Finite Group Databasebeta

← #498456

Subgroups — #498456

Order 512.

650 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2207C211yesmaximal, characteristic
2#23185C211yesmaximal, characteristic
3#22809C211yesmaximal, characteristic, complemented
4#13238C211yesmaximal, characteristic, complemented
5#13237C211yesmaximal, characteristic, complemented
6#13206C211yesmaximal, characteristic
7#13178C211yesmaximal, characteristic, complemented
8#13131C211yesmaximal, characteristic, complemented
9#13130C211yesmaximal, characteristic, complemented
10#13014C211yesmaximal, characteristic, complemented
11#13014C211yesmaximal, characteristic, complemented
12#12663C211yesmaximal, characteristic, complemented
13#12638C211yesmaximal, characteristic, complemented
14#12637C211yesmaximal, characteristic, complemented
15C4×#357C211yesmaximal, characteristic, complemented
16C22×SD32C2211yescharacteristic
17C22×SD32C2211yescharacteristic
18C2×#493C2211yescharacteristic
19C2×C4×Q16C2211yescharacteristic
20C2×C4×D8C2211yescharacteristic
21C2×C4×D8C2211yescharacteristic
22#2366—21no—
23#2356—21no—
24#2207C418yescomplemented
25C2×#367C2211yescharacteristic
26C2×#366C2211yescharacteristic
27C2×C4⋊C16C2211yescharacteristic
28C2×#358C2211yescharacteristic
29C2×#358C2211yescharacteristic
30C2×#357C2211yescharacteristic
31C2×#357C2211yescharacteristic
32C2×#357C2211yescharacteristic
33C2×#357C2211yescharacteristic
34C2×#357C2211yescharacteristic
35C2×#357C2211yescharacteristic
36C2×#346C2211yescharacteristic
37C2×#346C2211yescharacteristic
38C2×C4×C16C2211yescharacteristic
39#1982—21no—
40#1970C2211yescharacteristic
41#1970C2211yescharacteristic
42#1969C2211yescharacteristic
43#1969C2211yescharacteristic
44#1927—21no—
45#1927—21no—
46#1879—21no—
47#1879—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×D8—21no—
62C22×D8C2311yescharacteristic
63C22×D8C2311yescharacteristic
64C2×#127—21no—
65C2×C4×Q8—21no—
66C2×C4×D4—21no—
67C2×C4×D4—21no—
68C2×C4×D4—41no—
69C2×SD32—41no—
70C2×SD32—41no—
71C2×SD32—41no—
72C2×SD32—41no—
73C2×SD32C2×C414yescomplemented
74C2×SD32C2×C414yescomplemented
75C22×C16C2311yescharacteristic
76C22×C16C2311yescharacteristic
77C22×C16C2311yescharacteristic
78C22×C16C2311yescharacteristic
79#493—22no—
80#493C2×C414yes—
81C4×Q16—22no—
82C4×Q16C2×C414yescomplemented
83C4×D8—22no—
84C4×D8—22no—
85C4×D8—41no—
86C4×D8—41no—
87C2×#107C2311yescharacteristic
88C2×#107C2311yescharacteristic
89C2×#107C2311yescharacteristic
90C2×#107C2311yescharacteristic
91C2×#107C2311yescharacteristic
92C2×#106—21no—
93C2×C4⋊C8D411yescharacteristic
94C2×#103—21no—
95C2×#103—21no—
96C2×#103—21no—
97C2×#102—21no—
98C2×#102—21no—
99C2×#102—21no—
100C2×#102—21no—
101C2×C4×C8C2311yescharacteristic
102C2×C4×C8C2311yescharacteristic
103C2×C4×C8C2311yescharacteristic
104#391—21no—
105#391—21no—
106#390—21no—
107#390—21no—
108#390—41no—
109#387—41no—
110#387—41no—
111#386—41no—
112#386—41no—
113#383—41no—
114#382—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4D411yescharacteristic
119#376—41no—
120#367—41no—
121#366—41no—
122C4⋊C16C2×C414yescomplemented
123#358—22no—
124#358—22no—
125#357—22no—
126#357—22no—
127#357—41no—
128#357—41no—
129#357C2×C414yescomplemented
130#357C2×C414yescomplemented
131#346—22no—
132#346—41no—
133C4×C16—41no—
134#340—21no—
135#340—21no—
136#340—21no—
137#336D411yescharacteristic
138#336D411yescharacteristic
139C22×Q8—21no—
140C22×D4—21no—
141C22×D4—21no—
142C22×D4—41no—
143C22×D4—41no—
144C23×C4—41no—
145C23×C4—41no—
146C2×Q16—21no—
147C2×Q16—21no—
148C2×Q16—21no—
149C2×Q16—21no—
150C2×Q16—21no—
151C2×Q16C22×C412yes—
152C2×D8—24no—
153C2×D8—21no—
154C2×D8—21no—
155C2×D8—21no—
156C2×D8—21no—
157C2×D8—21no—
158C2×D8—21no—
159C2×D8—21no—
160C2×D8—21no—
161C2×D8—21no—
162C2×D8—21no—
163C2×D8—81no—
164C2×D8—81no—
165C2×D8C22×C412yes—
166C2×D8C22×C412yes—
167C22×C8C2×D411yescharacteristic
168C22×C8C2×D411yescharacteristic
169C22×C8C2×D411yescharacteristic
170C22×C8C2411yescharacteristic
171#127—24no—
172#127—42no—
173C4×Q8—24no—
174C4×Q8—42no—
175C4×D4—42no—
176C4×D4—42no—
177C4×D4—81no—
178C4×D4—81no—
179C4×D4—84no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4D811yescharacteristic
187C2×C4⋊C4D811yescharacteristic
188C2×C22⋊C4—41no—
189C2×C22⋊C4—41no—
190C2×C22⋊C4—41no—
191C2×C22⋊C4—41no—
192C2×C22⋊C4—41no—
193C2×C22⋊C4—41no—
194C2×C42—41no—
195C2×C42—41no—
196C2×C42—41no—
197C2×C42—41no—
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201SD32—44no—
202SD32—44no—
203C2×C16—21no—
204C2×C16—21no—
205C2×C16—21no—
206C2×C16—21no—
207C2×C16—21no—
208C2×C16—21no—
209C2×C16—21no—
210C2×C16—21no—
211C2×C16—21no—
212C2×C16C22×C412yes—
213C2×C16C22×C412yes—
214C2×C16#4112yes—
215#107—21no—
216#107—21no—
217#107—21no—
218#107—21no—
219#107—21no—
220#107—21no—
221#107—21no—
222#107—21no—
223#107C22×C412yes—
224#107#4112yes—
225#106—81no—
226C4⋊C8—22no—
227#103—24no—
228#103—81no—
229#103—81no—
230#102—42no—
231#102—42no—
232#102—81no—
233#102—81no—
234C4×C8—21no—
235C4×C8—21no—
236C4×C8—21no—
237C4×C8—21no—
238C4×C8C22×C412yes—
239C4×C8#4112yes—
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#95—41no—
249#95D811yescharacteristic
250#95D811yescharacteristic
251C24—41no—
252C24—41no—
253C2×Q8—22no—
254C2×Q8—41no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×D4—22no—
260C2×D4—22no—
261C2×D4—41no—
262C2×D4—41no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—44no—
272C2×D4—44no—
273C2×D4—82no—
274C2×D4—82no—
275C2×D4—161no—
276C2×D4—161no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—82no—
284C22×C4—82no—
285C22×C4—82no—
286C22×C4—82no—
287C22×C4—161no—
288C22×C4—161no—
289C22×C4C2×D811yescharacteristic
290C22×C4C2×D811yescharacteristic
291C22×C4C2×D811yescharacteristic
292C22×C4C2×D811yescharacteristic
293C22×C4C2×D811yescharacteristic
294C22×C4C2×D811yescharacteristic
295C22×C4C22×D411yescharacteristic
296Q16—22no—
297Q16—22no—
298Q16—41no—
299Q16—41no—
300D8—22no—
301D8—22no—
302D8—22no—
303D8—22no—
304D8—41no—
305D8—41no—
306D8—41no—
307D8—41no—
308D8—84no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8C4×D412yes—
318C2×C8C23×C411yescharacteristic
319C2×C8C2×#4111yescharacteristic
320C2×C8C2×#4111yescharacteristic
321C2×C8C2×#4111yescharacteristic
322C2×C8H(5,𝔽2)11yescharacteristic
323C2×C8H(5,𝔽2)11yescharacteristic
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—44no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C22⋊C4—82no—
338C22⋊C4—82no—
339C22⋊C4—82no—
340C22⋊C4—82no—
341C22⋊C4—161no—
342C22⋊C4—161no—
343C42—21no—
344C42—21no—
345C42—21no—
346C42—21no—
347C42—44no—
348C42—82no—
349C42—82no—
350C42—82no—
351C42C4×D412yes—
352C42#13512yes—
353C16—22no—
354C16—22no—
355C16—22no—
356C16—41no—
357C23—42no—
358C23—42no—
359C23—81no—
360C23—81no—
361C23—81no—
362C23—81no—
363C23—81no—
364C23—81no—
365C23—81no—
366C23—81no—
367C23—81no—
368C23—81no—
369C23—81no—
370C23—81no—
371C23C22×D811yescharacteristic
372Q8—42no—
373Q8—42no—
374Q8—81no—
375Q8—81no—
376D4—42no—
377D4—42no—
378D4—42no—
379D4—42no—
380D4—88no—
381D4—81no—
382D4—81no—
383D4—81no—
384D4—81no—
385D4—164no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—164no—
420C2×C4—164no—
421C2×C4C4×D812yes—
422C2×C4C4×D812yes—
423C2×C4#50912yes—
424C2×C4C2×C4×D411yescharacteristic
425C2×C4#54611yescharacteristic
426C2×C4#55011yescharacteristic
427C2×C4C2×#13511yescharacteristic
428C2×C4#57611yescharacteristic
429C2×C4#57711yescharacteristic
430C8—22no—
431C8—41no—
432C8—41no—
433C8C4×#4111yescharacteristic
434C8#51811yescharacteristic
435C8#55311yescharacteristic
436C8#55511yescharacteristic
437C22—82no—
438C22—82no—
439C22—82no—
440C22—82no—
441C22—82no—
442C22—82no—
443C22—161no—
444C22—161no—
445C22—161no—
446C22—161no—
447C22—161no—
448C22—161no—
449C22—161no—
450C22—161no—
451C22C2×C4×D811yescharacteristic
452C22#329411yescharacteristic
453C22#339111yescharacteristic
454C22C2×#50911yescharacteristic
455C22#341411yescharacteristic
456C22#341611yescharacteristic
457C22#341611yescharacteristic
458C4—22no—
459C4—22no—
460C4—22no—
461C4—41no—
462C4—41no—
463C4—41no—
464C4—82no—
465C4—82no—
466C4—82no—
467C4—161no—
468C4—161no—
469C4—161no—
470C4#236311yescharacteristic
471C4#294611yescharacteristic
472C4#330511yescharacteristic
473C4#339411yescharacteristic
474C2—162no—
475C2—162no—
476C2#2309311yescharacteristic
477C2C4×#50911yescharacteristic
478C2#3345711yescharacteristic
479C2#3387711yescharacteristic
480C2#3388411yescharacteristic
481C2#3396411yescharacteristic
482C2#3396811yescharacteristic
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. #22809C2semidirect
4. #13238C2semidirect
5. #13237C2semidirect
7. #13178C2semidirect
8. #13131C2semidirect
9. #13130C2semidirect
10. #13014C2semidirect
11. #13014C2semidirect
12. #12663C2semidirect
13. #12638C2semidirect
14. #12637C2semidirect
15. C4×#357C2semidirect
24. #2207C4semidirect
73. C2×SD32C2×C4semidirect
74. C2×SD32C2×C4semidirect
82. C4×Q16C2×C4semidirect
122. C4⋊C16C2×C4semidirect
129. #357C2×C4semidirect
130. #357C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#22072
#231852
#228092
#132382
#132372
#132062
#131782
#131312
#131302
#130142
#126632
#126382
#126372
C4×#3572