Finite Group Databasebeta

← #510669

Subgroups — #510669

Order 512.

500 conjugacy classes of subgroups.

The subgroup lattice has 445 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28013C211yesmaximal, characteristic, complemented
2#26484C211yesmaximal, characteristic, complemented
3#13539C211yesmaximal, characteristic, complemented
4#13516C211yesmaximal, characteristic, complemented
5#13476C211yesmaximal, characteristic, complemented
6#13448C211yesmaximal, characteristic
7#13448C211yesmaximal, characteristic
8#13363C211yesmaximal, characteristic, complemented
9#13275C211yesmaximal, characteristic, complemented
10#13160C211yesmaximal, characteristic, complemented
11#13149C211yesmaximal, characteristic, complemented
12#13149C211yesmaximal, characteristic, complemented
13#13131C211yesmaximal, characteristic, complemented
14#13012C211yesmaximal, characteristic
15#12813C211yesmaximal, characteristic, complemented
16C22×D16C2211yescharacteristic
17C2×#500C2211yescharacteristic
18C2×C4×D8C2211yescharacteristic
19#2513—21no—
20#2452—21no—
21C2×#367C2211yescharacteristic
22C2×C4⋊C16C2211yescharacteristic
23C2×C4⋊C16C2211yescharacteristic
24C2×C4⋊C16C2211yescharacteristic
25C2×#357C2211yescharacteristic
26C2×#357C2211yescharacteristic
27C2×#357C2211yescharacteristic
28C2×#357C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×#357C2211yescharacteristic
31C2×#346C2211yescharacteristic
32#2073—21no—
33#2067—21no—
34#2065—21no—
35#2023—21no—
36#1970C2211yescharacteristic
37#1970C2211yescharacteristic
38#1970C2211yescharacteristic
39#1970C2211yescharacteristic
40#1970C2211yescharacteristic
41#1942C2211yescharacteristic
42#1942C2211yescharacteristic
43#1942C2211yescharacteristic
44#1926—21no—
45#1862—21no—
46#1846C2211yescharacteristic
47C4×#107C2211yescharacteristic
48C8×C4⋊C4C2211yescharacteristic
49#1380C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×D8C2311yescharacteristic
60C22×D8C2311yescharacteristic
61C2×C4⋊Q8—21no—
62C2×C22⋊Q8—41no—
63C2×C4×D4—21no—
64C2×D16—22no—
65C2×D16—41no—
66C2×D16—41no—
67C22×C16C2311yescharacteristic
68C22×C16C2311yescharacteristic
69C22×C16C2311yescharacteristic
70C22×C16C2311yescharacteristic
71#500—22no—
72#500—41no—
73C4×D8—22no—
74C4×D8—41no—
75C2×#107—21no—
76C2×#107C2311yescharacteristic
77C2×#107C2311yescharacteristic
78C2×#107C2311yescharacteristic
79C2×#107C2311yescharacteristic
80C2×#107C2311yescharacteristic
81C2×#107C2311yescharacteristic
82C2×#106—21no—
83C2×C4⋊C8D411yescharacteristic
84C2×#102—21no—
85C2×#102—21no—
86C2×#102—21no—
87C2×#102—21no—
88C2×#102—21no—
89C2×#102—21no—
90C2×C4×C8C2311yescharacteristic
91C2×C4×C8C2311yescharacteristic
92C2×C4×C8C2311yescharacteristic
93#397—41no—
94#397—41no—
95#396—41no—
96#393—41no—
97#390—21no—
98#390—21no—
99#390—21no—
100#390—21no—
101#390—21no—
102#389—21no—
103#389—21no—
104#389—21no—
105#388—41no—
106#388—41no—
107#385—41no—
108#382—21no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4D411yescharacteristic
111#367—41no—
112C4⋊C16—41no—
113C4⋊C16—41no—
114C4⋊C16—41no—
115#357—22no—
116#357—22no—
117#357—22no—
118#357—41no—
119#357—41no—
120#357—41no—
121#346—41no—
122#340—21no—
123#340—21no—
124#340—21no—
125#340—21no—
126#336D411yescharacteristic
127#336D411yescharacteristic
128C22×Q8—41no—
129C22×D4—21no—
130C22×D4—21no—
131C23×C4—41no—
132C2×D8—21no—
133C2×D8—21no—
134C2×D8—21no—
135C2×D8—21no—
136C2×D8—21no—
137C2×D8—21no—
138C2×D8—21no—
139C2×D8—21no—
140C2×D8—21no—
141C2×D8—21no—
142C2×D8—21no—
143C2×D8#4112yes—
144C22×C8C2×D411yescharacteristic
145C22×C8C2×D411yescharacteristic
146C22×C8C2×D411yescharacteristic
147C22×C8C2411yescharacteristic
148C4⋊Q8—42no—
149C4⋊Q8—81no—
150C22⋊Q8—84no—
151C4×D4—42no—
152C4×D4—81no—
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—41no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4D811yescharacteristic
164C2×C4⋊C4D811yescharacteristic
165C2×C22⋊C4—41no—
166C2×C22⋊C4—41no—
167C2×C22⋊C4—41no—
168C2×C22⋊C4—41no—
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×C42—41no—
177C2×C42C2×D411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C42C2×D411yescharacteristic
180D16—44no—
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—21no—
191C2×C16—21no—
192C2×C16—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—42no—
205#107#4112yes—
206#106—81no—
207C4⋊C8—41no—
208#102—42no—
209#102—42no—
210#102—42no—
211#102—81no—
212#102—81no—
213#102—81no—
214C4×C8—21no—
215C4×C8—21no—
216C4×C8—21no—
217C4×C8—21no—
218C4×C8—21no—
219C4×C8—21no—
220#95—41no—
221#95—41no—
222#95—41no—
223#95—41no—
224#95—41no—
225#95—41no—
226#95D811yescharacteristic
227#95D811yescharacteristic
228C24—41no—
229C24—41no—
230C2×Q8—82no—
231C2×Q8—82no—
232C2×Q8—161no—
233C2×D4—22no—
234C2×D4—41no—
235C2×D4—41no—
236C2×D4—41no—
237C2×D4—41no—
238C2×D4—41no—
239C2×D4—41no—
240C2×D4—41no—
241C2×D4—41no—
242C2×D4—41no—
243C2×D4—41no—
244C2×D4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—82no—
252C22×C4—82no—
253C22×C4—161no—
254C22×C4C2×D811yescharacteristic
255C22×C4C2×D811yescharacteristic
256C22×C4C2×D811yescharacteristic
257C22×C4C2×D811yescharacteristic
258C22×C4C2×D811yescharacteristic
259C22×C4C2×D811yescharacteristic
260C22×C4C22×D411yescharacteristic
261D8—22no—
262D8—22no—
263D8—41no—
264D8—41no—
265D8—41no—
266D8—41no—
267D8—41no—
268D8—41no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—21no—
272C2×C8—21no—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8C2×#4111yescharacteristic
279C2×C8C2×#4111yescharacteristic
280C2×C8H(5,𝔽2)11yescharacteristic
281C2×C8H(5,𝔽2)11yescharacteristic
282C2×C82−1+411yescharacteristic
283C2×C82−1+411yescharacteristic
284C4⋊C4—22no—
285C4⋊C4—22no—
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—161no—
302C22⋊C4—82no—
303C22⋊C4—82no—
304C22⋊C4—82no—
305C22⋊C4—82no—
306C22⋊C4—82no—
307C22⋊C4—82no—
308C22⋊C4—161no—
309C22⋊C4—161no—
310C22⋊C4—161no—
311C22⋊C4—161no—
312C22⋊C4—161no—
313C42—21no—
314C42—21no—
315C42—21no—
316C42—21no—
317C42—21no—
318C42—21no—
319C42—161no—
320C16—41no—
321C16—41no—
322C16—41no—
323C16—41no—
324C23—42no—
325C23—81no—
326C23—81no—
327C23—81no—
328C23—81no—
329C23—81no—
330C23—81no—
331C23—81no—
332C23—81no—
333C23—81no—
334C23—81no—
335C23—81no—
336C23—81no—
337C23—81no—
338C23C22×D811yescharacteristic
339Q8—164no—
340D4—42no—
341D4—42no—
342D4—81no—
343D4—81no—
344D4—81no—
345D4—81no—
346D4—81no—
347D4—81no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
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—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—42no—
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—164no—
384C2×C4#50912yes—
385C2×C4#53711yescharacteristic
386C2×C4#54611yescharacteristic
387C2×C4#54811yescharacteristic
388C2×C4#57511yescharacteristic
389C2×C4#57711yescharacteristic
390C2×C4#57711yescharacteristic
391C8—41no—
392C8—41no—
393C8—41no—
394C8#53811yescharacteristic
395C8#54211yescharacteristic
396C8#55211yescharacteristic
397C8#56311yescharacteristic
398C22—82no—
399C22—82no—
400C22—82no—
401C22—161no—
402C22—161no—
403C22—161no—
404C22—161no—
405C22—161no—
406C22—161no—
407C22—161no—
408C22—161no—
409C22—161no—
410C22—161no—
411C22—161no—
412C22#324511yescharacteristic
413C22#329411yescharacteristic
414C22#335811yescharacteristic
415C22C2×#50911yescharacteristic
416C22#341411yescharacteristic
417C22#341511yescharacteristic
418C22#341611yescharacteristic
419C4—22no—
420C4—41no—
421C4—41no—
422C4—41no—
423C4—41no—
424C4—41no—
425C4—82no—
426C4—161no—
427C4—161no—
428C4—161no—
429C4—161no—
430C4—161no—
431C4#270211yescharacteristic
432C4#326311yescharacteristic
433C4#332611yescharacteristic
434C4#337611yescharacteristic
435C2—162no—
436C2—321no—
437C2#3180211yescharacteristic
438C2#3381711yescharacteristic
439C2#3382311yescharacteristic
440C2#3388511yescharacteristic
441C2#3388811yescharacteristic
442C2#3393711yescharacteristic
443C2#3394011yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D16C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ C8×C4⋊C4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #28013C2semidirect
2. #26484C2semidirect
3. #13539C2semidirect
4. #13516C2semidirect
5. #13476C2semidirect
8. #13363C2semidirect
9. #13275C2semidirect
10. #13160C2semidirect
11. #13149C2semidirect
12. #13149C2semidirect
13. #13131C2semidirect
15. #12813C2semidirect

Maximal subgroups

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

subgroupindex
#280132
#264842
#135392
#135162
#134762
#134482
#133632
#132752
#131602
#131492
#131312
#130122
#128132