Finite Group Databasebeta

← #501461

Subgroups — #501461

Order 512.

544 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
1C2×#2237C211yesmaximal, characteristic
2C2×#2164C211yesmaximal, characteristic
3#33139C211yesmaximal, characteristic
4#21158C211yesmaximal, characteristic
5#13516C211yesmaximal, characteristic, complemented
6#13515C211yesmaximal, characteristic, complemented
7#13477C211yesmaximal, characteristic, complemented
8#13477C211yesmaximal, characteristic, complemented
9#13167C211yesmaximal, characteristic, complemented
10#13166C211yesmaximal, characteristic, complemented
11#13150C211yesmaximal, characteristic, complemented
12#13148C211yesmaximal, characteristic, complemented
13#12993C211yesmaximal, characteristic
14#12920C211yesmaximal, characteristic
15#12868C211yesmaximal, characteristic
16C2×#467C2211yescharacteristic
17C2×#427C2211yescharacteristic
18C22×#107C2211yescharacteristic
19C2×#405C2211yescharacteristic
20#2625—21no—
21#2316—21no—
22#2237—24no—
23#2164—24no—
24C2×#367C2211yescharacteristic
25C2×#367C2211yescharacteristic
26C2×#367C2211yescharacteristic
27C2×#366C2211yescharacteristic
28C2×C4⋊C16C2211yescharacteristic
29C2×C4⋊C16C2211yescharacteristic
30C2×#358C2211yescharacteristic
31C2×#358C2211yescharacteristic
32C2×#358C2211yescharacteristic
33C2×#358C2211yescharacteristic
34C2×C22⋊C16C2211yescharacteristic
35C2×C22⋊C16C2211yescharacteristic
36C2×#346C2211yescharacteristic
37C2×C4×C16C2211yescharacteristic
38#1969C2211yescharacteristic
39#1969C2211yescharacteristic
40#1943—21no—
41#1943—21no—
42#1942C2211yescharacteristic
43#1942C2211yescharacteristic
44#1942C2211yescharacteristic
45#1942C2211yescharacteristic
46#1942C2211yescharacteristic
47#1942C2211yescharacteristic
48#1935—21no—
49#1935—21no—
50#1894—21no—
51#1808C2211yescharacteristic
52#1736—21no—
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1380C2211yescharacteristic
60#1380C2211yescharacteristic
61C22×Q16C2311yescharacteristic
62C22×M16D411yescharacteristic
63C23×C8C2311yescharacteristic
64C2×C22⋊Q8—21no—
65C2×#117—21no—
66C2×#117D411yescharacteristic
67C22×C4⋊C4—21no—
68C22×C16C2311yescharacteristic
69C22×C16C2311yescharacteristic
70C22×C16C2311yescharacteristic
71C22×C16C2311yescharacteristic
72#467—22no—
73#467—22no—
74#427—22no—
75#427—22no—
76C2×#107—21no—
77C2×#107—22no—
78C2×#107—22no—
79C2×#107C2311yescharacteristic
80C2×#107C2311yescharacteristic
81C2×#107C2311yescharacteristic
82C2×#107C2311yescharacteristic
83C2×#107C2311yescharacteristic
84C2×#107C2311yescharacteristic
85C2×#107C2311yescharacteristic
86C2×#106—21no—
87C2×#103—21no—
88C2×#103—21no—
89C2×#103—21no—
90#405—22no—
91#405—22no—
92C2×#97D411yescharacteristic
93C2×#97D411yescharacteristic
94C2×C4×C8C2311yescharacteristic
95C2×C4×C8C2311yescharacteristic
96#400—41no—
97#398—41no—
98#397—41no—
99#397—41no—
100#391—21no—
101#391—21no—
102#389—21no—
103#389—21no—
104#389—21no—
105#389—21no—
106#389—21no—
107#389—21no—
108#389—41no—
109#389—41no—
110#380—21no—
111#380—41no—
112#380—41no—
113#367—22no—
114#367—22no—
115#367—22no—
116#366—22no—
117C4⋊C16—22no—
118C4⋊C16—22no—
119#358—22no—
120#358—22no—
121#358—22no—
122#358—22no—
123C22⋊C16—22no—
124C22⋊C16D414yes—
125#346—22no—
126C4×C16—22no—
127#340—21no—
128#340—21no—
129#340—21no—
130#340—21no—
131#340—21no—
132#340—21no—
133#340—21no—
134C22×Q8—21no—
135C23×C4—41no—
136C23×C4C2×D411yescharacteristic
137C2×Q16—21no—
138C2×Q16—21no—
139C2×Q16—21no—
140C2×Q16—21no—
141C2×Q16—21no—
142C2×Q16—21no—
143C2×M16—22no—
144C2×M16—22no—
145C2×M16—41no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8C2×D411yescharacteristic
151C22×C8C2×D411yescharacteristic
152C22×C8C2×D412yes—
153C22×C8C2×D411yescharacteristic
154C22×C8#4112yes—
155C22×C8C2411yescharacteristic
156C22⋊Q8—42no—
157C22⋊Q8—42no—
158#117—22no—
159#117—22no—
160#117—42no—
161#117—42no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—21no—
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—42no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—41no—
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⋊C4D811yescharacteristic
180C2×C4⋊C4D811yescharacteristic
181C2×C22⋊C4—41no—
182C2×C22⋊C4—41no—
183C2×C22⋊C4—41no—
184C2×C22⋊C4D811yescharacteristic
185C2×C22⋊C4D811yescharacteristic
186C2×C42—41no—
187C2×C42C2×D411yescharacteristic
188C2×C42C2×D411yescharacteristic
189C2×C16—21no—
190C2×C16—21no—
191C2×C16—21no—
192C2×C16—21no—
193C2×C16—21no—
194C2×C16—21no—
195C2×C16—21no—
196C2×C16—21no—
197C2×C16—21no—
198C2×C16—21no—
199C2×C16C2×D412yes—
200C2×C16C2×D412yes—
201#107—24no—
202#107—21no—
203#107—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#4112yes—
215#107#4112yes—
216#106—42no—
217#103—42no—
218#103—42no—
219#103—42no—
220#97—22no—
221#97—22no—
222C4×C8—21no—
223C4×C8—21no—
224C4×C8—21no—
225C4×C8—21no—
226#95—41no—
227#95—41no—
228#95—41no—
229#95—41no—
230#95—41no—
231#95—41no—
232#95—41no—
233#95—41no—
234#95—41no—
235#95—41no—
236#95—41no—
237C24C2×D811yescharacteristic
238C2×Q8—41no—
239C2×Q8—41no—
240C2×Q8—41no—
241C2×Q8—41no—
242C2×Q8—41no—
243C2×Q8—41no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—44no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—161no—
258C22×C4—161no—
259C22×C4#12112yes—
260C22×C4C2×D811yescharacteristic
261C22×C4C2×D811yescharacteristic
262C22×C4C2×D811yescharacteristic
263C22×C4C2×D811yescharacteristic
264C22×C4C2×D811yescharacteristic
265C22×C4#13512yes—
266C22×C4C22×D411yescharacteristic
267Q16—41no—
268Q16—41no—
269Q16—41no—
270Q16—41no—
271M16—44no—
272C2×C8—21no—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—22no—
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—22no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8#12112yes—
284C2×C8#13712yes—
285C2×C8C22×D411yescharacteristic
286C2×C8C2×#4111yescharacteristic
287C2×C8C2×#4111yescharacteristic
288C2×C8C2×#4111yescharacteristic
289C2×C82−1+411yescharacteristic
290C2×C82−1+411yescharacteristic
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—44no—
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—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—82no—
309C4⋊C4—161no—
310C4⋊C4—161no—
311C4⋊C4—161no—
312C4⋊C4—161no—
313C4⋊C4—161no—
314C4⋊C4—161no—
315C22⋊C4—22no—
316C22⋊C4—22no—
317C22⋊C4—82no—
318C22⋊C4—82no—
319C22⋊C4—161no—
320C42—21no—
321C42—21no—
322C42—21no—
323C42—21no—
324C42—82no—
325C16—22no—
326C16—22no—
327C16—41no—
328C16—41no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23#46612yes—
336C23C22×D811yescharacteristic
337Q8—81no—
338Q8—81no—
339Q8—81no—
340Q8—81no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
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—42no—
360C2×C4—42no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
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—81no—
383C2×C4—164no—
384C2×C4#46612yes—
385C2×C4C2×#12111yescharacteristic
386C2×C4#52511yescharacteristic
387C2×C4#53611yescharacteristic
388C2×C4C2×#13511yescharacteristic
389C2×C4C2×#13711yescharacteristic
390C2×C4#57511yescharacteristic
391C8—22no—
392C8—22no—
393C8—41no—
394C8#54711yescharacteristic
395C8#54711yescharacteristic
396C8#55211yescharacteristic
397C8#55211yescharacteristic
398C22—22no—
399C22—22no—
400C22—22no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22C2×#46611yescharacteristic
406C22#310011yescharacteristic
407C22#322611yescharacteristic
408C22#341611yescharacteristic
409C22#341611yescharacteristic
410C22#341611yescharacteristic
411C22#341711yescharacteristic
412C4—22no—
413C4—41no—
414C4—41no—
415C4—41no—
416C4—41no—
417C4—82no—
418C4—82no—
419C4—161no—
420C4—161no—
421C4—161no—
422C4—161no—
423C4—161no—
424C4—161no—
425C4#241311yescharacteristic
426C4#305611yescharacteristic
427C4#310811yescharacteristic
428C4#323211yescharacteristic
429C2—42no—
430C2#2460011yescharacteristic
431C2#3361111yescharacteristic
432C2#3361311yescharacteristic
433C2#3368111yescharacteristic
434C2#3368311yescharacteristic
435C2#3380811yescharacteristic
436C2#3381011yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC24C2×D8
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#405 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #13516C2semidirect
6. #13515C2semidirect
7. #13477C2semidirect
8. #13477C2semidirect
9. #13167C2semidirect
10. #13166C2semidirect
11. #13150C2semidirect
12. #13148C2semidirect

Maximal subgroups

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

subgroupindex
C2×#22372
C2×#21642
#331392
#211582
#135162
#135152
#134772
#131672
#131662
#131502
#131482
#129932
#129202
#128682