Finite Group Databasebeta

← #507420

Subgroups — #507420

Order 512.

574 conjugacy classes of subgroups.

The subgroup lattice has 451 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33319C211yesmaximal, characteristic, complemented
2#21696C211yesmaximal, characteristic
3#13366C211yesmaximal, characteristic, complemented
4#13365C211yesmaximal, characteristic
5#13364C211yesmaximal, characteristic, complemented
6#13363C211yesmaximal, characteristic
7#13205C211yesmaximal, characteristic
8#13204C211yesmaximal, characteristic
9#12890C211yesmaximal, characteristic, complemented
10#12888C211yesmaximal, characteristic, complemented
11#12814C211yesmaximal, characteristic, complemented
12#12813C211yesmaximal, characteristic
13#12812C211yesmaximal, characteristic, complemented
14#12811C211yesmaximal, characteristic
15#12654C211yesmaximal, characteristic, complemented
16C2×#501—21no—
17C2×#500C2211yescharacteristic
18C2×#499C2211yescharacteristic
19C2×#495C2211yescharacteristic
20C2×#440—21no—
21C2×C4×Q16C2211yescharacteristic
22C2×C4×D8C2211yescharacteristic
23#2660—21no—
24#2326—21no—
25C2×C4⋊C16C2211yescharacteristic
26C2×C4⋊C16C2211yescharacteristic
27C2×#358C2211yescharacteristic
28C2×#358C2211yescharacteristic
29C2×#358C2211yescharacteristic
30C2×#358C2211yescharacteristic
31C2×#357C2211yescharacteristic
32C2×#357C2211yescharacteristic
33C2×#357C2211yescharacteristic
34C2×#357C2211yescharacteristic
35C2×#346C2211yescharacteristic
36C2×C4×C16C2211yescharacteristic
37#1973—21no—
38#1970C2211yescharacteristic
39#1969C2211yescharacteristic
40#1959—21no—
41#1847—21no—
42#1847—21no—
43#1846C2211yescharacteristic
44#1846C2211yescharacteristic
45#1845C2211yescharacteristic
46#1845C2211yescharacteristic
47#1845C2211yescharacteristic
48#1726C2211yescharacteristic
49#1380C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×Q16C2311yescharacteristic
60C22×SD16—21no—
61C22×D8C2311yescharacteristic
62C2×C4⋊Q8—21no—
63C2×#125—21no—
64C2×#124—21no—
65C2×C4×Q8—21no—
66C2×C4×D4—21no—
67C22×C16C2311yescharacteristic
68C22×C16C2311yescharacteristic
69C22×C16C2311yescharacteristic
70C22×C16C2311yescharacteristic
71#501—44no—
72#500—41no—
73#500—41no—
74#499—41no—
75#499—41no—
76#495—22no—
77#495Q814yescomplemented
78#440—44no—
79C4×Q16—22no—
80C4×Q16—22no—
81C4×D8—22no—
82C4×D8—22no—
83C2×#107—21no—
84C2×#107—21no—
85C2×#107—21no—
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×#103—21no—
97C2×#103—21no—
98C2×#102—21no—
99C2×#102—21no—
100C2×#97D411yescharacteristic
101C2×#97D411yescharacteristic
102C2×#97D411yescharacteristic
103C2×C4×C8C2311yescharacteristic
104C2×C4×C8C2311yescharacteristic
105C2×C4×C8C2311yescharacteristic
106#398—41no—
107#398—41no—
108#395—41no—
109#395—41no—
110#391—21no—
111#390—21no—
112#388—41no—
113#388—41no—
114#385—41no—
115#385—41no—
116#384—21no—
117#384—21no—
118#384—21no—
119#382—21no—
120#382—21no—
121#376D411yescharacteristic
122C4⋊C16—22no—
123C4⋊C16—22no—
124#358—41no—
125#358—41no—
126#358—41no—
127#358—41no—
128#357—41no—
129#357—41no—
130#357—41no—
131#357—41no—
132#346—22no—
133C4×C16—22no—
134C22×Q8—21no—
135C22×Q8—41no—
136C22×D4—21no—
137C23×C4—41no—
138C2×Q16—21no—
139C2×Q16—21no—
140C2×Q16—21no—
141C2×Q16—21no—
142C2×Q16—21no—
143C2×Q16C2×Q812yes—
144C2×SD16—24no—
145C2×SD16—42no—
146C2×SD16—42no—
147C2×D8—21no—
148C2×D8—21no—
149C2×D8—21no—
150C2×D8—21no—
151C2×D8—21no—
152C2×D8C2×Q812yes—
153C22×C8C2×D411yescharacteristic
154C22×C8C2×D411yescharacteristic
155C22×C8C2×D411yescharacteristic
156C22×C8C2411yescharacteristic
157C4⋊Q8—42no—
158C4⋊Q8—42no—
159#125—42no—
160#125—42no—
161#124—24no—
162#124—42no—
163C4×Q8—42no—
164C4×Q8—42no—
165C4×D4—42no—
166C4×D4—42no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—41no—
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×C22⋊C4—41no—
184C2×C22⋊C4—41no—
185C2×C22⋊C4—41no—
186C2×C22⋊C4—41no—
187C2×C22⋊C4—41no—
188C2×C42—41no—
189C2×C42C2×D411yescharacteristic
190C2×C42C2×D411yescharacteristic
191C2×C42C2×D411yescharacteristic
192C2×C16—21no—
193C2×C16—21no—
194C2×C16—21no—
195C2×C16—21no—
196C2×C16—21no—
197C2×C16—21no—
198C2×C16—21no—
199C2×C16—21no—
200C2×C16—21no—
201C2×C16—21no—
202C2×C16—21no—
203C2×C16—21no—
204#107—21no—
205#107—21no—
206#107—21no—
207#107—21no—
208#107—21no—
209#107—21no—
210#107—21no—
211#107—21no—
212#107—21no—
213#107—21no—
214#107—21no—
215#107—21no—
216#107—42no—
217#107—42no—
218#107—42no—
219#107—42no—
220#106—42no—
221#106—42no—
222#106—42no—
223#103—42no—
224#103—42no—
225#102—42no—
226#102—42no—
227#97—22no—
228#97—22no—
229#97SD1614yes—
230C4×C8—21no—
231C4×C8—21no—
232C4×C8C2×Q812yes—
233C4×C8#4112yes—
234C4×C8#4112yes—
235C4×C8#4112yes—
236#95—41no—
237#95—41no—
238#95—41no—
239#95—41no—
240#95—41no—
241#95—41no—
242#95D811yescharacteristic
243#95D811yescharacteristic
244#95D811yescharacteristic
245#95D811yescharacteristic
246C24—41no—
247C2×Q8—22no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×Q8—41no—
251C2×Q8—41no—
252C2×Q8—41no—
253C2×Q8—82no—
254C2×Q8—161no—
255C2×Q8—161no—
256C2×D4—22no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—41no—
261C2×D4—41no—
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—161no—
271C22×C4—161no—
272C22×C4C2×D811yescharacteristic
273C22×C4C2×D811yescharacteristic
274C22×C4C2×D811yescharacteristic
275C22×C4C2×D811yescharacteristic
276C22×C4C2×D811yescharacteristic
277C22×C4C2×D811yescharacteristic
278C22×C4C22×D411yescharacteristic
279Q16—22no—
280Q16—22no—
281Q16—41no—
282Q16—41no—
283SD16—48no—
284D8—22no—
285D8—22no—
286D8—41no—
287D8—41no—
288C2×C8—21no—
289C2×C8—21no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8C22⋊Q812yes—
293C2×C8#12412yes—
294C2×C8C2×SD1612yes—
295C2×C8C2×SD1612yes—
296C2×C8AGL(1,ℤ/8)12yes—
297C2×C8C22×Q811yescharacteristic
298C2×C8C2×#4111yescharacteristic
299C2×C8C2×#4111yescharacteristic
300C2×C8C2×#4111yescharacteristic
301C2×C8H(5,𝔽2)11yescharacteristic
302C2×C8H(5,𝔽2)11yescharacteristic
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—82no—
316C4⋊C4—82no—
317C4⋊C4—82no—
318C4⋊C4—82no—
319C4⋊C4—161no—
320C4⋊C4—161no—
321C4⋊C4—161no—
322C4⋊C4—161no—
323C4⋊C4—161no—
324C4⋊C4—161no—
325C22⋊C4—44no—
326C22⋊C4—44no—
327C22⋊C4—82no—
328C22⋊C4—161no—
329C22⋊C4—161no—
330C42—21no—
331C42—21no—
332C42—82no—
333C42C22⋊Q812yes—
334C42#12412yes—
335C42C2×SD1612yes—
336C42#13712yes—
337C16—41no—
338C16—41no—
339C16—41no—
340C16—41no—
341C23—42no—
342C23—81no—
343C23—81no—
344C23—81no—
345C23—81no—
346C23—81no—
347C23—81no—
348C23C22×D811yescharacteristic
349Q8—42no—
350Q8—42no—
351Q8—81no—
352Q8—81no—
353Q8—164no—
354D4—42no—
355D4—42no—
356D4—81no—
357D4—81no—
358C2×C4—21no—
359C2×C4—21no—
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—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—42no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
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—164no—
394C2×C4#47412yes—
395C2×C4#47412yes—
396C2×C4#48612yes—
397C2×C4#48612yes—
398C2×C4C2×C22⋊Q811yescharacteristic
399C2×C4C2×#12411yescharacteristic
400C2×C4#53511yescharacteristic
401C2×C4C22×SD1611yescharacteristic
402C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
403C2×C4C2×#13711yescharacteristic
404C8—22no—
405C8—22no—
406C8#47614yes—
407C8#55411yescharacteristic
408C8#55411yescharacteristic
409C8#55511yescharacteristic
410C8#55511yescharacteristic
411C22—82no—
412C22—82no—
413C22—82no—
414C22—161no—
415C22—161no—
416C22—161no—
417C22—161no—
418C22C2×#47411yescharacteristic
419C22C2×#48611yescharacteristic
420C22#320511yescharacteristic
421C22#341511yescharacteristic
422C22#341611yescharacteristic
423C22#341611yescharacteristic
424C22#341711yescharacteristic
425C4—22no—
426C4—22no—
427C4—22no—
428C4—22no—
429C4—41no—
430C4—41no—
431C4—82no—
432C4—161no—
433C4—161no—
434C4—161no—
435C4—161no—
436C4—161no—
437C4—161no—
438C4#256311yescharacteristic
439C4C2×#47611yescharacteristic
440C4#314111yescharacteristic
441C4#320611yescharacteristic
442C2—162no—
443C2#2913911yescharacteristic
444C2#3363311yescharacteristic
445C2#3363411yescharacteristic
446C2#3370211yescharacteristic
447C2#3370311yescharacteristic
448C2#3379011yescharacteristic
449C2#3379111yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D8C23
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #376 ◁ #1726 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33319C2semidirect
3. #13366C2semidirect
5. #13364C2semidirect
9. #12890C2semidirect
10. #12888C2semidirect
11. #12814C2semidirect
13. #12812C2semidirect
15. #12654C2semidirect
77. #495Q8semidirect

Maximal subgroups

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

subgroupindex
#333192
#216962
#133662
#133652
#133642
#133632
#132052
#132042
#128902
#128882
#128142
#128132
#128122
#128112
#126542