Finite Group Databasebeta

← #504087

Subgroups — #504087

Order 512.

642 conjugacy classes of subgroups.

The subgroup lattice has 471 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2226C211yesmaximal, characteristic, complemented
2C22×#367C211yesmaximal, characteristic, complemented
3#26065C211yesmaximal, characteristic, complemented
4#23025C211yesmaximal, characteristic
5#13527C211yesmaximal, characteristic, complemented
6#13516C211yesmaximal, characteristic, complemented
7#13476C211yesmaximal, characteristic, complemented
8#13452C211yesmaximal, characteristic, complemented
9#13177C211yesmaximal, characteristic, complemented
10#13111C211yesmaximal, characteristic, complemented
11#13059C211yesmaximal, characteristic
12#12991C211yesmaximal, characteristic
13#12937C211yesmaximal, characteristic, complemented
14#12920C211yesmaximal, characteristic, complemented
15#12912C211yesmaximal, characteristic, complemented
16C23×C16C2211yescharacteristic
17C2×#500C2211yescharacteristic
18C2×#466C2211yescharacteristic
19C2×C4×D8C2211yescharacteristic
20C22×#107C2211yescharacteristic
21C22×#107C2211yescharacteristic
22#2440—21no—
23#2361—21no—
24#2226—24no—
25C2×#367—24no—
26C2×#367C2211yescharacteristic
27C2×#367C2211yescharacteristic
28C2×#367C2211yescharacteristic
29C2×#367C2211yescharacteristic
30C2×#367C2211yescharacteristic
31C2×#367C2211yescharacteristic
32C2×C4⋊C16C2211yescharacteristic
33C2×C4⋊C16C2211yescharacteristic
34C2×#357C2211yescharacteristic
35C2×#357C2211yescharacteristic
36C2×#357C2211yescharacteristic
37C2×#357C2211yescharacteristic
38C2×C22⋊C16C2211yescharacteristic
39#2083—21no—
40#2057—21no—
41#1970C2211yescharacteristic
42#1942C2211yescharacteristic
43#1942C2211yescharacteristic
44#1942C2211yescharacteristic
45#1925—21no—
46#1874—21no—
47#1862—21no—
48#1852—21no—
49#1845C2211yescharacteristic
50#1808C2211yescharacteristic
51C4×#107C2211yescharacteristic
52C8×C22⋊C4C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1374C2211yescharacteristic
60#1374C2211yescharacteristic
61C22×D8C2311yescharacteristic
62C23×C8C2311yescharacteristic
63C2×C4⋊Q8—21no—
64C2×C22⋊Q8—41no—
65C2×#121—21no—
66C2×C4×D4—21no—
67C2×C4×D4—41no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C16—22no—
71C22×C16—22no—
72C22×C16—22no—
73C22×C16C2311yescharacteristic
74C22×C16C2311yescharacteristic
75C22×C16C2311yescharacteristic
76C22×C16C2311yescharacteristic
77#500—22no—
78#500—41no—
79#466—22no—
80#466Q814yescomplemented
81C4×D8—22no—
82C4×D8—41no—
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×#107C2311yescharacteristic
92C2×#107C2311yescharacteristic
93C2×#107C2311yescharacteristic
94C2×#107C2311yescharacteristic
95C2×#107C2311yescharacteristic
96C2×#107C2311yescharacteristic
97C2×#107C2311yescharacteristic
98C2×#102—21no—
99C2×#102—21no—
100C2×#102—21no—
101C2×C22⋊C8D411yescharacteristic
102C2×C4×C8C2311yescharacteristic
103C2×C4×C8C2311yescharacteristic
104#398—41no—
105#395—41no—
106#390—21no—
107#389—21no—
108#389—21no—
109#389—21no—
110#388—41no—
111#386—41no—
112#385—41no—
113#385—41no—
114#384—21no—
115#380—21no—
116C4×C4⋊C4—21no—
117C4×C22⋊C4D411yescharacteristic
118#367—22no—
119#367—22no—
120#367—22no—
121#367—22no—
122#367—41no—
123#367D414yescomplemented
124C4⋊C16—22no—
125C4⋊C16—22no—
126#357—22no—
127#357—22no—
128#357—22no—
129#357—41no—
130C22⋊C16—22no—
131#340—21no—
132#340—21no—
133#340—21no—
134#340—21no—
135#340—21no—
136#336D411yescharacteristic
137#336D411yescharacteristic
138C22×Q8—41no—
139C22×D4—21no—
140C22×D4—41no—
141C23×C4—41no—
142C23×C4—41no—
143C23×C4—41no—
144C23×C4C2×D411yescharacteristic
145C2×D8—21no—
146C2×D8—21no—
147C2×D8—21no—
148C2×D8—21no—
149C2×D8—21no—
150C2×D8C2×Q812yes—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8C2×D411yescharacteristic
156C22×C8C2×D411yescharacteristic
157C22×C8C2×D411yescharacteristic
158C22×C8C2×Q812yes—
159C22×C8#4112yes—
160C22×C8C2411yescharacteristic
161C4⋊Q8—42no—
162C4⋊Q8—81no—
163C22⋊Q8—84no—
164#121—24no—
165#121—42no—
166C4×D4—42no—
167C4×D4—81no—
168C4×D4—84no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—81no—
187C2×C4⋊C4—81no—
188C2×C22⋊C4—41no—
189C2×C22⋊C4—41no—
190C2×C22⋊C4—41no—
191C2×C22⋊C4—41no—
192C2×C22⋊C4—41no—
193C2×C22⋊C4D811yescharacteristic
194C2×C22⋊C4D811yescharacteristic
195C2×C42—41no—
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D411yescharacteristic
198C2×C16—28no—
199C2×C16—21no—
200C2×C16—21no—
201C2×C16—21no—
202C2×C16—21no—
203C2×C16—21no—
204C2×C16—21no—
205C2×C16—21no—
206C2×C16—21no—
207C2×C16—21no—
208C2×C16—21no—
209C2×C16C2×D412yes—
210C2×C16#4112yes—
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—42no—
223#107—42no—
224#107—42no—
225#107—81no—
226#107C2×D412yes—
227#107C2×D412yes—
228#107C2×Q812yes—
229#102—24no—
230#102—42no—
231#102—81no—
232C22⋊C8—22no—
233C4×C8—21no—
234C4×C8—21no—
235C4×C8—21no—
236C4×C8—21no—
237#95—41no—
238#95—41no—
239#95—41no—
240#95—41no—
241#95—41no—
242#95D811yescharacteristic
243#95D811yescharacteristic
244C24—41no—
245C24C2×D811yescharacteristic
246C2×Q8—82no—
247C2×Q8—82no—
248C2×Q8—161no—
249C2×D4—22no—
250C2×D4—44no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—82no—
257C2×D4—82no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—82no—
270C22×C4—82no—
271C22×C4—82no—
272C22×C4—82no—
273C22×C4—82no—
274C22×C4—82no—
275C22×C4—161no—
276C22×C4—161no—
277C22×C4—161no—
278C22×C4C22⋊Q812yes—
279C22×C4C2×D811yescharacteristic
280C22×C4C2×D811yescharacteristic
281C22×C4C2×D811yescharacteristic
282C22×C4C2×D811yescharacteristic
283C22×C4C2×D811yescharacteristic
284C22×C4#13712yes—
285C22×C4C22×D411yescharacteristic
286D8—22no—
287D8—22no—
288D8—41no—
289D8—41no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—22no—
293C2×C8—22no—
294C2×C8—22no—
295C2×C8—22no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8C22⋊Q812yes—
303C2×C8C22×D411yescharacteristic
304C2×C8C22×Q811yescharacteristic
305C2×C8C2×#4111yescharacteristic
306C2×C8C2×#4111yescharacteristic
307C2×C8H(5,𝔽2)11yescharacteristic
308C2×C82−1+411yescharacteristic
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—22no—
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—41no—
323C4⋊C4—82no—
324C4⋊C4—82no—
325C4⋊C4—82no—
326C4⋊C4—82no—
327C4⋊C4—82no—
328C4⋊C4—82no—
329C4⋊C4—82no—
330C4⋊C4—161no—
331C22⋊C4—22no—
332C22⋊C4—44no—
333C22⋊C4—82no—
334C22⋊C4—82no—
335C22⋊C4—82no—
336C22⋊C4—161no—
337C22⋊C4SD3214yes—
338C42—21no—
339C42—21no—
340C42—21no—
341C42—21no—
342C42—82no—
343C16—22no—
344C16—22no—
345C16—41no—
346C16—41no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—21no—
351C23—42no—
352C23—81no—
353C23—81no—
354C23—81no—
355C23—81no—
356C23—81no—
357C23—81no—
358C23#47412yes—
359C23C2×SD3212yes—
360C23#50912yes—
361C23C22×D811yescharacteristic
362Q8—164no—
363D4—42no—
364D4—42no—
365D4—88no—
366D4—81no—
367D4—81no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
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—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—164no—
406C2×C4—164no—
407C2×C4—164no—
408C2×C4#47412yes—
409C2×C4C2×SD3212yes—
410C2×C4C2×SD3212yes—
411C2×C4C2×C22⋊Q811yescharacteristic
412C2×C4D4211yescharacteristic
413C2×C4#54711yescharacteristic
414C2×C4C2×#13711yescharacteristic
415C2×C4#57611yescharacteristic
416C2×C4#57811yescharacteristic
417C8—22no—
418C8—41no—
419C8—41no—
420C8#52511yescharacteristic
421C8#53911yescharacteristic
422C8D4×Q811yescharacteristic
423C8#55411yescharacteristic
424C22—22no—
425C22—22no—
426C22—22no—
427C22—22no—
428C22—22no—
429C22—22no—
430C22—82no—
431C22—82no—
432C22—82no—
433C22—161no—
434C22—161no—
435C22—161no—
436C22—161no—
437C22#222614yescomplemented
438C22C2×#47411yescharacteristic
439C22D4×D811yescharacteristic
440C22#333411yescharacteristic
441C22C22×SD3211yescharacteristic
442C22C2×#50911yescharacteristic
443C22#341611yescharacteristic
444C22#341611yescharacteristic
445C4—22no—
446C4—22no—
447C4—22no—
448C4—41no—
449C4—41no—
450C4—82no—
451C4—82no—
452C4—82no—
453C4—161no—
454C4—161no—
455C4—161no—
456C4—161no—
457C4#244911yescharacteristic
458C4#308311yescharacteristic
459C4#327011yescharacteristic
460C4#334611yescharacteristic
461C2—24no—
462C2—162no—
463C2#2633611yescharacteristic
464C2C2×#222611yescharacteristic
465C2#3363411yescharacteristic
466C2#3383111yescharacteristic
467C2D4×SD3211yescharacteristic
468C2#3390211yescharacteristic
469C2#3391211yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC2×#466C22
LCSG ▷ C22×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
1. C2×#2226C2semidirect
2. C22×#367C2semidirect
3. #26065C2semidirect
5. #13527C2semidirect
6. #13516C2semidirect
7. #13476C2semidirect
8. #13452C2semidirect
9. #13177C2semidirect
10. #13111C2semidirect
13. #12937C2semidirect
14. #12920C2semidirect
15. #12912C2semidirect
80. #466Q8semidirect
123. #367D4semidirect
437. C22#2226semidirect

Maximal subgroups

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

subgroupindex
C2×#22262
C22×#3672
#260652
#230252
#135272
#135162
#134762
#134522
#131772
#131112
#130592
#129912
#129372
#129202
#129122