Finite Group Databasebeta

← #505487

Subgroups — #505487

Order 512.

598 conjugacy classes of subgroups.

The subgroup lattice has 463 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27838C211yesmaximal, characteristic
2C22⋊C4×Q16C211yesmaximal, characteristic
3#13494C211yesmaximal, characteristic
4#13492C211yesmaximal, characteristic
5#13483C211yesmaximal, characteristic, complemented
6#13475C211yesmaximal, characteristic, complemented
7#13148C211yesmaximal, characteristic, complemented
8#13132C211yesmaximal, characteristic, complemented
9#13120C211yesmaximal, characteristic
10#12938C211yesmaximal, characteristic, complemented
11#12936C211yesmaximal, characteristic, complemented
12#12814C211yesmaximal, characteristic, complemented
13#12732C211yesmaximal, characteristic
14#12732C211yesmaximal, characteristic
15C4×#358C211yesmaximal, characteristic, complemented
16C23×Q16C2211yescharacteristic
17C2×C4×Q16C2211yescharacteristic
18C2×C4×Q16C2211yescharacteristic
19#2495—21no—
20Q8×C22⋊C4—21no—
21C2×#367C2211yescharacteristic
22C2×#366C2211yescharacteristic
23C2×C4⋊C16C2211yescharacteristic
24C2×#358C2211yescharacteristic
25C2×#358C2211yescharacteristic
26C2×#358C2211yescharacteristic
27C2×#358C2211yescharacteristic
28C2×#358C2211yescharacteristic
29C2×#358C2211yescharacteristic
30C2×#358C2211yescharacteristic
31C2×#358C2211yescharacteristic
32C2×C22⋊C16C2211yescharacteristic
33C2×C22⋊C16C2211yescharacteristic
34C2×C4×C16C2211yescharacteristic
35#2101—21no—
36#2077—21no—
37#1969C2211yescharacteristic
38#1969C2211yescharacteristic
39#1942C2211yescharacteristic
40#1942C2211yescharacteristic
41#1920—21no—
42#1878—21no—
43#1878—21no—
44#1846C2211yescharacteristic
45#1808C2211yescharacteristic
46#1808C2211yescharacteristic
47#1808C2211yescharacteristic
48#1787—21no—
49C4×#107C2211yescharacteristic
50C8×C22⋊C4C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C23×Q8—21no—
60C22×Q16—22no—
61C22×Q16—22no—
62C22×Q16—22no—
63C22×Q16—22no—
64C22×Q16—21no—
65C22×Q16—41no—
66C22×Q16—41no—
67C22×Q16C2311yescharacteristic
68C22×Q16C2311yescharacteristic
69C23×C8C2311yescharacteristic
70C2×C4×Q8—21no—
71C2×C4×Q8—21no—
72C22×C16C2311yescharacteristic
73C22×C16C2311yescharacteristic
74C22×C16C2311yescharacteristic
75C22×C16C2311yescharacteristic
76C4×Q16—22no—
77C4×Q16—22no—
78C4×Q16—41no—
79C4×Q16—41no—
80C2×#107C2311yescharacteristic
81C2×#107C2311yescharacteristic
82C2×#107C2311yescharacteristic
83C2×#107C2311yescharacteristic
84C2×#107C2311yescharacteristic
85C2×#107C2311yescharacteristic
86C2×#106—21no—
87C2×#103—21no—
88C2×#103—21no—
89C2×#103—21no—
90C2×#103—21no—
91C2×C22⋊C8D411yescharacteristic
92C2×C4×C8C2311yescharacteristic
93C2×C4×C8C2311yescharacteristic
94#400—41no—
95#397—41no—
96#391—21no—
97#391—21no—
98#391—41no—
99#391—41no—
100#389—21no—
101#389—21no—
102#388—41no—
103#383—41no—
104#382—21no—
105#380—21no—
106#380—21no—
107#380—21no—
108#380—41no—
109C4×C4⋊C4—21no—
110C4×C22⋊C4—41no—
111C4×C22⋊C4D411yescharacteristic
112#367—41no—
113#366—41no—
114C4⋊C16—41no—
115#358—22no—
116#358—22no—
117#358—22no—
118#358—22no—
119#358—22no—
120#358—22no—
121#358—41no—
122#358—41no—
123C22⋊C16—41no—
124C22⋊C16—41no—
125C4×C16—41no—
126#340—21no—
127#340—21no—
128#340—21no—
129#340—21no—
130#340—21no—
131#340—21no—
132#336D411yescharacteristic
133#336D411yescharacteristic
134C22×Q8—24no—
135C22×Q8—21no—
136C22×Q8—21no—
137C22×Q8—42no—
138C22×Q8—42no—
139C22×Q8—42no—
140C22×Q8—41no—
141C22×Q8—81no—
142C22×Q8—81no—
143C23×C4—41no—
144C23×C4C2×D411yescharacteristic
145C2×Q16—24no—
146C2×Q16—21no—
147C2×Q16—21no—
148C2×Q16—21no—
149C2×Q16—21no—
150C2×Q16—21no—
151C2×Q16—21no—
152C2×Q16—21no—
153C2×Q16—21no—
154C2×Q16—42no—
155C2×Q16—42no—
156C2×Q16—44no—
157C2×Q16—44no—
158C2×Q16—44no—
159C2×Q16—44no—
160C2×Q16#4112yes—
161C2×Q16#4112yes—
162C2×Q16#4112yes—
163C2×Q16#4112yes—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8C2×D411yescharacteristic
171C22×C8C2×D411yescharacteristic
172C22×C8C2×D411yescharacteristic
173C22×C8C2411yescharacteristic
174C4×Q8—42no—
175C4×Q8—42no—
176C4×Q8—81no—
177C4×Q8—81no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—41no—
188C2×C22⋊C4—41no—
189C2×C22⋊C4—41no—
190C2×C22⋊C4—41no—
191C2×C22⋊C4D811yescharacteristic
192C2×C22⋊C4D811yescharacteristic
193C2×C42—41no—
194C2×C42—41no—
195C2×C42—41no—
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D411yescharacteristic
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—21no—
220#107—21no—
221#107—21no—
222#106—42no—
223#103—42no—
224#103—42no—
225#103—81no—
226#103—81no—
227C22⋊C8—41no—
228C4×C8—21no—
229C4×C8—21no—
230C4×C8—21no—
231C4×C8—21no—
232#95—41no—
233#95—41no—
234#95—41no—
235#95—41no—
236#95—41no—
237#95—41no—
238#95—41no—
239#95—41no—
240#95—41no—
241#95—41no—
242#95—41no—
243#95—41no—
244#95D811yescharacteristic
245#95D811yescharacteristic
246C24C2×D811yescharacteristic
247C2×Q8—22no—
248C2×Q8—22no—
249C2×Q8—22no—
250C2×Q8—22no—
251C2×Q8—44no—
252C2×Q8—41no—
253C2×Q8—41no—
254C2×Q8—41no—
255C2×Q8—41no—
256C2×Q8—48no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—82no—
262C2×Q8—82no—
263C2×Q8—84no—
264C2×Q8—84no—
265C2×Q8—84no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—41no—
273C22×C4—44no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—82no—
282C22×C4—82no—
283C22×C4C2×D811yescharacteristic
284C22×C4C2×D811yescharacteristic
285C22×C4C2×D811yescharacteristic
286C22×C4C2×D811yescharacteristic
287C22×C4C2×D811yescharacteristic
288C22×C4C22×D411yescharacteristic
289Q16—22no—
290Q16—22no—
291Q16—22no—
292Q16—22no—
293Q16—22no—
294Q16—48no—
295Q16—41no—
296Q16—41no—
297Q16#12614yes—
298C2×C8—21no—
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—41no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8C2×#4111yescharacteristic
312C2×C8C2×#4111yescharacteristic
313C2×C8C2×#4111yescharacteristic
314C2×C8C2×#4111yescharacteristic
315C2×C8H(5,𝔽2)11yescharacteristic
316C2×C8H(5,𝔽2)11yescharacteristic
317C4⋊C4—41no—
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—41no—
328C4⋊C4—41no—
329C4⋊C4—82no—
330C4⋊C4—82no—
331C4⋊C4—161no—
332C4⋊C4—161no—
333C22⋊C4—22no—
334C22⋊C4—22no—
335C22⋊C4—82no—
336C22⋊C4—82no—
337C22⋊C4—161no—
338C42—21no—
339C42—21no—
340C42—21no—
341C42—21no—
342C42—82no—
343C42—82no—
344C42—82no—
345C16—41no—
346C16—41no—
347C16—41no—
348C16—41no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23#50812yes—
356C23C22×D811yescharacteristic
357Q8—24no—
358Q8—42no—
359Q8—42no—
360Q8—42no—
361Q8—42no—
362Q8—42no—
363Q8—88no—
364Q8—81no—
365Q8—81no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
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—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—88no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
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#51012yes—
410C2×C4#52511yescharacteristic
411C2×C4#54611yescharacteristic
412C2×C4#54611yescharacteristic
413C2×C4#57511yescharacteristic
414C2×C4#57711yescharacteristic
415C2×C4#57711yescharacteristic
416C8—41no—
417C8—41no—
418C8—41no—
419C8C2×#12611yescharacteristic
420C8#53811yescharacteristic
421C8#55111yescharacteristic
422C8#55111yescharacteristic
423C22—22no—
424C22—22no—
425C22—22no—
426C22—41no—
427C22—41no—
428C22—41no—
429C22—41no—
430C22#310011yescharacteristic
431C22#329411yescharacteristic
432C22#329411yescharacteristic
433C22C2×#50811yescharacteristic
434C22C2×#51011yescharacteristic
435C22#341611yescharacteristic
436C22#341711yescharacteristic
437C4—22no—
438C4—41no—
439C4—41no—
440C4—41no—
441C4—41no—
442C4—44no—
443C4—82no—
444C4—161no—
445C4—161no—
446C4—161no—
447C4—161no—
448C4—161no—
449C4—161no—
450C4#248011yescharacteristic
451C4#312311yescharacteristic
452C4#331111yescharacteristic
453C4#332311yescharacteristic
454C2—42no—
455C2#2751811yescharacteristic
456C2#3366811yescharacteristic
457C2#3368111yescharacteristic
458C2#3386711yescharacteristic
459C2#3386811yescharacteristic
460C2#3387911yescharacteristic
461C2#3389611yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC24C2×D8
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
5. #13483C2semidirect
6. #13475C2semidirect
7. #13148C2semidirect
8. #13132C2semidirect
10. #12938C2semidirect
11. #12936C2semidirect
12. #12814C2semidirect
15. C4×#358C2semidirect

Maximal subgroups

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

subgroupindex
#278382
C22⋊C4×Q162
#134942
#134922
#134832
#134752
#131482
#131322
#131202
#129382
#129362
#128142
#127322
C4×#3582