Finite Group Databasebeta

← #497602

Subgroups — #497602

Order 512.

837 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#39617C211yesmaximal, characteristic
2Q8×#101C212yesmaximal, complemented
3#13303C214yesmaximal, complemented
4#13302C212yesmaximal, complemented
5#12803C212yesmaximal
6#12801C211yesmaximal, characteristic
7#12632C212yesmaximal
8C4×#355C211yesmaximal, characteristic
9C2×Q82C2211yescharacteristic
10C2×#554C411yescharacteristic
11Q8×M16—22no—
12#2868C411yescharacteristic
13#2677—24no—
14#2660C412yes—
15#2567C2211yescharacteristic
16#2565C2212yes—
17#2563C411yescharacteristic
18#2542C412yes—
19#2440—22no—
20#2423—22no—
21#2326C2212yes—
22C4×#124C2211yescharacteristic
23#2295C411yescharacteristic
24C2×#355C2212yes—
25C2×#355C2211yescharacteristic
26C2×#355C2211yescharacteristic
27#1997—24no—
28#1997—22no—
29#1801C2212yes—
30#1801C2214yes—
31C4×#101C2212yes—
32C4×#101C2214yes—
33#1394C2212yes—
34#1394C2214yes—
35#1394C2212yes—
36#1394C2214yes—
37Q82—42no—
38Q82—42no—
39#554—42no—
40#554—42no—
41C2×C4⋊Q8—22no—
42C2×C4⋊Q8C2×C411yescharacteristic
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×C4⋊Q8C2×C412yes—
45C2×C4⋊Q8D414yes—
46C2×C4⋊Q8D412yes—
47C2×#125C2×C412yes—
48C2×#124C2311yescharacteristic
49C2×C22⋊Q8—21no—
50C2×C22⋊Q8—24no—
51C2×C22⋊Q8—22no—
52C2×C4×Q8C2×C411yescharacteristic
53C2×C4×Q8D412yes—
54C2×C4×Q8C2312yes—
55C2×C4×Q8C2311yescharacteristic
56C2×C4×D4C2×C411yescharacteristic
57C2×C4×D4C2×C412yes—
58C22×C4⋊C4—21no—
59#446—42no—
60#446—44no—
61C8×Q8—42no—
62#423—22no—
63#423—24no—
64C2×#101C2314yes—
65C2×#101C2312yes—
66C2×#101C2312yes—
67C4×M16—22no—
68C4×M16—24no—
69#399—24no—
70#398—24no—
71#398—24no—
72#395—24no—
73#395—22no—
74#393—22no—
75#391—21no—
76#391C2311yescharacteristic
77#391C2312yes—
78#390C2×C412yes—
79#390C2×C411yescharacteristic
80#389—21no—
81#389—22no—
82#388—22no—
83#388—24no—
84#387—22no—
85#387—24no—
86#385—24no—
87#384—22no—
88#384C2×C412yes—
89#384C2×C412yes—
90#384C2×C412yes—
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#355—24no—
100#355—22no—
101#355Q814yescomplemented
102#343—24no—
103#343—22no—
104#343—24no—
105#343—22no—
106C22×Q8—21no—
107C22×Q8C22⋊C414yes—
108C22×Q8C22⋊C412yes—
109C22×Q8C22⋊C411yescharacteristic
110C22×Q8C2411yescharacteristic
111C22×D4C22×C411yescharacteristic
112C23×C4—21no—
113C23×C4—22no—
114C2×M16—22no—
115C2×M16—22no—
116C2×M16—24no—
117C4⋊Q8—44no—
118C4⋊Q8—41no—
119C4⋊Q8—41no—
120C4⋊Q8—42no—
121C4⋊Q8—44no—
122C4⋊Q8—42no—
123C4⋊Q8—44no—
124C4⋊Q8—41no—
125C4⋊Q8—41no—
126C4⋊Q8—84no—
127#125—44no—
128#124—22no—
129#124C2×Q812yes—
130#124#4112yes—
131C22⋊Q8—48no—
132C22⋊Q8—42no—
133C22⋊Q8—42no—
134C22⋊Q8—88no—
135C4×Q8—22no—
136C4×Q8—22no—
137C4×Q8—22no—
138C4×Q8—22no—
139C4×Q8—24no—
140C4×Q8—41no—
141C4×Q8—44no—
142C4×Q8—41no—
143C4×D4—41no—
144C4×D4—41no—
145C4×D4—44no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—42no—
155C2×C4⋊C4—42no—
156C2×C4⋊C4C22⋊C411yescharacteristic
157C2×C4⋊C4C22⋊C412yes—
158C2×C4⋊C4C22⋊C412yes—
159C2×C4⋊C4C22⋊C414yes—
160C2×C4⋊C4C22×C411yescharacteristic
161C2×C4⋊C4C22×C412yes—
162C2×C4⋊C4C22×C412yes—
163C2×C4⋊C4C22×C411yescharacteristic
164C2×C4⋊C4C2×D414yes—
165C2×C4⋊C4C2×D412yes—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—21no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—24no—
171C2×C42—21no—
172C2×C42—21no—
173C2×C42C22×C412yes—
174C2×C42C22×C411yescharacteristic
175C2×C42C22×C411yescharacteristic
176C2×C42C22×C411yescharacteristic
177C2×C42C22×C412yes—
178C2×C42C2×D412yes—
179C2×C42C2×D414yes—
180C4⋊C8—44no—
181C4⋊C8—42no—
182C4⋊C8—44no—
183C4⋊C8—42no—
184#101—22no—
185#101—24no—
186#101—22no—
187#101—24no—
188#101—22no—
189#101C2×Q814yes—
190#97—44no—
191#97—42no—
192C4×C8—44no—
193C4×C8—42no—
194#95—22no—
195#95—22no—
196#95—22no—
197#95—22no—
198#95—21no—
199#95—24no—
200#95—22no—
201#95—24no—
202C24—21no—
203C2×Q8—41no—
204C2×Q8—41no—
205C2×Q8—42no—
206C2×Q8—44no—
207C2×Q8—42no—
208C2×Q8—42no—
209C2×Q8—84no—
210C2×Q8—82no—
211C2×Q8—82no—
212C2×Q8#9912yes—
213C2×Q8#9912yes—
214C2×Q8C22×Q811yescharacteristic
215C2×Q8C2×#4111yescharacteristic
216C2×Q82−1+412yes—
217C2×D4—21no—
218C2×D4—41no—
219C2×D4—41no—
220C2×D4C4×Q812yes—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—21no—
225C22×C4—22no—
226C22×C4—44no—
227C22×C4—44no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—84no—
231C22×C4C2×C22⋊C411yescharacteristic
232C22×C4C2×C22⋊C411yescharacteristic
233C22×C4C2×C22⋊C412yes—
234C22×C4C2×C22⋊C412yes—
235C22×C4C2×C22⋊C414yes—
236C22×C4C2×C22⋊C412yes—
237C22×C4C23×C411yescharacteristic
238C22×C4C22×D412yes—
239M16—24no—
240M16—42no—
241M16—42no—
242M16—44no—
243M16—44no—
244M16—42no—
245C2×C8—42no—
246C2×C8—44no—
247C2×C8—42no—
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—24no—
251C4⋊C4—24no—
252C4⋊C4—24no—
253C4⋊C4—22no—
254C4⋊C4—22no—
255C4⋊C4—42no—
256C4⋊C4—41no—
257C4⋊C4—42no—
258C4⋊C4—44no—
259C4⋊C4—41no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—44no—
263C4⋊C4—44no—
264C4⋊C4—48no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—44no—
268C4⋊C4—84no—
269C4⋊C4—84no—
270C22⋊C4—24no—
271C22⋊C4—24no—
272C22⋊C4—42no—
273C22⋊C4—44no—
274C22⋊C4—44no—
275C22⋊C4—84no—
276C42—21no—
277C42—24no—
278C42—22no—
279C42—24no—
280C42—22no—
281C42—24no—
282C42—21no—
283C42—21no—
284C42—41no—
285C42—42no—
286C42—42no—
287C42—42no—
288C42C4×Q812yes—
289C23—22no—
290C23—42no—
291C23—42no—
292C23—41no—
293C23—41no—
294C23C22×C22⋊C411yescharacteristic
295Q8—24no—
296Q8—42no—
297Q8—41no—
298Q8—41no—
299Q8—42no—
300Q8—164no—
301Q8—162no—
302Q8—162no—
303D4—42no—
304D4—42no—
305C2×C4—21no—
306C2×C4—22no—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—21no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—24no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—42no—
323C2×C4—44no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—44no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—41no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—82no—
333C2×C4—82no—
334C2×C4—164no—
335C2×C4C2×#9914yes—
336C2×C4C2×#9911yescharacteristic
337C2×C4C2×#9911yescharacteristic
338C2×C4C2×C4×Q811yescharacteristic
339C2×C4#51911yescharacteristic
340C2×C4#54812yes—
341C2×C4D4×Q812yes—
342C8—44no—
343C8—84no—
344C8—82no—
345C22—21no—
346C22—21no—
347C22—44no—
348C22—42no—
349C22—82no—
350C22—81no—
351C22—81no—
352C22Q8×C22⋊C411yescharacteristic
353C22C22×#9911yescharacteristic
354C22#288411yescharacteristic
355C4—21no—
356C4—24no—
357C4—22no—
358C4—22no—
359C4—21no—
360C4—21no—
361C4—42no—
362C4—42no—
363C4—42no—
364C4—42no—
365C4—44no—
366C4—41no—
367C4—82no—
368C4—81no—
369C4—82no—
370C4#213312yes—
371C2—21no—
372C2—21no—
373C2—82no—
374C2Q8×#9911yescharacteristic
375C2C2×#213311yescharacteristic
376C2#3337311yescharacteristic
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
3. #13303C2semidirect
4. #13302C2semidirect
101. #355Q8semidirect

Maximal subgroups

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

subgroupindex
#396172
Q8×#1012
#133032
#133022
#128032
#128012
#126322
C4×#3552