Finite Group Databasebeta

← #501447

Subgroups — #501447

Order 512.

560 conjugacy classes of subgroups.

The subgroup lattice has 441 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2237C211yesmaximal, characteristic
2C2×#2163C211yesmaximal, characteristic
3#33130C211yesmaximal, characteristic
4#21151C211yesmaximal, characteristic
5#13516C211yesmaximal, characteristic, complemented
6#13515C211yesmaximal, characteristic, complemented
7#13477C211yesmaximal, characteristic, complemented
8#13475C211yesmaximal, characteristic, complemented
9#13168C211yesmaximal, characteristic, complemented
10#13167C211yesmaximal, characteristic, complemented
11#13150C211yesmaximal, characteristic, complemented
12#13150C211yesmaximal, characteristic, complemented
13#12994C211yesmaximal, characteristic
14#12921C211yesmaximal, characteristic
15#12868C211yesmaximal, characteristic
16C2×#467C2211yescharacteristic
17C2×#427C2211yescharacteristic
18C2×#427C2211yescharacteristic
19C22×C4×C8C2211yescharacteristic
20#2855—21no—
21#2296—21no—
22#2237—24no—
23#2163—24no—
24C2×#367C2211yescharacteristic
25C2×#367C2211yescharacteristic
26C2×#366C2211yescharacteristic
27C2×#366C2211yescharacteristic
28C2×C4⋊C16C2211yescharacteristic
29C2×C4⋊C16C2211yescharacteristic
30C2×#358C2211yescharacteristic
31C2×#358C2211yescharacteristic
32C2×#358C2211yescharacteristic
33C2×#358C2211yescharacteristic
34C2×C22⋊C16C2211yescharacteristic
35C2×C22⋊C16C2211yescharacteristic
36C2×#346C2211yescharacteristic
37C2×#346C2211yescharacteristic
38#1969C2211yescharacteristic
39#1969C2211yescharacteristic
40#1942C2211yescharacteristic
41#1942C2211yescharacteristic
42#1942C2211yescharacteristic
43#1942C2211yescharacteristic
44#1942C2211yescharacteristic
45#1942C2211yescharacteristic
46#1941—21no—
47#1941—21no—
48#1933—21no—
49#1933—21no—
50#1892—21no—
51#1808C2211yescharacteristic
52#1735—21no—
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1380C2211yescharacteristic
60#1380C2211yescharacteristic
61C22×Q16C2311yescharacteristic
62C23×C8D411yescharacteristic
63C23×C8C2311yescharacteristic
64C2×C22⋊Q8—21no—
65C2×#117—21no—
66C2×#117—21no—
67C22×C42D411yescharacteristic
68C22×C16C2311yescharacteristic
69C22×C16C2311yescharacteristic
70C22×C16C2311yescharacteristic
71C22×C16C2311yescharacteristic
72#467—22no—
73#467—22no—
74#427—22no—
75#427—22no—
76#427—22no—
77#427—22no—
78C2×#107C2311yescharacteristic
79C2×#107C2311yescharacteristic
80C2×#107C2311yescharacteristic
81C2×#107C2311yescharacteristic
82C2×#107C2311yescharacteristic
83C2×#107C2311yescharacteristic
84C2×#107C2311yescharacteristic
85C2×#106—21no—
86C2×#106—21no—
87C2×#103—21no—
88C2×#103—21no—
89C2×#103—21no—
90C2×C4×C8—22no—
91C2×C4×C8—22no—
92C2×C4×C8D411yescharacteristic
93C2×C4×C8D411yescharacteristic
94C2×C4×C8C2311yescharacteristic
95C2×C4×C8C2311yescharacteristic
96#391—21no—
97#391—21no—
98#389—21no—
99#389—21no—
100#389—21no—
101#389—21no—
102#389—21no—
103#389—21no—
104#387—41no—
105#387—41no—
106#385—41no—
107#385—41no—
108#385—41no—
109#381—41no—
110#380—21no—
111C4×C4⋊C4—41no—
112C4×C4⋊C4—41no—
113#367—22no—
114#367—22no—
115#366—22no—
116#366—22no—
117C4⋊C16—22no—
118C4⋊C16—22no—
119#358—22no—
120#358—22no—
121#358—22no—
122#358—22no—
123C22⋊C16—22no—
124C22⋊C16D414yes—
125#346—22no—
126#346—22no—
127#340—21no—
128#340—21no—
129#340—21no—
130#340—21no—
131#340—21no—
132#340—21no—
133#340—21no—
134C22×Q8—21no—
135C23×C4D811yescharacteristic
136C23×C4D811yescharacteristic
137C23×C4C2×D411yescharacteristic
138C2×Q16—21no—
139C2×Q16—21no—
140C2×Q16—21no—
141C2×Q16—21no—
142C2×Q16—21no—
143C2×Q16—21no—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—41no—
151C22×C8C2×D411yescharacteristic
152C22×C8C2×D411yescharacteristic
153C22×C8C2×D412yes—
154C22×C8C2×D411yescharacteristic
155C22×C8#4112yes—
156C22×C8C2411yescharacteristic
157C22⋊Q8—42no—
158C22⋊Q8—42no—
159#117—24no—
160#117—42no—
161#117—42no—
162#117—42no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—41no—
177C2×C22⋊C4—41no—
178C2×C22⋊C4—41no—
179C2×C22⋊C4—41no—
180C2×C22⋊C4—41no—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—41no—
184C2×C42—41no—
185C2×C42D811yescharacteristic
186C2×C42D811yescharacteristic
187C2×C42C2×D411yescharacteristic
188C2×C42C2×D411yescharacteristic
189C2×C16—21no—
190C2×C16—21no—
191C2×C16—21no—
192C2×C16—21no—
193C2×C16—21no—
194C2×C16—21no—
195C2×C16—21no—
196C2×C16—21no—
197C2×C16—21no—
198C2×C16—21no—
199C2×C16C2×D412yes—
200C2×C16C2×D412yes—
201#107—21no—
202#107—21no—
203#107—21no—
204#107—21no—
205#107—21no—
206#107—21no—
207#107—21no—
208#107—21no—
209#107—21no—
210#107—21no—
211#107—21no—
212#107—21no—
213#107#4112yes—
214#107#4112yes—
215#106—24no—
216#106—42no—
217#103—42no—
218#103—42no—
219#103—42no—
220C4×C8—22no—
221C4×C8—22no—
222C4×C8—21no—
223C4×C8—21no—
224C4×C8—21no—
225C4×C8—21no—
226#95—41no—
227#95—41no—
228#95—41no—
229#95—41no—
230#95—41no—
231#95—41no—
232#95—41no—
233#95—41no—
234#95—41no—
235#95—41no—
236#95—41no—
237C24C2×D811yescharacteristic
238C2×Q8—41no—
239C2×Q8—41no—
240C2×Q8—41no—
241C2×Q8—41no—
242C2×Q8—41no—
243C2×Q8—41no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4#12112yes—
263C22×C4C2×D811yescharacteristic
264C22×C4C2×D811yescharacteristic
265C22×C4C2×D811yescharacteristic
266C22×C4C2×D811yescharacteristic
267C22×C4C2×D811yescharacteristic
268C22×C4#13512yes—
269C22×C4C22×D411yescharacteristic
270Q16—41no—
271Q16—41no—
272Q16—41no—
273Q16—41no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—22no—
279C2×C8—22no—
280C2×C8—22no—
281C2×C8—22no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—44no—
286C2×C8#12112yes—
287C2×C8#13712yes—
288C2×C8C22×D411yescharacteristic
289C2×C8C2×#4111yescharacteristic
290C2×C8C2×#4111yescharacteristic
291C2×C8C2×#4111yescharacteristic
292C2×C82−1+411yescharacteristic
293C2×C82−1+411yescharacteristic
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—82no—
309C4⋊C4—82no—
310C4⋊C4—82no—
311C4⋊C4—82no—
312C4⋊C4—82no—
313C4⋊C4—82no—
314C4⋊C4—82no—
315C22⋊C4—44no—
316C22⋊C4—82no—
317C22⋊C4—82no—
318C22⋊C4—82no—
319C42—21no—
320C42—21no—
321C42—21no—
322C42—21no—
323C42—22no—
324C42—22no—
325C42—44no—
326C42—82no—
327C16—22no—
328C16—22no—
329C16—41no—
330C16—41no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23#46612yes—
338C23C22×D811yescharacteristic
339Q8—81no—
340Q8—81no—
341Q8—81no—
342Q8—81no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—42no—
363C2×C4—42no—
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—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4#46612yes—
387C2×C4#51012yes—
388C2×C4C2×#12111yescharacteristic
389C2×C4#52511yescharacteristic
390C2×C4#53611yescharacteristic
391C2×C4C2×#13511yescharacteristic
392C2×C4C2×#13711yescharacteristic
393C2×C4#57511yescharacteristic
394C8—22no—
395C8—22no—
396C8—41no—
397C8#54711yescharacteristic
398C8#54711yescharacteristic
399C8#55211yescharacteristic
400C8#55211yescharacteristic
401C22—22no—
402C22—22no—
403C22—22no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22C2×#46611yescharacteristic
409C22#310011yescharacteristic
410C22#322611yescharacteristic
411C22C2×#51011yescharacteristic
412C22#341411yescharacteristic
413C22#341411yescharacteristic
414C22#341411yescharacteristic
415C4—22no—
416C4—22no—
417C4—41no—
418C4—41no—
419C4—41no—
420C4—82no—
421C4—82no—
422C4—161no—
423C4—161no—
424C4—161no—
425C4—161no—
426C4—161no—
427C4—161no—
428C4#241311yescharacteristic
429C4#305611yescharacteristic
430C4#310811yescharacteristic
431C4#323211yescharacteristic
432C2—42no—
433C2#2460011yescharacteristic
434C2#3360411yescharacteristic
435C2#3360611yescharacteristic
436C2#3367111yescharacteristic
437C2#3367611yescharacteristic
438C2#3380311yescharacteristic
439C2#3380611yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC24C2×D8
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #13516C2semidirect
6. #13515C2semidirect
7. #13477C2semidirect
8. #13475C2semidirect
9. #13168C2semidirect
10. #13167C2semidirect
11. #13150C2semidirect
12. #13150C2semidirect

Maximal subgroups

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

subgroupindex
C2×#22372
C2×#21632
#331302
#211512
#135162
#135152
#134772
#134752
#131682
#131672
#131502
#129942
#129212
#128682