Finite Group Databasebeta

← #510301

Subgroups — #510301

Order 512.

508 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27840C211yesmaximal, characteristic
2#25415C211yesmaximal, characteristic
3#13515C211yesmaximal, characteristic, complemented
4#13481C211yesmaximal, characteristic, complemented
5#13475C211yesmaximal, characteristic, complemented
6#13409C211yesmaximal, characteristic, complemented
7#13383C211yesmaximal, characteristic
8#13383C211yesmaximal, characteristic
9#13253C211yesmaximal, characteristic, complemented
10#13112C211yesmaximal, characteristic, complemented
11#13060C211yesmaximal, characteristic
12#13060C211yesmaximal, characteristic
13#12936C211yesmaximal, characteristic, complemented
14#12912C211yesmaximal, characteristic
15#12812C211yesmaximal, characteristic, complemented
16C23×C16C2211yescharacteristic
17C2×#494C2211yescharacteristic
18C2×#467C2211yescharacteristic
19C2×#467C2211yescharacteristic
20C2×C4×Q16C2211yescharacteristic
21#2497—21no—
22#2429—21no—
23C2×#366C2211yescharacteristic
24C2×#366C2211yescharacteristic
25C2×C4⋊C16C2211yescharacteristic
26C2×C4⋊C16C2211yescharacteristic
27C2×#358C2211yescharacteristic
28C2×#358C2211yescharacteristic
29C2×#358C2211yescharacteristic
30C2×#358C2211yescharacteristic
31C2×#358C2211yescharacteristic
32C2×#358C2211yescharacteristic
33C2×#358C2211yescharacteristic
34C2×#358C2211yescharacteristic
35C2×C22⋊C16C2211yescharacteristic
36#2100—21no—
37#2076—21no—
38#2043—21no—
39#2001—21no—
40#1969C2211yescharacteristic
41#1942C2211yescharacteristic
42#1942C2211yescharacteristic
43#1942C2211yescharacteristic
44#1921—21no—
45#1878—21no—
46#1845C2211yescharacteristic
47#1808C2211yescharacteristic
48#1808C2211yescharacteristic
49C4×#107C2211yescharacteristic
50C8×C22⋊C4C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×Q16—21no—
60C22×Q16C2311yescharacteristic
61C22×Q16C2311yescharacteristic
62C23×C8C2311yescharacteristic
63C2×C4⋊Q8—21no—
64C2×C22⋊Q8—21no—
65C2×C22⋊Q8—21no—
66C2×C4×Q8—21no—
67C22×C16—41no—
68C22×C16—41no—
69C22×C16—41no—
70C22×C16C2311yescharacteristic
71C22×C16C2311yescharacteristic
72C22×C16C2311yescharacteristic
73C22×C16C2311yescharacteristic
74#494—22no—
75#494—41no—
76#467—22no—
77#467—22no—
78#467—41no—
79#467—41no—
80C4×Q16—22no—
81C4×Q16—41no—
82C2×#107—21no—
83C2×#107C2311yescharacteristic
84C2×#107C2311yescharacteristic
85C2×#107C2311yescharacteristic
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#107C2311yescharacteristic
89C2×#103—21no—
90C2×#103—21no—
91C2×#103—21no—
92C2×#103—21no—
93C2×C22⋊C8D411yescharacteristic
94C2×C4×C8C2311yescharacteristic
95C2×C4×C8C2311yescharacteristic
96#400—41no—
97#398—41no—
98#396—41no—
99#396—41no—
100#391—21no—
101#389—21no—
102#389—21no—
103#389—21no—
104#389—41no—
105#388—41no—
106#388—41no—
107#387—41no—
108#384—21no—
109#380—21no—
110#380—21no—
111C4×C4⋊C4—21no—
112C4×C22⋊C4D411yescharacteristic
113#366—22no—
114#366—41no—
115C4⋊C16—41no—
116C4⋊C16—41no—
117#358—22no—
118#358—41no—
119#358—41no—
120#358—41no—
121#358—41no—
122#358—41no—
123#358—41no—
124#358—41no—
125C22⋊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—21no—
135C22×Q8—21no—
136C23×C4C2×D411yescharacteristic
137C2×Q16—21no—
138C2×Q16—21no—
139C2×Q16—21no—
140C2×Q16—21no—
141C2×Q16—21no—
142C2×Q16—21no—
143C2×Q16—21no—
144C2×Q16—21no—
145C2×Q16—21no—
146C2×Q16—21no—
147C2×Q16—21no—
148C2×Q16—42no—
149C2×Q16—42no—
150C2×Q16—81no—
151C2×Q16#4112yes—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8C2×D411yescharacteristic
159C22×C8C2×D411yescharacteristic
160C22×C8C2×D411yescharacteristic
161C22×C8C2411yescharacteristic
162C4⋊Q8—42no—
163C4⋊Q8—81no—
164C22⋊Q8—42no—
165C22⋊Q8—42no—
166C22⋊Q8—81no—
167C22⋊Q8—81no—
168C4×Q8—42no—
169C4×Q8—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×C4⋊C4—41no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—41no—
186C2×C22⋊C4—41no—
187C2×C22⋊C4—41no—
188C2×C22⋊C4—41no—
189C2×C22⋊C4—41no—
190C2×C22⋊C4D811yescharacteristic
191C2×C22⋊C4D811yescharacteristic
192C2×C42—41no—
193C2×C42—41no—
194C2×C42C2×D411yescharacteristic
195C2×C42C2×D411yescharacteristic
196C2×C16—21no—
197C2×C16—21no—
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—44no—
209#107—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—42no—
221#107#4112yes—
222#103—42no—
223#103—42no—
224#103—42no—
225#103—81no—
226C22⋊C8—41no—
227C4×C8—21no—
228C4×C8—21no—
229C4×C8—21no—
230C4×C8—21no—
231#95—41no—
232#95—41no—
233#95—41no—
234#95—41no—
235#95—41no—
236#95—41no—
237#95—41no—
238#95—41no—
239#95D811yescharacteristic
240#95D811yescharacteristic
241C24C2×D811yescharacteristic
242C2×Q8—22no—
243C2×Q8—41no—
244C2×Q8—41no—
245C2×Q8—41no—
246C2×Q8—41no—
247C2×Q8—41no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×Q8—41no—
251C2×Q8—41no—
252C2×Q8—41no—
253C2×Q8—41no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4C2×D811yescharacteristic
269C22×C4C2×D811yescharacteristic
270C22×C4C2×D811yescharacteristic
271C22×C4C2×D811yescharacteristic
272C22×C4C2×D811yescharacteristic
273C22×C4C22×D411yescharacteristic
274Q16—22no—
275Q16—22no—
276Q16—41no—
277Q16—41no—
278Q16—41no—
279Q16—41no—
280Q16—41no—
281Q16—41no—
282Q16—84no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—21no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8C2×#4111yescharacteristic
297C2×C8C2×#4111yescharacteristic
298C2×C8H(5,𝔽2)11yescharacteristic
299C2×C8H(5,𝔽2)11yescharacteristic
300C2×C8H(5,𝔽2)11yescharacteristic
301C2×C82−1+411yescharacteristic
302C4⋊C4—22no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—82no—
315C4⋊C4—82no—
316C4⋊C4—82no—
317C4⋊C4—82no—
318C4⋊C4—82no—
319C4⋊C4—82no—
320C4⋊C4—82no—
321C4⋊C4—161no—
322C4⋊C4—161no—
323C4⋊C4—161no—
324C22⋊C4—41no—
325C22⋊C4—82no—
326C22⋊C4—82no—
327C22⋊C4—161no—
328C22⋊C4—161no—
329C22⋊C4Q3214yes—
330C42—21no—
331C42—21no—
332C42—21no—
333C42—21no—
334C42—82no—
335C42—161no—
336C16—41no—
337C16—41no—
338C16—41no—
339C16—41no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23C2×Q3212yes—
347C23C22×D811yescharacteristic
348Q8—42no—
349Q8—42no—
350Q8—81no—
351Q8—81no—
352Q8—81no—
353Q8—81no—
354Q8—81no—
355Q8—81no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—42no—
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—81no—
388C2×C4—81no—
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×C4C2×Q3212yes—
398C2×C4C2×Q3212yes—
399C2×C4#53711yescharacteristic
400C2×C4#54611yescharacteristic
401C2×C4#54611yescharacteristic
402C2×C4#57511yescharacteristic
403C2×C4#57711yescharacteristic
404C2×C4#57711yescharacteristic
405C8—41no—
406C8—41no—
407C8—41no—
408C8#53911yescharacteristic
409C8#54211yescharacteristic
410C8#55311yescharacteristic
411C8#56011yescharacteristic
412C22—22no—
413C22—22no—
414C22—22no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22#324511yescharacteristic
420C22#329411yescharacteristic
421C22#329411yescharacteristic
422C22C22×Q3211yescharacteristic
423C22#341411yescharacteristic
424C22#341611yescharacteristic
425C22#341611yescharacteristic
426C4—22no—
427C4—22no—
428C4—41no—
429C4—41no—
430C4—41no—
431C4—82no—
432C4—82no—
433C4—161no—
434C4—161no—
435C4—161no—
436C4—161no—
437C4—161no—
438C4—161no—
439C4#269111yescharacteristic
440C4#326011yescharacteristic
441C4#331111yescharacteristic
442C4#331711yescharacteristic
443C2—42no—
444C2#3148611yescharacteristic
445C2#3381511yescharacteristic
446C2#3382611yescharacteristic
447C2#3386611yescharacteristic
448C2#3386611yescharacteristic
449C2#3387411yescharacteristic
450C2#3388811yescharacteristic
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
3. #13515C2semidirect
4. #13481C2semidirect
5. #13475C2semidirect
6. #13409C2semidirect
9. #13253C2semidirect
10. #13112C2semidirect
13. #12936C2semidirect
15. #12812C2semidirect

Maximal subgroups

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

subgroupindex
#278402
#254152
#135152
#134812
#134752
#134092
#133832
#132532
#131122
#130602
#129362
#129122
#128122