Finite Group Databasebeta

← #510344

Subgroups — #510344

Order 512.

516 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#28532C211yesmaximal, characteristic, complemented
2#26482C211yesmaximal, characteristic
3#13528C211yesmaximal, characteristic
4#13507C211yesmaximal, characteristic, complemented
5#13477C211yesmaximal, characteristic, complemented
6#13452C211yesmaximal, characteristic, complemented
7#13366C211yesmaximal, characteristic, complemented
8#13363C211yesmaximal, characteristic
9#13281C211yesmaximal, characteristic, complemented
10#13237C211yesmaximal, characteristic
11#13104C211yesmaximal, characteristic, complemented
12#13014C211yesmaximal, characteristic
13#13013C211yesmaximal, characteristic
14#12936C211yesmaximal, characteristic, complemented
15#12813C211yesmaximal, characteristic
16C22×SD32C2211yescharacteristic
17C2×#500C2211yescharacteristic
18C2×#499C2211yescharacteristic
19C2×#475—21no—
20C2×C4×Q16C2211yescharacteristic
21C2×C4×D8C2211yescharacteristic
22#2568—21no—
23#2454—21no—
24C2×#367C2211yescharacteristic
25C2×#367C2211yescharacteristic
26C2×#366C2211yescharacteristic
27C2×C4⋊C16C2211yescharacteristic
28C2×#358C2211yescharacteristic
29C2×#358C2211yescharacteristic
30C2×#358C2211yescharacteristic
31C2×#357C2211yescharacteristic
32C2×#357C2211yescharacteristic
33C2×#357C2211yescharacteristic
34C2×C4×C16C2211yescharacteristic
35#2097—21no—
36#2097—21no—
37#2065—21no—
38#2025—21no—
39#1970C2211yescharacteristic
40#1970C2211yescharacteristic
41#1969C2211yescharacteristic
42#1969C2211yescharacteristic
43#1942C2211yescharacteristic
44#1942C2211yescharacteristic
45#1864—21no—
46#1846C2211yescharacteristic
47#1846C2211yescharacteristic
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#475—44no—
77C4×Q16—41no—
78C4×Q16—41no—
79C4×D8—22no—
80C4×D8—41no—
81C2×#107—21no—
82C2×#107C2311yescharacteristic
83C2×#107C2311yescharacteristic
84C2×#107C2311yescharacteristic
85C2×#107C2311yescharacteristic
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#106—21no—
89C2×#106—21no—
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#400—41no—
102#400—41no—
103#399—41no—
104#396—41no—
105#391—21no—
106#391—21no—
107#391—41no—
108#391—41no—
109#390—21no—
110#390—21no—
111#389—21no—
112#389—21no—
113#385—41no—
114#382—21no—
115#382—21no—
116#380—41no—
117C4×C4⋊C4D411yescharacteristic
118#367—41no—
119#367—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—
128C4×C16—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⋊C4D811yescharacteristic
174C2×C4⋊C4D811yescharacteristic
175C2×C22⋊C4—41no—
176C2×C22⋊C4—41no—
177C2×C22⋊C4—41no—
178C2×C22⋊C4—41no—
179C2×C22⋊C4—41no—
180C2×C42—41no—
181C2×C42C2×D411yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C42C2×D411yescharacteristic
184SD32—44no—
185C2×C16—21no—
186C2×C16—21no—
187C2×C16—21no—
188C2×C16—21no—
189C2×C16—21no—
190C2×C16—21no—
191C2×C16—21no—
192C2×C16—21no—
193C2×C16—21no—
194C2×C16—21no—
195C2×C16—21no—
196C2×C16C2×Q812yes—
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#107#4112yes—
210#106—42no—
211#106—42no—
212#106—42no—
213C4⋊C8—22no—
214#103—42no—
215#103—42no—
216#103—42no—
217#102—42no—
218#102—42no—
219#102—81no—
220C4×C8—21no—
221C4×C8—21no—
222C4×C8—21no—
223C4×C8—21no—
224C4×C8—21no—
225C4×C8—21no—
226#95—41no—
227#95—41no—
228#95—41no—
229#95—41no—
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—82no—
247C2×Q8—82no—
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—41no—
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—
314C22⋊C4—82no—
315C22⋊C4—82no—
316C22⋊C4—161no—
317C22⋊C4—161no—
318C22⋊C4—161no—
319C42—21no—
320C42—21no—
321C42—21no—
322C42—21no—
323C42—21no—
324C42—21no—
325C42—82no—
326C16—22no—
327C16—41no—
328C16—41no—
329C16—41no—
330C23—42no—
331C23—81no—
332C23—81no—
333C23—81no—
334C23—81no—
335C23—81no—
336C23—81no—
337C23C22×D811yescharacteristic
338Q8—42no—
339Q8—42no—
340Q8—81no—
341Q8—81no—
342Q8—164no—
343D4—42no—
344D4—42no—
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—42no—
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—81no—
384C2×C4—81no—
385C2×C4—164no—
386C2×C4#50812yes—
387C2×C4#53511yescharacteristic
388C2×C4#54611yescharacteristic
389C2×C4D4×Q811yescharacteristic
390C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
391C2×C4#57711yescharacteristic
392C2×C4#57811yescharacteristic
393C8—22no—
394C8—41no—
395C8—41no—
396C8#54211yescharacteristic
397C8#54311yescharacteristic
398C8#55411yescharacteristic
399C8#56211yescharacteristic
400C22—82no—
401C22—82no—
402C22—82no—
403C22—161no—
404C22—161no—
405C22—161no—
406C22—161no—
407C22#320511yescharacteristic
408C22#329411yescharacteristic
409C22Q8×D811yescharacteristic
410C22C2×#50811yescharacteristic
411C22#341411yescharacteristic
412C22#341611yescharacteristic
413C22#341611yescharacteristic
414C4—22no—
415C4—41no—
416C4—41no—
417C4—41no—
418C4—41no—
419C4—41no—
420C4—82no—
421C4—82no—
422C4—161no—
423C4—161no—
424C4—161no—
425C4—161no—
426C4—161no—
427C4#269211yescharacteristic
428C4#322511yescharacteristic
429C4#330811yescharacteristic
430C4#338411yescharacteristic
431C2—162no—
432C2#3151811yescharacteristic
433C2#3377711yescharacteristic
434C2#3378711yescharacteristic
435C2#3387111yescharacteristic
436C2#3387811yescharacteristic
437C2#3394811yescharacteristic
438C2#3395511yescharacteristic
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. #28532C2semidirect
4. #13507C2semidirect
5. #13477C2semidirect
6. #13452C2semidirect
7. #13366C2semidirect
9. #13281C2semidirect
11. #13104C2semidirect
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
#285322
#264822
#135282
#135072
#134772
#134522
#133662
#133632
#132812
#132372
#131042
#130142
#130132
#129362
#128132