Finite Group Databasebeta

← #507478

Subgroups — #507478

Order 512.

546 conjugacy classes of subgroups.

The subgroup lattice has 437 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33257C211yesmaximal, characteristic, complemented
2C4×#495C211yesmaximal, characteristic
3#13477C211yesmaximal, characteristic, complemented
4#13476C211yesmaximal, characteristic
5#13475C211yesmaximal, characteristic, complemented
6#13474C211yesmaximal, characteristic
7#13204C211yesmaximal, characteristic
8#13204C211yesmaximal, characteristic
9#12938C211yesmaximal, characteristic, complemented
10#12937C211yesmaximal, characteristic
11#12936C211yesmaximal, characteristic, complemented
12#12935C211yesmaximal, characteristic
13#12907C211yesmaximal, characteristic, complemented
14#12889C211yesmaximal, characteristic, complemented
15C4×C4⋊C16C211yesmaximal, characteristic, complemented
16C2×#495C2211yescharacteristic
17C2×C4×Q16C2211yescharacteristic
18C2×C4×SD16—21no—
19C2×C4×D8C2211yescharacteristic
20#2864—21no—
21C4×#124—21no—
22C2×#367C2211yescharacteristic
23C2×#367C2211yescharacteristic
24C2×#366C2211yescharacteristic
25C2×#366C2211yescharacteristic
26C2×C4⋊C16C2211yescharacteristic
27C2×C4⋊C16C2211yescharacteristic
28C2×#358C2211yescharacteristic
29C2×#358C2211yescharacteristic
30C2×#358C2211yescharacteristic
31C2×#358C2211yescharacteristic
32C2×#357C2211yescharacteristic
33C2×#357C2211yescharacteristic
34C2×#357C2211yescharacteristic
35C2×#357C2211yescharacteristic
36C2×C4×C16C2211yescharacteristic
37C2×C4×C16C2211yescharacteristic
38#1970C2211yescharacteristic
39#1969C2211yescharacteristic
40#1968—21no—
41#1958—21no—
42#1942C2211yescharacteristic
43#1942C2211yescharacteristic
44#1942C2211yescharacteristic
45#1942C2211yescharacteristic
46#1941—21no—
47#1941—21no—
48#1846C2211yescharacteristic
49#1846C2211yescharacteristic
50#1846C2211yescharacteristic
51#1835—21no—
52C42×C8C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×Q16C2311yescharacteristic
60C22×SD16—21no—
61C22×D8C2311yescharacteristic
62C2×#124—21no—
63C2×C4×Q8—21no—
64C2×C4×D4—21no—
65C22×C16C2311yescharacteristic
66C22×C16C2311yescharacteristic
67C22×C16C2311yescharacteristic
68C22×C16C2311yescharacteristic
69#495—22no—
70#495Q814yes—
71C4×Q16—41no—
72C4×Q16—41no—
73C4×SD16—44no—
74C4×D8—41no—
75C4×D8—41no—
76C2×#107C2311yescharacteristic
77C2×#107C2311yescharacteristic
78C2×#107C2311yescharacteristic
79C2×#107C2311yescharacteristic
80C2×#107C2311yescharacteristic
81C2×#107C2311yescharacteristic
82C2×#106—21no—
83C2×#106—21no—
84C2×#106—21no—
85C2×#103—21no—
86C2×#103—21no—
87C2×#102—21no—
88C2×#102—21no—
89C2×C4×C8D411yescharacteristic
90C2×C4×C8D411yescharacteristic
91C2×C4×C8D411yescharacteristic
92C2×C4×C8C2311yescharacteristic
93C2×C4×C8C2311yescharacteristic
94C2×C4×C8C2311yescharacteristic
95#391—21no—
96#390—21no—
97#389—21no—
98#389—21no—
99#389—21no—
100#389—21no—
101#388—41no—
102#388—41no—
103#387—41no—
104#387—41no—
105#387—41no—
106#387—41no—
107#382—21no—
108#382—21no—
109#382—21no—
110C4×C22⋊C4—41no—
111C4×C22⋊C4—41no—
112C43D411yescharacteristic
113#367—41no—
114#367—41no—
115#366—41no—
116#366—41no—
117C4⋊C16—22no—
118C4⋊C16—22no—
119#358—22no—
120#358—22no—
121#358—41no—
122#358—41no—
123#357—22no—
124#357—22no—
125#357—41no—
126#357—41no—
127C4×C16—22no—
128C4×C16—22no—
129#340—21no—
130#340—21no—
131#340—21no—
132#340—21no—
133C22×Q8—21no—
134C22×D4—21no—
135C23×C4—41no—
136C2×Q16—21no—
137C2×Q16—21no—
138C2×Q16—21no—
139C2×Q16—21no—
140C2×Q16—21no—
141C2×Q16C2×Q812yes—
142C2×SD16—24no—
143C2×SD16—42no—
144C2×SD16—42no—
145C2×D8—21no—
146C2×D8—21no—
147C2×D8—21no—
148C2×D8—21no—
149C2×D8—21no—
150C2×D8C2×Q812yes—
151C22×C8C2×D411yescharacteristic
152C22×C8C2×D411yescharacteristic
153C22×C8C2×D411yescharacteristic
154C22×C8C2411yescharacteristic
155#124—24no—
156#124—42no—
157C4×Q8—42no—
158C4×Q8—42no—
159C4×D4—42no—
160C4×D4—42no—
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×C22⋊C4—41no—
173C2×C22⋊C4—41no—
174C2×C22⋊C4—41no—
175C2×C22⋊C4—41no—
176C2×C22⋊C4—41no—
177C2×C42—41no—
178C2×C42D811yescharacteristic
179C2×C42D811yescharacteristic
180C2×C42D811yescharacteristic
181C2×C42D811yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C42C2×D411yescharacteristic
184C2×C42C2×D411yescharacteristic
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#4112yes—
196C2×C16#4112yes—
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—21no—
209#106—42no—
210#106—42no—
211#106—42no—
212#103—42no—
213#103—42no—
214#102—42no—
215#102—42no—
216C4×C8—22no—
217C4×C8—22no—
218C4×C8—21no—
219C4×C8—21no—
220C4×C8—21no—
221C4×C8—21no—
222C4×C8—22no—
223C4×C8C2×Q812yes—
224C4×C8#4112yes—
225#95—41no—
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#95—41no—
237C24—41no—
238C2×Q8—22no—
239C2×Q8—41no—
240C2×Q8—41no—
241C2×Q8—41no—
242C2×Q8—41no—
243C2×Q8—41no—
244C2×D4—22no—
245C2×D4—41no—
246C2×D4—41no—
247C2×D4—41no—
248C2×D4—41no—
249C2×D4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—82no—
258C22×C4—82no—
259C22×C4—82no—
260C22×C4C2×D811yescharacteristic
261C22×C4C2×D811yescharacteristic
262C22×C4C2×D811yescharacteristic
263C22×C4C2×D811yescharacteristic
264C22×C4C2×D811yescharacteristic
265C22×C4C2×D811yescharacteristic
266C22×C4C22×D411yescharacteristic
267Q16—22no—
268Q16—22no—
269Q16—41no—
270Q16—41no—
271SD16—48no—
272D8—22no—
273D8—22no—
274D8—41no—
275D8—41no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8C22⋊Q812yes—
283C2×C8#13512yes—
284C2×C8AGL(1,ℤ/8)12yes—
285C2×C8C22×Q811yescharacteristic
286C2×C8C2×#4111yescharacteristic
287C2×C8C2×#4111yescharacteristic
288C2×C8C2×#4111yescharacteristic
289C2×C8H(5,𝔽2)11yescharacteristic
290C2×C8H(5,𝔽2)11yescharacteristic
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—82no—
304C4⋊C4—82no—
305C4⋊C4—82no—
306C4⋊C4—82no—
307C4⋊C4—82no—
308C22⋊C4—44no—
309C22⋊C4—44no—
310C22⋊C4—82no—
311C22⋊C4—82no—
312C22⋊C4—82no—
313C42—21no—
314C42—21no—
315C42—21no—
316C42—21no—
317C42—22no—
318C42—22no—
319C42—41no—
320C42—41no—
321C42—82no—
322C42C22⋊Q812yes—
323C42#13712yes—
324C16—22no—
325C16—22no—
326C16—41no—
327C16—41no—
328C23—42no—
329C23—81no—
330C23—81no—
331C23—81no—
332C23—81no—
333C23—81no—
334C23—81no—
335C23C22×D811yescharacteristic
336Q8—42no—
337Q8—42no—
338Q8—81no—
339Q8—81no—
340D4—42no—
341D4—42no—
342D4—81no—
343D4—81no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
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—42no—
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—81no—
379C2×C4—164no—
380C2×C4#47412yes—
381C2×C4#47412yes—
382C2×C4#50812yes—
383C2×C4#50812yes—
384C2×C4C2×C22⋊Q811yescharacteristic
385C2×C4#53211yescharacteristic
386C2×C4#53511yescharacteristic
387C2×C4C2×#13511yescharacteristic
388C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
389C2×C4C2×#13711yescharacteristic
390C8—41no—
391C8—41no—
392C8#47814yes—
393C8#55311yescharacteristic
394C8#55311yescharacteristic
395C8#55411yescharacteristic
396C8#55411yescharacteristic
397C22—82no—
398C22—82no—
399C22—82no—
400C22—161no—
401C22—161no—
402C22—161no—
403C22—161no—
404C22C2×#47411yescharacteristic
405C22#316911yescharacteristic
406C22#320511yescharacteristic
407C22C2×#50811yescharacteristic
408C22C2×#50811yescharacteristic
409C22#341411yescharacteristic
410C22#341411yescharacteristic
411C4—22no—
412C4—22no—
413C4—22no—
414C4—22no—
415C4—41no—
416C4—41no—
417C4—82no—
418C4—161no—
419C4—161no—
420C4—161no—
421C4—161no—
422C4—161no—
423C4—161no—
424C4#256611yescharacteristic
425C4C2×#47811yescharacteristic
426C4#318211yescharacteristic
427C4#321011yescharacteristic
428C2—162no—
429C2#2919211yescharacteristic
430C2#3362611yescharacteristic
431C2#3362611yescharacteristic
432C2#3373511yescharacteristic
433C2#3375011yescharacteristic
434C2#3377711yescharacteristic
435C2#3377811yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D8C23
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ C42×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33257C2semidirect
3. #13477C2semidirect
5. #13475C2semidirect
9. #12938C2semidirect
11. #12936C2semidirect
13. #12907C2semidirect
14. #12889C2semidirect
15. C4×C4⋊C16C2semidirect

Maximal subgroups

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

subgroupindex
#332572
C4×#4952
#134772
#134762
#134752
#134742
#132042
#129382
#129372
#129362
#129352
#129072
#128892
C4×C4⋊C162