Finite Group Databasebeta

← #505129

Subgroups — #505129

Order 512.

570 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26254C211yesmaximal, characteristic
2#22808C211yesmaximal, characteristic, complemented
3#13526C211yesmaximal, characteristic, complemented
4#13480C211yesmaximal, characteristic, complemented
5#13475C211yesmaximal, characteristic, complemented
6#13328C211yesmaximal, characteristic
7#13252C211yesmaximal, characteristic
8#13238C211yesmaximal, characteristic, complemented
9#13130C211yesmaximal, characteristic, complemented
10#13112C211yesmaximal, characteristic, complemented
11#13105C211yesmaximal, characteristic, complemented
12#13014C211yesmaximal, characteristic, complemented
13#13013C211yesmaximal, characteristic, complemented
14#13013C211yesmaximal, characteristic, complemented
15#12637C211yesmaximal, characteristic, complemented
16C22×SD32C2211yescharacteristic
17C22×SD32C2211yescharacteristic
18C2×#493C2211yescharacteristic
19C2×C4×Q16C2211yescharacteristic
20C2×C4×D8C2211yescharacteristic
21#2445—21no—
22#2355—21no—
23C2×#366C2211yescharacteristic
24C2×#366C2211yescharacteristic
25C2×C4⋊C16C2211yescharacteristic
26C2×#358C2211yescharacteristic
27C2×#358C2211yescharacteristic
28C2×#357C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×#357C2211yescharacteristic
31C2×#357C2211yescharacteristic
32C2×#357C2211yescharacteristic
33C2×#357C2211yescharacteristic
34C2×#346C2211yescharacteristic
35#2054—21no—
36#2041—21no—
37#1970C2211yescharacteristic
38#1970C2211yescharacteristic
39#1970C2211yescharacteristic
40#1969C2211yescharacteristic
41#1969C2211yescharacteristic
42#1942C2211yescharacteristic
43#1928—21no—
44#1928—21no—
45#1879—21no—
46#1845C2211yescharacteristic
47#1803—21no—
48C4×#107C2211yescharacteristic
49C4×#107C2211yescharacteristic
50C8×C4⋊C4C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1374C2211yescharacteristic
56#1374C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×Q16C2311yescharacteristic
60C22×D8—21no—
61C22×D8C2311yescharacteristic
62C22×D8C2311yescharacteristic
63C2×#127—21no—
64C2×#123—41no—
65C2×C4×Q8—21no—
66C2×C4×D4—21no—
67C2×SD32—22no—
68C2×SD32—22no—
69C2×SD32—22no—
70C2×SD32—22no—
71C2×SD32—41no—
72C2×SD32—41no—
73C22×C16C2311yescharacteristic
74C22×C16C2311yescharacteristic
75C22×C16C2311yescharacteristic
76C22×C16C2311yescharacteristic
77#493—22no—
78#493—41no—
79C4×Q16—22no—
80C4×Q16—22no—
81C4×D8—22no—
82C4×D8—41no—
83C2×#107—21no—
84C2×#107C2311yescharacteristic
85C2×#107C2311yescharacteristic
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#107C2311yescharacteristic
89C2×C4⋊C8D411yescharacteristic
90C2×#103—21no—
91C2×#103—21no—
92C2×#103—21no—
93C2×#102—21no—
94C2×#102—21no—
95C2×#102—21no—
96C2×#102—21no—
97C2×C4×C8C2311yescharacteristic
98C2×C4×C8C2311yescharacteristic
99C2×C4×C8C2311yescharacteristic
100#394—41no—
101#394—41no—
102#391—21no—
103#391—21no—
104#390—21no—
105#390—21no—
106#390—21no—
107#389—21no—
108#387—41no—
109#387—41no—
110#386—41no—
111#384—21no—
112#382—41no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—41no—
116C4×C4⋊C4D411yescharacteristic
117#366—22no—
118#366—41no—
119C4⋊C16—41no—
120#358—41no—
121#358—41no—
122#357—22no—
123#357—22no—
124#357—22no—
125#357—41no—
126#357—41no—
127#357—41no—
128#346—41no—
129#340—21no—
130#340—21no—
131#340—21no—
132#336D411yescharacteristic
133#336D411yescharacteristic
134C22×Q8—21no—
135C22×D4—21no—
136C22×D4—21no—
137C22×D4—41no—
138C22×D4—41no—
139C23×C4—41no—
140C2×Q16—21no—
141C2×Q16—21no—
142C2×Q16—21no—
143C2×Q16—21no—
144C2×Q16—21no—
145C2×Q16#4112yes—
146C2×D8—21no—
147C2×D8—21no—
148C2×D8—21no—
149C2×D8—21no—
150C2×D8—21no—
151C2×D8—21no—
152C2×D8—21no—
153C2×D8—21no—
154C2×D8—21no—
155C2×D8—21no—
156C2×D8—21no—
157C2×D8—42no—
158C2×D8—42no—
159C2×D8—81no—
160C2×D8#4112yes—
161C22×C8C2×D411yescharacteristic
162C22×C8C2×D411yescharacteristic
163C22×C8C2×D411yescharacteristic
164C22×C8C2411yescharacteristic
165#127—42no—
166#127—81no—
167#123—84no—
168C4×Q8—24no—
169C4×Q8—42no—
170C4×D4—42no—
171C4×D4—81no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4D811yescharacteristic
182C2×C4⋊C4D811yescharacteristic
183C2×C22⋊C4—41no—
184C2×C22⋊C4—41no—
185C2×C22⋊C4—41no—
186C2×C22⋊C4—41no—
187C2×C22⋊C4—41no—
188C2×C22⋊C4—41no—
189C2×C22⋊C4—41no—
190C2×C42—41no—
191C2×C42—41no—
192C2×C42—41no—
193C2×C42C2×D411yescharacteristic
194C2×C42C2×D411yescharacteristic
195C2×C42C2×D411yescharacteristic
196SD32—44no—
197SD32—44no—
198C2×C16—21no—
199C2×C16—21no—
200C2×C16—21no—
201C2×C16—21no—
202C2×C16—21no—
203C2×C16—21no—
204C2×C16—21no—
205C2×C16—21no—
206C2×C16—21no—
207C2×C16—21no—
208C2×C16—21no—
209C2×C16#4112yes—
210#107—21no—
211#107—21no—
212#107—21no—
213#107—21no—
214#107—21no—
215#107—21no—
216#107—21no—
217#107—21no—
218#107—21no—
219#107—81no—
220#107#4112yes—
221C4⋊C8—41no—
222#103—24no—
223#103—81no—
224#103—81no—
225#102—42no—
226#102—81no—
227#102—81no—
228#102—81no—
229C4×C8—21no—
230C4×C8—21no—
231C4×C8—21no—
232C4×C8—21no—
233C4×C8—21no—
234C4×C8—21no—
235#95—41no—
236#95—41no—
237#95—41no—
238#95—41no—
239#95—41no—
240#95—41no—
241#95—41no—
242#95D811yescharacteristic
243#95D811yescharacteristic
244C24—41no—
245C24—41no—
246C2×Q8—22no—
247C2×Q8—41no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×Q8—41no—
251C2×Q8—41no—
252C2×D4—22no—
253C2×D4—41no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—41no—
261C2×D4—41no—
262C2×D4—41no—
263C2×D4—41no—
264C2×D4—82no—
265C2×D4—82no—
266C2×D4—82no—
267C2×D4—82no—
268C2×D4—161no—
269C2×D4—161no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—82no—
277C22×C4—82no—
278C22×C4—161no—
279C22×C4C2×D811yescharacteristic
280C22×C4C2×D811yescharacteristic
281C22×C4C2×D811yescharacteristic
282C22×C4C2×D811yescharacteristic
283C22×C4C2×D811yescharacteristic
284C22×C4C2×D811yescharacteristic
285C22×C4C22×D411yescharacteristic
286Q16—22no—
287Q16—22no—
288Q16—41no—
289Q16—41no—
290D8—22no—
291D8—22no—
292D8—41no—
293D8—41no—
294D8—41no—
295D8—41no—
296D8—41no—
297D8—41no—
298D8—84no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8C2×#4111yescharacteristic
309C2×C8C2×#4111yescharacteristic
310C2×C8C2×#4111yescharacteristic
311C2×C8C2×#4111yescharacteristic
312C2×C8H(5,𝔽2)11yescharacteristic
313C2×C8H(5,𝔽2)11yescharacteristic
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—44no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—82no—
328C4⋊C4—82no—
329C4⋊C4—161no—
330C22⋊C4—82no—
331C22⋊C4—82no—
332C22⋊C4—82no—
333C22⋊C4—82no—
334C22⋊C4—161no—
335C22⋊C4—161no—
336C22⋊C4—161no—
337C42—21no—
338C42—21no—
339C42—21no—
340C42—21no—
341C42—21no—
342C42—21no—
343C42—44no—
344C42—161no—
345C42—161no—
346C16—22no—
347C16—41no—
348C16—41no—
349C16—41no—
350C23—42no—
351C23—81no—
352C23—81no—
353C23—81no—
354C23—81no—
355C23—81no—
356C23—81no—
357C23—81no—
358C23—81no—
359C23—81no—
360C23—81no—
361C23—81no—
362C23—81no—
363C23—81no—
364C23C22×D811yescharacteristic
365Q8—42no—
366Q8—42no—
367Q8—81no—
368Q8—81no—
369D4—42no—
370D4—42no—
371D4—81no—
372D4—81no—
373D4—81no—
374D4—81no—
375D4—81no—
376D4—81no—
377D4—164no—
378D4—164no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—164no—
414C2×C4#50912yes—
415C2×C4#51012yes—
416C2×C4#52511yescharacteristic
417C2×C4#54611yescharacteristic
418C2×C4#55011yescharacteristic
419C2×C4#57511yescharacteristic
420C2×C4#57611yescharacteristic
421C2×C4#57711yescharacteristic
422C8—41no—
423C8—41no—
424C8—41no—
425C8#52911yescharacteristic
426C8#54011yescharacteristic
427C8#55111yescharacteristic
428C8#55111yescharacteristic
429C22—82no—
430C22—82no—
431C22—82no—
432C22—161no—
433C22—161no—
434C22—161no—
435C22—161no—
436C22—161no—
437C22—161no—
438C22—161no—
439C22—161no—
440C22—161no—
441C22—161no—
442C22—161no—
443C22#310011yescharacteristic
444C22#329411yescharacteristic
445C22#339111yescharacteristic
446C22C2×#50911yescharacteristic
447C22C2×#51011yescharacteristic
448C22#341511yescharacteristic
449C22#341611yescharacteristic
450C4—22no—
451C4—22no—
452C4—41no—
453C4—41no—
454C4—41no—
455C4—41no—
456C4—82no—
457C4—82no—
458C4—161no—
459C4—161no—
460C4—161no—
461C4—161no—
462C4#247111yescharacteristic
463C4#311911yescharacteristic
464C4#331911yescharacteristic
465C4#340111yescharacteristic
466C2—162no—
467C2—321no—
468C2#2719011yescharacteristic
469C2#3367211yescharacteristic
470C2#3367911yescharacteristic
471C2#3388111yescharacteristic
472C2#3389111yescharacteristic
473C2#3396611yescharacteristic
474C2#3396711yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC2×#493C22
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. #22808C2semidirect
3. #13526C2semidirect
4. #13480C2semidirect
5. #13475C2semidirect
8. #13238C2semidirect
9. #13130C2semidirect
10. #13112C2semidirect
11. #13105C2semidirect
12. #13014C2semidirect
13. #13013C2semidirect
14. #13013C2semidirect
15. #12637C2semidirect

Maximal subgroups

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

subgroupindex
#262542
#228082
#135262
#134802
#134752
#133282
#132522
#132382
#131302
#131122
#131052
#130142
#130132
#126372