Finite Group Databasebeta

← #498214

Subgroups — #498214

Order 512.

624 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2233C211yesmaximal, characteristic
2#23025C211yesmaximal, characteristic
3#22612C211yesmaximal, characteristic, complemented
4#13167C211yesmaximal, characteristic, complemented
5#13149C211yesmaximal, characteristic, complemented
6#13130C211yesmaximal, characteristic, complemented
7#13111C211yesmaximal, characteristic, complemented
8#13110C211yesmaximal, characteristic, complemented
9#13059C211yesmaximal, characteristic
10#12935C211yesmaximal, characteristic, complemented
11#12919C211yesmaximal, characteristic, complemented
12#12674C211yesmaximal, characteristic, complemented
13#12637C211yesmaximal, characteristic, complemented
14#12611C211yesmaximal, characteristic, complemented
15C2×#1380C211yesmaximal, characteristic, complemented
16C23×C16C2211yescharacteristic
17C2×#466C2211yescharacteristic
18C2×C4×D8C2211yescharacteristic
19C22×#107C2211yescharacteristic
20C22×#107C2211yescharacteristic
21#2361—21no—
22#2345—21no—
23#2233C418yescomplemented
24C2×#367C2211yescharacteristic
25C2×#367C2211yescharacteristic
26C2×#366C2211yescharacteristic
27C2×#366C2211yescharacteristic
28C2×#357C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×#357C2211yescharacteristic
31C2×#357C2211yescharacteristic
32C2×C22⋊C16C2211yescharacteristic
33C2×#346C2211yescharacteristic
34C2×#346C2211yescharacteristic
35#1970C2211yescharacteristic
36#1942C2211yescharacteristic
37#1942C2211yescharacteristic
38#1942C2211yescharacteristic
39#1925—21no—
40#1921—21no—
41#1921—21no—
42#1874—21no—
43#1862—21no—
44#1824—21no—
45#1808C2211yescharacteristic
46C4×#107C2211yescharacteristic
47C4×#107C2211yescharacteristic
48C4×#107C2211yescharacteristic
49C8×C22⋊C4C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1374C2211yescharacteristic
59#1374C2211yescharacteristic
60C22×D8C2311yescharacteristic
61C23×C8C2311yescharacteristic
62C2×#121—21no—
63C2×C4×D4—21no—
64C2×C4×D4—41no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C16—22no—
68C22×C16—22no—
69C22×C16—41no—
70C22×C16C2311yescharacteristic
71C22×C16C2311yescharacteristic
72C22×C16C2311yescharacteristic
73C22×C16C2311yescharacteristic
74#466—22no—
75#466C2×C414yes—
76C4×D8—22no—
77C4×D8—41no—
78C2×#107—21no—
79C2×#107—22no—
80C2×#107—22no—
81C2×#107—21no—
82C2×#107—41no—
83C2×#107C2×C414yescomplemented
84C2×#107C2311yescharacteristic
85C2×#107C2311yescharacteristic
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#107C2311yescharacteristic
89C2×#107C2311yescharacteristic
90C2×#107C2311yescharacteristic
91C2×#102—21no—
92C2×#102—21no—
93C2×#102—21no—
94C2×C22⋊C8D411yescharacteristic
95C2×C4×C8C2311yescharacteristic
96C2×C4×C8C2311yescharacteristic
97#390—21no—
98#389—21no—
99#389—21no—
100#389—21no—
101#388—41no—
102#387—41no—
103#387—41no—
104#386—41no—
105#385—41no—
106#380—21no—
107#380—41no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—21no—
111C4×C22⋊C4—41no—
112C4×C22⋊C4D411yescharacteristic
113#367C2×C414yescomplemented
114#367C2×C414yescomplemented
115#366—22no—
116#366—41no—
117#357—22no—
118#357—41no—
119#357C2×C414yescomplemented
120#357C2×C414yescomplemented
121C22⋊C16C2×C414yescomplemented
122#346—22no—
123#346—41no—
124#340—21no—
125#340—21no—
126#340—21no—
127#340—21no—
128#340—21no—
129#340—21no—
130#340—21no—
131#336D411yescharacteristic
132#336D411yescharacteristic
133C22×D4—21no—
134C22×D4—41no—
135C23×C4—41no—
136C23×C4—41no—
137C23×C4—41no—
138C23×C4C2×D411yescharacteristic
139C2×D8—21no—
140C2×D8—21no—
141C2×D8—21no—
142C2×D8—21no—
143C2×D8—21no—
144C2×D8C22×C412yes—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8C22×C412yes—
150C22×C8C2×D411yescharacteristic
151C22×C8C2×D411yescharacteristic
152C22×C8C2×D411yescharacteristic
153C22×C8#4112yes—
154C22×C8C2411yescharacteristic
155#121—24no—
156#121—42no—
157C4×D4—42no—
158C4×D4—81no—
159C4×D4—84no—
160C2×C4⋊C4—24no—
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—21no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—81no—
176C2×C22⋊C4—41no—
177C2×C22⋊C4—41no—
178C2×C22⋊C4—41no—
179C2×C22⋊C4—41no—
180C2×C22⋊C4—41no—
181C2×C22⋊C4D811yescharacteristic
182C2×C22⋊C4D811yescharacteristic
183C2×C42—41no—
184C2×C42—41no—
185C2×C42—41no—
186C2×C42C2×D411yescharacteristic
187C2×C42C2×D411yescharacteristic
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×C16—21no—
197C2×C16—44no—
198C2×C16C22×C412yes—
199C2×C16C22×C412yes—
200C2×C16#4112yes—
201#107—24no—
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#107C22×C412yes—
214#107C22×C412yes—
215#107C22×C412yes—
216#107#4112yes—
217#102—24no—
218#102—42no—
219#102—81no—
220C22⋊C8—22no—
221C4×C8—21no—
222C4×C8—21no—
223C4×C8—21no—
224C4×C8—21no—
225#95—41no—
226#95—41no—
227#95—41no—
228#95—41no—
229#95—41no—
230#95—41no—
231#95—41no—
232#95D811yescharacteristic
233#95D811yescharacteristic
234C24—41no—
235C24C2×D811yescharacteristic
236C2×D4—22no—
237C2×D4—44no—
238C2×D4—41no—
239C2×D4—41no—
240C2×D4—41no—
241C2×D4—41no—
242C2×D4—41no—
243C2×D4—82no—
244C2×D4—82no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—44no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—82no—
258C22×C4—82no—
259C22×C4—82no—
260C22×C4—82no—
261C22×C4—161no—
262C22×C4—161no—
263C22×C4—161no—
264C22×C4—161no—
265C22×C4C4×D412yes—
266C22×C4C2×D811yescharacteristic
267C22×C4C2×D811yescharacteristic
268C22×C4C2×D811yescharacteristic
269C22×C4C2×D811yescharacteristic
270C22×C4C2×D811yescharacteristic
271C22×C4#13512yes—
272C22×C4C22×D411yescharacteristic
273D8—22no—
274D8—22no—
275D8—41no—
276D8—41no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—22no—
280C2×C8—22no—
281C2×C8—22no—
282C2×C8—22no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8C4×D412yes—
290C2×C8C23×C411yescharacteristic
291C2×C8C2×#4111yescharacteristic
292C2×C8C2×#4111yescharacteristic
293C2×C8C2×#4111yescharacteristic
294C2×C8H(5,𝔽2)11yescharacteristic
295C2×C82−1+411yescharacteristic
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—44no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—82no—
312C4⋊C4—82no—
313C4⋊C4—161no—
314C4⋊C4—161no—
315C22⋊C4—22no—
316C22⋊C4—22no—
317C22⋊C4—44no—
318C22⋊C4—82no—
319C22⋊C4—82no—
320C22⋊C4—82no—
321C22⋊C4—161no—
322C42—21no—
323C42—21no—
324C42—21no—
325C42—21no—
326C42—82no—
327C42—161no—
328C42—161no—
329C16—22no—
330C16—22no—
331C16—22no—
332C16—41no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23—42no—
338C23—81no—
339C23—81no—
340C23—81no—
341C23—81no—
342C23—81no—
343C23—81no—
344C23C4×D812yes—
345C23#50912yes—
346C23#51012yes—
347C23C22×D811yescharacteristic
348D4—42no—
349D4—42no—
350D4—88no—
351D4—81no—
352D4—81no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
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—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
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—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—164no—
393C2×C4—164no—
394C2×C4—164no—
395C2×C4C4×D812yes—
396C2×C4C2×C4×D411yescharacteristic
397C2×C4#54611yescharacteristic
398C2×C4#54711yescharacteristic
399C2×C4C2×#13511yescharacteristic
400C2×C4#57711yescharacteristic
401C2×C4#57811yescharacteristic
402C8—22no—
403C8—41no—
404C8—41no—
405C8C4×#4111yescharacteristic
406C8#52011yescharacteristic
407C8#55111yescharacteristic
408C8#55211yescharacteristic
409C22—22no—
410C22—22no—
411C22—22no—
412C22—22no—
413C22—22no—
414C22—22no—
415C22—82no—
416C22—82no—
417C22—82no—
418C22—161no—
419C22—161no—
420C22—161no—
421C22—161no—
422C22#217514yes—
423C22C2×C4×D811yescharacteristic
424C22#329411yescharacteristic
425C22#333411yescharacteristic
426C22C2×#50911yescharacteristic
427C22C2×#51011yescharacteristic
428C22#341511yescharacteristic
429C22#341611yescharacteristic
430C4—22no—
431C4—41no—
432C4—41no—
433C4—41no—
434C4—41no—
435C4—82no—
436C4—82no—
437C4—82no—
438C4—82no—
439C4—161no—
440C4—161no—
441C4—161no—
442C4#235911yescharacteristic
443C4#294711yescharacteristic
444C4#330011yescharacteristic
445C4#333711yescharacteristic
446C2—24no—
447C2—162no—
448C2#2291611yescharacteristic
449C2C2×#217511yescharacteristic
450C2#3345411yescharacteristic
451C2#3388311yescharacteristic
452C2#3388511yescharacteristic
453C2#3391211yescharacteristic
454C2#3391311yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC2×#466C22
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ C8×C22⋊C4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #22612C2semidirect
4. #13167C2semidirect
5. #13149C2semidirect
6. #13130C2semidirect
7. #13111C2semidirect
8. #13110C2semidirect
10. #12935C2semidirect
11. #12919C2semidirect
12. #12674C2semidirect
13. #12637C2semidirect
14. #12611C2semidirect
15. C2×#1380C2semidirect
23. #2233C4semidirect
83. C2×#107C2×C4semidirect
113. #367C2×C4semidirect
114. #367C2×C4semidirect
119. #357C2×C4semidirect
120. #357C2×C4semidirect
121. C22⋊C16C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#22332
#230252
#226122
#131672
#131492
#131302
#131112
#131102
#130592
#129352
#129192
#126742
#126372
#126112
C2×#13802