Finite Group Databasebeta

← #499677

Subgroups — #499677

Order 512.

558 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21725C211yesmaximal, characteristic, complemented
2#21696C211yesmaximal, characteristic
3#13538C211yesmaximal, characteristic, complemented
4#13537C211yesmaximal, characteristic, complemented
5#13483C211yesmaximal, characteristic, complemented
6#13482C211yesmaximal, characteristic
7#13408C211yesmaximal, characteristic
8#13408C211yesmaximal, characteristic
9#13364C211yesmaximal, characteristic, complemented
10#13363C211yesmaximal, characteristic
11#12638C211yesmaximal, characteristic, complemented
12#12637C211yesmaximal, characteristic
13C4×#358C211yesmaximal, characteristic, complemented
14C4×#357C211yesmaximal, characteristic
15#12592C211yesmaximal, characteristic, complemented
16C2×#500C2211yescharacteristic
17C2×#495C2211yescharacteristic
18C2×#440—21no—
19C2×C4×Q16C2211yescharacteristic
20C2×C4×D8C2211yescharacteristic
21#2331—21no—
22#2326—21no—
23C2×#358C2211yescharacteristic
24C2×#358C2211yescharacteristic
25C2×#358C2211yescharacteristic
26C2×#358C2211yescharacteristic
27C2×#357C2211yescharacteristic
28C2×#357C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×#357C2211yescharacteristic
31C2×#346C2211yescharacteristic
32C2×C4×C16C2211yescharacteristic
33#1993—21no—
34#1973—21no—
35#1970C2211yescharacteristic
36#1969C2211yescharacteristic
37#1959—21no—
38#1943—21no—
39#1942C2211yescharacteristic
40#1942C2211yescharacteristic
41#1846C2211yescharacteristic
42#1845C2211yescharacteristic
43#1776—21no—
44C4×#107C2211yescharacteristic
45C4×#107C2211yescharacteristic
46#1726C2211yescharacteristic
47#1380C2211yescharacteristic
48#1380C2211yescharacteristic
49#1380C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1374C2211yescharacteristic
56#1374C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×Q16C2311yescharacteristic
60C22×SD16—21no—
61C22×D8C2311yescharacteristic
62C2×C4⋊Q8—21no—
63C2×#124—21no—
64C2×C4×Q8—21no—
65C2×C4×D4—21no—
66C22×C16C2311yescharacteristic
67C22×C16C2311yescharacteristic
68C22×C16C2311yescharacteristic
69C22×C16C2311yescharacteristic
70#500—22no—
71#500—22no—
72#495—22no—
73#495—22no—
74#440—44no—
75C4×Q16—22no—
76C4×Q16—22no—
77C4×D8—22no—
78C4×D8—22no—
79C2×#107—21no—
80C2×#107—21no—
81C2×#107C2311yescharacteristic
82C2×#107C2311yescharacteristic
83C2×#107C2311yescharacteristic
84C2×#107C2311yescharacteristic
85C2×#107C2311yescharacteristic
86C2×#107C2311yescharacteristic
87C2×#106—21no—
88C2×#103—21no—
89C2×#103—21no—
90C2×#102—21no—
91C2×#102—21no—
92C2×#97D411yescharacteristic
93C2×#97D411yescharacteristic
94C2×#97D411yescharacteristic
95C2×C4×C8C2311yescharacteristic
96C2×C4×C8C2311yescharacteristic
97C2×C4×C8C2311yescharacteristic
98#391—21no—
99#391—41no—
100#391—41no—
101#390—21no—
102#389—21no—
103#389—21no—
104#388—41no—
105#388—41no—
106#387—41no—
107#387—41no—
108#385—41no—
109#385—41no—
110#384—21no—
111#382—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—21no—
114#376D411yescharacteristic
115#358—22no—
116#358—22no—
117#358—22no—
118#358—22no—
119#357—22no—
120#357—22no—
121#357—22no—
122#357—22no—
123#346—22no—
124C4×C16—22no—
125#340—21no—
126#340—21no—
127#340—21no—
128#340—21no—
129C22×Q8—21no—
130C22×Q8—41no—
131C22×D4—21no—
132C23×C4—41no—
133C2×Q16—21no—
134C2×Q16—21no—
135C2×Q16—21no—
136C2×Q16—21no—
137C2×Q16—21no—
138C2×Q16#4112yes—
139C2×SD16—24no—
140C2×SD16—42no—
141C2×SD16—42no—
142C2×D8—21no—
143C2×D8—21no—
144C2×D8—21no—
145C2×D8—21no—
146C2×D8—21no—
147C2×D8#4112yes—
148C22×C8C2×D411yescharacteristic
149C22×C8C2×D411yescharacteristic
150C22×C8C2×D411yescharacteristic
151C22×C8C2411yescharacteristic
152C4⋊Q8—24no—
153C4⋊Q8—42no—
154#124—24no—
155#124—42no—
156C4×Q8—42no—
157C4×Q8—42no—
158C4×D4—42no—
159C4×D4—42no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—41no—
172C2×C22⋊C4—41no—
173C2×C22⋊C4—41no—
174C2×C22⋊C4—41no—
175C2×C22⋊C4—41no—
176C2×C22⋊C4—41no—
177C2×C42—41no—
178C2×C42—41no—
179C2×C42—41no—
180C2×C42C2×D411yescharacteristic
181C2×C42C2×D411yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C16—21no—
184C2×C16—21no—
185C2×C16—21no—
186C2×C16—21no—
187C2×C16—21no—
188C2×C16—21no—
189C2×C16—21no—
190C2×C16—21no—
191C2×C16—21no—
192C2×C16—21no—
193C2×C16—21no—
194C2×C16—21no—
195#107—24no—
196#107—21no—
197#107—21no—
198#107—21no—
199#107—21no—
200#107—21no—
201#107—21no—
202#107—21no—
203#107—21no—
204#107—21no—
205#107—21no—
206#107—42no—
207#107#4112yes—
208#107#4112yes—
209#106—42no—
210#103—42no—
211#103—42no—
212#102—42no—
213#102—42no—
214#97—41no—
215#97—41no—
216#97SD1614yes—
217C4×C8—21no—
218C4×C8—21no—
219C4×C8—21no—
220C4×C8—21no—
221C4×C8#4112yes—
222C4×C8#4112yes—
223#95—41no—
224#95—41no—
225#95—41no—
226#95—41no—
227#95—41no—
228#95—41no—
229#95—41no—
230#95—41no—
231#95D811yescharacteristic
232#95D811yescharacteristic
233#95D811yescharacteristic
234#95D811yescharacteristic
235C24—41no—
236C2×Q8—22no—
237C2×Q8—44no—
238C2×Q8—41no—
239C2×Q8—41no—
240C2×Q8—41no—
241C2×Q8—41no—
242C2×Q8—41no—
243C2×Q8—161no—
244C2×Q8—161no—
245C2×D4—22no—
246C2×D4—41no—
247C2×D4—41no—
248C2×D4—41no—
249C2×D4—41no—
250C2×D4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—82no—
259C22×C4—161no—
260C22×C4—161no—
261C22×C4C2×D811yescharacteristic
262C22×C4C2×D811yescharacteristic
263C22×C4C2×D811yescharacteristic
264C22×C4C2×D811yescharacteristic
265C22×C4C2×D811yescharacteristic
266C22×C4C2×D811yescharacteristic
267C22×C4C22×D411yescharacteristic
268Q16—22no—
269Q16—22no—
270Q16—41no—
271Q16—41no—
272SD16—48no—
273D8—22no—
274D8—22no—
275D8—41no—
276D8—41no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8#12412yes—
284C2×C8C2×SD1612yes—
285C2×C8C2×SD1612yes—
286C2×C8C2×#4111yescharacteristic
287C2×C8C2×#4111yescharacteristic
288C2×C8C2×#4111yescharacteristic
289C2×C8C2×#4111yescharacteristic
290C2×C8C2×#4111yescharacteristic
291C2×C8C2×#4111yescharacteristic
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—44no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—82no—
306C4⋊C4—161no—
307C4⋊C4—161no—
308C4⋊C4—161no—
309C4⋊C4—161no—
310C22⋊C4—44no—
311C22⋊C4—44no—
312C22⋊C4—82no—
313C22⋊C4—161no—
314C22⋊C4—161no—
315C42—21no—
316C42—21no—
317C42—21no—
318C42—21no—
319C42—82no—
320C42—161no—
321C42—161no—
322C42#12412yes—
323C42C2×SD1612yes—
324C16—41no—
325C16—41no—
326C16—41no—
327C16—41no—
328C23—42no—
329C23—81no—
330C23—81no—
331C23—81no—
332C23—81no—
333C23—81no—
334C23—81no—
335C23C22×D811yescharacteristic
336Q8—42no—
337Q8—42no—
338Q8—81no—
339Q8—81no—
340Q8—164no—
341D4—42no—
342D4—42no—
343D4—81no—
344D4—81no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
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—164no—
383C2×C4#48612yes—
384C2×C4#48612yes—
385C2×C4#52511yescharacteristic
386C2×C4#52511yescharacteristic
387C2×C4C2×#12411yescharacteristic
388C2×C4C22×SD1611yescharacteristic
389C2×C4#57511yescharacteristic
390C2×C4#57511yescharacteristic
391C8—22no—
392C8—22no—
393C8—22no—
394C8#52911yescharacteristic
395C8#52911yescharacteristic
396C8#52911yescharacteristic
397C8#52911yescharacteristic
398C22—82no—
399C22—82no—
400C22—82no—
401C22—161no—
402C22—161no—
403C22—161no—
404C22—161no—
405C22#310011yescharacteristic
406C22#310011yescharacteristic
407C22C2×#48611yescharacteristic
408C22#341511yescharacteristic
409C22#341611yescharacteristic
410C22#341611yescharacteristic
411C22#341711yescharacteristic
412C4—22no—
413C4—22no—
414C4—41no—
415C4—41no—
416C4—41no—
417C4—41no—
418C4—82no—
419C4—82no—
420C4—82no—
421C4—161no—
422C4—161no—
423C4—161no—
424C4—161no—
425C4#239811yescharacteristic
426C4#310111yescharacteristic
427C4#310311yescharacteristic
428C4#313511yescharacteristic
429C2—162no—
430C2#2381511yescharacteristic
431C2#3367811yescharacteristic
432C2#3368111yescharacteristic
433C2#3368411yescharacteristic
434C2#3368511yescharacteristic
435C2#3370211yescharacteristic
436C2#3370311yescharacteristic
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. #21725C2semidirect
3. #13538C2semidirect
4. #13537C2semidirect
5. #13483C2semidirect
9. #13364C2semidirect
11. #12638C2semidirect
13. C4×#358C2semidirect
15. #12592C2semidirect

Maximal subgroups

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

subgroupindex
#217252
#216962
#135382
#135372
#134832
#134822
#134082
#133642
#133632
#126382
#126372
C4×#3582
C4×#3572
#125922