Finite Group Databasebeta

← #498495

Subgroups — #498495

Order 512.

636 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2217C211yesmaximal, characteristic
2#23027C211yesmaximal, characteristic
3#23026C211yesmaximal, characteristic, complemented
4#13150C211yesmaximal, characteristic, complemented
5#13149C211yesmaximal, characteristic, complemented
6#13111C211yesmaximal, characteristic, complemented
7#13110C211yesmaximal, characteristic, complemented
8#13014C211yesmaximal, characteristic, complemented
9#13013C211yesmaximal, characteristic, complemented
10#12994C211yesmaximal, characteristic, complemented
11#12992C211yesmaximal, characteristic
12#12638C211yesmaximal, characteristic, complemented
13#12637C211yesmaximal, characteristic, complemented
14#12611C211yesmaximal, characteristic, complemented
15#12556C211yesmaximal, characteristic, complemented
16C22×SD32C2211yescharacteristic
17C23×C16C2211yescharacteristic
18C2×#467C2211yescharacteristic
19C2×#466C2211yescharacteristic
20C2×C4×Q16C2211yescharacteristic
21C2×C4×D8C2211yescharacteristic
22C2×#427C2211yescharacteristic
23C2×#427C2211yescharacteristic
24#2363—21no—
25#2362—21no—
26#2217C418yescomplemented
27C2×#367C2211yescharacteristic
28C2×#358C2211yescharacteristic
29C2×#358C2211yescharacteristic
30C2×#358C2211yescharacteristic
31C2×#357C2211yescharacteristic
32C2×#357C2211yescharacteristic
33C2×#357C2211yescharacteristic
34C2×C22⋊C16C2211yescharacteristic
35C2×#346C2211yescharacteristic
36C2×#346C2211yescharacteristic
37#1970C2211yescharacteristic
38#1969C2211yescharacteristic
39#1942C2211yescharacteristic
40#1942C2211yescharacteristic
41#1927—21no—
42#1926—21no—
43#1875—21no—
44#1873—21no—
45#1865—21no—
46#1864—21no—
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×Q16C2311yescharacteristic
61C22×D8C2311yescharacteristic
62C23×C8C2311yescharacteristic
63C2×C22⋊Q8—21no—
64C2×#121—21no—
65C2×C4×Q8—21no—
66C2×C4×D4—21no—
67C2×#117—21no—
68C2×#117—21no—
69C2×SD32—41no—
70C2×SD32—41no—
71C2×SD32C2×C414yes—
72C22×C16—41no—
73C22×C16—41no—
74C22×C16C2×C414yescomplemented
75C22×C16C2311yescharacteristic
76C22×C16C2311yescharacteristic
77C22×C16C2311yescharacteristic
78C22×C16C2311yescharacteristic
79#467—22no—
80#467C2×C414yescomplemented
81#466—22no—
82#466C2×C414yes—
83C4×Q16—22no—
84C4×Q16—41no—
85C4×D8—22no—
86C4×D8—41no—
87#427—22no—
88#427—22no—
89#427—41no—
90#427—41no—
91C2×#107C2311yescharacteristic
92C2×#107C2311yescharacteristic
93C2×#107C2311yescharacteristic
94C2×#107C2311yescharacteristic
95C2×#107C2311yescharacteristic
96C2×#107C2311yescharacteristic
97C2×#106—21no—
98C2×#106—21no—
99C2×#103—21no—
100C2×#103—21no—
101C2×#103—21no—
102C2×#102—21no—
103C2×#102—21no—
104C2×#102—21no—
105C2×C22⋊C8D411yescharacteristic
106C2×C4×C8C2311yescharacteristic
107C2×C4×C8C2311yescharacteristic
108#391—21no—
109#390—21no—
110#390—41no—
111#389—21no—
112#389—21no—
113#388—41no—
114#388—41no—
115#388—41no—
116#387—41no—
117#386—41no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—41no—
121C4×C22⋊C4—41no—
122C4×C22⋊C4D411yescharacteristic
123#367C2×C414yescomplemented
124#358—22no—
125#358—41no—
126#358C2×C414yescomplemented
127#357—22no—
128#357—41no—
129#357C2×C414yes—
130C22⋊C16—22no—
131#346—41no—
132#346—41no—
133#340—21no—
134#340—21no—
135#340—21no—
136#340—21no—
137#336D411yescharacteristic
138#336D411yescharacteristic
139C22×Q8—21no—
140C22×D4—21no—
141C22×D4—41no—
142C23×C4—41no—
143C23×C4C2×D411yescharacteristic
144C2×Q16—21no—
145C2×Q16—21no—
146C2×Q16—21no—
147C2×Q16—21no—
148C2×Q16—21no—
149C2×Q16C22×C412yes—
150C2×D8—21no—
151C2×D8—21no—
152C2×D8—21no—
153C2×D8—21no—
154C2×D8—21no—
155C2×D8C22×C412yes—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8C22×C412yes—
161C22×C8C2×D411yescharacteristic
162C22×C8C2×D411yescharacteristic
163C22×C8C2×D411yescharacteristic
164C22×C8#4112yes—
165C22×C8C2411yescharacteristic
166C22⋊Q8—24no—
167C22⋊Q8—42no—
168#121—24no—
169#121—42no—
170C4×Q8—42no—
171C4×Q8—81no—
172C4×D4—42no—
173C4×D4—81no—
174#117—42no—
175#117—42no—
176#117—81no—
177#117—81no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—41no—
187C2×C4⋊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×C22⋊C4D811yescharacteristic
196C2×C22⋊C4D811yescharacteristic
197C2×C42—41no—
198C2×C42—41no—
199C2×C42—41no—
200C2×C42—41no—
201C2×C42—41no—
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204SD32—44no—
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×C16—21no—
214C2×C16—21no—
215C2×C16—44no—
216C2×C16C22×C412yes—
217C2×C16C22×C412yes—
218#107—21no—
219#107—21no—
220#107—21no—
221#107—21no—
222#107—21no—
223#107—21no—
224#107—21no—
225#107—21no—
226#107C22×C412yes—
227#107C22×C412yes—
228#107#4112yes—
229#107#4112yes—
230#106—42no—
231#106—42no—
232#103—24no—
233#103—42no—
234#103—81no—
235#102—24no—
236#102—42no—
237#102—81no—
238C22⋊C8—22no—
239C4×C8—21no—
240C4×C8—21no—
241C4×C8—21no—
242C4×C8—21no—
243#95—41no—
244#95—41no—
245#95—41no—
246#95—41no—
247#95—41no—
248#95—41no—
249#95D811yescharacteristic
250#95D811yescharacteristic
251C24—41no—
252C24C2×D811yescharacteristic
253C2×Q8—22no—
254C2×Q8—41no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×D4—22no—
260C2×D4—44no—
261C2×D4—41no—
262C2×D4—41no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—161no—
267C2×D4—161no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—82no—
280C22×C4—82no—
281C22×C4—161no—
282C22×C4C4×D412yes—
283C22×C4C2×D811yescharacteristic
284C22×C4C2×D811yescharacteristic
285C22×C4C2×D811yescharacteristic
286C22×C4C2×D811yescharacteristic
287C22×C4C2×D811yescharacteristic
288C22×C4#13512yes—
289C22×C4C22×D411yescharacteristic
290Q16—22no—
291Q16—22no—
292Q16—41no—
293Q16—41no—
294D8—22no—
295D8—22no—
296D8—41no—
297D8—41no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—22no—
301C2×C8—22no—
302C2×C8—22no—
303C2×C8—22no—
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—22no—
320C4⋊C4—22no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—44no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—82no—
331C4⋊C4—82no—
332C4⋊C4—161no—
333C22⋊C4—41no—
334C22⋊C4—41no—
335C22⋊C4—44no—
336C22⋊C4—44no—
337C22⋊C4—82no—
338C22⋊C4—82no—
339C22⋊C4—82no—
340C22⋊C4—82no—
341C22⋊C4—161no—
342C42—21no—
343C42—21no—
344C42—21no—
345C42—21no—
346C42—82no—
347C42—82no—
348C42—82no—
349C42—82no—
350C42—161no—
351C16—22no—
352C16—22no—
353C16—41no—
354C16—41no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—42no—
362C23—81no—
363C23—81no—
364C23—81no—
365C23—81no—
366C23—81no—
367C23—81no—
368C23C4×D812yes—
369C23C22×D811yescharacteristic
370Q8—42no—
371Q8—42no—
372Q8—81no—
373Q8—81no—
374D4—42no—
375D4—42no—
376D4—81no—
377D4—81no—
378D4—164no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
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—42no—
401C2×C4—42no—
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—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—164no—
419C2×C4C4×D812yes—
420C2×C4C2×C4×D411yescharacteristic
421C2×C4#54611yescharacteristic
422C2×C4#54611yescharacteristic
423C2×C4C2×#13511yescharacteristic
424C2×C4#57711yescharacteristic
425C2×C4#57711yescharacteristic
426C8—22no—
427C8—41no—
428C8—41no—
429C8C4×#4111yescharacteristic
430C8#51811yescharacteristic
431C8#55311yescharacteristic
432C8#55311yescharacteristic
433C22—22no—
434C22—22no—
435C22—22no—
436C22—41no—
437C22—41no—
438C22—41no—
439C22—41no—
440C22—82no—
441C22—82no—
442C22—82no—
443C22—161no—
444C22—161no—
445C22—161no—
446C22—161no—
447C22C2×C4×D811yescharacteristic
448C22#329411yescharacteristic
449C22#329411yescharacteristic
450C22#341411yescharacteristic
451C22#341411yescharacteristic
452C22#341611yescharacteristic
453C22#341611yescharacteristic
454C4—22no—
455C4—41no—
456C4—41no—
457C4—41no—
458C4—41no—
459C4—82no—
460C4—82no—
461C4—82no—
462C4—82no—
463C4—82no—
464C4—161no—
465C4—161no—
466C4#236411yescharacteristic
467C4#295211yescharacteristic
468C4#330111yescharacteristic
469C4#330111yescharacteristic
470C2—42no—
471C2—162no—
472C2#2312711yescharacteristic
473C2#3345111yescharacteristic
474C2#3345711yescharacteristic
475C2#3387411yescharacteristic
476C2#3387811yescharacteristic
477C2#3388411yescharacteristic
478C2#3388811yescharacteristic
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. #23026C2semidirect
4. #13150C2semidirect
5. #13149C2semidirect
6. #13111C2semidirect
7. #13110C2semidirect
8. #13014C2semidirect
9. #13013C2semidirect
10. #12994C2semidirect
12. #12638C2semidirect
13. #12637C2semidirect
14. #12611C2semidirect
15. #12556C2semidirect
26. #2217C4semidirect
74. C22×C16C2×C4semidirect
80. #467C2×C4semidirect
123. #367C2×C4semidirect
126. #358C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#22172
#230272
#230262
#131502
#131492
#131112
#131102
#130142
#130132
#129942
#129922
#126382
#126372
#126112
#125562