Finite Group Databasebeta

← #190285

Subgroups — #190285

Order 512.

999 conjugacy classes of subgroups.

The subgroup lattice has 393 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34182C211yesmaximal, characteristic
2#20727C211yesmaximal, characteristic
3#17365C211yesmaximal, characteristic
4#8767C214yesmaximal, complemented
5#8766C214yesmaximal, complemented
6#8094C212yesmaximal
7#7961C212yesmaximal
8C2×#519—21no—
9C22×C4×Q8C2211yescharacteristic
10#2920C2212yes—
11#2920C2212yes—
12#2351C414yes—
13#2351C2212yes—
14#2351C2212yes—
15Q8×C4⋊C4—22no—
16Q8×C4⋊C4—21no—
17Q8×C4⋊C4—21no—
18#2316—21no—
19#2316—22no—
20#2285C412yes—
21#2285C412yes—
22#2285C2211yescharacteristic
23#2285C2211yescharacteristic
24#1821—21no—
25#1801C2212yes—
26#1801C2212yes—
27#1784—21no—
28C4×#101—21no—
29C4×#101—21no—
30#1467—22no—
31#1463—22no—
32#1461C2212yes—
33#1461C2212yes—
34#1303C2218yes—
35#1286C2218yes—
36C23×Q8D411yescharacteristic
37#519—48no—
38C2×C4×Q8—24no—
39C2×C4×Q8—22no—
40C2×C4×Q8—22no—
41C2×C4×Q8—22no—
42C2×C4×Q8—22no—
43C2×C4×Q8—22no—
44C2×C4×Q8—22no—
45C2×C4×Q8C2×C412yes—
46C2×C4×Q8D411yescharacteristic
47C2×C4×Q8D411yescharacteristic
48C2×C4×Q8D412yes—
49C2×C4×Q8C2311yescharacteristic
50C2×C4×Q8C2311yescharacteristic
51C2×#117—22no—
52C2×#117—22no—
53C2×#117—21no—
54C2×#117—21no—
55C22×C4⋊C4D411yescharacteristic
56C22×C4⋊C4D412yes—
57C22×C42C2×C412yes—
58C22×C42C2311yescharacteristic
59#432—22no—
60#432—22no—
61#432—22no—
62#432—22no—
63#431—22no—
64#431—22no—
65#431—22no—
66#431C2314yes—
67#423—22no—
68#423C2314yes—
69C2×#101—22no—
70C2×#101—22no—
71#407—22no—
72#407—22no—
73C4×M16—22no—
74C4×M16—22no—
75#389—24no—
76#389—24no—
77#389—24no—
78#389—22no—
79#389—22no—
80#389—24no—
81#389—24no—
82#384—22no—
83#384—22no—
84#384—21no—
85#384—21no—
86#382C2×C412yes—
87#382C2×C414yes—
88#382C2×C414yes—
89#382C2312yes—
90#380—22no—
91#380—22no—
92#380—21no—
93#380—21no—
94#380C2×C412yes—
95#380C2×C411yescharacteristic
96#380C2×C411yescharacteristic
97C4×C4⋊C4—22no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4—21no—
100C4×C4⋊C4—21no—
101C4×C4⋊C4C2×C412yes—
102C4×C4⋊C4C2×C414yes—
103C4×C4⋊C4C2×C414yes—
104C4×C4⋊C4C2312yes—
105#343—22no—
106#343—22no—
107#324—24no—
108#324—24no—
109C22×Q8—24no—
110C22×Q8—24no—
111C22×Q8—22no—
112C22×Q8—22no—
113C22×Q8C22⋊C412yes—
114C22×Q8C2×D411yescharacteristic
115C22×Q8C2×D411yescharacteristic
116C23×C4C22⋊C412yes—
117C23×C4C22⋊C412yes—
118C23×C4C22×C411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D411yescharacteristic
121C2×M16—24no—
122C2×M16—42no—
123C4×Q8—24no—
124C4×Q8—24no—
125C4×Q8—22no—
126C4×Q8—22no—
127C4×Q8—24no—
128C4×Q8—22no—
129C4×Q8—22no—
130C4×Q8—24no—
131C4×Q8—22no—
132C4×Q8—22no—
133C4×Q8—22no—
134C4×Q8—22no—
135C4×Q8—48no—
136C4×Q8—416no—
137C4×Q8—48no—
138#117—44no—
139#117—48no—
140#117—44no—
141#117—48no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4C22⋊C412yes—
158C2×C4⋊C4C22⋊C412yes—
159C2×C4⋊C4C22⋊C412yes—
160C2×C4⋊C4C2×D412yes—
161C2×C4⋊C4C2×D412yes—
162C2×C4⋊C4C2×D411yescharacteristic
163C2×C4⋊C4C2×D411yescharacteristic
164C2×C22⋊C4—21no—
165C2×C22⋊C4—21no—
166C2×C22⋊C4—22no—
167C2×C42—22no—
168C2×C42—22no—
169C2×C42—22no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—24no—
173C2×C42—24no—
174C2×C42—22no—
175C2×C42C22⋊C412yes—
176C2×C42C22⋊C412yes—
177C2×C42C22×C412yes—
178C2×C42C22×C412yes—
179C2×C42C22×C411yescharacteristic
180C2×C42C2×D411yescharacteristic
181C2×C42C2×D411yescharacteristic
182C2×C42#4112yes—
183C2×C42#4112yes—
184C2×C42#4112yes—
185C2×C42#4112yes—
186C2×C42C2411yescharacteristic
187C4⋊C8—24no—
188C4⋊C8—24no—
189C4⋊C8—24no—
190C4⋊C8—42no—
191#101—44no—
192#101—44no—
193C22⋊C8—24no—
194C22⋊C8—42no—
195C22⋊C8—42no—
196C22⋊C8—42no—
197#97—42no—
198#97—42no—
199C4×C8—24no—
200C4×C8—42no—
201#95—24no—
202#95—24no—
203#95—22no—
204#95—22no—
205#95C22×C414yes—
206#95C22×C412yes—
207#95C22×C412yes—
208C24C2×C22⋊C411yescharacteristic
209C2×Q8—216no—
210C2×Q8—216no—
211C2×Q8—22no—
212C2×Q8—22no—
213C2×Q8—22no—
214C2×Q8—22no—
215C2×Q8—24no—
216C2×Q8—24no—
217C2×Q8—24no—
218C2×Q8AGL(1,ℤ/8)14yes—
219C2×Q8#13714yes—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—24no—
225C22×C4—24no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—24no—
231C22×C4—42no—
232C22×C4—81no—
233C22×C4—82no—
234C22×C4C2×C22⋊C411yescharacteristic
235C22×C4C2×C22⋊C411yescharacteristic
236C22×C4C2×C22⋊C412yes—
237C22×C4C2×C22⋊C412yes—
238C22×C4C2×C22⋊C412yes—
239C22×C4C2×C22⋊C412yes—
240C22×C4C2×C22⋊C411yescharacteristic
241C22×C4#11711yescharacteristic
242C22×C4#11711yescharacteristic
243C22×C4#11711yescharacteristic
244C22×C4#11711yescharacteristic
245C22×C4#12312yes—
246C22×C4#12312yes—
247C22×C4#12312yes—
248C22×C4#12312yes—
249C22×C4C23×C411yescharacteristic
250C22×C4C22×D411yescharacteristic
251C22×C4C22×D411yescharacteristic
252M16—44no—
253M16—44no—
254M16—84no—
255C2×C8—44no—
256C2×C8—44no—
257C4⋊C4—24no—
258C4⋊C4—24no—
259C4⋊C4—22no—
260C4⋊C4—22no—
261C4⋊C4—24no—
262C4⋊C4—24no—
263C4⋊C4—24no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—44no—
267C4⋊C4—44no—
268C4⋊C4—48no—
269C4⋊C4—44no—
270C4⋊C4—48no—
271C4⋊C4—82no—
272C4⋊C4—82no—
273C4⋊C4—84no—
274C4⋊C4—84no—
275C22⋊C4—42no—
276C22⋊C4—44no—
277C22⋊C4—42no—
278C42—22no—
279C42—22no—
280C42—22no—
281C42—22no—
282C42—22no—
283C42—22no—
284C42—22no—
285C42—22no—
286C42—44no—
287C42—48no—
288C42—44no—
289C42—48no—
290C42#11712yes—
291C42#11712yes—
292C42#12312yes—
293C42#12312yes—
294C42#12312yes—
295C42#12312yes—
296C42C2×#4112yes—
297C42C2×#4112yes—
298C23—21no—
299C23—21no—
300C23—41no—
301C23—41no—
302C23#38111yescharacteristic
303C23#38111yescharacteristic
304C23C22×C22⋊C411yescharacteristic
305Q8—216no—
306Q8—28no—
307Q8—28no—
308C2×C4—28no—
309C2×C4—28no—
310C2×C4—21no—
311C2×C4—22no—
312C2×C4—21no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—24no—
317C2×C4—24no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—42no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—44no—
340C2×C4—42no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4#38111yescharacteristic
344C2×C4#38112yes—
345C2×C4#38112yes—
346C2×C4#38111yescharacteristic
347C2×C4#41814yes—
348C2×C4#41914yes—
349C2×C4C2×#11711yescharacteristic
350C2×C4C2×#11711yescharacteristic
351C2×C4C2×#12312yes—
352C2×C4C2×#12312yes—
353C2×C4C2×AGL(1,ℤ/8)12yes—
354C2×C4C2×#13712yes—
355C8—84no—
356C8—84no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—42no—
366C22—42no—
367C22C2×#38111yescharacteristic
368C22C2×#41811yescharacteristic
369C22C2×#41911yescharacteristic
370C4—22no—
371C4—24no—
372C4—24no—
373C4—22no—
374C4—22no—
375C4—21no—
376C4—21no—
377C4—21no—
378C4—42no—
379C4—42no—
380C4—82no—
381C4—84no—
382C4—82no—
383C4#183012yes—
384C4#309214yes—
385C2—21no—
386C2—21no—
387C2—41no—
388C2—41no—
389C2#2070011yescharacteristic
390C2#2070811yescharacteristic
391C2C2×#183011yescharacteristic
subgroupG/H
center, socleC22C2×#381
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8767C2semidirect
5. #8766C2semidirect

Maximal subgroups

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

subgroupindex
#341822
#207272
#173652
#87672
#87662
#80942
#79612