Finite Group Databasebeta

← #497658

Subgroups — #497658

Order 512.

809 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#39172C211yesmaximal, characteristic
2Q8×#101C212yesmaximal, complemented
3#13228C214yesmaximal
4#13227C212yesmaximal, complemented
5#12804C212yesmaximal, complemented
6#12801C211yesmaximal, characteristic
7#12631C212yesmaximal, complemented
8C4×#355C211yesmaximal, characteristic
9C2×Q82C2211yescharacteristic
10C2×#556C411yescharacteristic
11Q8×M16—22no—
12#2862C411yescharacteristic
13#2716—24no—
14#2640C412yes—
15#2568C412yes—
16#2563C411yescharacteristic
17#2546C2211yescharacteristic
18#2545C2212yes—
19#2439—22no—
20#2421—22no—
21#2327C2212yes—
22C4×#124C2211yescharacteristic
23#2295C411yescharacteristic
24C2×#356C2214yes—
25C2×#356C2214yes—
26C2×#355C2212yes—
27C2×#355C2212yes—
28#1997—22no—
29#1997—24no—
30#1801C2212yes—
31#1801C2214yes—
32C4×#101C2212yes—
33C4×#101C2214yes—
34#1394C2212yes—
35#1394C2211yescharacteristic
36#1394C2211yescharacteristic
37Q82—42no—
38Q82—42no—
39#556—42no—
40#556—42no—
41C2×C4⋊Q8C2×C411yescharacteristic
42C2×C4⋊Q8C2×C411yescharacteristic
43C2×C4⋊Q8C2×C412yes—
44C2×C4⋊Q8D412yes—
45C2×C4⋊Q8D414yes—
46C2×C4⋊Q8C2312yes—
47C2×#126—22no—
48C2×#124C2×C412yes—
49C2×#124C2311yescharacteristic
50C2×C22⋊Q8—21no—
51C2×C4×Q8C2×C412yes—
52C2×C4×Q8C2×C411yescharacteristic
53C2×C4×Q8D412yes—
54C2×C4×Q8C2312yes—
55C2×C4×Q8C2311yescharacteristic
56C2×C4×D4C2×C411yescharacteristic
57C2×#117—21no—
58#446—42no—
59#446—44no—
60C8×Q8—42no—
61#423—22no—
62#423—24no—
63C2×#101C2312yes—
64C2×#101C2314yes—
65C2×#101C2312yes—
66C4×M16—22no—
67C4×M16—24no—
68#400—24no—
69#398—22no—
70#397—24no—
71#396—24no—
72#393—22no—
73#393—24no—
74#391—24no—
75#391—22no—
76#391—21no—
77#391C2×C412yes—
78#391C2×C412yes—
79#391C2×C412yes—
80#391C2311yescharacteristic
81#390C2×C412yes—
82#390C2×C411yescharacteristic
83#389—21no—
84#389—22no—
85#388—22no—
86#388—24no—
87#387—22no—
88#387—24no—
89#387—24no—
90#382—22no—
91#382C2×C412yes—
92#382C2×C411yescharacteristic
93#380—22no—
94C4×C4⋊C4C2×C411yescharacteristic
95C4×C4⋊C4C2×C412yes—
96C4×C22⋊C4—22no—
97#376C2×C412yes—
98C43C2×C411yescharacteristic
99#356—28no—
100#356—44no—
101#355—24no—
102#355—42no—
103#343—24no—
104#343—22no—
105#343—24no—
106#343—22no—
107C22×Q8C22⋊C412yes—
108C22×Q8C22⋊C414yes—
109C22×Q8C22⋊C411yescharacteristic
110C22×Q8C22×C412yes—
111C22×Q8C2411yescharacteristic
112C22×D4C22×C411yescharacteristic
113C23×C4—21no—
114C2×M16—22no—
115C2×M16—22no—
116C2×M16—24no—
117C4⋊Q8—24no—
118C4⋊Q8—22no—
119C4⋊Q8—22no—
120C4⋊Q8—41no—
121C4⋊Q8—41no—
122C4⋊Q8—44no—
123C4⋊Q8—42no—
124C4⋊Q8—44no—
125C4⋊Q8—42no—
126C4⋊Q8—44no—
127C4⋊Q8—41no—
128C4⋊Q8—41no—
129#126—48no—
130#124—22no—
131#124—21no—
132#124—21no—
133#124—44no—
134C22⋊Q8—42no—
135C22⋊Q8—42no—
136C4×Q8—22no—
137C4×Q8—22no—
138C4×Q8—22no—
139C4×Q8—22no—
140C4×Q8—24no—
141C4×Q8—41no—
142C4×Q8—44no—
143C4×Q8—41no—
144C4×Q8—44no—
145C4×D4—41no—
146C4×D4—41no—
147#117—42no—
148#117—42no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4C22⋊C411yescharacteristic
156C2×C4⋊C4C22⋊C412yes—
157C2×C4⋊C4C22⋊C412yes—
158C2×C4⋊C4C22⋊C414yes—
159C2×C4⋊C4C22×C411yescharacteristic
160C2×C4⋊C4C22×C412yes—
161C2×C4⋊C4C22×C411yescharacteristic
162C2×C4⋊C4C2×D414yes—
163C2×C4⋊C4C2×D412yes—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—21no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—24no—
170C2×C42—22no—
171C2×C42—21no—
172C2×C42—21no—
173C2×C42C22×C411yescharacteristic
174C2×C42C22×C412yes—
175C2×C42C22×C411yescharacteristic
176C2×C42C22×C412yes—
177C2×C42C22×C411yescharacteristic
178C2×C42C2×D412yes—
179C2×C42C2×D414yes—
180C4⋊C8—44no—
181C4⋊C8—42no—
182C4⋊C8—44no—
183C4⋊C8—42no—
184#101—22no—
185#101—22no—
186#101—24no—
187#101—24no—
188#101—22no—
189#101#4114yes—
190#97—44no—
191#97—42no—
192C4×C8—44no—
193C4×C8—42no—
194#95—22no—
195#95—22no—
196#95—22no—
197#95—21no—
198#95—22no—
199#95—24no—
200#95—24no—
201#95—24no—
202#95—24no—
203#95—22no—
204C24—21no—
205C2×Q8—22no—
206C2×Q8—22no—
207C2×Q8—41no—
208C2×Q8—41no—
209C2×Q8—44no—
210C2×Q8—42no—
211C2×Q8—42no—
212C2×Q8—42no—
213C2×Q8—42no—
214C2×Q8—82no—
215C2×Q8—84no—
216C2×Q8#9912yes—
217C2×Q8#9912yes—
218C2×Q8C2×#4112yes—
219C2×Q8H(5,𝔽2)11yescharacteristic
220C2×Q82−1+411yescharacteristic
221C2×D4—21no—
222C2×D4—21no—
223C2×D4—41no—
224C2×D4—41no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—44no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4C2×C22⋊C411yescharacteristic
234C22×C4C2×C22⋊C412yes—
235C22×C4C2×C22⋊C414yes—
236C22×C4C2×C22⋊C412yes—
237C22×C4C2×C22⋊C411yescharacteristic
238C22×C4C2×C22⋊C412yes—
239C22×C4C23×C411yescharacteristic
240C22×C4C22×D412yes—
241M16—24no—
242M16—42no—
243M16—42no—
244M16—44no—
245M16—42no—
246M16—44no—
247C2×C8—42no—
248C2×C8—42no—
249C2×C8—44no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—24no—
253C4⋊C4—24no—
254C4⋊C4—24no—
255C4⋊C4—22no—
256C4⋊C4—22no—
257C4⋊C4—42no—
258C4⋊C4—41no—
259C4⋊C4—44no—
260C4⋊C4—42no—
261C4⋊C4—41no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4—44no—
265C4⋊C4—44no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—44no—
269C4⋊C4—82no—
270C22⋊C4—44no—
271C22⋊C4—42no—
272C22⋊C4—42no—
273C22⋊C4—42no—
274C22⋊C4—44no—
275C22⋊C4—82no—
276C22⋊C4—84no—
277C42—24no—
278C42—22no—
279C42—24no—
280C42—22no—
281C42—21no—
282C42—21no—
283C42—24no—
284C42—21no—
285C42—22no—
286C42—21no—
287C42—22no—
288C42—41no—
289C42—42no—
290C42—44no—
291C42—81no—
292C23—42no—
293C23—42no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23C22×C22⋊C411yescharacteristic
298Q8—24no—
299Q8—42no—
300Q8—42no—
301Q8—41no—
302Q8—41no—
303Q8—82no—
304Q8—82no—
305Q8—164no—
306Q8—162no—
307D4—81no—
308D4—81no—
309C2×C4—21no—
310C2×C4—22no—
311C2×C4—21no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—21no—
316C2×C4—24no—
317C2×C4—22no—
318C2×C4—42no—
319C2×C4—41no—
320C2×C4—42no—
321C2×C4—44no—
322C2×C4—42no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—41no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—41no—
334C2×C4—42no—
335C2×C4—41no—
336C2×C4—42no—
337C2×C4—82no—
338C2×C4—82no—
339C2×C4C2×#9914yes—
340C2×C4C2×#9911yescharacteristic
341C2×C4C2×#9911yescharacteristic
342C2×C4C4×#4111yescharacteristic
343C2×C4#52011yescharacteristic
344C2×C4#54812yes—
345C2×C4#55012yes—
346C8—44no—
347C8—82no—
348C8—84no—
349C22—21no—
350C22—21no—
351C22—82no—
352C22—82no—
353C22—81no—
354C22—81no—
355C22—81no—
356C22#234111yescharacteristic
357C22C22×#9911yescharacteristic
358C22#288411yescharacteristic
359C4—22no—
360C4—22no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—24no—
365C4—42no—
366C4—41no—
367C4—44no—
368C4—42no—
369C4—81no—
370C4—82no—
371C4—81no—
372C4—82no—
373C4—81no—
374C4#213312yes—
375C2—21no—
376C2—21no—
377C2—161no—
378C2#2253411yescharacteristic
379C2C2×#213311yescharacteristic
380C2#3337111yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×Q8C24
prime radicalC22×D4C22×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×Q8 ◁ C2×Q82 ◁ G
LECSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. Q8×#101C2semidirect
4. #13227C2semidirect
5. #12804C2semidirect
7. #12631C2semidirect

Maximal subgroups

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

subgroupindex
#391722
Q8×#1012
#132282
#132272
#128042
#128012
#126312
C4×#3552