Finite Group Databasebeta

← #495876

Subgroups — #495876

Order 512.

704 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2220C211yesmaximal, characteristic
2C2×#2175C211yesmaximal, characteristic, complemented
3C2×#2108C211yesmaximal, characteristic, complemented
4#21525C211yesmaximal, characteristic, complemented
5#21524C211yesmaximal, characteristic
6#13237C211yesmaximal, characteristic, complemented
7#13165C211yesmaximal, characteristic, complemented
8#13150C211yesmaximal, characteristic, complemented
9#13149C211yesmaximal, characteristic, complemented
10#13060C211yesmaximal, characteristic, complemented
11#13059C211yesmaximal, characteristic
12#12836C211yesmaximal, characteristic, complemented
13#12674C211yesmaximal, characteristic, complemented
14C4×#358C211yesmaximal, characteristic, complemented
15C4×#357C211yesmaximal, characteristic, complemented
16C22×SD32C2211yescharacteristic
17C22×M32C2211yescharacteristic
18C23×C16C2211yescharacteristic
19C2×#467C2211yescharacteristic
20C2×#466C2211yescharacteristic
21C2×C4×Q16C2211yescharacteristic
22C2×C4×D8C2211yescharacteristic
23C2×#405C2211yescharacteristic
24#2324—21no—
25#2323—21no—
26#2220C418yescomplemented
27#2175—24no—
28C2×#367C2211yescharacteristic
29C2×#366C2211yescharacteristic
30C2×#366C2211yescharacteristic
31C2×#358C2211yescharacteristic
32C2×#358C2211yescharacteristic
33C2×#358C2211yescharacteristic
34C2×#357C2211yescharacteristic
35C2×#357C2211yescharacteristic
36C2×#357C2211yescharacteristic
37#2108—24no—
38C2×#346C2211yescharacteristic
39C2×#346C2211yescharacteristic
40C2×C4×C16C2211yescharacteristic
41C2×C4×C16C2211yescharacteristic
42#1970C2211yescharacteristic
43#1969C2211yescharacteristic
44#1942C2211yescharacteristic
45#1942C2211yescharacteristic
46#1936—21no—
47#1935—21no—
48#1895—21no—
49#1894—21no—
50#1808C2211yescharacteristic
51#1808C2211yescharacteristic
52C4×#107C2211yescharacteristic
53C4×#107C2211yescharacteristic
54#1763—21no—
55#1762—21no—
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1380C2211yescharacteristic
60#1380C2211yescharacteristic
61#1380C2211yescharacteristic
62C22×Q16C2311yescharacteristic
63C22×D8C2311yescharacteristic
64C22×M16D411yescharacteristic
65C23×C8C2311yescharacteristic
66C2×C22⋊Q8—21no—
67C2×#121—21no—
68C2×C4×Q8—21no—
69C2×C4×D4—21no—
70C2×#117D411yescharacteristic
71C2×SD32—22no—
72C2×SD32—22no—
73C2×SD32C2×C414yes—
74C2×M32—22no—
75C2×M32—22no—
76C2×M32C2×C414yescomplemented
77C22×C16—22no—
78C22×C16—22no—
79C22×C16—22no—
80C22×C16C2311yescharacteristic
81C22×C16C2311yescharacteristic
82C22×C16C2311yescharacteristic
83C22×C16C2311yescharacteristic
84#467—22no—
85#467C2×C414yescomplemented
86#466—22no—
87#466C2×C414yes—
88C4×Q16—22no—
89C4×Q16—22no—
90C4×D8—22no—
91C4×D8—22no—
92C2×#107C2311yescharacteristic
93C2×#107C2311yescharacteristic
94C2×#107C2311yescharacteristic
95C2×#107C2311yescharacteristic
96C2×#107C2311yescharacteristic
97C2×#107C2311yescharacteristic
98C2×#103—21no—
99C2×#103—21no—
100C2×#103—21no—
101C2×#102—21no—
102C2×#102—21no—
103C2×#102—21no—
104#405—22no—
105#405—22no—
106C2×#97D411yescharacteristic
107C2×#97D411yescharacteristic
108C2×C4×C8C2311yescharacteristic
109C2×C4×C8C2311yescharacteristic
110#391—21no—
111#390—21no—
112#390—41no—
113#389—21no—
114#389—21no—
115#389—41no—
116#388—41no—
117#388—41no—
118#388—41no—
119#387—41no—
120#386—41no—
121#385—41no—
122#380—21no—
123#380—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126#367—22no—
127#366—22no—
128#366C2×C414yescomplemented
129#358—22no—
130#358—22no—
131#358C2×C414yescomplemented
132#357—22no—
133#357—22no—
134#357C2×C414yes—
135#346—22no—
136#346D414yescomplemented
137C4×C16—22no—
138C4×C16—22no—
139#340—21no—
140#340—21no—
141#340—21no—
142#340—21no—
143#340—21no—
144#340—21no—
145C22×Q8—21no—
146C22×D4—21no—
147C22×D4—41no—
148C23×C4—41no—
149C23×C4C2×D411yescharacteristic
150C2×Q16—21no—
151C2×Q16—21no—
152C2×Q16—21no—
153C2×Q16—21no—
154C2×Q16—21no—
155C2×Q16C22×C412yes—
156C2×D8—21no—
157C2×D8—21no—
158C2×D8—21no—
159C2×D8—21no—
160C2×D8—21no—
161C2×D8C22×C412yes—
162C2×M16—22no—
163C2×M16—22no—
164C2×M16—22no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8C22×C412yes—
170C22×C8C2×D411yescharacteristic
171C22×C8C2×D411yescharacteristic
172C22×C8C2×D411yescharacteristic
173C22×C8#4112yes—
174C22×C8C2411yescharacteristic
175C22⋊Q8—24no—
176C22⋊Q8—42no—
177#121—24no—
178#121—42no—
179C4×Q8—42no—
180C4×Q8—42no—
181C4×D4—42no—
182C4×D4—42no—
183#117—22no—
184#117—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4D811yescharacteristic
196C2×C4⋊C4D811yescharacteristic
197C2×C22⋊C4—41no—
198C2×C22⋊C4—41no—
199C2×C22⋊C4—41no—
200C2×C22⋊C4—41no—
201C2×C22⋊C4—41no—
202C2×C22⋊C4—41no—
203C2×C22⋊C4—41no—
204C2×C22⋊C4D811yescharacteristic
205C2×C22⋊C4D811yescharacteristic
206C2×C42—41no—
207C2×C42—41no—
208C2×C42—41no—
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D411yescharacteristic
211SD32—28no—
212M32—28no—
213C2×C16—28no—
214C2×C16—21no—
215C2×C16—21no—
216C2×C16—21no—
217C2×C16—21no—
218C2×C16—21no—
219C2×C16—21no—
220C2×C16C22×C412yes—
221C2×C16C22×C412yes—
222C2×C16C2×D412yes—
223C2×C16C2×D412yes—
224C2×C16#4112yes—
225C2×C16#4112yes—
226#107—21no—
227#107—21no—
228#107—21no—
229#107—21no—
230#107—21no—
231#107—21no—
232#107—21no—
233#107—21no—
234#107—21no—
235#107—21no—
236#107C22×C412yes—
237#107C22×C412yes—
238#103—24no—
239#103—42no—
240#103—42no—
241#102—24no—
242#102—42no—
243#102—42no—
244#97—22no—
245#97—22no—
246C4×C8—21no—
247C4×C8—21no—
248C4×C8C2×D412yes—
249C4×C8#4112yes—
250#95—41no—
251#95—41no—
252#95—41no—
253#95—41no—
254#95—41no—
255#95—41no—
256#95—41no—
257#95—41no—
258C24—41no—
259C24C2×D811yescharacteristic
260C2×Q8—22no—
261C2×Q8—41no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×D4—22no—
267C2×D4—44no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—161no—
274C2×D4—161no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—82no—
287C22×C4—161no—
288C22×C4—161no—
289C22×C4C4×D412yes—
290C22×C4C2×D811yescharacteristic
291C22×C4C2×D811yescharacteristic
292C22×C4C2×D811yescharacteristic
293C22×C4C2×D811yescharacteristic
294C22×C4C2×D811yescharacteristic
295C22×C4#13512yes—
296C22×C4C22×D411yescharacteristic
297Q16—22no—
298Q16—22no—
299Q16—41no—
300Q16—41no—
301D8—22no—
302D8—22no—
303D8—41no—
304D8—41no—
305M16—28no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—22no—
315C2×C8—22no—
316C2×C8C4×D412yes—
317C2×C8#12112yes—
318C2×C8#13512yes—
319C2×C8C23×C411yescharacteristic
320C2×C8C22×D411yescharacteristic
321C2×C8C2×#4111yescharacteristic
322C2×C8C2×#4111yescharacteristic
323C2×C8C2×#4111yescharacteristic
324C2×C8C2×#4111yescharacteristic
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—44no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—82no—
341C4⋊C4—161no—
342C4⋊C4—161no—
343C22⋊C4—22no—
344C22⋊C4—22no—
345C22⋊C4—44no—
346C22⋊C4—44no—
347C22⋊C4—82no—
348C22⋊C4—161no—
349C22⋊C4—161no—
350C22⋊C4—161no—
351C22⋊C4—161no—
352C42—21no—
353C42—21no—
354C42—82no—
355C42—161no—
356C42—161no—
357C42#12112yes—
358C42#13512yes—
359C16—22no—
360C16—22no—
361C16—22no—
362C16C4×D414yes—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—42no—
370C23—81no—
371C23—81no—
372C23—81no—
373C23—81no—
374C23—81no—
375C23—81no—
376C23C4×D812yes—
377C23C22×D811yescharacteristic
378Q8—42no—
379Q8—42no—
380Q8—81no—
381Q8—81no—
382D4—42no—
383D4—42no—
384D4—81no—
385D4—81no—
386D4—164no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
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—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—164no—
425C2×C4C4×D812yes—
426C2×C4#46612yes—
427C2×C4#46612yes—
428C2×C4C2×C4×D411yescharacteristic
429C2×C4C2×#12111yescharacteristic
430C2×C4#53211yescharacteristic
431C2×C4C2×#13511yescharacteristic
432C2×C4C2×#13511yescharacteristic
433C2×C4C2×#13511yescharacteristic
434C8—22no—
435C8—22no—
436C8—22no—
437C8C2×C4×D411yescharacteristic
438C8C4×#4111yescharacteristic
439C8#52511yescharacteristic
440C8#52911yescharacteristic
441C22—22no—
442C22—22no—
443C22—22no—
444C22—41no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22—82no—
449C22—82no—
450C22—82no—
451C22—161no—
452C22—161no—
453C22—161no—
454C22—161no—
455C22C2×C4×D811yescharacteristic
456C22C2×#46611yescharacteristic
457C22#316911yescharacteristic
458C22#341511yescharacteristic
459C22#341611yescharacteristic
460C22#341611yescharacteristic
461C22#341711yescharacteristic
462C4—22no—
463C4—22no—
464C4—22no—
465C4—41no—
466C4—41no—
467C4—82no—
468C4—82no—
469C4—82no—
470C4—161no—
471C4—161no—
472C4—161no—
473C4—161no—
474C4C4×#12111yescharacteristic
475C4C4×#13511yescharacteristic
476C4#305011yescharacteristic
477C4#316811yescharacteristic
478C2—42no—
479C2—162no—
480C2C4×#46611yescharacteristic
481C2#3345411yescharacteristic
482C2#3345511yescharacteristic
483C2#3361211yescharacteristic
484C2#3361311yescharacteristic
485C2#3373911yescharacteristic
486C2#3374311yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC2×#466C22
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#405 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#2175C2semidirect
3. C2×#2108C2semidirect
4. #21525C2semidirect
6. #13237C2semidirect
7. #13165C2semidirect
8. #13150C2semidirect
9. #13149C2semidirect
10. #13060C2semidirect
12. #12836C2semidirect
13. #12674C2semidirect
14. C4×#358C2semidirect
15. C4×#357C2semidirect
26. #2220C4semidirect
76. C2×M32C2×C4semidirect
85. #467C2×C4semidirect
128. #366C2×C4semidirect
131. #358C2×C4semidirect
136. #346D4semidirect

Maximal subgroups

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

subgroupindex
C2×#22202
C2×#21752
C2×#21082
#215252
#215242
#132372
#131652
#131502
#131492
#130602
#130592
#128362
#126742
C4×#3582
C4×#3572