Finite Group Databasebeta

← #498445

Subgroups — #498445

Order 512.

606 conjugacy classes of subgroups.

The subgroup lattice has 461 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2211C211yesmaximal, characteristic
2#23186C211yesmaximal, characteristic
3#22809C211yesmaximal, characteristic, complemented
4#13237C211yesmaximal, characteristic
5#13236C211yesmaximal, characteristic, complemented
6#13204C211yesmaximal, characteristic
7#13178C211yesmaximal, characteristic, complemented
8#13132C211yesmaximal, characteristic, complemented
9#13130C211yesmaximal, characteristic
10#13015C211yesmaximal, characteristic, complemented
11#13014C211yesmaximal, characteristic
12C16×C4⋊C4C211yesmaximal, characteristic, complemented
13C4×#358C211yesmaximal, characteristic, complemented
14C4×#358C211yesmaximal, characteristic, complemented
15C4×#357C211yesmaximal, characteristic
16C22×Q32C2211yescharacteristic
17C22×SD32C2211yescharacteristic
18C2×#495C2211yescharacteristic
19C2×C4×Q16C2211yescharacteristic
20C2×C4×Q16C2211yescharacteristic
21C2×C4×D8C2211yescharacteristic
22#2367—21no—
23#2356—21no—
24#2211C418yescomplemented
25C2×#367C2211yescharacteristic
26C2×#366C2211yescharacteristic
27C2×C4⋊C16C2211yescharacteristic
28C2×#358C2211yescharacteristic
29C2×#358C2211yescharacteristic
30C2×#358C2211yescharacteristic
31C2×#358C2211yescharacteristic
32C2×#358C2211yescharacteristic
33C2×#357C2211yescharacteristic
34C2×#357C2211yescharacteristic
35C2×#357C2211yescharacteristic
36C2×C4×C16C2211yescharacteristic
37C2×C4×C16C2211yescharacteristic
38C2×C4×C16C2211yescharacteristic
39#1983—21no—
40#1970C2211yescharacteristic
41#1969C2211yescharacteristic
42#1969C2211yescharacteristic
43#1969C2211yescharacteristic
44#1927—21no—
45#1927—21no—
46#1880—21no—
47#1877—21no—
48#1846C2211yescharacteristic
49#1805—21no—
50C4×#107C2211yescharacteristic
51C4×#107C2211yescharacteristic
52C4×#107C2211yescharacteristic
53C8×C4⋊C4C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1374C2211yescharacteristic
59#1374C2211yescharacteristic
60C22×Q16C2311yescharacteristic
61C22×Q16C2311yescharacteristic
62C22×SD16—21no—
63C22×D8C2311yescharacteristic
64C2×#124—21no—
65C2×C4×Q8—21no—
66C2×C4×Q8—21no—
67C2×C4×D4—21no—
68C2×Q32—41no—
69C2×Q32—41no—
70C2×Q32C2×C414yescomplemented
71C2×SD32—41no—
72C2×SD32—41no—
73C2×SD32C2×C414yes—
74C22×C16C2311yescharacteristic
75C22×C16C2311yescharacteristic
76C22×C16C2311yescharacteristic
77C22×C16C2311yescharacteristic
78#495—22no—
79#495C2×C414yes—
80C4×Q16—22no—
81C4×Q16—22no—
82C4×Q16—41no—
83C4×Q16C2×C414yescomplemented
84C4×D8—22no—
85C4×D8—41no—
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#107C2311yescharacteristic
89C2×#107C2311yescharacteristic
90C2×#107C2311yescharacteristic
91C2×#106—21no—
92C2×C4⋊C8D411yescharacteristic
93C2×#103—21no—
94C2×#103—21no—
95C2×#103—21no—
96C2×#103—21no—
97C2×#103—21no—
98C2×#102—21no—
99C2×#102—21no—
100C2×C4×C8C2311yescharacteristic
101C2×C4×C8C2311yescharacteristic
102C2×C4×C8C2311yescharacteristic
103#391—21no—
104#391—21no—
105#391—21no—
106#390—21no—
107#388—41no—
108#388—41no—
109#387—41no—
110#387—41no—
111#385—41no—
112#383—41no—
113#382—21no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4D411yescharacteristic
118C4×C22⋊C4—41no—
119#376—41no—
120#367—41no—
121#366—41no—
122C4⋊C16C2×C414yescomplemented
123#358—22no—
124#358—22no—
125#358—22no—
126#358—41no—
127#358C2×C414yescomplemented
128#357—22no—
129#357—41no—
130#357C2×C414yes—
131C4×C16—22no—
132C4×C16—41no—
133C4×C16—41no—
134#340—21no—
135#340—21no—
136#340—21no—
137#336D411yescharacteristic
138#336D411yescharacteristic
139C22×Q8—21no—
140C22×Q8—21no—
141C22×D4—21no—
142C23×C4—41no—
143C2×Q16—21no—
144C2×Q16—21no—
145C2×Q16—21no—
146C2×Q16—21no—
147C2×Q16—21no—
148C2×Q16—21no—
149C2×Q16—21no—
150C2×Q16—21no—
151C2×Q16—21no—
152C2×Q16—21no—
153C2×Q16C22×C412yes—
154C2×Q16C22×C412yes—
155C2×SD16—24no—
156C2×SD16—81no—
157C2×SD16—81no—
158C2×D8—21no—
159C2×D8—21no—
160C2×D8—21no—
161C2×D8—21no—
162C2×D8—21no—
163C2×D8C22×C412yes—
164C22×C8C2×D411yescharacteristic
165C22×C8C2×D411yescharacteristic
166C22×C8C2×D411yescharacteristic
167C22×C8C2411yescharacteristic
168#124—24no—
169#124—42no—
170C4×Q8—24no—
171C4×Q8—42no—
172C4×Q8—42no—
173C4×Q8—81no—
174C4×D4—42no—
175C4×D4—81no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4D811yescharacteristic
184C2×C4⋊C4D811yescharacteristic
185C2×C22⋊C4—41no—
186C2×C22⋊C4—41no—
187C2×C22⋊C4—41no—
188C2×C22⋊C4—41no—
189C2×C22⋊C4—41no—
190C2×C42—41no—
191C2×C42—41no—
192C2×C42—41no—
193C2×C42—41no—
194C2×C42—41no—
195C2×C42C2×D411yescharacteristic
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D411yescharacteristic
198Q32—44no—
199SD32—44no—
200C2×C16—21no—
201C2×C16—21no—
202C2×C16—21no—
203C2×C16—21no—
204C2×C16—21no—
205C2×C16—21no—
206C2×C16—21no—
207C2×C16—21no—
208C2×C16—21no—
209C2×C16C22×C412yes—
210C2×C16C22×C412yes—
211C2×C16#4112yes—
212#107—21no—
213#107—21no—
214#107—21no—
215#107—21no—
216#107—21no—
217#107—21no—
218#107—21no—
219#107—21no—
220#107C22×C412yes—
221#107#4112yes—
222#106—81no—
223C4⋊C8—22no—
224#103—24no—
225#103—42no—
226#103—81no—
227#103—81no—
228#103—81no—
229#102—42no—
230#102—81no—
231C4×C8—21no—
232C4×C8—21no—
233C4×C8—21no—
234C4×C8—21no—
235C4×C8C22×C412yes—
236C4×C8#4112yes—
237#95—41no—
238#95—41no—
239#95—41no—
240#95—41no—
241#95—41no—
242#95—41no—
243#95—41no—
244#95—41no—
245#95—41no—
246#95—41no—
247#95—41no—
248#95D811yescharacteristic
249#95D811yescharacteristic
250C24—41no—
251C2×Q8—22no—
252C2×Q8—22no—
253C2×Q8—41no—
254C2×Q8—41no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×Q8—41no—
263C2×D4—22no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—82no—
277C22×C4—82no—
278C22×C4—161no—
279C22×C4C2×D811yescharacteristic
280C22×C4C2×D811yescharacteristic
281C22×C4C2×D811yescharacteristic
282C22×C4C2×D811yescharacteristic
283C22×C4C2×D811yescharacteristic
284C22×C4C2×D811yescharacteristic
285C22×C4C22×D411yescharacteristic
286Q16—22no—
287Q16—22no—
288Q16—22no—
289Q16—22no—
290Q16—41no—
291Q16—41no—
292Q16—41no—
293Q16—41no—
294SD16—84no—
295D8—22no—
296D8—22no—
297D8—41no—
298D8—41no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8C4×D412yes—
308C2×C8C23×C411yescharacteristic
309C2×C8C2×#4111yescharacteristic
310C2×C8C2×#4111yescharacteristic
311C2×C8C2×#4111yescharacteristic
312C2×C8H(5,𝔽2)11yescharacteristic
313C2×C8H(5,𝔽2)11yescharacteristic
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—22no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—44no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—82no—
328C22⋊C4—44no—
329C22⋊C4—44no—
330C22⋊C4—82no—
331C22⋊C4—82no—
332C22⋊C4—161no—
333C42—21no—
334C42—21no—
335C42—21no—
336C42—21no—
337C42—44no—
338C42—82no—
339C42—82no—
340C42—82no—
341C42—161no—
342C42C4×D412yes—
343C42#13512yes—
344C16—22no—
345C16—22no—
346C16—22no—
347C16—41no—
348C23—42no—
349C23—81no—
350C23—81no—
351C23—81no—
352C23—81no—
353C23—81no—
354C23—81no—
355C23C22×D811yescharacteristic
356Q8—42no—
357Q8—42no—
358Q8—42no—
359Q8—42no—
360Q8—81no—
361Q8—81no—
362Q8—81no—
363Q8—81no—
364D4—42no—
365D4—42no—
366D4—81no—
367D4—81no—
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—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
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—81no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—164no—
404C2×C4C4×D812yes—
405C2×C4C4×D812yes—
406C2×C4#50812yes—
407C2×C4#50812yes—
408C2×C4C2×C4×D411yescharacteristic
409C2×C4#54611yescharacteristic
410C2×C4#55011yescharacteristic
411C2×C4C2×#13511yescharacteristic
412C2×C4#57611yescharacteristic
413C2×C4#57711yescharacteristic
414C8—22no—
415C8—41no—
416C8—41no—
417C8C4×#4111yescharacteristic
418C8#51811yescharacteristic
419C8#55311yescharacteristic
420C8#55511yescharacteristic
421C22—82no—
422C22—82no—
423C22—82no—
424C22—161no—
425C22—161no—
426C22—161no—
427C22—161no—
428C22C2×C4×D811yescharacteristic
429C22#329411yescharacteristic
430C22#339111yescharacteristic
431C22C2×#50811yescharacteristic
432C22C2×#50811yescharacteristic
433C22#341611yescharacteristic
434C22#341711yescharacteristic
435C4—22no—
436C4—22no—
437C4—22no—
438C4—22no—
439C4—41no—
440C4—41no—
441C4—82no—
442C4—82no—
443C4—82no—
444C4—82no—
445C4—161no—
446C4—161no—
447C4—161no—
448C4#236311yescharacteristic
449C4#294611yescharacteristic
450C4#330511yescharacteristic
451C4#339411yescharacteristic
452C2—162no—
453C2#2309311yescharacteristic
454C2C4×#50811yescharacteristic
455C2#3345511yescharacteristic
456C2#3386311yescharacteristic
457C2#3386711yescharacteristic
458C2#3396011yescharacteristic
459C2#3396111yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC22×D8C23
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. #22809C2semidirect
5. #13236C2semidirect
7. #13178C2semidirect
8. #13132C2semidirect
10. #13015C2semidirect
12. C16×C4⋊C4C2semidirect
13. C4×#358C2semidirect
14. C4×#358C2semidirect
24. #2211C4semidirect
70. C2×Q32C2×C4semidirect
83. C4×Q16C2×C4semidirect
122. C4⋊C16C2×C4semidirect
127. #358C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#22112
#231862
#228092
#132372
#132362
#132042
#131782
#131322
#131302
#130152
#130142
C16×C4⋊C42
C4×#3582
C4×#3572