescharacteristic
253C22×C4C2×D811yescharacteristic
254C22×C4C22×D411yescharacteristic
255D8—22no—
256D8—22no—
257D8—41no—
258D8—41no—
259D8—41no—
260D8—41no—
261D8—41no—
262D8—41no—
263C2×C8—21no—
264C2×C8—21no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8#13512yes—
272C2×C8C2×#4111yescharacteristic
273C2×C8C2×#4111yescharacteristic
274C2×C8C2×#4111yescharacteristic
275C2×C8C2×#4111yescharacteristic
276C2×C8H(5,𝔽2)11yescharacteristic
277C2×C82−1+411yescharacteristic
278C4⋊C4—22no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—41no—
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—161no—
291C4⋊C4—161no—
292C4⋊C4—161no—
293C4⋊C4—161no—
294C4⋊C4—161no—
295C22⋊C4—44no—
296C22⋊C4—161no—
297C22⋊C4—161no—
298C22⋊C4—161no—
299C22⋊C4—161no—
300C22⋊C4—161no—
301C22⋊C4—161no—
302C22⋊C4—161no—
303C22⋊C4—161no—
304C22⋊C4—161no—
305C22⋊C4—161no—
306C42—21no—
307C42—21no—
308C42—21no—
309C42—21no—
310C42—21no—
311C42—21no—
312C42—161no—
313C16—22no—
314C16—22no—
315C16—41no—
316C16—41no—
317C23—42no—
318C23—81no—
319C23—81no—
320C23—81no—
321C23—81no—
322C23—81no—
323C23—81no—
324C23—81no—
325C23—81no—
326C23—81no—
327C23—81no—
328C23—81no—
329C23—81no—
330C23—81no—
331C23C22×D811yescharacteristic
332D4—42no—
333D4—42no—
334D4—81no—
335D4—81no—
336D4—81no—
337D4—81no—
338D4—81no—
339D4—81no—
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—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—42no—
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—81no—
375C2×C4—81no—
376C2×C4—164no—
377C2×C4#52511yescharacteristic
378C2×C4#53211yescharacteristic
379C2×C4#53711yescharacteristic
380C2×C4C2×#13511yescharacteristic
381C2×C4#57511yescharacteristic
382C2×C4#57511yescharacteristic
383C8—22no—
384C8—41no—
385C8—41no—
386C8#55111yescharacteristic
387C8#55211yescharacteristic
388C8#55311yescharacteristic
389C8#55611yescharacteristic
390C22—82no—
391C22—82no—
392C22—82no—
393C22—161no—
394C22—161no—
395C22—161no—
396C22—161no—
397C22—161no—
398C22—161no—
399C22—161no—
400C22—161no—
401C22—161no—
402C22—161no—
403C22—161no—
404C22#310011yescharacteristic
405C22#316911yescharacteristic
406C22#324511yescharacteristic
407C22#341511yescharacteristic
408C22#341511yescharacteristic
409C22#341611yescharacteristic
410C22#341611yescharacteristic
411C4—41no—
412C4—41no—
413C4—41no—
414C4—41no—
415C4—41no—
416C4—41no—
417C4—82no—
418C4—161no—
419C4—161no—
420C4—161no—
421C4—161no—
422C4—161no—
423C4#260111yescharacteristic
424C4#310211yescharacteristic
425C4#318311yescharacteristic
426C4#325011yescharacteristic
427C2—162no—
428C2—321no—
429C2#2942811yescharacteristic
430C2#3367811yescharacteristic
431C2#3367911yescharacteristic
432C2#3374011yescharacteristic
433C2#3374211yescharacteristic
434C2#3382311yescharacteristic
435C2#3382311yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D16C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #376 ◁ #1726 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33272C2semidirect
2. #21282C2semidirect
3. #13514C2semidirect
4. #13476C2semidirect
5. #13476C2semidirect
8. #13166C2semidirect
9. #13149C2semidirect
10. #13149C2semidirect
11. #13131C2semidirect
12. #13109C2semidirect
14. #12937C2semidirect
15. #12654C2semidirect

Maximal subgroups

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

subgroupindex
#332722
#212822
#135142
#134762
#134482
#132352
#131662
#131492
#131312
#131092
#130122
#129372
#126542