Finite Group Databasebeta

← #504591

Subgroups — #504591

Order 512.

541 conjugacy classes of subgroups.

The subgroup lattice has 469 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27790C211yesmaximal, characteristic
2#23026C211yesmaximal, characteristic, complemented
3#13449C211yesmaximal, characteristic, complemented
4#13382C211yesmaximal, characteristic, complemented
5#13366C211yesmaximal, characteristic, complemented
6#13305C211yesmaximal, characteristic
7#13277C211yesmaximal, characteristic, complemented
8#13149C211yesmaximal, characteristic, complemented
9#13131C211yesmaximal, characteristic, complemented
10#13121C211yesmaximal, characteristic, complemented
11#13111C211yesmaximal, characteristic, complemented
12#13013C211yesmaximal, characteristic, complemented
13#12992C211yesmaximal, characteristic
14#12868C211yesmaximal, characteristic, complemented
15#12638C211yesmaximal, characteristic, complemented
16C22×SD32C2211yescharacteristic
17C2×#499C2211yescharacteristic
18C2×#467C2211yescharacteristic
19C2×#466C2211yescharacteristic
20C2×C4×Q16C2211yescharacteristic
21C2×#427C2211yescharacteristic
22#2490—21no—
23#2362—21no—
24C2×#367C2211yescharacteristic
25C2×C4⋊C16C2211yescharacteristic
26C2×#358C2211yescharacteristic
27C2×#358C2211yescharacteristic
28C2×#358C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×#357C2211yescharacteristic
31C2×#357C2211yescharacteristic
32C2×C22⋊C16C2211yescharacteristic
33C2×C22⋊C16C2211yescharacteristic
34C2×#346C2211yescharacteristic
35#2034—21no—
36#2025—21no—
37#1970C2211yescharacteristic
38#1970C2211yescharacteristic
39#1969C2211yescharacteristic
40#1942C2211yescharacteristic
41#1927—21no—
42#1926—21no—
43#1873—21no—
44#1865—21no—
45#1808C2211yescharacteristic
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×Q16C2311yescharacteristic
60C22×D8C2311yescharacteristic
61C23×C8C2311yescharacteristic
62C2×#125—21no—
63C2×C22⋊Q8—21no—
64C2×#121—21no—
65C2×C4×Q8—21no—
66C2×#117—21no—
67C2×SD32—22no—
68C2×SD32—41no—
69C2×SD32—41no—
70C22×C16C2311yescharacteristic
71C22×C16C2311yescharacteristic
72C22×C16C2311yescharacteristic
73C22×C16C2311yescharacteristic
74#499—22no—
75#499—41no—
76#467—22no—
77#467—22no—
78#466—22no—
79#466—41no—
80C4×Q16—22no—
81C4×Q16—41no—
82#427—22no—
83#427—41no—
84C2×#107C2311yescharacteristic
85C2×#107C2311yescharacteristic
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#107C2311yescharacteristic
89C2×#107C2311yescharacteristic
90C2×#106—21no—
91C2×#106—21no—
92C2×#103—21no—
93C2×#103—21no—
94C2×#103—21no—
95C2×#102—21no—
96C2×#102—21no—
97C2×#102—21no—
98C2×C22⋊C8D411yescharacteristic
99C2×C4×C8C2311yescharacteristic
100C2×C4×C8C2311yescharacteristic
101#397—41no—
102#394—41no—
103#391—21no—
104#390—21no—
105#390—21no—
106#390—41no—
107#389—21no—
108#388—41no—
109#388—41no—
110#388—41no—
111#387—41no—
112#380—21no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—41no—
116C4×C22⋊C4D411yescharacteristic
117#367—22no—
118C4⋊C16—41no—
119#358—22no—
120#358—22no—
121#358—41no—
122#357—22no—
123#357—22no—
124#357—41no—
125C22⋊C16—41no—
126C22⋊C16—41no—
127#346—41no—
128#340—21no—
129#340—21no—
130#340—21no—
131#340—21no—
132#336D411yescharacteristic
133#336D411yescharacteristic
134C22×Q8—21no—
135C22×D4—21no—
136C22×D4—41no—
137C23×C4C2×D411yescharacteristic
138C2×Q16—21no—
139C2×Q16—21no—
140C2×Q16—21no—
141C2×Q16—21no—
142C2×Q16—21no—
143C2×Q16#4112yes—
144C2×D8—21no—
145C2×D8—21no—
146C2×D8—21no—
147C2×D8—21no—
148C2×D8—21no—
149C2×D8—21no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8C2×D411yescharacteristic
157C22×C8C2×D411yescharacteristic
158C22×C8C2×D411yescharacteristic
159C22×C8C2411yescharacteristic
160#125—42no—
161#125—81no—
162C22⋊Q8—24no—
163C22⋊Q8—42no—
164#121—42no—
165#121—81no—
166C4×Q8—42no—
167C4×Q8—81no—
168#117—42no—
169#117—81no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—41no—
181C2×C22⋊C4—41no—
182C2×C22⋊C4—41no—
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⋊C4D811yescharacteristic
190C2×C22⋊C4D811yescharacteristic
191C2×C42—41no—
192C2×C42—41no—
193C2×C42—41no—
194C2×C42—41no—
195C2×C42C2×D411yescharacteristic
196C2×C42C2×D411yescharacteristic
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—21no—
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#4112yes—
220#107#4112yes—
221#107#4112yes—
222#106—42no—
223#106—42no—
224#103—24no—
225#103—42no—
226#103—81no—
227#102—42no—
228#102—42no—
229#102—81no—
230C22⋊C8—41no—
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#95D811yescharacteristic
242#95D811yescharacteristic
243C24—41no—
244C24C2×D811yescharacteristic
245C2×Q8—22no—
246C2×Q8—41no—
247C2×Q8—41no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×Q8—41no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4—82no—
258C2×D4—82no—
259C2×D4—161no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4C2×D811yescharacteristic
274C22×C4C2×D811yescharacteristic
275C22×C4C2×D811yescharacteristic
276C22×C4C2×D811yescharacteristic
277C22×C4C2×D811yescharacteristic
278C22×C4C22×D411yescharacteristic
279Q16—22no—
280Q16—22no—
281Q16—41no—
282Q16—41no—
283D8—41no—
284D8—41no—
285D8—41no—
286D8—41no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—21no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8C2×#4111yescharacteristic
301C2×C8C2×#4111yescharacteristic
302C2×C8C2×#4111yescharacteristic
303C2×C8C2×#4111yescharacteristic
304C2×C8H(5,𝔽2)11yescharacteristic
305C2×C8H(5,𝔽2)11yescharacteristic
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—44no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—82no—
320C4⋊C4—82no—
321C4⋊C4—82no—
322C4⋊C4—82no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C22⋊C4—44no—
326C22⋊C4—82no—
327C22⋊C4—82no—
328C22⋊C4—82no—
329C22⋊C4—161no—
330C22⋊C4—161no—
331C22⋊C4—161no—
332C22⋊C4—161no—
333C42—21no—
334C42—21no—
335C42—21no—
336C42—21no—
337C42—82no—
338C42—82no—
339C42—82no—
340C42—161no—
341C16—41no—
342C16—41no—
343C16—41no—
344C16—41no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—81no—
353C23—81no—
354C23—81no—
355C23—81no—
356C23—81no—
357C23—81no—
358C23—81no—
359C23C22×D811yescharacteristic
360Q8—42no—
361Q8—42no—
362Q8—81no—
363Q8—81no—
364D4—81no—
365D4—81no—
366D4—81no—
367D4—81no—
368D4—164no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—81no—
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#52511yescharacteristic
410C2×C4#54611yescharacteristic
411C2×C4#54611yescharacteristic
412C2×C4#57511yescharacteristic
413C2×C4#57711yescharacteristic
414C2×C4#57711yescharacteristic
415C8—41no—
416C8—41no—
417C8—41no—
418C8#52911yescharacteristic
419C8#53811yescharacteristic
420C8#55511yescharacteristic
421C8#55911yescharacteristic
422C22—41no—
423C22—41no—
424C22—41no—
425C22—41no—
426C22—41no—
427C22—41no—
428C22—41no—
429C22—161no—
430C22—161no—
431C22—161no—
432C22—161no—
433C22—161no—
434C22—161no—
435C22—161no—
436C22#310011yescharacteristic
437C22#329411yescharacteristic
438C22#329411yescharacteristic
439C22#341411yescharacteristic
440C22#341411yescharacteristic
441C22#341611yescharacteristic
442C22#341611yescharacteristic
443C4—41no—
444C4—41no—
445C4—41no—
446C4—41no—
447C4—41no—
448C4—82no—
449C4—82no—
450C4—82no—
451C4—82no—
452C4—161no—
453C4—161no—
454C4—161no—
455C4#245911yescharacteristic
456C4#312511yescharacteristic
457C4#332111yescharacteristic
458C4#332611yescharacteristic
459C2—81no—
460C2—321no—
461C2#2672011yescharacteristic
462C2#3367111yescharacteristic
463C2#3368611yescharacteristic
464C2#3387411yescharacteristic
465C2#3387811yescharacteristic
466C2#3388411yescharacteristic
467C2#3389311yescharacteristic
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. #23026C2semidirect
3. #13449C2semidirect
4. #13382C2semidirect
5. #13366C2semidirect
7. #13277C2semidirect
8. #13149C2semidirect
9. #13131C2semidirect
10. #13121C2semidirect
11. #13111C2semidirect
12. #13013C2semidirect
14. #12868C2semidirect
15. #12638C2semidirect

Maximal subgroups

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

subgroupindex
#277902
#230262
#134492
#133822
#133662
#133052
#132772
#131492
#131312
#131212
#131112
#130132
#129922
#128682
#126382