Finite Group Databasebeta

← #190277

Subgroups — #190277

Order 512.

923 conjugacy classes of subgroups.

The subgroup lattice has 395 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34121C211yesmaximal, characteristic
2#20728C211yesmaximal, characteristic
3#17366C211yesmaximal, characteristic
4#8770C214yesmaximal, complemented
5#8769C214yesmaximal, complemented
6#8098C212yesmaximal, complemented
7#7961C212yesmaximal, complemented
8C2×C4×#41C2211yescharacteristic
9#2920C2212yes—
10#2920C2212yes—
11#2357C2212yes—
12#2356C2212yes—
13#2355—21no—
14#2354—21no—
15#2353C414yes—
16#2352—22no—
17#2298—22no—
18#2297—21no—
19#2285C2211yescharacteristic
20#2283C412yes—
21C4×#117—21no—
22C4×#117C412yes—
23C2×C4×C4⋊C4C2211yescharacteristic
24#1822—21no—
25#1801C2212yes—
26#1800C2212yes—
27#1785—21no—
28C4×#101—21no—
29C4×#100—21no—
30#1465—22no—
31#1464—22no—
32#1461C2212yes—
33#1460C2212yes—
34#1303C2218yes—
35#1286C2218yes—
36C22×#41D411yescharacteristic
37C4×#41—22no—
38C4×#41—22no—
39C4×#41—22no—
40C4×#41—22no—
41C2×C4×Q8D411yescharacteristic
42C2×C4×Q8C2311yescharacteristic
43C2×C4×D4C2×C412yes—
44C2×C4×D4D412yes—
45C2×C4×D4D411yescharacteristic
46C2×C4×D4C2311yescharacteristic
47C2×#117—22no—
48C2×#117—21no—
49C2×#117D411yescharacteristic
50C2×#117D412yes—
51C22×C4⋊C4—21no—
52C22×C42—21no—
53C22×C42C2×C412yes—
54C22×C42C2311yescharacteristic
55#432—22no—
56#432—22no—
57#432—22no—
58#432—22no—
59#431—22no—
60#431—22no—
61#431—22no—
62#431C2314yes—
63#423—22no—
64#423C2314yes—
65C2×#101—22no—
66C2×#100—22no—
67#407—22no—
68#407—22no—
69C4×M16—22no—
70C4×M16—22no—
71#391—22no—
72#390—22no—
73#388—24no—
74#388—24no—
75#387—24no—
76#387—24no—
77#386—24no—
78#384—21no—
79#383C2×C414yes—
80#382—22no—
81#382C2312yes—
82#381C2×C414yes—
83#380—22no—
84#380—21no—
85#380C2×C412yes—
86C4×C4⋊C4—22no—
87C4×C4⋊C4—22no—
88C4×C4⋊C4—22no—
89C4×C4⋊C4—21no—
90C4×C4⋊C4—21no—
91C4×C4⋊C4—21no—
92C4×C4⋊C4C2×C411yescharacteristic
93C4×C4⋊C4C2×C411yescharacteristic
94C4×C4⋊C4C2×C412yes—
95C4×C4⋊C4C2312yes—
96C4×C22⋊C4—22no—
97C4×C22⋊C4—22no—
98C4×C22⋊C4—24no—
99C4×C22⋊C4C2×C414yes—
100#376—22no—
101#376C2×C414yes—
102C2×#95C2×C412yes—
103C43—22no—
104#343—22no—
105#343—22no—
106#324—24no—
107#323—24no—
108C2×#41—22no—
109C2×#41—22no—
110C2×#41—42no—
111C2×#41—42no—
112C22×Q8C2×D411yescharacteristic
113C22×D4C22⋊C412yes—
114C22×D4C2×D411yescharacteristic
115C23×C4—22no—
116C23×C4C22⋊C412yes—
117C23×C4C22⋊C412yes—
118C23×C4C22×C412yes—
119C23×C4C22×C411yescharacteristic
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C2×M16—24no—
123C2×M16—42no—
124C4×Q8—22no—
125C4×Q8—22no—
126C4×Q8—22no—
127C4×Q8—22no—
128C4×D4—24no—
129C4×D4—24no—
130C4×D4—24no—
131C4×D4—24no—
132C4×D4—22no—
133C4×D4—22no—
134C4×D4—22no—
135C4×D4—22no—
136#117—22no—
137#117—22no—
138#117—24no—
139#117—24no—
140#117—48no—
141#117—44no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—44no—
149C2×C4⋊C4C22⋊C412yes—
150C2×C4⋊C4C2×D412yes—
151C2×C4⋊C4C2×D411yescharacteristic
152C2×C4⋊C4C2×D411yescharacteristic
153C2×C22⋊C4—22no—
154C2×C22⋊C4—24no—
155C2×C22⋊C4—24no—
156C2×C22⋊C4C22⋊C412yes—
157C2×C22⋊C4C22⋊C412yes—
158C2×C22⋊C4C2×D412yes—
159C2×C42—22no—
160C2×C42—22no—
161C2×C42—22no—
162C2×C42—22no—
163C2×C42—22no—
164C2×C42—24no—
165C2×C42—21no—
166C2×C42—21no—
167C2×C42—22no—
168C2×C42—24no—
169C2×C42—44no—
170C2×C42C22⋊C412yes—
171C2×C42C22⋊C412yes—
172C2×C42C22×C412yes—
173C2×C42C22×C411yescharacteristic
174C2×C42C2×D411yescharacteristic
175C2×C42C2×D411yescharacteristic
176C2×C42#4112yes—
177C2×C42#4112yes—
178C2×C42#4112yes—
179C2×C42#4112yes—
180C2×C42C2411yescharacteristic
181C4⋊C8—24no—
182C4⋊C8—24no—
183C4⋊C8—24no—
184C4⋊C8—42no—
185#101—44no—
186#100—44no—
187C22⋊C8—24no—
188C22⋊C8—42no—
189C22⋊C8—42no—
190C22⋊C8—42no—
191#97—42no—
192#97—42no—
193C4×C8—24no—
194C4×C8—42no—
195#95—22no—
196#95—24no—
197#95—24no—
198#95—24no—
199#95—24no—
200#95—22no—
201#95C22×C412yes—
202#95C22×C412yes—
203#95C22×C414yes—
204C24C2×C22⋊C412yes—
205C24C2×C22⋊C411yescharacteristic
206#41—48no—
207#41—48no—
208C2×Q8—22no—
209C2×Q8—22no—
210C2×Q8—22no—
211C2×D4—24no—
212C2×D4—22no—
213C2×D4—22no—
214C2×D4—22no—
215C2×D4—42no—
216C2×D4—42no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—24no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—44no—
232C22×C4—44no—
233C22×C4—44no—
234C22×C4—81no—
235C22×C4—82no—
236C22×C4—82no—
237C22×C4C2×C22⋊C412yes—
238C22×C4C2×C22⋊C411yescharacteristic
239C22×C4C2×C22⋊C411yescharacteristic
240C22×C4C2×C22⋊C412yes—
241C22×C4C2×C22⋊C412yes—
242C22×C4C2×C22⋊C411yescharacteristic
243C22×C4#11712yes—
244C22×C4#11712yes—
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—22no—
259C4⋊C4—22no—
260C4⋊C4—24no—
261C4⋊C4—24no—
262C4⋊C4—42no—
263C4⋊C4—44no—
264C4⋊C4—44no—
265C4⋊C4—44no—
266C4⋊C4—82no—
267C22⋊C4—24no—
268C22⋊C4—24no—
269C22⋊C4—24no—
270C22⋊C4—48no—
271C22⋊C4—44no—
272C22⋊C4—84no—
273C42—22no—
274C42—22no—
275C42—22no—
276C42—22no—
277C42—22no—
278C42—22no—
279C42—42no—
280C42—42no—
281C42—44no—
282C42—48no—
283C42—44no—
284C42—84no—
285C42#11711yescharacteristic
286C42#11711yescharacteristic
287C42#11711yescharacteristic
288C42#11711yescharacteristic
289C42#12312yes—
290C42#12312yes—
291C42#12312yes—
292C42#12312yes—
293C42C2×#4112yes—
294C42C2×#4112yes—
295C23—21no—
296C23—21no—
297C23—22no—
298C23—22no—
299C23—22no—
300C23—41no—
301C23—41no—
302C23—42no—
303C23—42no—
304C23#38112yes—
305C23C22×C22⋊C411yescharacteristic
306Q8—44no—
307D4—48no—
308D4—44no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—24no—
312C2×C4—24no—
313C2×C4—22no—
314C2×C4—24no—
315C2×C4—24no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—42no—
324C2×C4—41no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—44no—
328C2×C4—44no—
329C2×C4—44no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—44no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—44no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4—88no—
344C2×C4#38111yescharacteristic
345C2×C4#38111yescharacteristic
346C2×C4#38111yescharacteristic
347C2×C4#38112yes—
348C2×C4#38111yescharacteristic
349C2×C4C2×#11711yescharacteristic
350C2×C4C2×#11711yescharacteristic
351C2×C4C2×#12312yes—
352C2×C4C2×#12312yes—
353C2×C4#57512yes—
354C2×C4#57512yes—
355C8—84no—
356C8—84no—
357C22—21no—
358C22—21no—
359C22—22no—
360C22—22no—
361C22—22no—
362C22—42no—
363C22—42no—
364C22—42no—
365C22—42no—
366C22—42no—
367C22—82no—
368C22—82no—
369C22C2×#38111yescharacteristic
370C22#289611yescharacteristic
371C22#289611yescharacteristic
372C4—21no—
373C4—21no—
374C4—22no—
375C4—21no—
376C4—21no—
377C4—21no—
378C4—22no—
379C4—44no—
380C4—42no—
381C4—42no—
382C4—42no—
383C4—84no—
384C4—82no—
385C4#182712yes—
386C4#309314yes—
387C2—21no—
388C2—21no—
389C2—42no—
390C2—82no—
391C2#2070111yescharacteristic
392C2C2×#182711yescharacteristic
393C2#2071311yescharacteristic
subgroupG/H
center, socleC22C2×#381
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41D4
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. #8770C2semidirect
5. #8769C2semidirect
6. #8098C2semidirect
7. #7961C2semidirect

Maximal subgroups

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

subgroupindex
#341212
#207282
#173662
#87702
#87692
#80982
#79612