Finite Group Databasebeta

← #505284

Subgroups — #505284

Order 512.

496 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27790C211yesmaximal, characteristic
2#22614C211yesmaximal, characteristic, complemented
3#13494C211yesmaximal, characteristic, complemented
4#13476C211yesmaximal, characteristic, complemented
5#13453C211yesmaximal, characteristic, complemented
6#13380C211yesmaximal, characteristic
7#13277C211yesmaximal, characteristic, complemented
8#13147C211yesmaximal, characteristic, complemented
9#13131C211yesmaximal, characteristic, complemented
10#13130C211yesmaximal, characteristic, complemented
11#13120C211yesmaximal, characteristic, complemented
12#13110C211yesmaximal, characteristic, complemented
13#12992C211yesmaximal, characteristic
14#12921C211yesmaximal, characteristic, complemented
15C4×#366C211yesmaximal, characteristic, complemented
16C2×#499C2211yescharacteristic
17C2×#466C2211yescharacteristic
18C2×#427C2211yescharacteristic
19#2490—21no—
20#2347—21no—
21C2×#367C2211yescharacteristic
22C2×#367C2211yescharacteristic
23C2×#366C2211yescharacteristic
24C2×#366C2211yescharacteristic
25C2×C4⋊C16C2211yescharacteristic
26C2×#357C2211yescharacteristic
27C2×#357C2211yescharacteristic
28C2×#357C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×C22⋊C16C2211yescharacteristic
31C2×C22⋊C16C2211yescharacteristic
32C2×C4×C16C2211yescharacteristic
33#2034—21no—
34#2025—21no—
35#1970C2211yescharacteristic
36#1970C2211yescharacteristic
37#1942C2211yescharacteristic
38#1942C2211yescharacteristic
39#1926—21no—
40#1919—21no—
41#1919—21no—
42#1825—21no—
43#1808C2211yescharacteristic
44#1808C2211yescharacteristic
45C4×#107C2211yescharacteristic
46C4×#107C2211yescharacteristic
47C4×#107C2211yescharacteristic
48C8×C22⋊C4C2211yescharacteristic
49#1380C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×D8C2311yescharacteristic
60C23×C8C2311yescharacteristic
61C2×#125—21no—
62C2×#121—21no—
63C2×#117—21no—
64C22×C16C2311yescharacteristic
65C22×C16C2311yescharacteristic
66C22×C16C2311yescharacteristic
67C22×C16C2311yescharacteristic
68#499—22no—
69#499—41no—
70#466—22no—
71#466—41no—
72#427—22no—
73#427—22no—
74C2×#107C2311yescharacteristic
75C2×#107C2311yescharacteristic
76C2×#107C2311yescharacteristic
77C2×#107C2311yescharacteristic
78C2×#107C2311yescharacteristic
79C2×#107C2311yescharacteristic
80C2×#107C2311yescharacteristic
81C2×#106—21no—
82C2×#106—21no—
83C2×#102—21no—
84C2×#102—21no—
85C2×#102—21no—
86C2×C22⋊C8D411yescharacteristic
87C2×C4×C8C2311yescharacteristic
88C2×C4×C8C2311yescharacteristic
89#397—41no—
90#394—41no—
91#390—21no—
92#390—21no—
93#390—41no—
94#389—21no—
95#389—21no—
96#388—41no—
97#388—41no—
98#388—41no—
99#383—41no—
100#380—21no—
101#380—21no—
102C4×C4⋊C4—21no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4—41no—
106C4×C22⋊C4D411yescharacteristic
107#367—22no—
108#367—41no—
109#366—22no—
110#366—22no—
111C4⋊C16—41no—
112#357—22no—
113#357—22no—
114#357—22no—
115#357—41no—
116C22⋊C16—22no—
117C22⋊C16—41no—
118C4×C16—22no—
119#340—21no—
120#340—21no—
121#340—21no—
122#340—21no—
123#340—21no—
124#340—21no—
125#340—21no—
126#336D411yescharacteristic
127#336D411yescharacteristic
128C22×D4—21no—
129C22×D4—41no—
130C23×C4C2×D411yescharacteristic
131C2×D8—21no—
132C2×D8—21no—
133C2×D8—21no—
134C2×D8—21no—
135C2×D8—21no—
136C2×D8—21no—
137C22×C8—21no—
138C22×C8—21no—
139C22×C8—21no—
140C22×C8—21no—
141C22×C8—21no—
142C22×C8—21no—
143C22×C8C2×D411yescharacteristic
144C22×C8C2×D411yescharacteristic
145C22×C8C2×D411yescharacteristic
146C22×C8C2411yescharacteristic
147#125—42no—
148#125—81no—
149#121—42no—
150#121—81no—
151#117—24no—
152#117—42no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊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×C22⋊C4—41no—
172C2×C22⋊C4D811yescharacteristic
173C2×C22⋊C4D811yescharacteristic
174C2×C42—41no—
175C2×C42—41no—
176C2×C42—41no—
177C2×C42—41no—
178C2×C42C2×D411yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C16—21no—
181C2×C16—21no—
182C2×C16—21no—
183C2×C16—21no—
184C2×C16—21no—
185C2×C16—21no—
186C2×C16—21no—
187C2×C16—21no—
188C2×C16—21no—
189C2×C16—21no—
190C2×C16—21no—
191C2×C16#4112yes—
192#107—21no—
193#107—21no—
194#107—21no—
195#107—21no—
196#107—21no—
197#107—21no—
198#107—21no—
199#107—21no—
200#107—21no—
201#107—21no—
202#107—21no—
203#107#4112yes—
204#107#4112yes—
205#107#4112yes—
206#106—24no—
207#106—42no—
208#102—42no—
209#102—42no—
210#102—81no—
211C22⋊C8—41no—
212C4×C8—21no—
213C4×C8—21no—
214C4×C8—21no—
215C4×C8—21no—
216#95—41no—
217#95—41no—
218#95—41no—
219#95—41no—
220#95—41no—
221#95—41no—
222#95—41no—
223#95D811yescharacteristic
224#95D811yescharacteristic
225C24—41no—
226C24C2×D811yescharacteristic
227C2×D4—41no—
228C2×D4—41no—
229C2×D4—41no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4—41no—
233C2×D4—82no—
234C2×D4—82no—
235C2×D4—161no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4C2×D811yescharacteristic
250C22×C4C2×D811yescharacteristic
251C22×C4C2×D811yescharacteristic
252C22×C4C2×D811yescharacteristic
253C22×C4C2×D811yescharacteristic
254C22×C4C22×D411yescharacteristic
255D8—41no—
256D8—41no—
257D8—41no—
258D8—41no—
259C2×C8—21no—
260C2×C8—21no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8—21no—
264C2×C8—21no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—41no—
269C2×C8—41no—
270C2×C8—41no—
271C2×C8—41no—
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—22no—
280C4⋊C4—22no—
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—41no—
291C4⋊C4—41no—
292C4⋊C4—82no—
293C4⋊C4—82no—
294C4⋊C4—82no—
295C4⋊C4—82no—
296C22⋊C4—22no—
297C22⋊C4—41no—
298C22⋊C4—44no—
299C22⋊C4—82no—
300C22⋊C4—82no—
301C22⋊C4—161no—
302C22⋊C4—161no—
303C22⋊C4—161no—
304C22⋊C4—161no—
305C22⋊C4—161no—
306C42—21no—
307C42—21no—
308C42—21no—
309C42—21no—
310C42—44no—
311C42—161no—
312C42—161no—
313C42—161no—
314C16—22no—
315C16—41no—
316C16—41no—
317C16—41no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—81no—
326C23—81no—
327C23—81no—
328C23—81no—
329C23—81no—
330C23—81no—
331C23—81no—
332C23C22×D811yescharacteristic
333D4—81no—
334D4—81no—
335D4—81no—
336D4—81no—
337D4—164no—
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—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—42no—
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—81no—
377C2×C4#50812yes—
378C2×C4#52511yescharacteristic
379C2×C4#54611yescharacteristic
380C2×C4#54711yescharacteristic
381C2×C4#57511yescharacteristic
382C2×C4#57711yescharacteristic
383C2×C4#57811yescharacteristic
384C8—41no—
385C8—41no—
386C8—41no—
387C8#52911yescharacteristic
388C8#54211yescharacteristic
389C8#55511yescharacteristic
390C8#55611yescharacteristic
391C22—41no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—161no—
399C22—161no—
400C22—161no—
401C22—161no—
402C22—161no—
403C22—161no—
404C22—161no—
405C22#310011yescharacteristic
406C22#329411yescharacteristic
407C22#333411yescharacteristic
408C22C2×#50811yescharacteristic
409C22#341411yescharacteristic
410C22#341511yescharacteristic
411C22#341611yescharacteristic
412C4—22no—
413C4—41no—
414C4—41no—
415C4—41no—
416C4—41no—
417C4—82no—
418C4—82no—
419C4—82no—
420C4—161no—
421C4—161no—
422C4—161no—
423C4—161no—
424C4#247411yescharacteristic
425C4#312111yescharacteristic
426C4#332711yescharacteristic
427C4#334811yescharacteristic
428C2—81no—
429C2—321no—
430C2#2732611yescharacteristic
431C2#3366211yescharacteristic
432C2#3367911yescharacteristic
433C2#3387111yescharacteristic
434C2#3388911yescharacteristic
435C2#3390611yescharacteristic
436C2#3391711yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC2×#466C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ C8×C22⋊C4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22614C2semidirect
3. #13494C2semidirect
4. #13476C2semidirect
5. #13453C2semidirect
7. #13277C2semidirect
8. #13147C2semidirect
9. #13131C2semidirect
10. #13130C2semidirect
11. #13120C2semidirect
12. #13110C2semidirect
14. #12921C2semidirect
15. C4×#366C2semidirect

Maximal subgroups

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

subgroupindex
#277902
#226142
#134942
#134762
#134532
#133802
#132772
#131472
#131312
#131302
#131202
#131102
#129922
#129212
C4×#3662