Finite Group Databasebeta

← #510347

Subgroups — #510347

Order 512.

521 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#28531C211yesmaximal, characteristic, complemented
2#26482C211yesmaximal, characteristic
3#13526C211yesmaximal, characteristic
4#13505C211yesmaximal, characteristic, complemented
5#13475C211yesmaximal, characteristic, complemented
6#13451C211yesmaximal, characteristic, complemented
7#13366C211yesmaximal, characteristic, complemented
8#13363C211yesmaximal, characteristic
9#13284C211yesmaximal, characteristic, complemented
10#13238C211yesmaximal, characteristic
11#13105C211yesmaximal, characteristic, complemented
12#13014C211yesmaximal, characteristic
13#13013C211yesmaximal, characteristic
14#12936C211yesmaximal, characteristic, complemented
15#12811C211yesmaximal, characteristic
16C22×SD32C2211yescharacteristic
17C2×#500C2211yescharacteristic
18C2×#499C2211yescharacteristic
19C2×#477—21no—
20C2×C4×Q16C2211yescharacteristic
21C2×C4×D8C2211yescharacteristic
22#2567—21no—
23#2454—21no—
24C2×#366C2211yescharacteristic
25C2×#366C2211yescharacteristic
26C2×#366C2211yescharacteristic
27C2×C4⋊C16C2211yescharacteristic
28C2×#358C2211yescharacteristic
29C2×#358C2211yescharacteristic
30C2×#358C2211yescharacteristic
31C2×#357C2211yescharacteristic
32C2×#357C2211yescharacteristic
33C2×#357C2211yescharacteristic
34C2×#346C2211yescharacteristic
35#2095—21no—
36#2095—21no—
37#2065—21no—
38#2025—21no—
39#1970C2211yescharacteristic
40#1970C2211yescharacteristic
41#1969C2211yescharacteristic
42#1969C2211yescharacteristic
43#1942C2211yescharacteristic
44#1942C2211yescharacteristic
45#1864—21no—
46#1845C2211yescharacteristic
47#1845C2211yescharacteristic
48C8×C4⋊C4C2211yescharacteristic
49#1380C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1374C2211yescharacteristic
56#1374C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×Q16C2311yescharacteristic
60C22×D8C2311yescharacteristic
61C2×C4⋊Q8—21no—
62C2×#125—21no—
63C2×C4×Q8—21no—
64C2×C4×D4—21no—
65C2×SD32—41no—
66C2×SD32—41no—
67C2×SD32Q814yescomplemented
68C22×C16C2311yescharacteristic
69C22×C16C2311yescharacteristic
70C22×C16C2311yescharacteristic
71C22×C16C2311yescharacteristic
72#500—41no—
73#500—41no—
74#499—22no—
75#499—41no—
76#477—44no—
77C4×Q16—41no—
78C4×Q16—41no—
79C4×D8—22no—
80C4×D8—41no—
81C2×#107—21no—
82C2×#107—21no—
83C2×#107—21no—
84C2×#107C2311yescharacteristic
85C2×#107C2311yescharacteristic
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#107C2311yescharacteristic
89C2×#107C2311yescharacteristic
90C2×#106—21no—
91C2×C4⋊C8D411yescharacteristic
92C2×#103—21no—
93C2×#103—21no—
94C2×#103—21no—
95C2×#102—21no—
96C2×#102—21no—
97C2×#102—21no—
98C2×C4×C8C2311yescharacteristic
99C2×C4×C8C2311yescharacteristic
100C2×C4×C8C2311yescharacteristic
101#399—41no—
102#396—41no—
103#395—41no—
104#395—41no—
105#391—21no—
106#391—21no—
107#390—21no—
108#390—21no—
109#389—21no—
110#389—21no—
111#389—41no—
112#389—41no—
113#385—41no—
114#384—21no—
115#384—21no—
116#380—41no—
117C4×C4⋊C4D411yescharacteristic
118#366—41no—
119#366—41no—
120#366—41no—
121C4⋊C16—41no—
122#358—22no—
123#358—41no—
124#358—41no—
125#357—41no—
126#357—41no—
127#357—41no—
128#346—41no—
129#340—21no—
130#340—21no—
131#336D411yescharacteristic
132#336D411yescharacteristic
133C22×Q8—21no—
134C22×Q8—41no—
135C22×D4—21no—
136C23×C4—41no—
137C2×Q16—21no—
138C2×Q16—21no—
139C2×Q16—21no—
140C2×Q16—21no—
141C2×Q16—21no—
142C2×Q16C2×Q812yes—
143C2×D8—21no—
144C2×D8—21no—
145C2×D8—21no—
146C2×D8—21no—
147C2×D8—21no—
148C2×D8C2×Q812yes—
149C22×C8C2×D411yescharacteristic
150C22×C8C2×D411yescharacteristic
151C22×C8C2×D411yescharacteristic
152C22×C8C2411yescharacteristic
153C4⋊Q8—42no—
154C4⋊Q8—42no—
155#125—42no—
156#125—81no—
157C4×Q8—42no—
158C4×Q8—42no—
159C4×D4—42no—
160C4×D4—81no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—41no—
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×C4⋊C4D811yescharacteristic
178C2×C4⋊C4D811yescharacteristic
179C2×C22⋊C4—41no—
180C2×C22⋊C4—41no—
181C2×C22⋊C4—41no—
182C2×C22⋊C4—41no—
183C2×C22⋊C4—41no—
184C2×C42—41no—
185C2×C42C2×D411yescharacteristic
186C2×C42C2×D411yescharacteristic
187C2×C42C2×D411yescharacteristic
188SD32—44no—
189C2×C16—21no—
190C2×C16—21no—
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×C16C2×Q812yes—
201#107—21no—
202#107—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—42no—
213#107—42no—
214#107—42no—
215#107#4112yes—
216#106—42no—
217C4⋊C8—22no—
218#103—42no—
219#103—42no—
220#103—42no—
221#102—42no—
222#102—42no—
223#102—81no—
224C4×C8—21no—
225C4×C8—21no—
226C4×C8—21no—
227C4×C8—21no—
228C4×C8—21no—
229C4×C8—21no—
230#95—41no—
231#95—41no—
232#95—41no—
233#95—41no—
234#95—41no—
235#95—41no—
236#95D811yescharacteristic
237#95D811yescharacteristic
238C24—41no—
239C2×Q8—22no—
240C2×Q8—41no—
241C2×Q8—41no—
242C2×Q8—41no—
243C2×Q8—41no—
244C2×Q8—41no—
245C2×Q8—82no—
246C2×Q8—161no—
247C2×Q8—161no—
248C2×D4—22no—
249C2×D4—41no—
250C2×D4—41no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—82no—
262C22×C4—82no—
263C22×C4—161no—
264C22×C4C2×D811yescharacteristic
265C22×C4C2×D811yescharacteristic
266C22×C4C2×D811yescharacteristic
267C22×C4C2×D811yescharacteristic
268C22×C4C2×D811yescharacteristic
269C22×C4C2×D811yescharacteristic
270C22×C4C22×D411yescharacteristic
271Q16—22no—
272Q16—22no—
273Q16—41no—
274Q16—41no—
275D8—22no—
276D8—22no—
277D8—41no—
278D8—41no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8AGL(1,ℤ/8)12yes—
288C2×C8C22×Q811yescharacteristic
289C2×C8C2×#4111yescharacteristic
290C2×C8H(5,𝔽2)11yescharacteristic
291C2×C8H(5,𝔽2)11yescharacteristic
292C2×C8H(5,𝔽2)11yescharacteristic
293C2×C82−1+411yescharacteristic
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—82no—
309C4⋊C4—82no—
310C4⋊C4—82no—
311C4⋊C4—82no—
312C4⋊C4—161no—
313C4⋊C4—161no—
314C4⋊C4—161no—
315C4⋊C4—161no—
316C4⋊C4—161no—
317C4⋊C4—161no—
318C22⋊C4—82no—
319C22⋊C4—82no—
320C22⋊C4—161no—
321C22⋊C4—161no—
322C22⋊C4—161no—
323C42—21no—
324C42—21no—
325C42—21no—
326C42—21no—
327C42—21no—
328C42—21no—
329C42—82no—
330C16—22no—
331C16—41no—
332C16—41no—
333C16—41no—
334C23—42no—
335C23—81no—
336C23—81no—
337C23—81no—
338C23—81no—
339C23—81no—
340C23—81no—
341C23C22×D811yescharacteristic
342Q8—42no—
343Q8—42no—
344Q8—81no—
345Q8—81no—
346Q8—164no—
347D4—42no—
348D4—42no—
349D4—81no—
350D4—81no—
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—21no—
366C2×C4—21no—
367C2×C4—42no—
368C2×C4—42no—
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—81no—
388C2×C4—164no—
389C2×C4#51012yes—
390C2×C4#51012yes—
391C2×C4#53511yescharacteristic
392C2×C4#54611yescharacteristic
393C2×C4D4×Q811yescharacteristic
394C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
395C2×C4#57711yescharacteristic
396C2×C4#57811yescharacteristic
397C8—22no—
398C8—41no—
399C8—41no—
400C8#54211yescharacteristic
401C8#54311yescharacteristic
402C8#55411yescharacteristic
403C8#56211yescharacteristic
404C22—82no—
405C22—82no—
406C22—82no—
407C22—161no—
408C22—161no—
409C22—161no—
410C22—161no—
411C22#320511yescharacteristic
412C22#329411yescharacteristic
413C22Q8×D811yescharacteristic
414C22C2×#51011yescharacteristic
415C22C2×#51011yescharacteristic
416C22#341511yescharacteristic
417C22#341511yescharacteristic
418C4—22no—
419C4—22no—
420C4—41no—
421C4—41no—
422C4—41no—
423C4—41no—
424C4—82no—
425C4—82no—
426C4—161no—
427C4—161no—
428C4—161no—
429C4—161no—
430C4—161no—
431C4#269211yescharacteristic
432C4#322511yescharacteristic
433C4#330811yescharacteristic
434C4#338411yescharacteristic
435C2—162no—
436C2#3151811yescharacteristic
437C2#3378411yescharacteristic
438C2#3378611yescharacteristic
439C2#3388111yescharacteristic
440C2#3390011yescharacteristic
441C2#3395211yescharacteristic
442C2#3395211yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D8C23
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. #28531C2semidirect
4. #13505C2semidirect
5. #13475C2semidirect
6. #13451C2semidirect
7. #13366C2semidirect
9. #13284C2semidirect
11. #13105C2semidirect
14. #12936C2semidirect
67. C2×SD32Q8semidirect

Maximal subgroups

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

subgroupindex
#285312
#264822
#135262
#135052
#134752
#134512
#133662
#133632
#132842
#132382
#131052
#130142
#130132
#129362
#128112