Finite Group Databasebeta

← #498213

Subgroups — #498213

Order 512.

588 conjugacy classes of subgroups.

The subgroup lattice has 449 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2234C211yesmaximal, characteristic
2#23027C211yesmaximal, characteristic
3#22613C211yesmaximal, characteristic, complemented
4#13168C211yesmaximal, characteristic, complemented
5#13149C211yesmaximal, characteristic, complemented
6#13131C211yesmaximal, characteristic, complemented
7#13110C211yesmaximal, characteristic, complemented
8#13109C211yesmaximal, characteristic, complemented
9#13059C211yesmaximal, characteristic
10#12935C211yesmaximal, characteristic, complemented
11#12921C211yesmaximal, characteristic, complemented
12#12674C211yesmaximal, characteristic, complemented
13#12637C211yesmaximal, characteristic, complemented
14#12611C211yesmaximal, characteristic, complemented
15#12556C211yesmaximal, characteristic, complemented
16C23×C16C2211yescharacteristic
17C2×#466C2211yescharacteristic
18C2×C4×D8C2211yescharacteristic
19C2×#427C2211yescharacteristic
20C2×#427C2211yescharacteristic
21#2363—21no—
22#2346—21no—
23#2234C418yescomplemented
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#1926—21no—
40#1920—21no—
41#1920—21no—
42#1875—21no—
43#1864—21no—
44#1825—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×#117—21no—
65C2×#117—21no—
66C22×C16—22no—
67C22×C16—22no—
68C22×C16—41no—
69C22×C16C2311yescharacteristic
70C22×C16C2311yescharacteristic
71C22×C16C2311yescharacteristic
72C22×C16C2311yescharacteristic
73#466—22no—
74#466C2×C414yes—
75C4×D8—22no—
76C4×D8—41no—
77#427—22no—
78#427—22no—
79#427—41no—
80#427C2×C414yescomplemented
81C2×#107C2311yescharacteristic
82C2×#107C2311yescharacteristic
83C2×#107C2311yescharacteristic
84C2×#107C2311yescharacteristic
85C2×#107C2311yescharacteristic
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#106—21no—
89C2×#106—21no—
90C2×#102—21no—
91C2×#102—21no—
92C2×#102—21no—
93C2×C22⋊C8D411yescharacteristic
94C2×C4×C8C2311yescharacteristic
95C2×C4×C8C2311yescharacteristic
96#390—21no—
97#390—41no—
98#389—21no—
99#389—21no—
100#389—21no—
101#388—41no—
102#388—41no—
103#388—41no—
104#386—41no—
105#382—41no—
106#380—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—21no—
110C4×C22⋊C4—41no—
111C4×C22⋊C4D411yescharacteristic
112#376—41no—
113#367—41no—
114#367C2×C414yescomplemented
115#366—22no—
116#366C2×C414yescomplemented
117#357—22no—
118#357—41no—
119#357C2×C414yes—
120#357C2×C414yes—
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×C4C2×D411yescharacteristic
137C2×D8—21no—
138C2×D8—21no—
139C2×D8—21no—
140C2×D8—21no—
141C2×D8—21no—
142C2×D8C22×C412yes—
143C22×C8—21no—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8C22×C412yes—
148C22×C8C2×D411yescharacteristic
149C22×C8C2×D411yescharacteristic
150C22×C8C2×D411yescharacteristic
151C22×C8#4112yes—
152C22×C8C2411yescharacteristic
153#121—24no—
154#121—42no—
155C4×D4—42no—
156C4×D4—81no—
157#117—24no—
158#117—42no—
159#117—42no—
160#117—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—21no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—41no—
171C2×C22⋊C4—41no—
172C2×C22⋊C4—41no—
173C2×C22⋊C4—41no—
174C2×C22⋊C4—41no—
175C2×C22⋊C4—41no—
176C2×C22⋊C4—41no—
177C2×C22⋊C4—41no—
178C2×C22⋊C4D811yescharacteristic
179C2×C22⋊C4D811yescharacteristic
180C2×C42—41no—
181C2×C42—41no—
182C2×C42—41no—
183C2×C42—41no—
184C2×C42—41no—
185C2×C42C2×D411yescharacteristic
186C2×C42C2×D411yescharacteristic
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×C16—44no—
197C2×C16C22×C412yes—
198C2×C16C22×C412yes—
199C2×C16#4112yes—
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#107—21no—
210#107C22×C412yes—
211#107C22×C412yes—
212#107C22×C412yes—
213#107#4112yes—
214#106—24no—
215#106—42no—
216#102—24no—
217#102—42no—
218#102—81no—
219C22⋊C8—22no—
220C4×C8—21no—
221C4×C8—21no—
222C4×C8—21no—
223C4×C8—21no—
224#95—41no—
225#95—41no—
226#95—41no—
227#95—41no—
228#95—41no—
229#95—41no—
230#95—41no—
231#95D811yescharacteristic
232#95D811yescharacteristic
233C24—41no—
234C24C2×D811yescharacteristic
235C2×D4—22no—
236C2×D4—44no—
237C2×D4—41no—
238C2×D4—41no—
239C2×D4—41no—
240C2×D4—41no—
241C2×D4—41no—
242C2×D4—161no—
243C2×D4—161no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—82no—
256C22×C4—82no—
257C22×C4—161no—
258C22×C4C4×D412yes—
259C22×C4C2×D811yescharacteristic
260C22×C4C2×D811yescharacteristic
261C22×C4C2×D811yescharacteristic
262C22×C4C2×D811yescharacteristic
263C22×C4C2×D811yescharacteristic
264C22×C4#13512yes—
265C22×C4C22×D411yescharacteristic
266D8—22no—
267D8—22no—
268D8—41no—
269D8—41no—
270C2×C8—21no—
271C2×C8—21no—
272C2×C8—22no—
273C2×C8—22no—
274C2×C8—22no—
275C2×C8—22no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8C4×D412yes—
283C2×C8C23×C411yescharacteristic
284C2×C8C2×#4111yescharacteristic
285C2×C8C2×#4111yescharacteristic
286C2×C8C2×#4111yescharacteristic
287C2×C8H(5,𝔽2)11yescharacteristic
288C2×C82−1+411yescharacteristic
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—22no—
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—161no—
304C4⋊C4—161no—
305C4⋊C4—161no—
306C22⋊C4—41no—
307C22⋊C4—41no—
308C22⋊C4—44no—
309C22⋊C4—44no—
310C22⋊C4—82no—
311C22⋊C4—82no—
312C22⋊C4—82no—
313C22⋊C4—161no—
314C22⋊C4—161no—
315C42—21no—
316C42—21no—
317C42—21no—
318C42—21no—
319C42—44no—
320C42—82no—
321C42—82no—
322C42—82no—
323C42—161no—
324C16—22no—
325C16—22no—
326C16—22no—
327C16—41no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—42no—
335C23—81no—
336C23—81no—
337C23—81no—
338C23—81no—
339C23—81no—
340C23—81no—
341C23C4×D812yes—
342C23C22×D811yescharacteristic
343D4—42no—
344D4—42no—
345D4—81no—
346D4—81no—
347D4—164no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
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—21no—
365C2×C4—21no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
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—81no—
386C2×C4—81no—
387C2×C4—164no—
388C2×C4C4×D812yes—
389C2×C4C2×C4×D411yescharacteristic
390C2×C4#54611yescharacteristic
391C2×C4#54711yescharacteristic
392C2×C4C2×#13511yescharacteristic
393C2×C4#57711yescharacteristic
394C2×C4#57811yescharacteristic
395C8—22no—
396C8—41no—
397C8—41no—
398C8C4×#4111yescharacteristic
399C8#52011yescharacteristic
400C8#55111yescharacteristic
401C8#55211yescharacteristic
402C22—22no—
403C22—22no—
404C22—22no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22—82no—
410C22—82no—
411C22—82no—
412C22—161no—
413C22—161no—
414C22—161no—
415C22—161no—
416C22C2×C4×D811yescharacteristic
417C22#329411yescharacteristic
418C22#333411yescharacteristic
419C22#341411yescharacteristic
420C22#341411yescharacteristic
421C22#341511yescharacteristic
422C22#341611yescharacteristic
423C4—22no—
424C4—41no—
425C4—41no—
426C4—41no—
427C4—41no—
428C4—82no—
429C4—82no—
430C4—82no—
431C4—82no—
432C4—161no—
433C4—161no—
434C4—161no—
435C4#235911yescharacteristic
436C4#294711yescharacteristic
437C4#330011yescharacteristic
438C4#333711yescharacteristic
439C2—42no—
440C2—162no—
441C2#2291611yescharacteristic
442C2#3345111yescharacteristic
443C2#3345411yescharacteristic
444C2#3388411yescharacteristic
445C2#3388611yescharacteristic
446C2#3391411yescharacteristic
447C2#3391511yescharacteristic
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. #22613C2semidirect
4. #13168C2semidirect
5. #13149C2semidirect
6. #13131C2semidirect
7. #13110C2semidirect
8. #13109C2semidirect
10. #12935C2semidirect
11. #12921C2semidirect
12. #12674C2semidirect
13. #12637C2semidirect
14. #12611C2semidirect
15. #12556C2semidirect
23. #2234C4semidirect
80. #427C2×C4semidirect
114. #367C2×C4semidirect
116. #366C2×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×#22342
#230272
#226132
#131682
#131492
#131312
#131102
#131092
#130592
#129352
#129212
#126742
#126372
#126112
#125562