Finite Group Databasebeta

← #511320

Subgroups — #511320

Order 512.

487 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27661C211yesmaximal, characteristic
2#27660C211yesmaximal, characteristic
3#13516C211yesmaximal, characteristic, complemented
4#13514C211yesmaximal, characteristic, complemented
5#13494C211yesmaximal, characteristic, complemented
6#13494C211yesmaximal, characteristic, complemented
7#13493C211yesmaximal, characteristic, complemented
8#13493C211yesmaximal, characteristic, complemented
9#13189C211yesmaximal, characteristic
10#13166C211yesmaximal, characteristic, complemented
11#13165C211yesmaximal, characteristic, complemented
12#12921C211yesmaximal, characteristic, complemented
13#12921C211yesmaximal, characteristic, complemented
14#12920C211yesmaximal, characteristic, complemented
15#12919C211yesmaximal, characteristic, complemented
16C2×C8×D4C2211yescharacteristic
17C2×#427C2211yescharacteristic
18C22×#107C2211yescharacteristic
19#2486—21no—
20#2485—21no—
21C2×#367C2211yescharacteristic
22C2×#367C2211yescharacteristic
23C2×#367C2211yescharacteristic
24C2×#366C2211yescharacteristic
25C2×#366C2211yescharacteristic
26C2×#366C2211yescharacteristic
27C2×C4⋊C16C2211yescharacteristic
28C2×C22⋊C16C2211yescharacteristic
29C2×C22⋊C16C2211yescharacteristic
30C2×C22⋊C16C2211yescharacteristic
31C2×C22⋊C16C2211yescharacteristic
32C2×C4×C16C2211yescharacteristic
33#2088—21no—
34#2086—21no—
35#2077—21no—
36#2075—21no—
37#1942C2211yescharacteristic
38#1942C2211yescharacteristic
39#1942C2211yescharacteristic
40#1942C2211yescharacteristic
41#1825—21no—
42#1823—21no—
43#1808C2211yescharacteristic
44#1808C2211yescharacteristic
45#1808C2211yescharacteristic
46#1808C2211yescharacteristic
47#1808C2211yescharacteristic
48#1808C2211yescharacteristic
49#1380C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59C23×C8C2311yescharacteristic
60C23×C8C2311yescharacteristic
61C2×#123—41no—
62C2×C4×D4D411yescharacteristic
63C2×#117—21no—
64C22×C4⋊C4—21no—
65C22×C16C2311yescharacteristic
66C22×C16C2311yescharacteristic
67C22×C16C2311yescharacteristic
68C22×C16C2311yescharacteristic
69C8×D4—41no—
70C8×D4—41no—
71#427—41no—
72#427—41no—
73C2×#107—21no—
74C2×#107—41no—
75C2×#107—41no—
76C2×#107C2311yescharacteristic
77C2×#107C2311yescharacteristic
78C2×#107C2311yescharacteristic
79C2×#107C2311yescharacteristic
80C2×#107C2311yescharacteristic
81C2×#107C2311yescharacteristic
82C2×#107C2311yescharacteristic
83C2×#107C2311yescharacteristic
84C2×#106—21no—
85C2×C4⋊C8D411yescharacteristic
86C2×C22⋊C8D411yescharacteristic
87C2×C22⋊C8D411yescharacteristic
88C2×C4×C8C2311yescharacteristic
89#398—41no—
90#397—41no—
91#397—41no—
92#394—41no—
93#389—21no—
94#389—21no—
95#389—21no—
96#389—21no—
97#388—41no—
98#387—41no—
99#381—41no—
100#380—21no—
101#380—21no—
102#380—21no—
103#380—21no—
104#380—21no—
105#380—21no—
106#367—22no—
107#367—41no—
108#367—41no—
109#366—22no—
110#366—41no—
111#366—41no—
112C4⋊C16—41no—
113C22⋊C16—22no—
114C22⋊C16—22no—
115C22⋊C16—22no—
116C22⋊C16—22no—
117C4×C16—41no—
118#340—21no—
119#340—21no—
120#340—21no—
121#340—21no—
122#340—21no—
123#340—21no—
124#340—21no—
125#340—21no—
126#340—21no—
127#340—21no—
128C22×D4D811yescharacteristic
129C23×C4—41no—
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C22×C8—21no—
133C22×C8—21no—
134C22×C8—21no—
135C22×C8—21no—
136C22×C8—21no—
137C22×C8—21no—
138C22×C8—21no—
139C22×C8—21no—
140C22×C8—21no—
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8C2×D411yescharacteristic
145C22×C8C2×D411yescharacteristic
146C22×C8C2×D411yescharacteristic
147C22×C8C2411yescharacteristic
148#123—84no—
149C4×D4—41no—
150C4×D4—41no—
151#117—42no—
152#117—42no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4D811yescharacteristic
169C2×C22⋊C4—41no—
170C2×C22⋊C4—41no—
171C2×C22⋊C4—41no—
172C2×C22⋊C4—41no—
173C2×C22⋊C4—41no—
174C2×C22⋊C4—41no—
175C2×C22⋊C4—41no—
176C2×C22⋊C4D811yescharacteristic
177C2×C22⋊C4D811yescharacteristic
178C2×C42—41no—
179C2×C42C2×D411yescharacteristic
180C2×C16—21no—
181C2×C16—21no—
182C2×C16—21no—
183C2×C16—21no—
184C2×C16—21no—
185C2×C16—21no—
186C2×C16—21no—
187C2×C16—21no—
188C2×C16—21no—
189C2×C16—21no—
190C2×C16#4112yes—
191C2×C16#4112yes—
192#107—21no—
193#107—21no—
194#107—21no—
195#107—21no—
196#107—21no—
197#107—21no—
198#107—21no—
199#107—21no—
200#107—21no—
201#107—21no—
202#107—21no—
203#107—21no—
204#107—21no—
205#107—21no—
206#107—21no—
207#107—21no—
208#107—42no—
209#106—42no—
210C4⋊C8—41no—
211C22⋊C8—41no—
212C22⋊C8—41no—
213C4×C8—21no—
214C4×C8—21no—
215#95—41no—
216#95—41no—
217#95—41no—
218#95—41no—
219#95—41no—
220#95—41no—
221#95—41no—
222#95—41no—
223#95—41no—
224#95—41no—
225C24C2×D811yescharacteristic
226C24C2×D811yescharacteristic
227C2×D4—22no—
228C2×D4—22no—
229C2×D4—41no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—82no—
251C22×C4—82no—
252C22×C4—82no—
253C22×C4C2×D811yescharacteristic
254C22×C4C2×D811yescharacteristic
255C22×C4C2×D811yescharacteristic
256C22×C4C2×D811yescharacteristic
257C22×C4C22×D411yescharacteristic
258C2×C8—21no—
259C2×C8—21no—
260C2×C8—21no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8—21no—
264C2×C8—21no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—41no—
268C2×C8—41no—
269C2×C8—41no—
270C2×C8—41no—
271C2×C8—41no—
272C2×C8—41no—
273C2×C8—41no—
274C2×C8—41no—
275C2×C8C2×#4111yescharacteristic
276C2×C8C2×#4111yescharacteristic
277C2×C8H(5,𝔽2)11yescharacteristic
278C2×C8H(5,𝔽2)11yescharacteristic
279C2×C82−1+411yescharacteristic
280C2×C82−1+411yescharacteristic
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—82no—
299C4⋊C4—82no—
300C4⋊C4—82no—
301C4⋊C4—82no—
302C4⋊C4—161no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C22⋊C4—82no—
306C22⋊C4—82no—
307C22⋊C4—82no—
308C22⋊C4—82no—
309C22⋊C4—82no—
310C22⋊C4—161no—
311C22⋊C4—161no—
312C42—21no—
313C42—21no—
314C42—82no—
315C16—22no—
316C16—22no—
317C16—41no—
318C16—41no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23#50812yes—
333C23C22×D811yescharacteristic
334D4—44no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
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—164no—
379C2×C4#50912yes—
380C2×C4#53411yescharacteristic
381C2×C4#54711yescharacteristic
382C2×C4#54711yescharacteristic
383C2×C4#57511yescharacteristic
384C2×C4#57811yescharacteristic
385C2×C4#57811yescharacteristic
386C8—41no—
387C8—41no—
388C8—41no—
389C8#53911yescharacteristic
390C8#53911yescharacteristic
391C8#55911yescharacteristic
392C8#56311yescharacteristic
393C22—22no—
394C22—22no—
395C22—22no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22#318611yescharacteristic
408C22#333411yescharacteristic
409C22#333411yescharacteristic
410C22C2×#50811yescharacteristic
411C22C2×#50911yescharacteristic
412C22#341611yescharacteristic
413C22#341711yescharacteristic
414C4—22no—
415C4—41no—
416C4—41no—
417C4—41no—
418C4—161no—
419C4—161no—
420C4—161no—
421C4—161no—
422C4—161no—
423C4—161no—
424C4—161no—
425C4—161no—
426C4#272411yescharacteristic
427C4#320211yescharacteristic
428C4#335211yescharacteristic
429C4#335211yescharacteristic
430C2—42no—
431C2—81no—
432C2#3245811yescharacteristic
433C2#3375411yescharacteristic
434C2#3376711yescharacteristic
435C2#3390311yescharacteristic
436C2#3390411yescharacteristic
437C2#3392011yescharacteristic
438C2#3392111yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D4D8
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #13516C2semidirect
4. #13514C2semidirect
5. #13494C2semidirect
6. #13494C2semidirect
7. #13493C2semidirect
8. #13493C2semidirect
10. #13166C2semidirect
11. #13165C2semidirect
12. #12921C2semidirect
13. #12921C2semidirect
14. #12920C2semidirect
15. #12919C2semidirect

Maximal subgroups

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

subgroupindex
#276612
#276602
#135162
#135142
#134942
#134932
#131892
#131662
#131652
#129212
#129202
#129192