Finite Group Databasebeta

← #511300

Subgroups — #511300

Order 512.

494 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28534C211yesmaximal, characteristic, complemented
2#28013C211yesmaximal, characteristic, complemented
3#13526C211yesmaximal, characteristic, complemented
4#13482C211yesmaximal, characteristic, complemented
5#13482C211yesmaximal, characteristic, complemented
6#13453C211yesmaximal, characteristic, complemented
7#13448C211yesmaximal, characteristic
8#13365C211yesmaximal, characteristic, complemented
9#13239C211yesmaximal, characteristic
10#13178C211yesmaximal, characteristic, complemented
11#13130C211yesmaximal, characteristic, complemented
12#13105C211yesmaximal, characteristic, complemented
13#13012C211yesmaximal, characteristic
14#12935C211yesmaximal, characteristic, complemented
15#12890C211yesmaximal, characteristic, complemented
16C22×D16C2211yescharacteristic
17C2×#499C2211yescharacteristic
18C2×#478—21no—
19C2×C4×D8C2211yescharacteristic
20#2566—21no—
21#2513—21no—
22C2×#367C2211yescharacteristic
23C2×#366C2211yescharacteristic
24C2×#366C2211yescharacteristic
25C2×C4⋊C16C2211yescharacteristic
26C2×#357C2211yescharacteristic
27C2×#357C2211yescharacteristic
28C2×#357C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×#357C2211yescharacteristic
31C2×#357C2211yescharacteristic
32C2×#346C2211yescharacteristic
33#2094—21no—
34#2092—21no—
35#2073—21no—
36#2065—21no—
37#1970C2211yescharacteristic
38#1970C2211yescharacteristic
39#1970C2211yescharacteristic
40#1970C2211yescharacteristic
41#1970C2211yescharacteristic
42#1942C2211yescharacteristic
43#1926—21no—
44#1846C2211yescharacteristic
45#1846C2211yescharacteristic
46#1845C2211yescharacteristic
47C4×#107C2211yescharacteristic
48C8×C4⋊C4C2211yescharacteristic
49#1380C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1374C2211yescharacteristic
56#1374C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×D8C2311yescharacteristic
60C22×D8C2311yescharacteristic
61C2×#125—21no—
62C2×C4×D4—21no—
63C2×D16—22no—
64C2×D16—22no—
65C2×D16—41no—
66C22×C16C2311yescharacteristic
67C22×C16C2311yescharacteristic
68C22×C16C2311yescharacteristic
69C22×C16C2311yescharacteristic
70#499—41no—
71#499—41no—
72#478—44no—
73C4×D8—41no—
74C4×D8—41no—
75C2×#107—21no—
76C2×#107C2311yescharacteristic
77C2×#107C2311yescharacteristic
78C2×#107C2311yescharacteristic
79C2×#107C2311yescharacteristic
80C2×#107C2311yescharacteristic
81C2×#107C2311yescharacteristic
82C2×#106—21no—
83C2×#106—21no—
84C2×#106—21no—
85C2×C4⋊C8D411yescharacteristic
86C2×#102—21no—
87C2×#102—21no—
88C2×#102—21no—
89C2×#102—21no—
90C2×#102—21no—
91C2×#102—21no—
92C2×C4×C8C2311yescharacteristic
93C2×C4×C8C2311yescharacteristic
94C2×C4×C8C2311yescharacteristic
95#397—41no—
96#397—41no—
97#396—41no—
98#396—41no—
99#390—21no—
100#390—21no—
101#390—21no—
102#390—21no—
103#390—21no—
104#389—21no—
105#388—41no—
106#388—41no—
107#385—41no—
108#385—41no—
109#384—21no—
110#382—21no—
111#382—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4D411yescharacteristic
114#367—41no—
115#366—41no—
116#366—41no—
117C4⋊C16—22no—
118#357—22no—
119#357—22no—
120#357—41no—
121#357—41no—
122#357—41no—
123#357—41no—
124#346—22no—
125#340—21no—
126#340—21no—
127#336D411yescharacteristic
128#336D411yescharacteristic
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
148#125—42no—
149#125—42no—
150C4×D4—42no—
151C4×D4—42no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—41no—
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#4112yes—
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—42no—
206#106—42no—
207#106—42no—
208#106—81no—
209C4⋊C8—22no—
210#102—42no—
211#102—42no—
212#102—42no—
213#102—42no—
214#102—81no—
215#102—81no—
216C4×C8—21no—
217C4×C8—21no—
218C4×C8—21no—
219C4×C8—21no—
220C4×C8—21no—
221C4×C8—21no—
222#95—41no—
223#95—41no—
224#95—41no—
225#95—41no—
226#95—41no—
227#95—41no—
228#95D811yescharacteristic
229#95D811yescharacteristic
230C24—41no—
231C24—41no—
232C2×D4—22no—
233C2×D4—41no—
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—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—82no—
251C22×C4—161no—
252C22×C4—161no—
253C22×C4C2×D811yescharacteristic
254C22×C4C2×D811yescharacteristic
255C22×C4C2×D811yescharacteristic
256C22×C4C2×D811yescharacteristic
257C22×C4C2×D811yescharacteristic
258C22×C4C2×D811yescharacteristic
259C22×C4C22×D411yescharacteristic
260D8—22no—
261D8—22no—
262D8—41no—
263D8—41no—
264D8—41no—
265D8—41no—
266D8—41no—
267D8—41no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—21no—
272C2×C8—21no—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8AGL(1,ℤ/8)12yes—
277C2×C8C2×#4111yescharacteristic
278C2×C8C2×#4111yescharacteristic
279C2×C8H(5,𝔽2)11yescharacteristic
280C2×C8H(5,𝔽2)11yescharacteristic
281C2×C82−1+411yescharacteristic
282C2×C82−1+411yescharacteristic
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—82no—
298C4⋊C4—82no—
299C4⋊C4—161no—
300C4⋊C4—161no—
301C4⋊C4—161no—
302C22⋊C4—82no—
303C22⋊C4—82no—
304C22⋊C4—82no—
305C22⋊C4—82no—
306C22⋊C4—82no—
307C22⋊C4—161no—
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—22no—
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
339D4—42no—
340D4—42no—
341D4—81no—
342D4—81no—
343D4—81no—
344D4—81no—
345D4—81no—
346D4—81no—
347C2×C4—21no—
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—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—164no—
384C2×C4#53511yescharacteristic
385C2×C4#54711yescharacteristic
386C2×C4#54811yescharacteristic
387C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
388C2×C4#57711yescharacteristic
389C2×C4#57811yescharacteristic
390C8—22no—
391C8—41no—
392C8—41no—
393C8#53911yescharacteristic
394C8#54211yescharacteristic
395C8#55111yescharacteristic
396C8#56311yescharacteristic
397C22—82no—
398C22—82no—
399C22—82no—
400C22—161no—
401C22—161no—
402C22—161no—
403C22—161no—
404C22—161no—
405C22—161no—
406C22—161no—
407C22—161no—
408C22—161no—
409C22—161no—
410C22—161no—
411C22#320511yescharacteristic
412C22#333411yescharacteristic
413C22#335811yescharacteristic
414C22#341411yescharacteristic
415C22#341411yescharacteristic
416C22#341511yescharacteristic
417C22#341511yescharacteristic
418C4—41no—
419C4—41no—
420C4—41no—
421C4—41no—
422C4—41no—
423C4—41no—
424C4—161no—
425C4—161no—
426C4—161no—
427C4—161no—
428C4—161no—
429C4—161no—
430C4#272311yescharacteristic
431C4#322511yescharacteristic
432C4#334511yescharacteristic
433C4#336711yescharacteristic
434C2—162no—
435C2—321no—
436C2#3242611yescharacteristic
437C2#3378511yescharacteristic
438C2#3378611yescharacteristic
439C2#3391411yescharacteristic
440C2#3392211yescharacteristic
441C2#3393711yescharacteristic
442C2#3393711yescharacteristic
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. #28534C2semidirect
2. #28013C2semidirect
3. #13526C2semidirect
4. #13482C2semidirect
5. #13482C2semidirect
6. #13453C2semidirect
8. #13365C2semidirect
10. #13178C2semidirect
11. #13130C2semidirect
12. #13105C2semidirect
14. #12935C2semidirect
15. #12890C2semidirect

Maximal subgroups

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

subgroupindex
#285342
#280132
#135262
#134822
#134532
#134482
#133652
#132392
#131782
#131302
#131052
#130122
#129352
#128902