Finite Group Databasebeta

← #498216

Subgroups — #498216

Order 512.

638 conjugacy classes of subgroups.

The subgroup lattice has 457 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2232C211yesmaximal, characteristic
2#23025C211yesmaximal, characteristic
3#22611C211yesmaximal, characteristic, complemented
4#13168C211yesmaximal, characteristic, complemented
5#13149C211yesmaximal, characteristic, complemented
6#13131C211yesmaximal, characteristic, complemented
7#13112C211yesmaximal, characteristic, complemented
8#13112C211yesmaximal, characteristic, complemented
9#13059C211yesmaximal, characteristic
10#12937C211yesmaximal, characteristic, complemented
11#12920C211yesmaximal, 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#2344—21no—
23#2232C418yescomplemented
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#1918—21no—
41#1918—21no—
42#1874—21no—
43#1862—21no—
44#1823—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—
65C2×#117—41no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C16—22no—
69C22×C16—22no—
70C22×C16—41no—
71C22×C16C2311yescharacteristic
72C22×C16C2311yescharacteristic
73C22×C16C2311yescharacteristic
74C22×C16C2311yescharacteristic
75#466—22no—
76#466C2×C414yes—
77C4×D8—22no—
78C4×D8—41no—
79C2×#107—21no—
80C2×#107—22no—
81C2×#107—22no—
82C2×#107—21no—
83C2×#107—41no—
84C2×#107C2×C414yescomplemented
85C2×#107C2311yescharacteristic
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#107C2311yescharacteristic
89C2×#107C2311yescharacteristic
90C2×#107C2311yescharacteristic
91C2×#107C2311yescharacteristic
92C2×#102—21no—
93C2×#102—21no—
94C2×#102—21no—
95C2×C22⋊C8D411yescharacteristic
96C2×C4×C8C2311yescharacteristic
97C2×C4×C8C2311yescharacteristic
98#390—21no—
99#389—21no—
100#389—21no—
101#389—21no—
102#388—41no—
103#387—41no—
104#387—41no—
105#386—41no—
106#385—41no—
107#381—41no—
108#380—21no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—21no—
112C4×C22⋊C4D411yescharacteristic
113#367—22no—
114#367—41no—
115#366C2×C414yescomplemented
116#366C2×C414yescomplemented
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—
160#117—84no—
161C2×C4⋊C4—24no—
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—21no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—81no—
177C2×C22⋊C4—41no—
178C2×C22⋊C4—41no—
179C2×C22⋊C4—41no—
180C2×C22⋊C4—41no—
181C2×C22⋊C4—41no—
182C2×C22⋊C4D811yescharacteristic
183C2×C22⋊C4D811yescharacteristic
184C2×C42—41no—
185C2×C42—41no—
186C2×C42—41no—
187C2×C42C2×D411yescharacteristic
188C2×C42C2×D411yescharacteristic
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—44no—
199C2×C16C22×C412yes—
200C2×C16C22×C412yes—
201C2×C16#4112yes—
202#107—24no—
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—21no—
213#107—42no—
214#107C22×C412yes—
215#107C22×C412yes—
216#107C22×C412yes—
217#107#4112yes—
218#102—24no—
219#102—42no—
220#102—81no—
221C22⋊C8—22no—
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#95D811yescharacteristic
234#95D811yescharacteristic
235C24—41no—
236C24C2×D811yescharacteristic
237C2×D4—22no—
238C2×D4—44no—
239C2×D4—41no—
240C2×D4—41no—
241C2×D4—41no—
242C2×D4—41no—
243C2×D4—41no—
244C2×D4—82no—
245C2×D4—82no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—44no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—82no—
259C22×C4—82no—
260C22×C4—82no—
261C22×C4—82no—
262C22×C4—82no—
263C22×C4—82no—
264C22×C4—161no—
265C22×C4—161no—
266C22×C4C4×D412yes—
267C22×C4C2×D811yescharacteristic
268C22×C4C2×D811yescharacteristic
269C22×C4C2×D811yescharacteristic
270C22×C4C2×D811yescharacteristic
271C22×C4C2×D811yescharacteristic
272C22×C4#13512yes—
273C22×C4C22×D411yescharacteristic
274D8—22no—
275D8—22no—
276D8—41no—
277D8—41no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—22no—
281C2×C8—22no—
282C2×C8—22no—
283C2×C8—22no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—21no—
290C2×C8C4×D412yes—
291C2×C8C23×C411yescharacteristic
292C2×C8C2×#4111yescharacteristic
293C2×C8C2×#4111yescharacteristic
294C2×C8C2×#4111yescharacteristic
295C2×C8H(5,𝔽2)11yescharacteristic
296C2×C82−1+411yescharacteristic
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—44no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—82no—
313C4⋊C4—82no—
314C4⋊C4—82no—
315C4⋊C4—82no—
316C22⋊C4—22no—
317C22⋊C4—22no—
318C22⋊C4—44no—
319C22⋊C4—82no—
320C22⋊C4—82no—
321C22⋊C4—82no—
322C22⋊C4—161no—
323C42—21no—
324C42—21no—
325C42—21no—
326C42—21no—
327C42—82no—
328C42—82no—
329C42—82no—
330C16—22no—
331C16—22no—
332C16—22no—
333C16—41no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23—42no—
339C23—81no—
340C23—81no—
341C23—81no—
342C23—81no—
343C23—81no—
344C23—81no—
345C23C4×D812yes—
346C23#50912yes—
347C23#50912yes—
348C23C22×D811yescharacteristic
349D4—42no—
350D4—42no—
351D4—88no—
352D4—81no—
353D4—81no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
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—21no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
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—88no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—164no—
395C2×C4—164no—
396C2×C4C4×D812yes—
397C2×C4C2×C4×D411yescharacteristic
398C2×C4#54611yescharacteristic
399C2×C4#54711yescharacteristic
400C2×C4C2×#13511yescharacteristic
401C2×C4#57711yescharacteristic
402C2×C4#57811yescharacteristic
403C8—22no—
404C8—41no—
405C8—41no—
406C8C4×#4111yescharacteristic
407C8#52011yescharacteristic
408C8#55111yescharacteristic
409C8#55211yescharacteristic
410C22—22no—
411C22—22no—
412C22—22no—
413C22—22no—
414C22—22no—
415C22—22no—
416C22—82no—
417C22—82no—
418C22—82no—
419C22—161no—
420C22—161no—
421C22—161no—
422C22—161no—
423C22#217714yes—
424C22C2×C4×D811yescharacteristic
425C22#329411yescharacteristic
426C22#333411yescharacteristic
427C22C2×#50911yescharacteristic
428C22C2×#50911yescharacteristic
429C22#341611yescharacteristic
430C22#341711yescharacteristic
431C4—22no—
432C4—41no—
433C4—41no—
434C4—41no—
435C4—41no—
436C4—82no—
437C4—82no—
438C4—82no—
439C4—82no—
440C4—161no—
441C4—161no—
442C4—161no—
443C4#235911yescharacteristic
444C4#294711yescharacteristic
445C4#330011yescharacteristic
446C4#333711yescharacteristic
447C2—24no—
448C2—162no—
449C2#2291611yescharacteristic
450C2C2×#217711yescharacteristic
451C2#3345511yescharacteristic
452C2#3387511yescharacteristic
453C2#3387711yescharacteristic
454C2#3391011yescharacteristic
455C2#3391211yescharacteristic
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. #22611C2semidirect
4. #13168C2semidirect
5. #13149C2semidirect
6. #13131C2semidirect
7. #13112C2semidirect
8. #13112C2semidirect
10. #12937C2semidirect
11. #12920C2semidirect
12. #12674C2semidirect
13. #12637C2semidirect
14. #12611C2semidirect
15. C2×#1380C2semidirect
23. #2232C4semidirect
84. C2×#107C2×C4semidirect
115. #366C2×C4semidirect
116. #366C2×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×#22322
#230252
#226112
#131682
#131492
#131312
#131122
#130592
#129372
#129202
#126742
#126372
#126112
C2×#13802