Finite Group Databasebeta

← #508526

Subgroups — #508526

Order 512.

581 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33173C211yesmaximal, characteristic
2#20817C211yesmaximal, characteristic, complemented
3#13530C211yesmaximal, characteristic, complemented
4#13527C211yesmaximal, characteristic, complemented
5#13481C211yesmaximal, characteristic, complemented
6#13480C211yesmaximal, characteristic, complemented
7#13382C212yesmaximal, complemented
8#13380C212yesmaximal
9#13254C212yesmaximal, complemented
10#13178C212yesmaximal, complemented
11#12783C211yesmaximal, characteristic, complemented
12C2×#495C2211yescharacteristic
13C2×#467C2211yescharacteristic
14C2×#466C2211yescharacteristic
15C2×#465—21no—
16C22×C4×C8C2211yescharacteristic
17#2850—21no—
18#2285—21no—
19C2×#367C2212yes—
20C2×#366C2212yes—
21C2×C4⋊C16C2212yes—
22C2×#358C2212yes—
23C2×#358C2212yes—
24C2×#357C2212yes—
25C2×#357C2212yes—
26C2×C22⋊C16C2212yes—
27#1970C2211yescharacteristic
28#1969C2211yescharacteristic
29#1968—21no—
30#1958—21no—
31#1846C2211yescharacteristic
32#1845C2211yescharacteristic
33#1845C2211yescharacteristic
34#1834—21no—
35#1808—21no—
36#1808C2212yes—
37C4×#107—21no—
38C4×#107C2212yes—
39#1380C2212yes—
40#1380C2212yes—
41#1374C2211yescharacteristic
42#1374C2211yescharacteristic
43C22×Q16C2311yescharacteristic
44C22×SD16—21no—
45C22×D8C2311yescharacteristic
46C23×C8D411yescharacteristic
47C23×C8C2311yescharacteristic
48C2×#124—21no—
49C2×C22⋊Q8—21no—
50C2×#121—21no—
51C2×#117—41no—
52C22×C42D411yescharacteristic
53C22×C16C2312yes—
54C22×C16C2312yes—
55#495—24no—
56#467—42no—
57#466—42no—
58#465—44no—
59C2×#107—21no—
60C2×#107—21no—
61C2×#107C2312yes—
62C2×#107C2312yes—
63C2×#107C2311yescharacteristic
64C2×#107C2311yescharacteristic
65C2×#106—21no—
66C2×#103—21no—
67C2×#103—21no—
68C2×#102—21no—
69C2×#102—21no—
70C2×C4×C8—24no—
71C2×C4×C8D411yescharacteristic
72C2×C4×C8D411yescharacteristic
73C2×C4×C8C2311yescharacteristic
74C2×C4×C8C2311yescharacteristic
75#391—21no—
76#390—21no—
77#388—41no—
78#388—41no—
79#386—41no—
80#386—41no—
81#384—21no—
82#384—21no—
83#382—21no—
84#382—41no—
85#380—22no—
86#380—41no—
87C4×C4⋊C4—22no—
88C4×C4⋊C4—41no—
89#367—42no—
90#366—42no—
91C4⋊C16—24no—
92#358—42no—
93#358—42no—
94#357—42no—
95#357—42no—
96C22⋊C16—42no—
97#340—22no—
98#340—22no—
99C22×Q8—21no—
100C22×D4—21no—
101C22×D4—41no—
102C23×C4D811yescharacteristic
103C23×C4D811yescharacteristic
104C23×C4C2×D411yescharacteristic
105C2×Q16—22no—
106C2×Q16—21no—
107C2×Q16—22no—
108C2×Q16—21no—
109C2×SD16—42no—
110C2×SD16—42no—
111C2×SD16—42no—
112C2×D8—22no—
113C2×D8—21no—
114C2×D8—22no—
115C2×D8—21no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—21no—
120C22×C8—22no—
121C22×C8—21no—
122C22×C8—41no—
123C22×C8C2×D411yescharacteristic
124C22×C8C2×D411yescharacteristic
125C22×C8C2×D411yescharacteristic
126C22×C8C2411yescharacteristic
127#124—42no—
128#124—42no—
129C22⋊Q8—42no—
130C22⋊Q8—42no—
131#121—42no—
132#121—42no—
133#117—84no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—41no—
139C2×C4⋊C4—41no—
140C2×C4⋊C4—41no—
141C2×C4⋊C4—41no—
142C2×C4⋊C4—41no—
143C2×C22⋊C4—42no—
144C2×C22⋊C4—41no—
145C2×C22⋊C4—41no—
146C2×C22⋊C4—41no—
147C2×C22⋊C4—41no—
148C2×C22⋊C4—41no—
149C2×C22⋊C4—41no—
150C2×C42—22no—
151C2×C42—22no—
152C2×C42—42no—
153C2×C42D811yescharacteristic
154C2×C42D811yescharacteristic
155C2×C42C2×D411yescharacteristic
156C2×C42C2×D411yescharacteristic
157C2×C16—22no—
158C2×C16—22no—
159C2×C16—22no—
160C2×C16—22no—
161C2×C16—22no—
162C2×C16—22no—
163#107—22no—
164#107—22no—
165#107—22no—
166#107—22no—
167#107—22no—
168#107—22no—
169#107—42no—
170#107—42no—
171#106—42no—
172#103—42no—
173#103—42no—
174#102—42no—
175#102—42no—
176C4×C8—22no—
177C4×C8—22no—
178C4×C8—22no—
179C4×C8#4114yes—
180#95—42no—
181#95—42no—
182#95—41no—
183#95—41no—
184#95—41no—
185#95—41no—
186C24—41no—
187C24C2×D811yescharacteristic
188C2×Q8—41no—
189C2×Q8—41no—
190C2×Q8—41no—
191C2×Q8—41no—
192C2×Q8—41no—
193C2×Q8—41no—
194C2×D4—41no—
195C2×D4—41no—
196C2×D4—41no—
197C2×D4—41no—
198C2×D4—41no—
199C2×D4—41no—
200C2×D4—82no—
201C2×D4—82no—
202C2×D4—82no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4C2×D811yescharacteristic
221C22×C4C2×D811yescharacteristic
222C22×C4C2×D811yescharacteristic
223C22×C4C2×D811yescharacteristic
224C22×C4C2×D811yescharacteristic
225C22×C4C22×D411yescharacteristic
226Q16—42no—
227Q16—42no—
228SD16—84no—
229D8—42no—
230D8—42no—
231C2×C8—21no—
232C2×C8—21no—
233C2×C8—21no—
234C2×C8—21no—
235C2×C8—21no—
236C2×C8—21no—
237C2×C8—21no—
238C2×C8—44no—
239C2×C8—42no—
240C2×C8—42no—
241C2×C8#12312yes—
242C2×C8AGL(1,ℤ/8)12yes—
243C2×C8C2×#4112yes—
244C2×C8H(5,𝔽2)11yescharacteristic
245C2×C8H(5,𝔽2)12yes—
246C2×C82−1+411yescharacteristic
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—82no—
256C4⋊C4—82no—
257C4⋊C4—82no—
258C4⋊C4—82no—
259C4⋊C4—82no—
260C22⋊C4—82no—
261C22⋊C4—82no—
262C22⋊C4—84no—
263C22⋊C4—82no—
264C22⋊C4—82no—
265C22⋊C4—82no—
266C22⋊C4—82no—
267C42—21no—
268C42—21no—
269C42—22no—
270C42—22no—
271C42—84no—
272C42#12312yes—
273C42#13712yes—
274C16—42no—
275C16—42no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—81no—
283C23—81no—
284C23—81no—
285C23—81no—
286C23—81no—
287C23—81no—
288C23—81no—
289C23#50912yes—
290C23C22×D811yescharacteristic
291Q8—81no—
292Q8—81no—
293Q8—81no—
294Q8—81no—
295D4—81no—
296D4—81no—
297D4—81no—
298D4—81no—
299D4—164no—
300C2×C4—28no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—44no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4—82no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—82no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—82no—
330C2×C4—82no—
331C2×C4—81no—
332C2×C4#48012yes—
333C2×C4#48012yes—
334C2×C4#50812yes—
335C2×C4#51012yes—
336C2×C4C2×#12311yescharacteristic
337C2×C4#53511yescharacteristic
338C2×C4#53711yescharacteristic
339C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
340C2×C4C2×#13711yescharacteristic
341C2×C4#57511yescharacteristic
342C8—22no—
343C8—22no—
344C8—41no—
345C8#54012yes—
346C8#54212yes—
347C22—22no—
348C22—22no—
349C22—22no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—161no—
355C22—161no—
356C22—161no—
357C22—161no—
358C22—161no—
359C22—161no—
360C22—161no—
361C22C2×#48011yescharacteristic
362C22#320511yescharacteristic
363C22#324511yescharacteristic
364C22C2×#50811yescharacteristic
365C22C2×#50911yescharacteristic
366C22C2×#51011yescharacteristic
367C22#341411yescharacteristic
368C4—22no—
369C4—22no—
370C4—22no—
371C4—22no—
372C4—41no—
373C4—162no—
374C4—161no—
375C4—161no—
376C4—162no—
377C4—161no—
378C4#262411yescharacteristic
379C4#309411yescharacteristic
380C4#321611yescharacteristic
381C4#325411yescharacteristic
382C2—42no—
383C2—321no—
384C2#2992011yescharacteristic
385C2#3364511yescharacteristic
386C2#3364611yescharacteristic
387C2#3377611yescharacteristic
388C2#3377911yescharacteristic
389C2#3381311yescharacteristic
390C2#3381811yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC2×#466C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20817C2semidirect
3. #13530C2semidirect
4. #13527C2semidirect
5. #13481C2semidirect
6. #13480C2semidirect
7. #13382C2semidirect
9. #13254C2semidirect
10. #13178C2semidirect
11. #12783C2semidirect

Maximal subgroups

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

subgroupindex
#331732
#208172
#135302
#135272
#134812
#134802
#133822
#133802
#132542
#131782
#127832