Finite Group Databasebeta

← #504076

Subgroups — #504076

Order 512.

504 conjugacy classes of subgroups.

The subgroup lattice has 435 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28740C211yesmaximal, characteristic
2Q8×#107C211yesmaximal, characteristic, complemented
3#13530C211yesmaximal, characteristic, complemented
4#13526C211yesmaximal, characteristic
5#13481C211yesmaximal, characteristic, complemented
6#13480C211yesmaximal, characteristic
7#13264C211yesmaximal, characteristic, complemented
8#13254C211yesmaximal, characteristic
9#13205C211yesmaximal, characteristic
10#13176C211yesmaximal, characteristic, complemented
11#13133C211yesmaximal, characteristic, complemented
12#13132C211yesmaximal, characteristic, complemented
13#13131C211yesmaximal, characteristic
14#13130C211yesmaximal, characteristic
15#12674C211yesmaximal, characteristic, complemented
16C2×#495C2211yescharacteristic
17C2×C8×Q8C2211yescharacteristic
18#2592—21no—
19Q8×C4⋊C4—21no—
20C2×#367C2211yescharacteristic
21C2×#366C2211yescharacteristic
22C2×#366C2211yescharacteristic
23C2×#366C2211yescharacteristic
24C2×C4⋊C16C2211yescharacteristic
25C2×#358C2211yescharacteristic
26C2×#358C2211yescharacteristic
27C2×#358C2211yescharacteristic
28C2×#358C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×#357C2211yescharacteristic
31C2×#357C2211yescharacteristic
32C2×#357C2211yescharacteristic
33C2×#346C2211yescharacteristic
34#2055—21no—
35#2055—21no—
36#1983—21no—
37#1970C2211yescharacteristic
38#1970C2211yescharacteristic
39#1969C2211yescharacteristic
40#1969C2211yescharacteristic
41#1947—21no—
42#1947—21no—
43#1947—21no—
44#1845C2211yescharacteristic
45#1845C2211yescharacteristic
46#1845C2211yescharacteristic
47C4×#107C2211yescharacteristic
48C4×#107C2211yescharacteristic
49C4×#107C2211yescharacteristic
50C4×#107C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1374C2211yescharacteristic
56#1374C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×Q16C2311yescharacteristic
60C22×SD16—21no—
61C22×D8C2311yescharacteristic
62C2×#124—21no—
63C2×C4×Q8—41no—
64C2×C4×Q8D411yescharacteristic
65C22×C16C2311yescharacteristic
66C22×C16C2311yescharacteristic
67C22×C16C2311yescharacteristic
68C22×C16C2311yescharacteristic
69#495—22no—
70#495—22no—
71C8×Q8—22no—
72C8×Q8—22no—
73C2×#107—21no—
74C2×#107—21no—
75C2×#107—21no—
76C2×#107C2311yescharacteristic
77C2×#107C2311yescharacteristic
78C2×#107C2311yescharacteristic
79C2×#107C2311yescharacteristic
80C2×#107C2311yescharacteristic
81C2×#107C2311yescharacteristic
82C2×C4⋊C8D411yescharacteristic
83C2×C4⋊C8D411yescharacteristic
84C2×C4⋊C8D411yescharacteristic
85C2×#103—21no—
86C2×#103—21no—
87C2×#102—21no—
88C2×#102—21no—
89C2×C4×C8C2311yescharacteristic
90C2×C4×C8C2311yescharacteristic
91C2×C4×C8C2311yescharacteristic
92#397—41no—
93#393—41no—
94#391—21no—
95#391—21no—
96#390—21no—
97#390—21no—
98#389—41no—
99#389—41no—
100#389—41no—
101#388—41no—
102#388—41no—
103#384—21no—
104#384—21no—
105#384—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—21no—
110#367—22no—
111#366—22no—
112#366—41no—
113#366—41no—
114C4⋊C16—22no—
115#358—22no—
116#358—22no—
117#358—41no—
118#358—41no—
119#357—22no—
120#357—22no—
121#357—41no—
122#357—41no—
123#346—22no—
124#340—21no—
125#340—21no—
126#340—21no—
127#340—21no—
128C22×Q8—21no—
129C22×Q8D811yescharacteristic
130C22×D4—21no—
131C2×Q16—21no—
132C2×Q16—21no—
133C2×Q16—21no—
134C2×Q16—21no—
135C2×Q16—21no—
136C2×Q16—21no—
137C2×SD16—42no—
138C2×SD16—81no—
139C2×SD16—81no—
140C2×D8—21no—
141C2×D8—21no—
142C2×D8—21no—
143C2×D8—21no—
144C2×D8—21no—
145C2×D8—21no—
146C22×C8C2×D411yescharacteristic
147C22×C8C2×D411yescharacteristic
148C22×C8C2×D411yescharacteristic
149C22×C8C2411yescharacteristic
150#124—42no—
151#124—42no—
152C4×Q8—22no—
153C4×Q8—22no—
154C4×Q8—84no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4D811yescharacteristic
168C2×C4⋊C4D811yescharacteristic
169C2×C4⋊C4D811yescharacteristic
170C2×C22⋊C4—41no—
171C2×C22⋊C4—41no—
172C2×C22⋊C4—41no—
173C2×C22⋊C4—41no—
174C2×C22⋊C4—41no—
175C2×C22⋊C4—41no—
176C2×C42—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#4112yes—
195#107—21no—
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—81no—
207#107—81no—
208#107—81no—
209#107#4112yes—
210C4⋊C8—22no—
211C4⋊C8—22no—
212C4⋊C8—22no—
213#103—81no—
214#103—81no—
215#102—81no—
216#102—81no—
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—
231C24—41no—
232C2×Q8—22no—
233C2×Q8—22no—
234C2×Q8—22no—
235C2×Q8—41no—
236C2×Q8—41no—
237C2×Q8—41no—
238C2×Q8—41no—
239C2×Q8—41no—
240C2×Q8—41no—
241C2×D4—41no—
242C2×D4—41no—
243C2×D4—41no—
244C2×D4—41no—
245C2×D4—41no—
246C2×D4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4C2×D811yescharacteristic
255C22×C4C2×D811yescharacteristic
256C22×C4C2×D811yescharacteristic
257C22×C4C2×D811yescharacteristic
258C22×C4C2×D811yescharacteristic
259C22×C4C2×D811yescharacteristic
260C22×C4C22×D411yescharacteristic
261Q16—41no—
262Q16—41no—
263Q16—41no—
264Q16—41no—
265SD16—84no—
266D8—41no—
267D8—41no—
268D8—41no—
269D8—41no—
270C2×C8—21no—
271C2×C8—21no—
272C2×C8—21no—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8#12312yes—
279C2×C8C2×#4111yescharacteristic
280C2×C8C2×#4111yescharacteristic
281C2×C8C2×#4111yescharacteristic
282C2×C8C2×#4111yescharacteristic
283C2×C8H(5,𝔽2)11yescharacteristic
284C2×C82−1+411yescharacteristic
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—82no—
301C4⋊C4—82no—
302C4⋊C4—82no—
303C4⋊C4—161no—
304C4⋊C4—161no—
305C4⋊C4—161no—
306C22⋊C4—82no—
307C22⋊C4—82no—
308C22⋊C4—82no—
309C22⋊C4—82no—
310C22⋊C4—161no—
311C22⋊C4—161no—
312C42—21no—
313C42—21no—
314C42—21no—
315C42—21no—
316C42—82no—
317C42—82no—
318C42—82no—
319C42—161no—
320C42#12312yes—
321C42#13712yes—
322C16—22no—
323C16—41no—
324C16—41no—
325C16—41no—
326C23—81no—
327C23—81no—
328C23—81no—
329C23—81no—
330C23—81no—
331C23—81no—
332C23—81no—
333C23C22×D811yescharacteristic
334Q8—28no—
335Q8—81no—
336Q8—81no—
337Q8—81no—
338Q8—81no—
339D4—81no—
340D4—81no—
341D4—81no—
342D4—81no—
343C2×C4—21no—
344C2×C4—21no—
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—42no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
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#48012yes—
379C2×C4#48012yes—
380C2×C4#50912yes—
381C2×C4#51012yes—
382C2×C4C2×#12311yescharacteristic
383C2×C4#54811yescharacteristic
384C2×C4#55011yescharacteristic
385C2×C4C2×#13711yescharacteristic
386C2×C4#57611yescharacteristic
387C2×C4#57711yescharacteristic
388C8—22no—
389C8—41no—
390C8—41no—
391C8#52911yescharacteristic
392C8#54211yescharacteristic
393C8#55611yescharacteristic
394C8#55911yescharacteristic
395C22—161no—
396C22—161no—
397C22—161no—
398C22—161no—
399C22—161no—
400C22—161no—
401C22—161no—
402C22C2×#48011yescharacteristic
403C22#335811yescharacteristic
404C22#339111yescharacteristic
405C22C2×#50911yescharacteristic
406C22C2×#51011yescharacteristic
407C22#341511yescharacteristic
408C22#341711yescharacteristic
409C4—22no—
410C4—22no—
411C4—22no—
412C4—22no—
413C4—41no—
414C4—41no—
415C4—82no—
416C4—161no—
417C4—161no—
418C4—161no—
419C4—161no—
420C4—161no—
421C4—161no—
422C4#244811yescharacteristic
423C4#309611yescharacteristic
424C4#337511yescharacteristic
425C4#340011yescharacteristic
426C2—321no—
427C2#2631611yescharacteristic
428C2#3364611yescharacteristic
429C2#3365211yescharacteristic
430C2#3393411yescharacteristic
431C2#3394211yescharacteristic
432C2#3396511yescharacteristic
433C2#3396611yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D8C23
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×C8×Q8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. Q8×#107C2semidirect
3. #13530C2semidirect
5. #13481C2semidirect
7. #13264C2semidirect
10. #13176C2semidirect
11. #13133C2semidirect
12. #13132C2semidirect
15. #12674C2semidirect

Maximal subgroups

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

subgroupindex
#287402
Q8×#1072
#135302
#135262
#134812
#134802
#132642
#132542
#132052
#131762
#131332
#131322
#131312
#131302
#126742