Finite Group Databasebeta

← #190080

Subgroups — #190080

Order 512.

1085 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4⋊C4×#41C211yesmaximal, characteristic
2#17366C211yesmaximal, characteristic
3#17366C211yesmaximal, characteristic
4#8743C218yesmaximal, complemented
5#7953C212yesmaximal, complemented
6#7950C212yesmaximal, complemented
7C2×C4×#41C412yes—
8C2×C4×#41C2211yescharacteristic
9#2920C2212yes—
10#2920C2212yes—
11#2351—21no—
12Q8×C4⋊C4—21no—
13#2349—22no—
14#2349—21no—
15D4×C4⋊C4—21no—
16D4×C4⋊C4—22no—
17#2296C412yes—
18#2296C2214yes—
19#2285C412yes—
20#2285C2211yescharacteristic
21C2×C4×C4⋊C4C412yes—
22C2×C4×C4⋊C4C2211yescharacteristic
23#1465—22no—
24#1465—22no—
25#1464C2214yes—
26#1464C2214yes—
27#1461—21no—
28#1461—21no—
29#1460—21no—
30#1460—21no—
31#1285C2218yes—
32#1285C2218yes—
33C22×#41C2×C411yescharacteristic
34C4×#41—22no—
35C4×#41—22no—
36C4×#41—22no—
37C4×#41—22no—
38C4×#41—48no—
39C2×C4×Q8—22no—
40C2×C4×Q8D411yescharacteristic
41C2×C4×Q8D411yescharacteristic
42C2×C4×D4—24no—
43C2×C4×D4—22no—
44C2×C4×D4D412yes—
45C2×C4×D4D412yes—
46C2×C4×D4D411yescharacteristic
47C2×C4×D4D411yescharacteristic
48C2×#117C2×C414yes—
49C2×#117C2×C412yes—
50C2×#117C2×C411yescharacteristic
51C2×#117C2312yes—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—21no—
54C22×C42C2×C412yes—
55C22×C42C2×C414yes—
56C22×C42C2×C412yes—
57C22×C42C2311yescharacteristic
58#432—22no—
59#432—22no—
60#432C2314yes—
61#432C2314yes—
62#431—22no—
63#431—22no—
64#431—22no—
65#431—22no—
66#423—22no—
67#423—22no—
68#407—22no—
69#407—22no—
70C4×M16—22no—
71C4×M16—22no—
72#389—22no—
73#389—22no—
74#389—24no—
75#389—24no—
76#385—24no—
77#385—24no—
78#385—24no—
79#384—22no—
80#384—21no—
81#382—22no—
82#382—21no—
83#380—22no—
84#380—22no—
85#380—21no—
86#380—21no—
87C4×C4⋊C4—22no—
88C4×C4⋊C4—22no—
89C4×C4⋊C4—21no—
90C4×C4⋊C4—21no—
91C4×C4⋊C4C2×C411yescharacteristic
92C4×C4⋊C4C2×C412yes—
93C4×C4⋊C4C2×C414yes—
94C4×C4⋊C4C2×C411yescharacteristic
95C4×C4⋊C4C2×C414yes—
96C4×C4⋊C4C2312yes—
97C4×C4⋊C4C2312yes—
98C2×#95—22no—
99C2×#95—21no—
100#324—24no—
101#324—24no—
102#323—24no—
103#323—24no—
104C2×#41—24no—
105C2×#41—24no—
106C2×#41—22no—
107C2×#41—22no—
108C22×Q8C22⋊C411yescharacteristic
109C22×D4C22⋊C412yes—
110C22×D4C22⋊C411yescharacteristic
111C23×C4—22no—
112C23×C4—24no—
113C23×C4C22⋊C412yes—
114C23×C4C22⋊C412yes—
115C23×C4C22×C412yes—
116C23×C4C22×C411yescharacteristic
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C2×M16—42no—
120C2×M16—42no—
121C4×Q8—22no—
122C4×Q8—22no—
123C4×Q8—22no—
124C4×Q8—22no—
125C4×Q8—48no—
126C4×D4—24no—
127C4×D4—24no—
128C4×D4—24no—
129C4×D4—24no—
130C4×D4—22no—
131C4×D4—22no—
132C4×D4—22no—
133C4×D4—22no—
134C4×D4—416no—
135C4×D4—48no—
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—22no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—44no—
152C2×C4⋊C4—48no—
153C2×C4⋊C4C22⋊C411yescharacteristic
154C2×C4⋊C4C22⋊C411yescharacteristic
155C2×C4⋊C4C2×D412yes—
156C2×C4⋊C4C2×D412yes—
157C2×C22⋊C4—22no—
158C2×C22⋊C4—24no—
159C2×C22⋊C4C22⋊C412yes—
160C2×C22⋊C4C2×D412yes—
161C2×C22⋊C4C2×D412yes—
162C2×C42—22no—
163C2×C42—22no—
164C2×C42—22no—
165C2×C42—22no—
166C2×C42—24no—
167C2×C42—22no—
168C2×C42—48no—
169C2×C42—44no—
170C2×C42C22⋊C412yes—
171C2×C42C22⋊C412yes—
172C2×C42C22×C412yes—
173C2×C42C22×C412yes—
174C2×C42C22×C414yes—
175C2×C42C22×C412yes—
176C2×C42C22×C411yescharacteristic
177C2×C42C2×D411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C42#4112yes—
180C2×C42#4112yes—
181C2×C42#4112yes—
182C2×C42#4112yes—
183C2×C42C2411yescharacteristic
184C4⋊C8—24no—
185C4⋊C8—24no—
186C4⋊C8—42no—
187C4⋊C8—42no—
188C22⋊C8—24no—
189C22⋊C8—24no—
190C22⋊C8—42no—
191C22⋊C8—42no—
192#97—24no—
193#97—24no—
194C4×C8—42no—
195C4×C8—42no—
196#95—21no—
197#95—22no—
198#95—21no—
199#95—24no—
200#95—24no—
201#95—22no—
202#95—22no—
203C24C2×C22⋊C411yescharacteristic
204C24C2×C22⋊C412yes—
205#41—216no—
206#41—216no—
207C2×Q8—22no—
208C2×Q8—22no—
209C2×Q8#10014yes—
210C2×D4—24no—
211C2×D4—24no—
212C2×D4—22no—
213C2×D4—24no—
214C2×D4—22no—
215C2×D4#10114yes—
216C22×C4—24no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—24no—
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—22no—
229C22×C4—42no—
230C22×C4—44no—
231C22×C4—44no—
232C22×C4—48no—
233C22×C4—44no—
234C22×C4—48no—
235C22×C4—81no—
236C22×C4—82no—
237C22×C4—84no—
238C22×C4—82no—
239C22×C4C2×C22⋊C411yescharacteristic
240C22×C4C2×C22⋊C412yes—
241C22×C4C2×C22⋊C412yes—
242C22×C4C2×C22⋊C412yes—
243C22×C4C2×C22⋊C411yescharacteristic
244C22×C4C2×C22⋊C411yescharacteristic
245C22×C4#11712yes—
246C22×C4#11712yes—
247C22×C4#12312yes—
248C22×C4#12312yes—
249C22×C4#12312yes—
250C22×C4#12312yes—
251C22×C4C23×C411yescharacteristic
252C22×C4C22×D411yescharacteristic
253C22×C4C22×D411yescharacteristic
254M16—84no—
255M16—84no—
256C2×C8—44no—
257C2×C8—44no—
258C4⋊C4—24no—
259C4⋊C4—24no—
260C4⋊C4—24no—
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—44no—
264C4⋊C4—42no—
265C4⋊C4—44no—
266C4⋊C4—48no—
267C4⋊C4—48no—
268C4⋊C4—44no—
269C4⋊C4—82no—
270C4⋊C4—84no—
271C22⋊C4—24no—
272C22⋊C4—24no—
273C22⋊C4—24no—
274C22⋊C4—44no—
275C22⋊C4—48no—
276C42—22no—
277C42—22no—
278C42—22no—
279C42—22no—
280C42—22no—
281C42—22no—
282C42—42no—
283C42—44no—
284C42—42no—
285C42—48no—
286C42—44no—
287C42#11711yescharacteristic
288C42#11711yescharacteristic
289C42#11711yescharacteristic
290C42#11711yescharacteristic
291C42#12312yes—
292C42#12312yes—
293C42#12312yes—
294C42#12312yes—
295C42C2×#4112yes—
296C42C2×#4112yes—
297C23—21no—
298C23—22no—
299C23—21no—
300C23—22no—
301C23—41no—
302C23—41no—
303C23—42no—
304C23—42no—
305C23#38112yes—
306C23C2×#10114yes—
307C23C22×C22⋊C411yescharacteristic
308Q8—28no—
309D4—216no—
310D4—28no—
311C2×C4—28no—
312C2×C4—28no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—24no—
318C2×C4—24no—
319C2×C4—22no—
320C2×C4—24no—
321C2×C4—24no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—41no—
329C2×C4—42no—
330C2×C4—44no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—44no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—44no—
338C2×C4—44no—
339C2×C4—42no—
340C2×C4—44no—
341C2×C4—42no—
342C2×C4—82no—
343C2×C4—82no—
344C2×C4—88no—
345C2×C4—816no—
346C2×C4#38111yescharacteristic
347C2×C4#38111yescharacteristic
348C2×C4#38111yescharacteristic
349C2×C4#38112yes—
350C2×C4#38111yescharacteristic
351C2×C4C2×#10014yes—
352C2×C4C2×#10012yes—
353C2×C4C2×#10112yes—
354C2×C4C2×#11711yescharacteristic
355C2×C4C2×#11711yescharacteristic
356C2×C4C2×#12312yes—
357C2×C4C2×#12312yes—
358C8—84no—
359C8—84no—
360C22—21no—
361C22—21no—
362C22—22no—
363C22—24no—
364C22—22no—
365C22—22no—
366C22—44no—
367C22—42no—
368C22—44no—
369C22—42no—
370C22—42no—
371C22—42no—
372C22C2×#38111yescharacteristic
373C22C22×#10011yescharacteristic
374C22C22×#10111yescharacteristic
375C4—21no—
376C4—21no—
377C4—24no—
378C4—22no—
379C4—21no—
380C4—22no—
381C4—21no—
382C4—21no—
383C4—44no—
384C4—42no—
385C4—42no—
386C4—82no—
387C4—84no—
388C4#182114yes—
389C4#182712yes—
390C2—21no—
391C2—21no—
392C2—42no—
393C2—44no—
394C2C2×#182111yescharacteristic
395C2C2×#182711yescharacteristic
396C2#2070811yescharacteristic
subgroupG/H
center, socleC22C2×#381
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41C2×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. #8743C2semidirect
5. #7953C2semidirect
6. #7950C2semidirect

Maximal subgroups

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

subgroupindex
C4⋊C4×#412
#173662
#87432
#79532
#79502