Finite Group Databasebeta

← #506244

Subgroups — #506244

Order 512.

686 conjugacy classes of subgroups.

The subgroup lattice has 389 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×SD32C211yesmaximal, characteristic, complemented
2C2×D4×C16C211yesmaximal, characteristic, complemented
3#28228C211yesmaximal, characteristic, complemented
4#28227C211yesmaximal, characteristic
5#13494C212yesmaximal, complemented
6#13305C212yesmaximal, complemented
7#13060C212yesmaximal, complemented
8#13059C212yesmaximal, complemented
9#12888C211yesmaximal, characteristic, complemented
10#12812C211yesmaximal, characteristic, complemented
11#12811C211yesmaximal, characteristic, complemented
12C22×SD32C2211yescharacteristic
13C23×C16C2212yes—
14C2×#467C2212yes—
15C2×#466C2212yes—
16C2×C4×Q16C2211yescharacteristic
17C2×C4×D8C2211yescharacteristic
18C2×C8×D4C2211yescharacteristic
19#2538—21no—
20#2537—21no—
21C4×SD32—24no—
22D4×C16—24no—
23C2×#367C2211yescharacteristic
24C2×C4⋊C16C2211yescharacteristic
25C2×#358C2211yescharacteristic
26C2×#358C2212yes—
27C2×#357C2211yescharacteristic
28C2×#357C2212yes—
29C2×C22⋊C16C2212yes—
30C2×C4×C16C2211yescharacteristic
31#2036—22no—
32#2033—22no—
33#1845C2211yescharacteristic
34#1845C2211yescharacteristic
35#1808C2212yes—
36#1808C2212yes—
37#1803—21no—
38#1802—21no—
39#1380C2212yes—
40#1380C2212yes—
41#1380C2212yes—
42#1380C2212yes—
43C22×Q16C2311yescharacteristic
44C22×D8C2311yescharacteristic
45C23×C8C2312yes—
46C2×C22⋊Q8—22no—
47C2×#121—22no—
48C2×#121—41no—
49C2×C4×Q8—21no—
50C2×C4×D4—21no—
51C2×C4×D4D411yescharacteristic
52C2×SD32—22no—
53C2×SD32—22no—
54C2×SD32—22no—
55C22×C16—24no—
56C22×C16—42no—
57C22×C16—42no—
58C22×C16C2312yes—
59C22×C16C2311yescharacteristic
60C22×C16C2311yescharacteristic
61#467—24no—
62#467—42no—
63#466—24no—
64#466—42no—
65C4×Q16—22no—
66C4×Q16—22no—
67C4×D8—22no—
68C4×D8—22no—
69C8×D4—22no—
70C8×D4—22no—
71C2×#107—21no—
72C2×#107—21no—
73C2×#107C2311yescharacteristic
74C2×#107C2311yescharacteristic
75C2×#107C2312yes—
76C2×#107C2312yes—
77C2×C4⋊C8D411yescharacteristic
78C2×#103—22no—
79C2×#103—21no—
80C2×#102—22no—
81C2×#102—21no—
82C2×C22⋊C8D412yes—
83C2×C4×C8C2311yescharacteristic
84#396—42no—
85#394—42no—
86#390—41no—
87#389—41no—
88#384—21no—
89#384—21no—
90#380—22no—
91#380—22no—
92#380—41no—
93#367—22no—
94C4⋊C16—22no—
95#358—22no—
96#358—42no—
97#357—22no—
98#357—42no—
99C22⋊C16—24no—
100C4×C16D414yescomplemented
101#340—22no—
102#340—22no—
103C22×Q8—21no—
104C22×D4—21no—
105C22×D4—42no—
106C22×D4D811yescharacteristic
107C23×C4—41no—
108C23×C4C2×D412yes—
109C2×Q16—21no—
110C2×Q16—21no—
111C2×Q16—21no—
112C2×Q16—21no—
113C2×Q16—21no—
114C2×Q16#4112yes—
115C2×D8—21no—
116C2×D8—21no—
117C2×D8—21no—
118C2×D8—21no—
119C2×D8—21no—
120C2×D8#4112yes—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8C2×D412yes—
128C22×C8C2×D411yescharacteristic
129C22×C8C2411yescharacteristic
130C22⋊Q8—44no—
131C22⋊Q8—82no—
132#121—44no—
133#121—82no—
134#121—84no—
135C4×Q8—42no—
136C4×Q8—42no—
137C4×D4—22no—
138C4×D4—22no—
139C4×D4—42no—
140C4×D4—42no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4—41no—
147C2×C4⋊C4—42no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4D811yescharacteristic
152C2×C22⋊C4—42no—
153C2×C22⋊C4—42no—
154C2×C22⋊C4—41no—
155C2×C22⋊C4—42no—
156C2×C22⋊C4—42no—
157C2×C22⋊C4D812yes—
158C2×C42—41no—
159C2×C42C2×D411yescharacteristic
160SD32—28no—
161C2×C16—21no—
162C2×C16—21no—
163C2×C16—21no—
164C2×C16—21no—
165C2×C16—22no—
166C2×C16—22no—
167C2×C16—22no—
168C2×C16—48no—
169C2×C16C2×D412yes—
170C2×C16C2×D412yes—
171#107—21no—
172#107—21no—
173#107—21no—
174#107—21no—
175#107—22no—
176#107—22no—
177#107—22no—
178#107—22no—
179#107—81no—
180#107—81no—
181C4⋊C8—22no—
182#103—44no—
183#103—42no—
184#102—44no—
185#102—42no—
186C22⋊C8—24no—
187C4×C8C2×D412yes—
188C4×C8#4112yes—
189#95—42no—
190#95—42no—
191C24—41no—
192C24C2×D812yes—
193C2×Q8—22no—
194C2×Q8—41no—
195C2×Q8—41no—
196C2×Q8—41no—
197C2×Q8—41no—
198C2×Q8—41no—
199C2×D4—22no—
200C2×D4—22no—
201C2×D4—22no—
202C2×D4—41no—
203C2×D4—41no—
204C2×D4—41no—
205C2×D4—41no—
206C2×D4—41no—
207C2×D4—84no—
208C2×D4—84no—
209C2×D4—162no—
210C2×D4SD3214yescomplemented
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—82no—
223C22×C4—82no—
224C22×C4—82no—
225C22×C4C2×D812yes—
226C22×C4C2×D811yescharacteristic
227C22×C4C2×D811yescharacteristic
228C22×C4C22×D411yescharacteristic
229Q16—22no—
230Q16—22no—
231Q16—41no—
232Q16—41no—
233D8—22no—
234D8—22no—
235D8—41no—
236D8—41no—
237C2×C8—21no—
238C2×C8—21no—
239C2×C8—22no—
240C2×C8—22no—
241C2×C8—22no—
242C2×C8—42no—
243C2×C8—42no—
244C2×C8—42no—
245C2×C8—42no—
246C2×C8#12112yes—
247C2×C8C22×D411yescharacteristic
248C2×C8C2×#4111yescharacteristic
249C2×C8C2×#4111yescharacteristic
250C2×C8C2×#4111yescharacteristic
251C2×C8H(5,𝔽2)11yescharacteristic
252C2×C8H(5,𝔽2)11yescharacteristic
253C4⋊C4—22no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—42no—
257C4⋊C4—42no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—84no—
263C4⋊C4—82no—
264C4⋊C4—82no—
265C4⋊C4—82no—
266C4⋊C4—161no—
267C4⋊C4—161no—
268C22⋊C4—24no—
269C22⋊C4—84no—
270C22⋊C4—84no—
271C22⋊C4—84no—
272C22⋊C4—82no—
273C22⋊C4—162no—
274C42—82no—
275C42#12112yes—
276C42#13512yes—
277C16—22no—
278C16—22no—
279C16—42no—
280C23—22no—
281C23—22no—
282C23—22no—
283C23—22no—
284C23—22no—
285C23—22no—
286C23—42no—
287C23—81no—
288C23—81no—
289C23—81no—
290C23—81no—
291C23—81no—
292C23—81no—
293C23C2×SD3214yes—
294C23C22×D811yescharacteristic
295Q8—42no—
296Q8—42no—
297Q8—81no—
298Q8—81no—
299D4—28no—
300D4—42no—
301D4—42no—
302D4—81no—
303D4—81no—
304D4—168no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—82no—
319C2×C4—82no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4—81no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—164no—
330C2×C4#46612yes—
331C2×C4#46612yes—
332C2×C4C2×SD3212yes—
333C2×C4#50812yes—
334C2×C4C2×#12111yescharacteristic
335C2×C4#54611yescharacteristic
336C2×C4#54611yescharacteristic
337C2×C4C2×#13511yescharacteristic
338C2×C4#57711yescharacteristic
339C2×C4#57711yescharacteristic
340C8—22no—
341C8—42no—
342C8#52511yescharacteristic
343C8#53511yescharacteristic
344C8#55311yescharacteristic
345C8#55311yescharacteristic
346C22—24no—
347C22—24no—
348C22—24no—
349C22—42no—
350C22—42no—
351C22—42no—
352C22—42no—
353C22—82no—
354C22—82no—
355C22—82no—
356C22—161no—
357C22—161no—
358C22—161no—
359C22—161no—
360C22C2×#46611yescharacteristic
361C22#329411yescharacteristic
362C22#329411yescharacteristic
363C22C22×SD3211yescharacteristic
364C22C2×#50811yescharacteristic
365C22#341511yescharacteristic
366C22#341711yescharacteristic
367C4—22no—
368C4—42no—
369C4—82no—
370C4—161no—
371C4—162no—
372C4—161no—
373C4—162no—
374C4#221714yes—
375C4#253511yescharacteristic
376C4#306011yescharacteristic
377C4#330311yescharacteristic
378C4#330311yescharacteristic
379C2—44no—
380C2—162no—
381C2#2814011yescharacteristic
382C2C2×#221711yescharacteristic
383C2#3361011yescharacteristic
384C2#3386211yescharacteristic
385C2#3386511yescharacteristic
386C2#3386911yescharacteristic
387C2#3387211yescharacteristic
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×C4×SD32C2semidirect
2. C2×D4×C16C2semidirect
3. #28228C2semidirect
5. #13494C2semidirect
6. #13305C2semidirect
7. #13060C2semidirect
8. #13059C2semidirect
9. #12888C2semidirect
10. #12812C2semidirect
11. #12811C2semidirect
100. C4×C16D4semidirect
210. C2×D4SD32semidirect

Maximal subgroups

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

subgroupindex
C2×C4×SD322
C2×D4×C162
#282282
#282272
#134942
#133052
#130602
#130592
#128882
#128122
#128112