Finite Group Databasebeta

← #507434

Subgroups — #507434

Order 512.

556 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
1#33208C211yesmaximal, characteristic
2C4×#467C211yesmaximal, characteristic
3#13539C211yesmaximal, characteristic, complemented
4#13529C211yesmaximal, characteristic, complemented
5#13453C211yesmaximal, characteristic, complemented
6#13366C211yesmaximal, characteristic, complemented
7#13111C211yesmaximal, characteristic, complemented
8#13109C211yesmaximal, characteristic, complemented
9#13060C211yesmaximal, characteristic
10#13060C211yesmaximal, characteristic
11#12938C211yesmaximal, characteristic, complemented
12#12936C211yesmaximal, characteristic, complemented
13#12835C211yesmaximal, characteristic
14#12834C211yesmaximal, characteristic
15C2×#1374C211yesmaximal, characteristic
16C23×C16C2211yescharacteristic
17C23×C16C2211yescharacteristic
18C2×#499C2211yescharacteristic
19C2×#467C2211yescharacteristic
20C2×C4×Q16C2211yescharacteristic
21C22×C4×C8C2211yescharacteristic
22#2851—21no—
23C4×C22⋊Q8—21no—
24C2×#367C2211yescharacteristic
25C2×#367C2211yescharacteristic
26C2×#366C2211yescharacteristic
27C2×#366C2211yescharacteristic
28C2×#358C2211yescharacteristic
29C2×#358C2211yescharacteristic
30C2×#358C2211yescharacteristic
31C2×#358C2211yescharacteristic
32#1991—21no—
33#1969C2211yescharacteristic
34#1942C2211yescharacteristic
35#1942C2211yescharacteristic
36#1942C2211yescharacteristic
37#1941—21no—
38#1933—21no—
39#1892—21no—
40#1846C2211yescharacteristic
41#1808C2211yescharacteristic
42#1808C2211yescharacteristic
43#1808C2211yescharacteristic
44#1807—21no—
45C4×#107C2211yescharacteristic
46C4×#103—21no—
47#1380C2211yescharacteristic
48#1380C2211yescharacteristic
49#1380C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1374C2211yescharacteristic
56#1374C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×Q16C2311yescharacteristic
60C23×C8D411yescharacteristic
61C23×C8C2311yescharacteristic
62C2×#125—21no—
63C2×C22⋊Q8—21no—
64C2×C4×Q8—21no—
65C22×C42D411yescharacteristic
66C22×C16—22no—
67C22×C16—22no—
68C22×C16—22no—
69C22×C16—22no—
70C22×C16—41no—
71C22×C16—41no—
72C22×C16C2311yescharacteristic
73C22×C16C2311yescharacteristic
74C22×C16C2311yescharacteristic
75C22×C16C2311yescharacteristic
76#499—41no—
77#499—41no—
78#467—22no—
79#467Q814yes—
80C4×Q16—41no—
81C4×Q16—41no—
82C2×#107C2311yescharacteristic
83C2×#107C2311yescharacteristic
84C2×#107C2311yescharacteristic
85C2×#107C2311yescharacteristic
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#107C2311yescharacteristic
89C2×#106—21no—
90C2×#106—21no—
91C2×#103—21no—
92C2×#103—21no—
93C2×#103—21no—
94C2×C4×C8—41no—
95C2×C4×C8—41no—
96C2×C4×C8D411yescharacteristic
97C2×C4×C8D411yescharacteristic
98C2×C4×C8C2311yescharacteristic
99C2×C4×C8C2311yescharacteristic
100#391—21no—
101#389—21no—
102#389—21no—
103#389—21no—
104#387—41no—
105#387—41no—
106#387—41no—
107#385—41no—
108#385—41no—
109#385—41no—
110#382—21no—
111#380—21no—
112#380—21no—
113#380—21no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—41no—
116C4×C22⋊C4—41no—
117#367—22no—
118#367—41no—
119#366—22no—
120#366—41no—
121#358—22no—
122#358—22no—
123#358—41no—
124#358—41no—
125#340—21no—
126#340—21no—
127#340—21no—
128#340—21no—
129#340—21no—
130#340—21no—
131#340—21no—
132C22×Q8—21no—
133C23×C4D811yescharacteristic
134C23×C4D811yescharacteristic
135C23×C4C2×D411yescharacteristic
136C2×Q16—21no—
137C2×Q16—21no—
138C2×Q16—21no—
139C2×Q16—21no—
140C2×Q16—21no—
141C2×Q16C2×Q812yes—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—41no—
149C22×C8C2×D411yescharacteristic
150C22×C8C2×D411yescharacteristic
151C22×C8C2×D411yescharacteristic
152C22×C8C2×Q812yes—
153C22×C8#4112yes—
154C22×C8C2411yescharacteristic
155#125—42no—
156#125—42no—
157C22⋊Q8—24no—
158C22⋊Q8—42no—
159C4×Q8—42no—
160C4×Q8—42no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—41no—
175C2×C22⋊C4—41no—
176C2×C22⋊C4—41no—
177C2×C22⋊C4—41no—
178C2×C22⋊C4—41no—
179C2×C42—41no—
180C2×C42—41no—
181C2×C42—41no—
182C2×C42—41no—
183C2×C42D811yescharacteristic
184C2×C42D811yescharacteristic
185C2×C42C2×D411yescharacteristic
186C2×C42C2×D411yescharacteristic
187C2×C16—21no—
188C2×C16—21no—
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—44no—
198C2×C16—44no—
199C2×C16#4112yes—
200C2×C16#4112yes—
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—21no—
214#107C2×Q812yes—
215#106—42no—
216#106—42no—
217#103—24no—
218#103—42no—
219#103—42no—
220C4×C8—21no—
221C4×C8—21no—
222C4×C8—21no—
223C4×C8—21no—
224C4×C8—41no—
225C4×C8—41no—
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—22no—
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×C4C22⋊Q812yes—
263C22×C4C2×D811yescharacteristic
264C22×C4C2×D811yescharacteristic
265C22×C4C2×D811yescharacteristic
266C22×C4C2×D811yescharacteristic
267C22×C4C2×D811yescharacteristic
268C22×C4#13712yes—
269C22×C4C22×D411yescharacteristic
270Q16—22no—
271Q16—22no—
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×C8C22⋊Q812yes—
287C2×C8#13512yes—
288C2×C8C22×Q811yescharacteristic
289C2×C8C2×#4111yescharacteristic
290C2×C8C2×#4111yescharacteristic
291C2×C8C2×#4111yescharacteristic
292C2×C8H(5,𝔽2)11yescharacteristic
293C2×C8H(5,𝔽2)11yescharacteristic
294C4⋊C4—22no—
295C4⋊C4—41no—
296C4⋊C4—44no—
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—41no—
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—41no—
324C42—41no—
325C42—82no—
326C42—82no—
327C16—22no—
328C16—22no—
329C16—41no—
330C16—41no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23#47412yes—
336C23#50812yes—
337C23#51012yes—
338C23C22×D811yescharacteristic
339Q8—42no—
340Q8—42no—
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#47412yes—
387C2×C4#50812yes—
388C2×C4C2×C22⋊Q811yescharacteristic
389C2×C4#53211yescharacteristic
390C2×C4#53411yescharacteristic
391C2×C4C2×#13511yescharacteristic
392C2×C4C2×#13711yescharacteristic
393C2×C4#57511yescharacteristic
394C8—22no—
395C8—22no—
396C8—41no—
397C8#55111yescharacteristic
398C8#55111yescharacteristic
399C8#55411yescharacteristic
400C8#55411yescharacteristic
401C22—22no—
402C22—22no—
403C22—22no—
404C22—22no—
405C22—22no—
406C22—22no—
407C22#222914yes—
408C22C2×#47411yescharacteristic
409C22#316911yescharacteristic
410C22#318611yescharacteristic
411C22C2×#50811yescharacteristic
412C22C2×#50811yescharacteristic
413C22C2×#51011yescharacteristic
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#256411yescharacteristic
429C4#307611yescharacteristic
430C4#318111yescharacteristic
431C4#319011yescharacteristic
432C2—24no—
433C2#2915711yescharacteristic
434C2C2×#222911yescharacteristic
435C2#3362611yescharacteristic
436C2#3373511yescharacteristic
437C2#3374711yescharacteristic
438C2#3375611yescharacteristic
439C2#3375711yescharacteristic
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
3. #13539C2semidirect
4. #13529C2semidirect
5. #13453C2semidirect
6. #13366C2semidirect
7. #13111C2semidirect
8. #13109C2semidirect
11. #12938C2semidirect
12. #12936C2semidirect

Maximal subgroups

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

subgroupindex
#332082
C4×#4672
#135392
#135292
#134532
#133662
#131112
#131092
#130602
#129382
#129362
#128352
#128342
C2×#13742