Finite Group Databasebeta

← #497921

Subgroups — #497921

Order 512.

660 conjugacy classes of subgroups.

The subgroup lattice has 467 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2168C211yesmaximal, characteristic
2#22656C211yesmaximal, characteristic
3D4×#107C211yesmaximal, characteristic
4#13178C211yesmaximal, characteristic, complemented
5#13177C211yesmaximal, characteristic, complemented
6#12921C211yesmaximal, characteristic, complemented
7#12920C211yesmaximal, characteristic, complemented
8#12920C211yesmaximal, characteristic, complemented
9#12919C211yesmaximal, characteristic, complemented
10#12890C211yesmaximal, characteristic, complemented
11#12836C211yesmaximal, characteristic, complemented
12#12835C211yesmaximal, characteristic, complemented
13#12674C211yesmaximal, characteristic, complemented
14#12557C211yesmaximal, characteristic, complemented
15C2×#1380C211yesmaximal, characteristic, complemented
16C22×M32C2211yescharacteristic
17C23×C16C2211yescharacteristic
18C2×C8×D4C2211yescharacteristic
19C2×#427C2211yescharacteristic
20C22×#107C2211yescharacteristic
21C22×#107C2211yescharacteristic
22C22×#107C2211yescharacteristic
23#2352—21no—
24D4×C4⋊C4—21no—
25#2168C418yescomplemented
26C2×#367C2211yescharacteristic
27C2×#367C2211yescharacteristic
28C2×#367C2211yescharacteristic
29C2×#367C2211yescharacteristic
30C2×#366C2211yescharacteristic
31C2×#366C2211yescharacteristic
32C2×C4⋊C16C2211yescharacteristic
33C2×C22⋊C16C2211yescharacteristic
34C2×C22⋊C16C2211yescharacteristic
35C2×#346C2211yescharacteristic
36#1949—21no—
37#1947—21no—
38#1854—21no—
39#1853—21no—
40#1852—21no—
41#1852—21no—
42#1846C2211yescharacteristic
43#1845C2211yescharacteristic
44#1808C2211yescharacteristic
45#1808C2211yescharacteristic
46#1808C2211yescharacteristic
47#1808C2211yescharacteristic
48C4×#107C2211yescharacteristic
49C4×#107C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1380C2211yescharacteristic
60C23×C8C2311yescharacteristic
61C23×C8C2311yescharacteristic
62C2×C4×D4—41no—
63C2×C4×D4D411yescharacteristic
64C2×#117—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C2×M32—41no—
69C2×M32—41no—
70C2×M32C2×C414yescomplemented
71C22×C16—41no—
72C22×C16—41no—
73C22×C16C2×C414yescomplemented
74C22×C16C2311yescharacteristic
75C22×C16C2311yescharacteristic
76C22×C16C2311yescharacteristic
77C22×C16C2311yescharacteristic
78C8×D4C2×C414yes—
79C8×D4Q814yes—
80#427—22no—
81#427—41no—
82C2×#107—22no—
83C2×#107—22no—
84C2×#107—22no—
85C2×#107—21no—
86C2×#107—21no—
87C2×#107—21no—
88C2×#107—21no—
89C2×#107—41no—
90C2×#107—41no—
91C2×#107—41no—
92C2×#107C2311yescharacteristic
93C2×#107C2311yescharacteristic
94C2×#107C2311yescharacteristic
95C2×#107C2311yescharacteristic
96C2×#107C2311yescharacteristic
97C2×#107C2311yescharacteristic
98C2×#107C2311yescharacteristic
99C2×#107C2311yescharacteristic
100C2×#106—21no—
101C2×#106—21no—
102C2×C4⋊C8D411yescharacteristic
103C2×C22⋊C8D411yescharacteristic
104C2×C22⋊C8D411yescharacteristic
105C2×C4×C8C2311yescharacteristic
106#389—41no—
107#388—41no—
108#387—41no—
109#386—41no—
110#385—41no—
111#385—41no—
112#384—21no—
113#382—21no—
114#380—21no—
115#380—21no—
116#380—21no—
117#380—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C22⋊C4—41no—
121#367—22no—
122#367—41no—
123#367—41no—
124#367—41no—
125#366—22no—
126#366—41no—
127C4⋊C16C2×C414yescomplemented
128C22⋊C16C2×C414yescomplemented
129C22⋊C16C2×C414yescomplemented
130#346C2×C414yescomplemented
131#340—21no—
132#340—21no—
133#340—21no—
134#340—21no—
135#340—21no—
136#340—21no—
137C22×D4D811yescharacteristic
138C23×C4—41no—
139C23×C4—41no—
140C23×C4—41no—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C22×C8—21no—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8C22×C412yes—
152C22×C8C22×C412yes—
153C22×C8C2×D411yescharacteristic
154C22×C8C2×D411yescharacteristic
155C22×C8C2×D411yescharacteristic
156C22×C8C2×Q812yes—
157C22×C8C2×Q812yes—
158C22×C8C2411yescharacteristic
159C4×D4—84no—
160C4×D4C4⋊C414yes—
161C4×D4Q1614yes—
162#117—42no—
163#117—81no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—81no—
181C2×C4⋊C4—81no—
182C2×C4⋊C4—81no—
183C2×C4⋊C4D811yescharacteristic
184C2×C22⋊C4—41no—
185C2×C22⋊C4—41no—
186C2×C22⋊C4—41no—
187C2×C22⋊C4—41no—
188C2×C22⋊C4—41no—
189C2×C22⋊C4D811yescharacteristic
190C2×C22⋊C4D811yescharacteristic
191C2×C42—41no—
192C2×C42—41no—
193C2×C42—41no—
194C2×C42C2×D411yescharacteristic
195M32—44no—
196C2×C16—21no—
197C2×C16—21no—
198C2×C16—21no—
199C2×C16—21no—
200C2×C16—21no—
201C2×C16—21no—
202C2×C16—21no—
203C2×C16—21no—
204C2×C16—44no—
205C2×C16C22×C412yes—
206C2×C16C22×C412yes—
207C2×C16C22×C412yes—
208C2×C16C22×C412yes—
209#107—21no—
210#107—21no—
211#107—21no—
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#107—21no—
221#107—21no—
222#107—21no—
223#107—42no—
224#107—42no—
225#107—42no—
226#107—81no—
227#107#4112yes—
228#107#4112yes—
229#106—42no—
230#106—81no—
231C4⋊C8C4⋊C414yes—
232C22⋊C8C4⋊C414yes—
233C22⋊C8C4⋊C414yes—
234C4×C8C22×C412yes—
235C4×C8C2×Q812yes—
236#95—41no—
237#95—41no—
238#95—41no—
239#95—41no—
240#95—41no—
241#95—41no—
242#95—41no—
243#95—41no—
244C24C2×D811yescharacteristic
245C24C2×D811yescharacteristic
246C2×D4—22no—
247C2×D4—22no—
248C2×D4#10714yes—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—82no—
266C22×C4—82no—
267C22×C4—82no—
268C22×C4—82no—
269C22×C4—82no—
270C22×C4—82no—
271C22×C4—161no—
272C22×C4—161no—
273C22×C4—161no—
274C22×C4C2×C4⋊C412yes—
275C22×C4C2×C4⋊C412yes—
276C22×C4C2×D811yescharacteristic
277C22×C4C2×D811yescharacteristic
278C22×C4C2×D811yescharacteristic
279C22×C4C2×D811yescharacteristic
280C22×C4C2×Q1612yes—
281C22×C4C2×Q1612yes—
282C22×C4C22×D411yescharacteristic
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—22no—
290C2×C8—22no—
291C2×C8—22no—
292C2×C8—22no—
293C2×C8—22no—
294C2×C8—22no—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8C2×C4⋊C412yes—
298C2×C8C2×C4⋊C412yes—
299C2×C8C2×C4⋊C412yes—
300C2×C8C23×C411yescharacteristic
301C2×C8C22×Q811yescharacteristic
302C2×C8C2×#4111yescharacteristic
303C2×C8C2×#4111yescharacteristic
304C2×C8H(5,𝔽2)11yescharacteristic
305C2×C8H(5,𝔽2)11yescharacteristic
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—82no—
323C4⋊C4—82no—
324C4⋊C4—82no—
325C4⋊C4—82no—
326C4⋊C4—161no—
327C4⋊C4#10714yes—
328C22⋊C4—82no—
329C22⋊C4—82no—
330C22⋊C4—82no—
331C22⋊C4—82no—
332C22⋊C4—161no—
333C22⋊C4#10714yes—
334C22⋊C4#10714yes—
335C42—82no—
336C42—82no—
337C42—161no—
338C42C2×C4⋊C412yes—
339C42C2×Q1612yes—
340C16—22no—
341C16—22no—
342C16—22no—
343C16—22no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23C2×#10712yes—
355C23C2×#10712yes—
356C23#50912yes—
357C23#51012yes—
358C23C22×D811yescharacteristic
359D4—28no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
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—164no—
401C2×C4—164no—
402C2×C4—164no—
403C2×C4C2×#10712yes—
404C2×C4C2×#10712yes—
405C2×C4C2×#10712yes—
406C2×C4C2×#10712yes—
407C2×C4C22×C4⋊C411yescharacteristic
408C2×C4#54611yescharacteristic
409C2×C4#54611yescharacteristic
410C2×C4C22×Q1611yescharacteristic
411C2×C4#57711yescharacteristic
412C2×C4#57711yescharacteristic
413C8—22no—
414C8—22no—
415C8—22no—
416C8C4×#4111yescharacteristic
417C8#51811yescharacteristic
418C8#55411yescharacteristic
419C8#55411yescharacteristic
420C22—22no—
421C22—22no—
422C22—22no—
423C22—22no—
424C22—22no—
425C22—22no—
426C22—22no—
427C22—22no—
428C22—22no—
429C22—41no—
430C22—41no—
431C22—41no—
432C22—41no—
433C22#215914yes—
434C22C22×#10711yescharacteristic
435C22#329411yescharacteristic
436C22#329411yescharacteristic
437C22C2×#50911yescharacteristic
438C22C2×#51011yescharacteristic
439C22#341611yescharacteristic
440C22#341611yescharacteristic
441C4—22no—
442C4—22no—
443C4—22no—
444C4—22no—
445C4—82no—
446C4—82no—
447C4—161no—
448C4—161no—
449C4—161no—
450C4—161no—
451C4—161no—
452C4—161no—
453C4#234911yescharacteristic
454C4#291211yescharacteristic
455C4#329911yescharacteristic
456C4#329911yescharacteristic
457C2—24no—
458C2—42no—
459C2#2270511yescharacteristic
460C2C2×#215911yescharacteristic
461C2#3341111yescharacteristic
462C2#3387311yescharacteristic
463C2#3387711yescharacteristic
464C2#3388311yescharacteristic
465C2#3388711yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC22×D4D8
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #13178C2semidirect
5. #13177C2semidirect
6. #12921C2semidirect
7. #12920C2semidirect
8. #12920C2semidirect
9. #12919C2semidirect
10. #12890C2semidirect
11. #12836C2semidirect
12. #12835C2semidirect
13. #12674C2semidirect
14. #12557C2semidirect
15. C2×#1380C2semidirect
25. #2168C4semidirect
70. C2×M32C2×C4semidirect
73. C22×C16C2×C4semidirect
127. C4⋊C16C2×C4semidirect
128. C22⋊C16C2×C4semidirect
129. C22⋊C16C2×C4semidirect
130. #346C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#21682
#226562
D4×#1072
#131782
#131772
#129212
#129202
#129192
#128902
#128362
#128352
#126742
#125572
C2×#13802