Finite Group Databasebeta

← #501704

Subgroups — #501704

Order 512.

674 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2224C211yesmaximal, characteristic, complemented
2C2×C4×D16C211yesmaximal, characteristic
3#28527C211yesmaximal, characteristic, complemented
4C4⋊C4×D8C211yesmaximal, characteristic, complemented
5#13505C212yesmaximal, complemented
6#13474C212yesmaximal, complemented
7#13282C211yesmaximal, characteristic, complemented
8#13159C211yesmaximal, characteristic, complemented
9#13012C212yesmaximal
10#12811C212yesmaximal, complemented
11#12811C211yesmaximal, characteristic, complemented
12C22×D16C2211yescharacteristic
13C2×#500C2211yescharacteristic
14C2×#474—21no—
15C2×C4×D8C2211yescharacteristic
16C2×C4×D8C2212yes—
17C2×C4×D8C2211yescharacteristic
18#2542—21no—
19D4×C4⋊C4—21no—
20#2224—24no—
21C4×D16—24no—
22C2×#366C2212yes—
23C2×#366C2211yescharacteristic
24C2×C4⋊C16C2211yescharacteristic
25C2×C4⋊C16C2212yes—
26C2×#357C2211yescharacteristic
27C2×#357C2212yes—
28C2×#357C2211yescharacteristic
29C2×#357C2212yes—
30C2×C4×C16C2211yescharacteristic
31#2093—22no—
32#1970C2212yes—
33#1942C2212yes—
34#1862—22no—
35#1845C2212yes—
36#1845C2211yescharacteristic
37#1802—21no—
38C8×C4⋊C4C2211yescharacteristic
39#1380C2212yes—
40#1380C2212yes—
41#1380C2212yes—
42#1374C2212yes—
43C22×D8C2311yescharacteristic
44C22×D8C2311yescharacteristic
45C2×C4⋊Q8—21no—
46C2×C22⋊Q8—41no—
47C2×C22⋊Q8—41no—
48C2×C4×D4—21no—
49C2×C4×D4—21no—
50C2×C4×D4—22no—
51C22×C4⋊C4—41no—
52C2×D16—22no—
53C2×D16—22no—
54C2×D16Q814yescomplemented
55C22×C16C2311yescharacteristic
56C22×C16C2312yes—
57C22×C16C2311yescharacteristic
58#500—22no—
59#500—22no—
60#474—44no—
61C4×D8—24no—
62C4×D8—22no—
63C4×D8—22no—
64C4×D8—22no—
65C4×D8—42no—
66C4×D8D414yes—
67C2×#107—21no—
68C2×#107—22no—
69C2×#107—21no—
70C2×#107C2311yescharacteristic
71C2×#107C2312yes—
72C2×#107C2312yes—
73C2×#107C2311yescharacteristic
74C2×C4⋊C8D411yescharacteristic
75C2×#102—21no—
76C2×#102—22no—
77C2×#102—21no—
78C2×#102—22no—
79C2×C4×C8C2312yes—
80C2×C4×C8C2311yescharacteristic
81#399—42no—
82#390—22no—
83#389—22no—
84#385—42no—
85#384—21no—
86#384—22no—
87#380—41no—
88C4×C4⋊C4D411yescharacteristic
89#366—22no—
90#366—24no—
91C4⋊C16—22no—
92C4⋊C16—42no—
93#357—22no—
94#357—24no—
95#357—22no—
96#357—42no—
97C4×C16—22no—
98#340—22no—
99#336D412yes—
100C22×Q8—41no—
101C22×D4—21no—
102C22×D4—21no—
103C23×C4—41no—
104C23×C4—41no—
105C23×C4—42no—
106C2×D8—21no—
107C2×D8—21no—
108C2×D8—21no—
109C2×D8—21no—
110C2×D8—21no—
111C2×D8—21no—
112C2×D8—21no—
113C2×D8—21no—
114C2×D8C2×D412yes—
115C2×D8C2×Q812yes—
116C2×D8C2×Q812yes—
117C2×D8#4112yes—
118C22×C8C2×D412yes—
119C22×C8C2×D411yescharacteristic
120C22×C8C2411yescharacteristic
121C4⋊Q8—42no—
122C4⋊Q8—42no—
123C22⋊Q8—84no—
124C22⋊Q8—84no—
125C4×D4—24no—
126C4×D4—44no—
127C4×D4—42no—
128C4×D4—42no—
129C4×D4—42no—
130C4×D4—82no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4—42no—
137C2×C4⋊C4—42no—
138C2×C4⋊C4—42no—
139C2×C4⋊C4—41no—
140C2×C4⋊C4—41no—
141C2×C4⋊C4—84no—
142C2×C4⋊C4D811yescharacteristic
143C2×C4⋊C4D811yescharacteristic
144C2×C22⋊C4—41no—
145C2×C22⋊C4—42no—
146C2×C22⋊C4—41no—
147C2×C22⋊C4—42no—
148C2×C22⋊C4—42no—
149C2×C42C2×D412yes—
150C2×C42C2×D411yescharacteristic
151D16—28no—
152C2×C16—21no—
153C2×C16—21no—
154C2×C16—21no—
155C2×C16—22no—
156C2×C16—22no—
157C2×C16—21no—
158C2×C16—21no—
159C2×C16—22no—
160C2×C16C2×Q812yes—
161#107—21no—
162#107—21no—
163#107—22no—
164#107—21no—
165#107—22no—
166#107—22no—
167#107—22no—
168#107—44no—
169#107—42no—
170#107—81no—
171#107C2×D412yes—
172C4⋊C8—22no—
173#102—24no—
174#102—44no—
175#102—44no—
176#102—42no—
177C4×C8—22no—
178C4×C8—22no—
179C4×C8C2×D412yes—
180C4×C8#4112yes—
181#95—42no—
182#95D812yes—
183C24—41no—
184C24—41no—
185C2×Q8—82no—
186C2×Q8—82no—
187C2×Q8—82no—
188C2×D4—22no—
189C2×D4—22no—
190C2×D4—22no—
191C2×D4—22no—
192C2×D4—41no—
193C2×D4—41no—
194C2×D4—41no—
195C2×D4—41no—
196C2×D4—41no—
197C2×D4—41no—
198C2×D4—41no—
199C2×D4—41no—
200C22×C4—41no—
201C22×C4—44no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—41no—
205C22×C4—84no—
206C22×C4—82no—
207C22×C4—82no—
208C22×C4—82no—
209C22×C4—84no—
210C22×C4—82no—
211C22×C4—82no—
212C22×C4—162no—
213C22×C4C2×D812yes—
214C22×C4C2×D811yescharacteristic
215C22×C4C2×D812yes—
216C22×C4C2×D811yescharacteristic
217C22×C4C22×D411yescharacteristic
218D8—22no—
219D8—22no—
220D8—22no—
221D8—22no—
222D8—22no—
223D8—41no—
224D8—41no—
225D8C22⋊Q814yescomplemented
226C2×C8—21no—
227C2×C8—21no—
228C2×C8—22no—
229C2×C8—22no—
230C2×C8—22no—
231C2×C8#12112yes—
232C2×C8C22×D411yescharacteristic
233C2×C8C22×Q811yescharacteristic
234C2×C8C2×#4111yescharacteristic
235C2×C8C2×#4111yescharacteristic
236C2×C8H(5,𝔽2)11yescharacteristic
237C2×C82−1+411yescharacteristic
238C4⋊C4—22no—
239C4⋊C4—22no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—41no—
246C4⋊C4—42no—
247C4⋊C4—84no—
248C4⋊C4—82no—
249C4⋊C4—84no—
250C4⋊C4—82no—
251C4⋊C4—84no—
252C4⋊C4—161no—
253C4⋊C4D1614yescomplemented
254C22⋊C4—44no—
255C22⋊C4—84no—
256C22⋊C4—84no—
257C22⋊C4—82no—
258C22⋊C4—84no—
259C42—22no—
260C42—22no—
261C42#12112yes—
262C42AGL(1,ℤ/8)12yes—
263C16—22no—
264C16—41no—
265C16—42no—
266C23—42no—
267C23—42no—
268C23—42no—
269C23—42no—
270C23—81no—
271C23—81no—
272C23—81no—
273C23—81no—
274C23—81no—
275C23—81no—
276C23—81no—
277C23—81no—
278C23—81no—
279C23—81no—
280C23C22×D811yescharacteristic
281Q8—164no—
282D4—24no—
283D4—42no—
284D4—42no—
285D4—42no—
286D4—42no—
287D4—42no—
288D4—81no—
289D4—81no—
290C2×C4—21no—
291C2×C4—22no—
292C2×C4—21no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—42no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4—88no—
301C2×C4—82no—
302C2×C4—81no—
303C2×C4—82no—
304C2×C4—82no—
305C2×C4—81no—
306C2×C4—82no—
307C2×C4—81no—
308C2×C4—82no—
309C2×C4—82no—
310C2×C4—168no—
311C2×C4—164no—
312C2×C4#45912yes—
313C2×C4#45912yes—
314C2×C4C2×D1614yes—
315C2×C4C2×D1612yes—
316C2×C4#50912yes—
317C2×C4C2×#12111yescharacteristic
318C2×C4#54611yescharacteristic
319C2×C4D4×Q811yescharacteristic
320C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
321C2×C4#57711yescharacteristic
322C2×C4#57811yescharacteristic
323C8—22no—
324C8—42no—
325C8C2×C22⋊Q811yescharacteristic
326C8#53711yescharacteristic
327C8#55211yescharacteristic
328C8#55411yescharacteristic
329C22—44no—
330C22—82no—
331C22—82no—
332C22—82no—
333C22—82no—
334C22—82no—
335C22—82no—
336C22—82no—
337C22—82no—
338C22—82no—
339C22—161no—
340C22—161no—
341C22—161no—
342C22—161no—
343C22C2×#45911yescharacteristic
344C22#329411yescharacteristic
345C22Q8×D811yescharacteristic
346C22C22×D1611yescharacteristic
347C22C2×#50911yescharacteristic
348C22#341611yescharacteristic
349C22#341711yescharacteristic
350C4—22no—
351C4—24no—
352C4—42no—
353C4—82no—
354C4—161no—
355C4—162no—
356C4—162no—
357C4#220614yes—
358C4#241711yescharacteristic
359C4#304611yescharacteristic
360C4#333011yescharacteristic
361C4#338411yescharacteristic
362C2—84no—
363C2—162no—
364C2#2481111yescharacteristic
365C2C2×#220611yescharacteristic
366C2#3359011yescharacteristic
367C2#3386111yescharacteristic
368C2#3389711yescharacteristic
369C2Q8×D1611yescharacteristic
370C2#3395111yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D16C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ C8×C4⋊C4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2224C2semidirect
3. #28527C2semidirect
4. C4⋊C4×D8C2semidirect
5. #13505C2semidirect
6. #13474C2semidirect
7. #13282C2semidirect
8. #13159C2semidirect
10. #12811C2semidirect
11. #12811C2semidirect
54. C2×D16Q8semidirect
225. D8C22⋊Q8semidirect
253. C4⋊C4D16semidirect

Maximal subgroups

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

subgroupindex
C2×#22242
C2×C4×D162
#285272
C4⋊C4×D82
#135052
#134742
#132822
#131592
#130122
#128112