Finite Group Databasebeta

← #498452

Subgroups — #498452

Order 512.

624 conjugacy classes of subgroups.

The subgroup lattice has 479 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2210C211yesmaximal, characteristic
2#23186C211yesmaximal, characteristic, complemented
3#22810C211yesmaximal, characteristic, complemented
4#13239C211yesmaximal, characteristic
5#13238C211yesmaximal, characteristic, complemented
6#13204C211yesmaximal, characteristic, complemented
7#13178C211yesmaximal, characteristic, complemented
8#13132C211yesmaximal, characteristic, complemented
9#13130C211yesmaximal, characteristic, complemented
10#13013C211yesmaximal, characteristic, complemented
11#13012C211yesmaximal, characteristic
12#12662C211yesmaximal, characteristic, complemented
13#12638C211yesmaximal, characteristic, complemented
14#12637C211yesmaximal, characteristic, complemented
15C4×#357C211yesmaximal, characteristic, complemented
16C22×SD32C2211yescharacteristic
17C22×D16C2211yescharacteristic
18C2×#495C2211yescharacteristic
19C2×C4×Q16C2211yescharacteristic
20C2×C4×D8C2211yescharacteristic
21C2×C4×D8C2211yescharacteristic
22#2367—21no—
23#2357—21no—
24#2210C418yescomplemented
25C2×#367C2211yescharacteristic
26C2×#366C2211yescharacteristic
27C2×C4⋊C16C2211yescharacteristic
28C2×#358C2211yescharacteristic
29C2×#358C2211yescharacteristic
30C2×#358C2211yescharacteristic
31C2×#357C2211yescharacteristic
32C2×#357C2211yescharacteristic
33C2×#357C2211yescharacteristic
34C2×#357C2211yescharacteristic
35C2×#357C2211yescharacteristic
36C2×#346C2211yescharacteristic
37C2×#346C2211yescharacteristic
38C2×C4×C16C2211yescharacteristic
39#1983—21no—
40#1970C2211yescharacteristic
41#1970C2211yescharacteristic
42#1970C2211yescharacteristic
43#1969C2211yescharacteristic
44#1926—21no—
45#1926—21no—
46#1880—21no—
47#1877—21no—
48#1846C2211yescharacteristic
49#1804—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×SD16—21no—
62C22×D8C2311yescharacteristic
63C22×D8C2311yescharacteristic
64C2×#124—21no—
65C2×C4×Q8—21no—
66C2×C4×D4—21no—
67C2×C4×D4—21no—
68C2×SD32—41no—
69C2×SD32—41no—
70C2×SD32C2×C414yescomplemented
71C2×D16—41no—
72C2×D16—41no—
73C2×D16C2×C414yes—
74C22×C16C2311yescharacteristic
75C22×C16C2311yescharacteristic
76C22×C16C2311yescharacteristic
77C22×C16C2311yescharacteristic
78#495—22no—
79#495C2×C414yes—
80C4×Q16—22no—
81C4×Q16—41no—
82C4×D8—22no—
83C4×D8—22no—
84C4×D8—41no—
85C4×D8C2×C414yescomplemented
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#107C2311yescharacteristic
89C2×#107C2311yescharacteristic
90C2×#107C2311yescharacteristic
91C2×#106—21no—
92C2×C4⋊C8D411yescharacteristic
93C2×#103—21no—
94C2×#103—21no—
95C2×#102—21no—
96C2×#102—21no—
97C2×#102—21no—
98C2×#102—21no—
99C2×#102—21no—
100C2×C4×C8C2311yescharacteristic
101C2×C4×C8C2311yescharacteristic
102C2×C4×C8C2311yescharacteristic
103#391—21no—
104#390—21no—
105#390—21no—
106#390—21no—
107#388—41no—
108#388—41no—
109#388—41no—
110#388—41no—
111#385—41no—
112#382—21no—
113#381—41no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4D411yescharacteristic
118C4×C22⋊C4—41no—
119C4×C22⋊C4—41no—
120#367—41no—
121#366—41no—
122C4⋊C16C2×C414yescomplemented
123#358—22no—
124#358—41no—
125#358C2×C414yescomplemented
126#357—22no—
127#357—22no—
128#357—22no—
129#357—41no—
130#357C2×C414yes—
131#346—41no—
132#346—41no—
133C4×C16—22no—
134#340—21no—
135#340—21no—
136#340—21no—
137#336D411yescharacteristic
138#336D411yescharacteristic
139C22×Q8—21no—
140C22×D4—21no—
141C22×D4—21no—
142C23×C4—41no—
143C23×C4—41no—
144C2×Q16—21no—
145C2×Q16—21no—
146C2×Q16—21no—
147C2×Q16—21no—
148C2×Q16—21no—
149C2×Q16C22×C412yes—
150C2×SD16—24no—
151C2×SD16—81no—
152C2×SD16—81no—
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×D8C22×C412yes—
164C2×D8C22×C412yes—
165C22×C8C2×D411yescharacteristic
166C22×C8C2×D411yescharacteristic
167C22×C8C2×D411yescharacteristic
168C22×C8C2411yescharacteristic
169#124—24no—
170#124—42no—
171C4×Q8—42no—
172C4×Q8—81no—
173C4×D4—24no—
174C4×D4—42no—
175C4×D4—42no—
176C4×D4—81no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
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×C22⋊C4—41no—
191C2×C22⋊C4—41no—
192C2×C22⋊C4—41no—
193C2×C22⋊C4—41no—
194C2×C22⋊C4—41no—
195C2×C42—41no—
196C2×C42—41no—
197C2×C42—41no—
198C2×C42—41no—
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202SD32—44no—
203D16—44no—
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×C16—21no—
213C2×C16C22×C412yes—
214C2×C16C22×C412yes—
215C2×C16#4112yes—
216#107—21no—
217#107—21no—
218#107—21no—
219#107—21no—
220#107—21no—
221#107—21no—
222#107—21no—
223#107—21no—
224#107C22×C412yes—
225#107#4112yes—
226#106—81no—
227C4⋊C8—22no—
228#103—42no—
229#103—81no—
230#102—24no—
231#102—42no—
232#102—81no—
233#102—81no—
234#102—81no—
235C4×C8—21no—
236C4×C8—21no—
237C4×C8—21no—
238C4×C8—21no—
239C4×C8C22×C412yes—
240C4×C8#4112yes—
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—
251C24—41no—
252C2×Q8—22no—
253C2×Q8—41no—
254C2×Q8—41no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×Q8—41no—
258C2×D4—22no—
259C2×D4—22no—
260C2×D4—41no—
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—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—44no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—82no—
278C22×C4—82no—
279C22×C4—161no—
280C22×C4—161no—
281C22×C4—161no—
282C22×C4C2×D811yescharacteristic
283C22×C4C2×D811yescharacteristic
284C22×C4C2×D811yescharacteristic
285C22×C4C2×D811yescharacteristic
286C22×C4C2×D811yescharacteristic
287C22×C4C2×D811yescharacteristic
288C22×C4C22×D411yescharacteristic
289Q16—22no—
290Q16—22no—
291Q16—41no—
292Q16—41no—
293SD16—84no—
294D8—22no—
295D8—22no—
296D8—22no—
297D8—22no—
298D8—41no—
299D8—41no—
300D8—41no—
301D8—41no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8C4×D412yes—
311C2×C8C23×C411yescharacteristic
312C2×C8C2×#4111yescharacteristic
313C2×C8C2×#4111yescharacteristic
314C2×C8C2×#4111yescharacteristic
315C2×C8H(5,𝔽2)11yescharacteristic
316C2×C8H(5,𝔽2)11yescharacteristic
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—82no—
330C22⋊C4—44no—
331C22⋊C4—44no—
332C22⋊C4—44no—
333C22⋊C4—82no—
334C22⋊C4—82no—
335C22⋊C4—82no—
336C22⋊C4—82no—
337C22⋊C4—161no—
338C22⋊C4—161no—
339C22⋊C4—161no—
340C42—21no—
341C42—21no—
342C42—21no—
343C42—21no—
344C42—82no—
345C42—161no—
346C42—161no—
347C42—161no—
348C42C4×D412yes—
349C42#13512yes—
350C16—22no—
351C16—22no—
352C16—22no—
353C16—41no—
354C23—42no—
355C23—42no—
356C23—81no—
357C23—81no—
358C23—81no—
359C23—81no—
360C23—81no—
361C23—81no—
362C23—81no—
363C23—81no—
364C23—81no—
365C23—81no—
366C23—81no—
367C23—81no—
368C23C22×D811yescharacteristic
369Q8—42no—
370Q8—42no—
371Q8—81no—
372Q8—81no—
373D4—42no—
374D4—42no—
375D4—42no—
376D4—42no—
377D4—81no—
378D4—81no—
379D4—81no—
380D4—81no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
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—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
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—164no—
416C2×C4—164no—
417C2×C4C4×D812yes—
418C2×C4C4×D812yes—
419C2×C4C2×C4×D411yescharacteristic
420C2×C4#54611yescharacteristic
421C2×C4#55011yescharacteristic
422C2×C4C2×#13511yescharacteristic
423C2×C4#57611yescharacteristic
424C2×C4#57711yescharacteristic
425C8—22no—
426C8—41no—
427C8—41no—
428C8C4×#4111yescharacteristic
429C8#51811yescharacteristic
430C8#55311yescharacteristic
431C8#55511yescharacteristic
432C22—82no—
433C22—82no—
434C22—82no—
435C22—82no—
436C22—82no—
437C22—82no—
438C22—161no—
439C22—161no—
440C22—161no—
441C22—161no—
442C22—161no—
443C22—161no—
444C22—161no—
445C22—161no—
446C22C2×C4×D811yescharacteristic
447C22#329411yescharacteristic
448C22#339111yescharacteristic
449C22#341411yescharacteristic
450C22#341411yescharacteristic
451C22#341511yescharacteristic
452C22#341611yescharacteristic
453C4—22no—
454C4—22no—
455C4—41no—
456C4—41no—
457C4—41no—
458C4—41no—
459C4—82no—
460C4—82no—
461C4—82no—
462C4—161no—
463C4—161no—
464C4—161no—
465C4#236311yescharacteristic
466C4#294611yescharacteristic
467C4#330511yescharacteristic
468C4#339411yescharacteristic
469C2—162no—
470C2—162no—
471C2#2309311yescharacteristic
472C2#3345111yescharacteristic
473C2#3345411yescharacteristic
474C2#3388911yescharacteristic
475C2#3389311yescharacteristic
476C2#3396811yescharacteristic
477C2#3397011yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC22×D16C22
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
2. #23186C2semidirect
3. #22810C2semidirect
5. #13238C2semidirect
6. #13204C2semidirect
7. #13178C2semidirect
8. #13132C2semidirect
9. #13130C2semidirect
10. #13013C2semidirect
12. #12662C2semidirect
13. #12638C2semidirect
14. #12637C2semidirect
15. C4×#357C2semidirect
24. #2210C4semidirect
70. C2×SD32C2×C4semidirect
85. C4×D8C2×C4semidirect
122. C4⋊C16C2×C4semidirect
125. #358C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#22102
#231862
#228102
#132392
#132382
#132042
#131782
#131322
#131302
#130132
#130122
#126622
#126382
#126372
C4×#3572