Finite Group Databasebeta

← #498618

Subgroups — #498618

Order 512.

618 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2244C211yesmaximal, characteristic
2#23185C211yesmaximal, characteristic
3#23160C211yesmaximal, characteristic, complemented
4#13277C211yesmaximal, characteristic, complemented
5#13252C211yesmaximal, characteristic
6#13149C211yesmaximal, characteristic, complemented
7#13147C211yesmaximal, characteristic, complemented
8#13130C211yesmaximal, characteristic, complemented
9#13130C211yesmaximal, characteristic, complemented
10#12937C211yesmaximal, characteristic, complemented
11#12937C211yesmaximal, characteristic, complemented
12#12674C211yesmaximal, characteristic, complemented
13#12663C211yesmaximal, characteristic, complemented
14#12637C211yesmaximal, characteristic, complemented
15C4×#357C211yesmaximal, characteristic, complemented
16C2×#499C2211yescharacteristic
17C2×#493C2211yescharacteristic
18C2×C4×D8C2211yescharacteristic
19C2×C4×D8C2211yescharacteristic
20#2370—21no—
21#2366—21no—
22#2244C418yescomplemented
23C2×#367C2211yescharacteristic
24C2×#366C2211yescharacteristic
25C2×C4⋊C16C2211yescharacteristic
26C2×#357C2211yescharacteristic
27C2×#357C2211yescharacteristic
28C2×#357C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×#357C2211yescharacteristic
31C2×#357C2211yescharacteristic
32C2×#357C2211yescharacteristic
33C2×#357C2211yescharacteristic
34C2×#346C2211yescharacteristic
35C2×#346C2211yescharacteristic
36C2×C4×C16C2211yescharacteristic
37#1989—21no—
38#1982—21no—
39#1970C2211yescharacteristic
40#1970C2211yescharacteristic
41#1942C2211yescharacteristic
42#1942C2211yescharacteristic
43#1920—21no—
44#1919—21no—
45#1879—21no—
46#1879—21no—
47C4×#107C2211yescharacteristic
48C4×#107C2211yescharacteristic
49C4×#107C2211yescharacteristic
50C4×#107C2211yescharacteristic
51C8×C4⋊C4C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1374C2211yescharacteristic
59#1374C2211yescharacteristic
60C22×D8—21no—
61C22×D8C2311yescharacteristic
62C22×D8C2311yescharacteristic
63C2×#127—21no—
64C2×#125—21no—
65C2×C4×D4—21no—
66C2×C4×D4—21no—
67C2×C4×D4—41no—
68C22×C16C2311yescharacteristic
69C22×C16C2311yescharacteristic
70C22×C16C2311yescharacteristic
71C22×C16C2311yescharacteristic
72#499—22no—
73#499C2×C414yescomplemented
74#493—22no—
75#493C2×C414yes—
76C4×D8—22no—
77C4×D8—22no—
78C4×D8—41no—
79C4×D8—41no—
80C2×#107C2311yescharacteristic
81C2×#107C2311yescharacteristic
82C2×#107C2311yescharacteristic
83C2×#107C2311yescharacteristic
84C2×#107C2311yescharacteristic
85C2×#107C2311yescharacteristic
86C2×#106—21no—
87C2×C4⋊C8D411yescharacteristic
88C2×#102—21no—
89C2×#102—21no—
90C2×#102—21no—
91C2×#102—21no—
92C2×C4×C8C2311yescharacteristic
93C2×C4×C8C2311yescharacteristic
94C2×C4×C8C2311yescharacteristic
95#390—21no—
96#390—21no—
97#390—41no—
98#389—21no—
99#389—21no—
100#389—41no—
101#387—41no—
102#387—41no—
103#386—41no—
104#386—41no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—41no—
110C4×C4⋊C4D411yescharacteristic
111#367—22no—
112#366—22no—
113C4⋊C16—22no—
114#357—22no—
115#357—22no—
116#357—41no—
117#357—41no—
118#357C2×C414yescomplemented
119#357C2×C414yescomplemented
120#357C2×C414yescomplemented
121#357C2×C414yescomplemented
122#346—41no—
123#346C2×C414yescomplemented
124C4×C16—41no—
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—21no—
135C22×D4—41no—
136C22×D4—41no—
137C23×C4—41no—
138C23×C4—41no—
139C2×D8—24no—
140C2×D8—21no—
141C2×D8—21no—
142C2×D8—21no—
143C2×D8—21no—
144C2×D8—21no—
145C2×D8—21no—
146C2×D8—21no—
147C2×D8—21no—
148C2×D8—21no—
149C2×D8—21no—
150C2×D8—81no—
151C2×D8—81no—
152C2×D8C22×C412yes—
153C2×D8C22×C412yes—
154C22×C8C2×D411yescharacteristic
155C22×C8C2×D411yescharacteristic
156C22×C8C2×D411yescharacteristic
157C22×C8C2411yescharacteristic
158#127—24no—
159#127—42no—
160#125—24no—
161#125—42no—
162C4×D4—42no—
163C4×D4—42no—
164C4×D4—81no—
165C4×D4—81no—
166C4×D4—84no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
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×C22⋊C4—41no—
185C2×C42—41no—
186C2×C42—41no—
187C2×C42—41no—
188C2×C42—41no—
189C2×C42C2×D411yescharacteristic
190C2×C42C2×D411yescharacteristic
191C2×C42C2×D411yescharacteristic
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×C16—21no—
201C2×C16—21no—
202C2×C16C22×C412yes—
203C2×C16C22×C412yes—
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#107C22×C412yes—
213#107C22×C412yes—
214#107#4112yes—
215#107#4112yes—
216#106—24no—
217C4⋊C8—22no—
218#102—42no—
219#102—42no—
220#102—81no—
221#102—81no—
222C4×C8—21no—
223C4×C8—21no—
224C4×C8—21no—
225C4×C8—21no—
226C4×C8C22×C412yes—
227C4×C8#4112yes—
228#95—41no—
229#95—41no—
230#95—41no—
231#95—41no—
232#95—41no—
233#95—41no—
234#95D811yescharacteristic
235#95D811yescharacteristic
236C24—41no—
237C24—41no—
238C2×D4—22no—
239C2×D4—22no—
240C2×D4—41no—
241C2×D4—41no—
242C2×D4—41no—
243C2×D4—41no—
244C2×D4—41no—
245C2×D4—41no—
246C2×D4—41no—
247C2×D4—41no—
248C2×D4—41no—
249C2×D4—41no—
250C2×D4—44no—
251C2×D4—44no—
252C2×D4—82no—
253C2×D4—82no—
254C2×D4—161no—
255C2×D4—161no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—82no—
263C22×C4—82no—
264C22×C4—82no—
265C22×C4—82no—
266C22×C4—161no—
267C22×C4—161no—
268C22×C4C2×D811yescharacteristic
269C22×C4C2×D811yescharacteristic
270C22×C4C2×D811yescharacteristic
271C22×C4C2×D811yescharacteristic
272C22×C4C2×D811yescharacteristic
273C22×C4C2×D811yescharacteristic
274C22×C4C22×D411yescharacteristic
275D8—22no—
276D8—22no—
277D8—22no—
278D8—22no—
279D8—41no—
280D8—41no—
281D8—41no—
282D8—41no—
283D8—84no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—21no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8C4×D412yes—
293C2×C8C23×C411yescharacteristic
294C2×C8C2×#4111yescharacteristic
295C2×C8C2×#4111yescharacteristic
296C2×C8C2×#4111yescharacteristic
297C2×C8H(5,𝔽2)11yescharacteristic
298C2×C82−1+411yescharacteristic
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—44no—
310C4⋊C4—44no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—82no—
316C4⋊C4—161no—
317C22⋊C4—82no—
318C22⋊C4—82no—
319C22⋊C4—82no—
320C22⋊C4—82no—
321C22⋊C4—161no—
322C22⋊C4—161no—
323C42—21no—
324C42—21no—
325C42—21no—
326C42—21no—
327C42—82no—
328C42—82no—
329C42—161no—
330C42—161no—
331C42C4×D412yes—
332C42#13512yes—
333C16—22no—
334C16—22no—
335C16—41no—
336C16—41no—
337C23—42no—
338C23—42no—
339C23—81no—
340C23—81no—
341C23—81no—
342C23—81no—
343C23—81no—
344C23—81no—
345C23—81no—
346C23—81no—
347C23—81no—
348C23—81no—
349C23—81no—
350C23—81no—
351C23C22×D811yescharacteristic
352D4—42no—
353D4—42no—
354D4—42no—
355D4—42no—
356D4—88no—
357D4—81no—
358D4—81no—
359D4—81no—
360D4—81no—
361D4—164no—
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—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
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—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—164no—
396C2×C4—164no—
397C2×C4C4×D812yes—
398C2×C4C4×D812yes—
399C2×C4#50912yes—
400C2×C4C2×C4×D411yescharacteristic
401C2×C4#54611yescharacteristic
402C2×C4#54811yescharacteristic
403C2×C4C2×#13511yescharacteristic
404C2×C4#57711yescharacteristic
405C2×C4#57711yescharacteristic
406C8—22no—
407C8—41no—
408C8—41no—
409C8C4×#4111yescharacteristic
410C8#52011yescharacteristic
411C8#55611yescharacteristic
412C8#55911yescharacteristic
413C22—82no—
414C22—82no—
415C22—82no—
416C22—82no—
417C22—82no—
418C22—82no—
419C22—161no—
420C22—161no—
421C22—161no—
422C22—161no—
423C22—161no—
424C22—161no—
425C22—161no—
426C22—161no—
427C22C2×C4×D811yescharacteristic
428C22#329411yescharacteristic
429C22#335811yescharacteristic
430C22C2×#50911yescharacteristic
431C22#341411yescharacteristic
432C22#341511yescharacteristic
433C22#341611yescharacteristic
434C4—22no—
435C4—22no—
436C4—22no—
437C4—41no—
438C4—41no—
439C4—41no—
440C4—82no—
441C4—82no—
442C4—82no—
443C4—82no—
444C4—161no—
445C4—161no—
446C4#236711yescharacteristic
447C4#294911yescharacteristic
448C4#330411yescharacteristic
449C4#336511yescharacteristic
450C2—162no—
451C2—162no—
452C2#2321611yescharacteristic
453C2C4×#50911yescharacteristic
454C2#3345411yescharacteristic
455C2#3387611yescharacteristic
456C2#3387711yescharacteristic
457C2#3393311yescharacteristic
458C2#3393811yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC2×#493C22
LCSG ▷ C2×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
3. #23160C2semidirect
4. #13277C2semidirect
6. #13149C2semidirect
7. #13147C2semidirect
8. #13130C2semidirect
9. #13130C2semidirect
10. #12937C2semidirect
11. #12937C2semidirect
12. #12674C2semidirect
13. #12663C2semidirect
14. #12637C2semidirect
15. C4×#357C2semidirect
22. #2244C4semidirect
73. #499C2×C4semidirect
118. #357C2×C4semidirect
119. #357C2×C4semidirect
120. #357C2×C4semidirect
121. #357C2×C4semidirect
123. #346C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#22442
#231852
#231602
#132772
#132522
#131492
#131472
#131302
#129372
#126742
#126632
#126372
C4×#3572