Finite Group Databasebeta

← #501950

Subgroups — #501950

Order 512.

491 conjugacy classes of subgroups.

The subgroup lattice has 413 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28267C211yesmaximal, characteristic
2Q8×#107C211yesmaximal, characteristic, complemented
3#13538C211yesmaximal, characteristic, complemented
4#13537C211yesmaximal, characteristic, complemented
5#13480C211yesmaximal, characteristic
6#13480C211yesmaximal, characteristic
7#13264C211yesmaximal, characteristic, complemented
8#13178C211yesmaximal, characteristic, complemented
9#13131C211yesmaximal, characteristic
10#13130C211yesmaximal, characteristic
11#13112C211yesmaximal, characteristic, complemented
12#13111C211yesmaximal, characteristic, complemented
13#12890C211yesmaximal, characteristic, complemented
14#12811C211yesmaximal, characteristic
15#12637C211yesmaximal, characteristic
16C2×C8×Q8C2211yescharacteristic
17C2×C4×D8C2211yescharacteristic
18#2580—21no—
19Q8×C4⋊C4—21no—
20C2×#367C2211yescharacteristic
21C2×#366C2211yescharacteristic
22C2×C4⋊C16C2211yescharacteristic
23C2×#357C2211yescharacteristic
24C2×#357C2211yescharacteristic
25C2×#357C2211yescharacteristic
26C2×#357C2211yescharacteristic
27C2×#346C2211yescharacteristic
28#2065—21no—
29#2065—21no—
30#1970C2211yescharacteristic
31#1970C2211yescharacteristic
32#1947—21no—
33#1947—21no—
34#1947—21no—
35#1942C2211yescharacteristic
36#1942C2211yescharacteristic
37#1846C2211yescharacteristic
38#1845C2211yescharacteristic
39#1845C2211yescharacteristic
40#1845C2211yescharacteristic
41#1804—21no—
42C4×#107C2211yescharacteristic
43C4×#107C2211yescharacteristic
44C4×#107C2211yescharacteristic
45#1380C2211yescharacteristic
46#1380C2211yescharacteristic
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×D8C2311yescharacteristic
60C2×C4×Q8—41no—
61C2×C4×Q8D411yescharacteristic
62C2×C4×D4—21no—
63C22×C16C2311yescharacteristic
64C22×C16C2311yescharacteristic
65C22×C16C2311yescharacteristic
66C22×C16C2311yescharacteristic
67C8×Q8—22no—
68C8×Q8—22no—
69C4×D8—22no—
70C4×D8—22no—
71C2×#107—21no—
72C2×#107—21no—
73C2×#107—21no—
74C2×#107C2311yescharacteristic
75C2×#107C2311yescharacteristic
76C2×#107C2311yescharacteristic
77C2×#107C2311yescharacteristic
78C2×#107C2311yescharacteristic
79C2×#107C2311yescharacteristic
80C2×#107C2311yescharacteristic
81C2×#106—21no—
82C2×C4⋊C8D411yescharacteristic
83C2×C4⋊C8D411yescharacteristic
84C2×C4⋊C8D411yescharacteristic
85C2×#102—21no—
86C2×#102—21no—
87C2×#102—21no—
88C2×C4×C8C2311yescharacteristic
89C2×C4×C8C2311yescharacteristic
90C2×C4×C8C2311yescharacteristic
91#399—41no—
92#393—41no—
93#390—21no—
94#390—21no—
95#389—21no—
96#389—21no—
97#389—41no—
98#389—41no—
99#389—41no—
100#385—41no—
101#385—41no—
102#384—21no—
103#384—21no—
104#384—21no—
105#382—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109#367—22no—
110#366—22no—
111C4⋊C16—22no—
112#357—22no—
113#357—22no—
114#357—22no—
115#357—22no—
116#346—22no—
117#340—21no—
118#340—21no—
119#340—21no—
120#340—21no—
121#340—21no—
122C22×Q8D811yescharacteristic
123C22×D4—21no—
124C23×C4—41no—
125C2×D8—21no—
126C2×D8—21no—
127C2×D8—21no—
128C2×D8—21no—
129C2×D8—21no—
130C2×D8#4112yes—
131C22×C8C2×D411yescharacteristic
132C22×C8C2×D411yescharacteristic
133C22×C8C2×D411yescharacteristic
134C22×C8C2411yescharacteristic
135C4×Q8—22no—
136C4×Q8—22no—
137C4×Q8—84no—
138C4×D4—42no—
139C4×D4—42no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—41no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4D811yescharacteristic
156C2×C4⋊C4D811yescharacteristic
157C2×C4⋊C4D811yescharacteristic
158C2×C22⋊C4—41no—
159C2×C22⋊C4—41no—
160C2×C22⋊C4—41no—
161C2×C22⋊C4—41no—
162C2×C22⋊C4—41no—
163C2×C42—41no—
164C2×C42—41no—
165C2×C42—41no—
166C2×C42C2×D411yescharacteristic
167C2×C42C2×D411yescharacteristic
168C2×C42C2×D411yescharacteristic
169C2×C16—21no—
170C2×C16—21no—
171C2×C16—21no—
172C2×C16—21no—
173C2×C16—21no—
174C2×C16—21no—
175C2×C16—21no—
176C2×C16—21no—
177C2×C16—21no—
178C2×C16—21no—
179C2×C16—21no—
180C2×C16—21no—
181#107—21no—
182#107—21no—
183#107—21no—
184#107—21no—
185#107—21no—
186#107—21no—
187#107—21no—
188#107—21no—
189#107—21no—
190#107—21no—
191#107—21no—
192#107—21no—
193#107—21no—
194#107—81no—
195#107—81no—
196#107—81no—
197#107#4112yes—
198#106—81no—
199C4⋊C8—22no—
200C4⋊C8—22no—
201C4⋊C8—22no—
202#102—42no—
203#102—81no—
204#102—81no—
205C4×C8—21no—
206C4×C8—21no—
207C4×C8—21no—
208C4×C8—21no—
209C4×C8#4112yes—
210C4×C8#4112yes—
211#95—41no—
212#95—41no—
213#95—41no—
214#95—41no—
215#95—41no—
216#95—41no—
217#95—41no—
218C24—41no—
219C2×Q8—22no—
220C2×Q8—22no—
221C2×Q8—22no—
222C2×D4—22no—
223C2×D4—41no—
224C2×D4—41no—
225C2×D4—41no—
226C2×D4—41no—
227C2×D4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—82no—
236C22×C4—161no—
237C22×C4—161no—
238C22×C4C2×D811yescharacteristic
239C22×C4C2×D811yescharacteristic
240C22×C4C2×D811yescharacteristic
241C22×C4C2×D811yescharacteristic
242C22×C4C2×D811yescharacteristic
243C22×C4C2×D811yescharacteristic
244C22×C4C22×D411yescharacteristic
245D8—22no—
246D8—22no—
247D8—41no—
248D8—41no—
249C2×C8—21no—
250C2×C8—21no—
251C2×C8—21no—
252C2×C8—21no—
253C2×C8—21no—
254C2×C8—21no—
255C2×C8—21no—
256C2×C8—21no—
257C2×C8#12312yes—
258C2×C8C2×#4111yescharacteristic
259C2×C8C2×#4111yescharacteristic
260C2×C8C2×#4111yescharacteristic
261C2×C8C2×#4111yescharacteristic
262C2×C82−1+411yescharacteristic
263C2×C82−1+411yescharacteristic
264C4⋊C4—22no—
265C4⋊C4—22no—
266C4⋊C4—22no—
267C4⋊C4—22no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—82no—
282C4⋊C4—82no—
283C4⋊C4—82no—
284C4⋊C4—161no—
285C4⋊C4—161no—
286C4⋊C4—161no—
287C4⋊C4—161no—
288C4⋊C4—161no—
289C22⋊C4—82no—
290C22⋊C4—82no—
291C22⋊C4—82no—
292C22⋊C4—161no—
293C22⋊C4—161no—
294C42—21no—
295C42—21no—
296C42—21no—
297C42—21no—
298C42—82no—
299C42—82no—
300C42—82no—
301C42#12312yes—
302C42#13712yes—
303C16—41no—
304C16—41no—
305C16—41no—
306C16—41no—
307C23—42no—
308C23—81no—
309C23—81no—
310C23—81no—
311C23—81no—
312C23—81no—
313C23—81no—
314C23C22×D811yescharacteristic
315Q8—28no—
316D4—42no—
317D4—42no—
318D4—81no—
319D4—81no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—42no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—164no—
356C2×C4#48012yes—
357C2×C4#48012yes—
358C2×C4#50912yes—
359C2×C4#51012yes—
360C2×C4C2×#12311yescharacteristic
361C2×C4#54811yescharacteristic
362C2×C4#54811yescharacteristic
363C2×C4C2×#13711yescharacteristic
364C2×C4#57711yescharacteristic
365C2×C4#57711yescharacteristic
366C8—22no—
367C8—41no—
368C8—41no—
369C8#52911yescharacteristic
370C8#54111yescharacteristic
371C8#55211yescharacteristic
372C8#55611yescharacteristic
373C22—82no—
374C22—82no—
375C22—82no—
376C22—161no—
377C22—161no—
378C22—161no—
379C22—161no—
380C22C2×#48011yescharacteristic
381C22#335811yescharacteristic
382C22#335811yescharacteristic
383C22C2×#50911yescharacteristic
384C22C2×#51011yescharacteristic
385C22#341511yescharacteristic
386C22#341611yescharacteristic
387C4—22no—
388C4—22no—
389C4—22no—
390C4—22no—
391C4—41no—
392C4—41no—
393C4—82no—
394C4—161no—
395C4—161no—
396C4—161no—
397C4—161no—
398C4—161no—
399C4—161no—
400C4#242111yescharacteristic
401C4#309911yescharacteristic
402C4#336911yescharacteristic
403C4#337211yescharacteristic
404C2—162no—
405C2#2499011yescharacteristic
406C2#3364611yescharacteristic
407C2#3365011yescharacteristic
408C2#3393311yescharacteristic
409C2#3393911yescharacteristic
410C2#3394011yescharacteristic
411C2#3394211yescharacteristic
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. #13538C2semidirect
4. #13537C2semidirect
7. #13264C2semidirect
8. #13178C2semidirect
11. #13112C2semidirect
12. #13111C2semidirect
13. #12890C2semidirect

Maximal subgroups

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

subgroupindex
#282672
Q8×#1072
#135382
#135372
#134802
#132642
#131782
#131312
#131302
#131122
#131112
#128902
#128112
#126372