Finite Group Databasebeta

← #501713

Subgroups — #501713

Order 512.

624 conjugacy classes of subgroups.

The subgroup lattice has 469 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2228C211yesmaximal, characteristic, complemented
2C2×C4×D16C211yesmaximal, characteristic
3#28534C211yesmaximal, characteristic, complemented
4#22682C211yesmaximal, characteristic, complemented
5#13507C211yesmaximal, characteristic, complemented
6#13505C211yesmaximal, characteristic, complemented
7#13476C211yesmaximal, characteristic, complemented
8#13474C211yesmaximal, characteristic, complemented
9#13281C211yesmaximal, characteristic, complemented
10#13159C211yesmaximal, characteristic, complemented
11#13012C211yesmaximal, characteristic
12#13012C211yesmaximal, characteristic
13#12813C211yesmaximal, characteristic, complemented
14#12813C211yesmaximal, characteristic, complemented
15#12811C211yesmaximal, characteristic, complemented
16C22×D16C2211yescharacteristic
17C2×#499C2211yescharacteristic
18C2×#478—21no—
19C2×C4×D8C2211yescharacteristic
20C2×C4×D8C2211yescharacteristic
21C2×C4×D8C2211yescharacteristic
22C2×C4×D8C2211yescharacteristic
23#2566—21no—
24#2349—21no—
25#2228—24no—
26C4×D16—24no—
27C2×#367C2211yescharacteristic
28C2×#366C2211yescharacteristic
29C2×#366C2211yescharacteristic
30C2×C4⋊C16C2211yescharacteristic
31C2×C4⋊C16C2211yescharacteristic
32C2×C4⋊C16C2211yescharacteristic
33C2×#357C2211yescharacteristic
34C2×#357C2211yescharacteristic
35C2×#357C2211yescharacteristic
36C2×#357C2211yescharacteristic
37C2×#357C2211yescharacteristic
38C2×#357C2211yescharacteristic
39C2×C4×C16C2211yescharacteristic
40#2094—21no—
41#2092—21no—
42#1970C2211yescharacteristic
43#1970C2211yescharacteristic
44#1942C2211yescharacteristic
45#1942C2211yescharacteristic
46#1864—21no—
47#1864—21no—
48#1846C2211yescharacteristic
49#1846C2211yescharacteristic
50#1845C2211yescharacteristic
51#1804—21no—
52C8×C4⋊C4C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1374C2211yescharacteristic
60#1374C2211yescharacteristic
61C22×D8C2311yescharacteristic
62C22×D8C2311yescharacteristic
63C2×#125—21no—
64C2×C4×D4—21no—
65C2×C4×D4—21no—
66C2×C4×D4—21no—
67C2×C4×D4—21no—
68C2×D16—22no—
69C2×D16—22no—
70C2×D16Q814yes—
71C22×C16C2311yescharacteristic
72C22×C16C2311yescharacteristic
73C22×C16C2311yescharacteristic
74C22×C16C2311yescharacteristic
75#499—22no—
76#499—22no—
77#478—44no—
78C4×D8—22no—
79C4×D8—22no—
80C4×D8—22no—
81C4×D8—22no—
82C4×D8—22no—
83C4×D8—41no—
84C4×D8—41no—
85C4×D8D414yes—
86C2×#107—21no—
87C2×#107C2311yescharacteristic
88C2×#107C2311yescharacteristic
89C2×#107C2311yescharacteristic
90C2×#107C2311yescharacteristic
91C2×#107C2311yescharacteristic
92C2×#107C2311yescharacteristic
93C2×#106—21no—
94C2×#106—21no—
95C2×#106—21no—
96C2×C4⋊C8D411yescharacteristic
97C2×#102—21no—
98C2×#102—21no—
99C2×#102—21no—
100C2×#102—21no—
101C2×#102—21no—
102C2×#102—21no—
103C2×C4×C8C2311yescharacteristic
104C2×C4×C8C2311yescharacteristic
105C2×C4×C8C2311yescharacteristic
106#396—41no—
107#396—41no—
108#390—21no—
109#390—21no—
110#389—21no—
111#389—21no—
112#385—41no—
113#385—41no—
114#385—41no—
115#385—41no—
116#384—21no—
117#382—21no—
118#382—21no—
119#380—41no—
120C4×C4⋊C4D411yescharacteristic
121C2×#95—41no—
122#367—22no—
123#366—22no—
124#366—22no—
125C4⋊C16—22no—
126C4⋊C16—41no—
127C4⋊C16—41no—
128#357—22no—
129#357—22no—
130#357—22no—
131#357—22no—
132#357—41no—
133#357—41no—
134C4×C16—22no—
135#340—21no—
136#340—21no—
137#336D411yescharacteristic
138#336D411yescharacteristic
139C22×D4—21no—
140C22×D4—21no—
141C23×C4—41no—
142C23×C4—41no—
143C23×C4—41no—
144C23×C4—41no—
145C2×D8—21no—
146C2×D8—21no—
147C2×D8—21no—
148C2×D8—21no—
149C2×D8—21no—
150C2×D8—21no—
151C2×D8—21no—
152C2×D8—21no—
153C2×D8C2×D412yes—
154C2×D8C2×Q812yes—
155C2×D8C2×Q812yes—
156C2×D8#4112yes—
157C22×C8C2×D411yescharacteristic
158C22×C8C2×D411yescharacteristic
159C22×C8C2×D411yescharacteristic
160C22×C8C2411yescharacteristic
161#125—42no—
162#125—42no—
163C4×D4—24no—
164C4×D4—42no—
165C4×D4—42no—
166C4×D4—42no—
167C4×D4—42no—
168C4×D4—42no—
169C4×D4—81no—
170C4×D4—81no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—41no—
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×C42C2×D411yescharacteristic
194C2×C42C2×D411yescharacteristic
195C2×C42C2×D411yescharacteristic
196D16—28no—
197C2×C16—21no—
198C2×C16—21no—
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×C16C2×Q812yes—
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—42no—
221#107C2×D412yes—
222#106—42no—
223#106—42no—
224#106—81no—
225C4⋊C8—22no—
226#102—24no—
227#102—42no—
228#102—42no—
229#102—42no—
230#102—42no—
231#102—42no—
232C4×C8—21no—
233C4×C8—21no—
234C4×C8—21no—
235C4×C8—21no—
236C4×C8C2×D412yes—
237C4×C8#4112yes—
238#95—41no—
239#95—41no—
240#95—41no—
241#95—41no—
242#95—41no—
243#95—41no—
244#95D811yescharacteristic
245#95D811yescharacteristic
246C24—41no—
247C24—41no—
248C2×D4—22no—
249C2×D4—22no—
250C2×D4—22no—
251C2×D4—22no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—41no—
260C22×C4—41no—
261C22×C4—44no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—82no—
268C22×C4—82no—
269C22×C4—82no—
270C22×C4—82no—
271C22×C4—82no—
272C22×C4—161no—
273C22×C4—161no—
274C22×C4—161no—
275C22×C4—161no—
276C22×C4—161no—
277C22×C4—161no—
278C22×C4C2×D811yescharacteristic
279C22×C4C2×D811yescharacteristic
280C22×C4C2×D811yescharacteristic
281C22×C4C2×D811yescharacteristic
282C22×C4C2×D811yescharacteristic
283C22×C4C2×D811yescharacteristic
284C22×C4C22×D411yescharacteristic
285D8—22no—
286D8—22no—
287D8—22no—
288D8—22no—
289D8—22no—
290D8—41no—
291D8—41no—
292D8C22⋊Q814yes—
293C2×C8—21no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8#12112yes—
302C2×C8C22×D411yescharacteristic
303C2×C8C22×Q811yescharacteristic
304C2×C8C2×#4111yescharacteristic
305C2×C8C2×#4111yescharacteristic
306C2×C8H(5,𝔽2)11yescharacteristic
307C2×C82−1+411yescharacteristic
308C4⋊C4—22no—
309C4⋊C4—22no—
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—161no—
325C4⋊C4—161no—
326C4⋊C4—161no—
327C4⋊C4—161no—
328C22⋊C4—44no—
329C22⋊C4—82no—
330C22⋊C4—82no—
331C22⋊C4—82no—
332C22⋊C4—82no—
333C22⋊C4—82no—
334C22⋊C4—161no—
335C22⋊C4—161no—
336C42—21no—
337C42—21no—
338C42—21no—
339C42—21no—
340C42#12112yes—
341C42AGL(1,ℤ/8)12yes—
342C16—22no—
343C16—41no—
344C16—41no—
345C16—41no—
346C23—42no—
347C23—42no—
348C23—42no—
349C23—42no—
350C23—81no—
351C23—81no—
352C23—81no—
353C23—81no—
354C23—81no—
355C23—81no—
356C23—81no—
357C23—81no—
358C23—81no—
359C23—81no—
360C23C22×D811yescharacteristic
361D4—24no—
362D4—42no—
363D4—42no—
364D4—42no—
365D4—42no—
366D4—42no—
367D4—81no—
368D4—81no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—42no—
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—81no—
401C2×C4—81no—
402C2×C4—164no—
403C2×C4—164no—
404C2×C4—164no—
405C2×C4—164no—
406C2×C4#45912yes—
407C2×C4#45912yes—
408C2×C4#50812yes—
409C2×C4C2×#12111yescharacteristic
410C2×C4#54611yescharacteristic
411C2×C4D4×Q811yescharacteristic
412C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
413C2×C4#57711yescharacteristic
414C2×C4#57811yescharacteristic
415C8—22no—
416C8—41no—
417C8—41no—
418C8C2×C22⋊Q811yescharacteristic
419C8#53711yescharacteristic
420C8#55211yescharacteristic
421C8#55411yescharacteristic
422C22—44no—
423C22—82no—
424C22—82no—
425C22—82no—
426C22—82no—
427C22—82no—
428C22—82no—
429C22—82no—
430C22—82no—
431C22—82no—
432C22—161no—
433C22—161no—
434C22—161no—
435C22—161no—
436C22C2×#45911yescharacteristic
437C22#329411yescharacteristic
438C22Q8×D811yescharacteristic
439C22C2×#50811yescharacteristic
440C22#341411yescharacteristic
441C22#341511yescharacteristic
442C22#341511yescharacteristic
443C4—22no—
444C4—22no—
445C4—22no—
446C4—41no—
447C4—41no—
448C4—41no—
449C4—82no—
450C4—161no—
451C4—161no—
452C4—161no—
453C4—161no—
454C4—161no—
455C4#241711yescharacteristic
456C4#304611yescharacteristic
457C4#333011yescharacteristic
458C4#338411yescharacteristic
459C2—84no—
460C2—162no—
461C2#2481111yescharacteristic
462C2#3357911yescharacteristic
463C2#3358511yescharacteristic
464C2#3386411yescharacteristic
465C2#3388911yescharacteristic
466C2#3394711yescharacteristic
467C2#3395611yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D16C22
LCSG ▷ C22×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
1. C2×#2228C2semidirect
3. #28534C2semidirect
4. #22682C2semidirect
5. #13507C2semidirect
6. #13505C2semidirect
7. #13476C2semidirect
8. #13474C2semidirect
9. #13281C2semidirect
10. #13159C2semidirect
13. #12813C2semidirect
14. #12813C2semidirect
15. #12811C2semidirect

Maximal subgroups

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

subgroupindex
C2×#22282
C2×C4×D162
#285342
#226822
#135072
#135052
#134762
#134742
#132812
#131592
#130122
#128132
#128112