Finite Group Databasebeta

← #506375

Subgroups — #506375

Order 512.

698 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
1C2×#2220C212yesmaximal, complemented
2C2×C4×SD32C211yesmaximal, characteristic, complemented
3C2×#2169C211yesmaximal, characteristic, complemented
4#28228C211yesmaximal, characteristic, complemented
5#28227C211yesmaximal, characteristic
6#13383C212yesmaximal, complemented
7#13380C212yesmaximal, complemented
8#12905C211yesmaximal, characteristic, complemented
9#12836C212yesmaximal, complemented
10#12812C211yesmaximal, characteristic, complemented
11#12811C211yesmaximal, characteristic, complemented
12C22×SD32C2211yescharacteristic
13C22×M32C2212yes—
14C2×#467C2212yes—
15C2×#466C2212yes—
16C2×C4×Q16C2211yescharacteristic
17C2×C4×D8C2211yescharacteristic
18C2×C8×D4C2211yescharacteristic
19#2538—21no—
20#2537—21no—
21#2220—28no—
22C4×SD32—24no—
23#2169—24no—
24C2×#367C2211yescharacteristic
25C2×#366C2212yes—
26C2×#366C2212yes—
27C2×C4⋊C16C2211yescharacteristic
28C2×#358C2212yes—
29C2×#358C2211yescharacteristic
30C2×#357C2212yes—
31C2×#357C2211yescharacteristic
32C2×C22⋊C16C2212yes—
33C2×C4×C16C2211yescharacteristic
34#2036—22no—
35#2033—22no—
36#1845C2211yescharacteristic
37#1845C2211yescharacteristic
38#1808C2212yes—
39#1808C2212yes—
40#1803—21no—
41#1802—21no—
42#1380C2212yes—
43#1380C2212yes—
44C22×Q16C2311yescharacteristic
45C22×D8C2311yescharacteristic
46C23×C8C2312yes—
47C2×C22⋊Q8—22no—
48C2×#121—22no—
49C2×#121—41no—
50C2×C4×Q8—21no—
51C2×C4×D4—21no—
52C2×C4×D4D411yescharacteristic
53C2×SD32—22no—
54C2×SD32—22no—
55C2×SD32D414yescomplemented
56C2×M32—24no—
57C2×M32—24no—
58C2×M32—24no—
59C22×C16C2312yes—
60C22×C16C2311yescharacteristic
61C22×C16C2311yescharacteristic
62#467—24no—
63#467—42no—
64#466—24no—
65#466—42no—
66C4×Q16—22no—
67C4×Q16—22no—
68C4×D8—22no—
69C4×D8—22no—
70C8×D4—22no—
71C8×D4—22no—
72C2×#107—21no—
73C2×#107—21no—
74C2×#107C2311yescharacteristic
75C2×#107C2311yescharacteristic
76C2×#107C2312yes—
77C2×#107C2312yes—
78C2×C4⋊C8D411yescharacteristic
79C2×#103—22no—
80C2×#103—21no—
81C2×#102—22no—
82C2×#102—21no—
83C2×C22⋊C8D412yes—
84C2×C4×C8C2311yescharacteristic
85#396—42no—
86#394—42no—
87#390—41no—
88#389—41no—
89#384—21no—
90#384—21no—
91#380—22no—
92#380—22no—
93#380—41no—
94#367—22no—
95#366—24no—
96#366—42no—
97C4⋊C16—22no—
98#358—24no—
99#358—22no—
100#357—24no—
101#357—22no—
102C22⋊C16—24no—
103C4×C16D414yescomplemented
104#340—22no—
105#340—22no—
106C22×Q8—21no—
107C22×D4—21no—
108C22×D4—42no—
109C22×D4D811yescharacteristic
110C23×C4—41no—
111C23×C4C2×D412yes—
112C2×Q16—21no—
113C2×Q16—21no—
114C2×Q16—21no—
115C2×Q16—21no—
116C2×Q16—21no—
117C2×Q16C2×D412yes—
118C2×D8—21no—
119C2×D8—21no—
120C2×D8—21no—
121C2×D8—21no—
122C2×D8—21no—
123C2×D8C2×D412yes—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8C2×D412yes—
131C22×C8C2×D411yescharacteristic
132C22×C8C2411yescharacteristic
133C22⋊Q8—44no—
134C22⋊Q8—82no—
135#121—44no—
136#121—82no—
137#121—84no—
138C4×Q8—42no—
139C4×Q8—42no—
140C4×D4—22no—
141C4×D4—22no—
142C4×D4—42no—
143C4×D4—42no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—42no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4D811yescharacteristic
155C2×C22⋊C4—42no—
156C2×C22⋊C4—42no—
157C2×C22⋊C4—41no—
158C2×C22⋊C4—42no—
159C2×C22⋊C4—42no—
160C2×C22⋊C4D812yes—
161C2×C42—41no—
162C2×C42C2×D411yescharacteristic
163SD32—28no—
164M32—48no—
165C2×C16—21no—
166C2×C16—21no—
167C2×C16—22no—
168C2×C16—22no—
169C2×C16—22no—
170C2×C16C2×D412yes—
171C2×C16C2×D412yes—
172C2×C16C2×D412yes—
173C2×C16#4112yes—
174#107—22no—
175#107—21no—
176#107—22no—
177#107—21no—
178#107—21no—
179#107—21no—
180#107—22no—
181#107—22no—
182#107—81no—
183#107—81no—
184C4⋊C8—22no—
185#103—44no—
186#103—42no—
187#102—44no—
188#102—42no—
189C22⋊C8—24no—
190C4×C8C2×D412yes—
191C4×C8#4112yes—
192#95—42no—
193#95—42no—
194C24—41no—
195C24C2×D812yes—
196C2×Q8—22no—
197C2×Q8—41no—
198C2×Q8—41no—
199C2×Q8—41no—
200C2×Q8—41no—
201C2×Q8—41no—
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—22no—
205C2×D4—22no—
206C2×D4—41no—
207C2×D4—41no—
208C2×D4—41no—
209C2×D4—41no—
210C2×D4—41no—
211C2×D4—84no—
212C2×D4—84no—
213C2×D4—162no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—82no—
226C22×C4—82no—
227C22×C4—82no—
228C22×C4C2×D812yes—
229C22×C4C2×D811yescharacteristic
230C22×C4C2×D811yescharacteristic
231C22×C4C22×D411yescharacteristic
232Q16—22no—
233Q16—22no—
234Q16—41no—
235Q16—41no—
236D8—22no—
237D8—22no—
238D8—41no—
239D8—41no—
240C2×C8—21no—
241C2×C8—21no—
242C2×C8—22no—
243C2×C8—22no—
244C2×C8—22no—
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—42no—
248C2×C8—42no—
249C2×C8#12112yes—
250C2×C8C22×D411yescharacteristic
251C2×C8C22×D411yescharacteristic
252C2×C8C2×#4111yescharacteristic
253C2×C8C2×#4111yescharacteristic
254C2×C8H(5,𝔽2)11yescharacteristic
255C2×C82−1+411yescharacteristic
256C4⋊C4—22no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C4⋊C4—84no—
266C4⋊C4—82no—
267C4⋊C4—82no—
268C4⋊C4—82no—
269C4⋊C4—161no—
270C4⋊C4—161no—
271C22⋊C4—24no—
272C22⋊C4—84no—
273C22⋊C4—84no—
274C22⋊C4—84no—
275C22⋊C4—82no—
276C22⋊C4—162no—
277C42—82no—
278C42#12112yes—
279C42#13512yes—
280C16—22no—
281C16—42no—
282C16#12114yescomplemented
283C23—22no—
284C23—22no—
285C23—22no—
286C23—22no—
287C23—22no—
288C23—22no—
289C23—22no—
290C23—42no—
291C23—81no—
292C23—81no—
293C23—81no—
294C23—81no—
295C23—81no—
296C23—81no—
297C23C22×D811yescharacteristic
298Q8—42no—
299Q8—42no—
300Q8—81no—
301Q8—81no—
302D4—44no—
303D4—42no—
304D4—42no—
305D4—81no—
306D4—81no—
307D4—168no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—81no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—164no—
333C2×C4#46612yes—
334C2×C4#46612yes—
335C2×C4#50912yes—
336C2×C4#51012yes—
337C2×C4C2×#12111yescharacteristic
338C2×C4D4211yescharacteristic
339C2×C4#54711yescharacteristic
340C2×C4C2×#13511yescharacteristic
341C2×C4#57611yescharacteristic
342C2×C4#57811yescharacteristic
343C8—22no—
344C8—42no—
345C8C2×#12111yescharacteristic
346C8#53711yescharacteristic
347C8#54711yescharacteristic
348C8#55511yescharacteristic
349C22—42no—
350C22—42no—
351C22—42no—
352C22—42no—
353C22—42no—
354C22—42no—
355C22—42no—
356C22—82no—
357C22—82no—
358C22—82no—
359C22—161no—
360C22—161no—
361C22—161no—
362C22—161no—
363C22C2×#46611yescharacteristic
364C22D4×D811yescharacteristic
365C22#333411yescharacteristic
366C22C2×#50911yescharacteristic
367C22C2×#51011yescharacteristic
368C22#341511yescharacteristic
369C22#341711yescharacteristic
370C4—22no—
371C4—42no—
372C4—82no—
373C4—162no—
374C4—162no—
375C4—161no—
376C4—161no—
377C4#222014yescomplemented
378C4#253711yescharacteristic
379C4#306111yescharacteristic
380C4#328311yescharacteristic
381C4#334111yescharacteristic
382C2—82no—
383C2—162no—
384C2#2822711yescharacteristic
385C2C2×#222011yescharacteristic
386C2#3361011yescharacteristic
387C2#3384011yescharacteristic
388C2#3384211yescharacteristic
389C2#3392011yescharacteristic
390C2#3392411yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radical#28227C2
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2220C2semidirect
2. C2×C4×SD32C2semidirect
3. C2×#2169C2semidirect
4. #28228C2semidirect
6. #13383C2semidirect
7. #13380C2semidirect
8. #12905C2semidirect
9. #12836C2semidirect
10. #12812C2semidirect
11. #12811C2semidirect
55. C2×SD32D4semidirect
103. C4×C16D4semidirect
282. C16#121semidirect
377. C4#2220semidirect

Maximal subgroups

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

subgroupindex
C2×#22202
C2×C4×SD322
C2×#21692
#282282
#282272
#133832
#133802
#129052
#128362
#128122
#128112