Finite Group Databasebeta

← #507535

Subgroups — #507535

Order 512.

556 conjugacy classes of subgroups.

The subgroup lattice has 443 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#467C211yesmaximal, characteristic
2#21151C211yesmaximal, characteristic
3#13516C211yesmaximal, characteristic, complemented
4#13515C211yesmaximal, characteristic, complemented
5#13477C211yesmaximal, characteristic, complemented
6#13475C211yesmaximal, characteristic, complemented
7#13111C211yesmaximal, characteristic, complemented
8#13109C211yesmaximal, characteristic, complemented
9#12994C211yesmaximal, characteristic
10#12994C211yesmaximal, characteristic
11#12938C211yesmaximal, characteristic, complemented
12#12936C211yesmaximal, characteristic, complemented
13#12921C211yesmaximal, characteristic
14#12921C211yesmaximal, characteristic
15#12783C211yesmaximal, characteristic
16C2×#467C2211yescharacteristic
17C2×C4×Q16C2211yescharacteristic
18C2×#427C2211yescharacteristic
19C2×#427C2211yescharacteristic
20C22×C4×C8C2211yescharacteristic
21C4×C22⋊Q8—21no—
22#2296—21no—
23C2×#367C2211yescharacteristic
24C2×#367C2211yescharacteristic
25C2×#366C2211yescharacteristic
26C2×#366C2211yescharacteristic
27C2×C4⋊C16C2211yescharacteristic
28C2×C4⋊C16C2211yescharacteristic
29C2×#358C2211yescharacteristic
30C2×#358C2211yescharacteristic
31C2×#358C2211yescharacteristic
32C2×#358C2211yescharacteristic
33C2×C22⋊C16C2211yescharacteristic
34C2×C22⋊C16C2211yescharacteristic
35#1969C2211yescharacteristic
36#1942C2211yescharacteristic
37#1942C2211yescharacteristic
38#1942C2211yescharacteristic
39#1942C2211yescharacteristic
40#1942C2211yescharacteristic
41#1941—21no—
42#1941—21no—
43#1933—21no—
44#1892—21no—
45#1808C2211yescharacteristic
46C4×#107C2211yescharacteristic
47C4×#103—21no—
48#1735—21no—
49#1380C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×Q16C2311yescharacteristic
60C23×C8D411yescharacteristic
61C23×C8C2311yescharacteristic
62C2×C22⋊Q8—21no—
63C2×C4×Q8—21no—
64C2×#117—21no—
65C2×#117—21no—
66C22×C42D411yescharacteristic
67C22×C16C2311yescharacteristic
68C22×C16C2311yescharacteristic
69C22×C16C2311yescharacteristic
70C22×C16C2311yescharacteristic
71#467—22no—
72#467Q814yes—
73C4×Q16—41no—
74C4×Q16—41no—
75#427—22no—
76#427—22no—
77#427—41no—
78#427—41no—
79C2×#107C2311yescharacteristic
80C2×#107C2311yescharacteristic
81C2×#107C2311yescharacteristic
82C2×#107C2311yescharacteristic
83C2×#107C2311yescharacteristic
84C2×#107C2311yescharacteristic
85C2×#107C2311yescharacteristic
86C2×#106—21no—
87C2×#106—21no—
88C2×#103—21no—
89C2×#103—21no—
90C2×#103—21no—
91C2×C4×C8—41no—
92C2×C4×C8—41no—
93C2×C4×C8D411yescharacteristic
94C2×C4×C8D411yescharacteristic
95C2×C4×C8C2311yescharacteristic
96C2×C4×C8C2311yescharacteristic
97#391—21no—
98#389—21no—
99#389—21no—
100#389—21no—
101#389—21no—
102#389—21no—
103#387—41no—
104#385—41no—
105#385—41no—
106#385—41no—
107#380—21no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—41no—
110C4×C4⋊C4—41no—
111C4×C4⋊C4—41no—
112C4×C22⋊C4—41no—
113#367—22no—
114#367—41no—
115#366—22no—
116#366—41no—
117C4⋊C16—41no—
118C4⋊C16—41no—
119#358—22no—
120#358—22no—
121#358—41no—
122#358—41no—
123C22⋊C16—22no—
124C22⋊C16—22no—
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—41no—
148C22×C8—41no—
149C22×C8C2×D411yescharacteristic
150C22×C8C2×D411yescharacteristic
151C22×C8C2×D411yescharacteristic
152C22×C8C2×Q812yes—
153C22×C8#4112yes—
154C22×C8C2411yescharacteristic
155C22⋊Q8—24no—
156C22⋊Q8—42no—
157C4×Q8—42no—
158C4×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—41no—
182C2×C42—41no—
183C2×C42—41no—
184C2×C42—41no—
185C2×C42—41no—
186C2×C42—41no—
187C2×C42D811yescharacteristic
188C2×C42D811yescharacteristic
189C2×C42C2×D411yescharacteristic
190C2×C42C2×D411yescharacteristic
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×C16—21no—
200C2×C16—21no—
201C2×C16—21no—
202C2×C16—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#107#4112yes—
216#107#4112yes—
217#106—24no—
218#106—42no—
219#103—24no—
220#103—42no—
221#103—42no—
222C4×C8—21no—
223C4×C8—21no—
224C4×C8—21no—
225C4×C8—21no—
226C4×C8—41no—
227C4×C8—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—41no—
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—22no—
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—22no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—44no—
287C2×C8C22⋊Q812yes—
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—22no—
296C4⋊C4—22no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—44no—
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—44no—
317C22⋊C4—82no—
318C22⋊C4—82no—
319C42—21no—
320C42—21no—
321C42—21no—
322C42—21no—
323C42—41no—
324C42—41no—
325C42—44no—
326C42—82no—
327C42—82no—
328C42—82no—
329C16—41no—
330C16—41no—
331C16—41no—
332C16—41no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23#47412yes—
340C23C22×D811yescharacteristic
341Q8—42no—
342Q8—42no—
343Q8—81no—
344Q8—81no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
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—81no—
387C2×C4#47412yes—
388C2×C4#50812yes—
389C2×C4#51012yes—
390C2×C4C2×C22⋊Q811yescharacteristic
391C2×C4#52511yescharacteristic
392C2×C4#53411yescharacteristic
393C2×C4C2×#13711yescharacteristic
394C2×C4#57511yescharacteristic
395C2×C4#57511yescharacteristic
396C8—22no—
397C8—41no—
398C8—41no—
399C8#55311yescharacteristic
400C8#55311yescharacteristic
401C8#55411yescharacteristic
402C8#55411yescharacteristic
403C22—22no—
404C22—22no—
405C22—22no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22—41no—
410C22C2×#47411yescharacteristic
411C22#310011yescharacteristic
412C22#318611yescharacteristic
413C22C2×#50811yescharacteristic
414C22C2×#51011yescharacteristic
415C22#341411yescharacteristic
416C22#341411yescharacteristic
417C4—22no—
418C4—22no—
419C4—22no—
420C4—41no—
421C4—41no—
422C4—82no—
423C4—82no—
424C4—82no—
425C4—82no—
426C4—161no—
427C4—161no—
428C4—161no—
429C4—161no—
430C4#257911yescharacteristic
431C4#308011yescharacteristic
432C4#311711yescharacteristic
433C4#319611yescharacteristic
434C2—42no—
435C2#2923411yescharacteristic
436C2#3362611yescharacteristic
437C2#3362811yescharacteristic
438C2#3366211yescharacteristic
439C2#3367611yescharacteristic
440C2#3375611yescharacteristic
441C2#3376311yescharacteristic
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. #13516C2semidirect
4. #13515C2semidirect
5. #13477C2semidirect
6. #13475C2semidirect
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
C4×#4672
#211512
#135162
#135152
#134772
#134752
#131112
#131092
#129942
#129382
#129362
#129212
#127832