Finite Group Databasebeta

← #190173

Subgroups — #190173

Order 512.

1167 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34181C211yesmaximal, characteristic
2#17364C211yesmaximal, characteristic
3#17364C211yesmaximal, characteristic
4#8756C218yesmaximal, complemented
5#8109C212yesmaximal, complemented
6#8105C212yesmaximal, complemented
7C2×#518C412yes—
8C22×C4×D4C2211yescharacteristic
9#2920C2212yes—
10#2920C2212yes—
11#2349—21no—
12#2349—21no—
13#2349—22no—
14D4×C4⋊C4—21no—
15D4×C4⋊C4—21no—
16D4×C4⋊C4—22no—
17#2316C412yes—
18#2315C2214yes—
19#2285C2211yescharacteristic
20#2285C2211yescharacteristic
21C2×#380C412yes—
22C2×#380C412yes—
23#1466—22no—
24#1466—22no—
25#1462C2214yes—
26#1462C2214yes—
27#1460—21no—
28#1460—21no—
29#1460—21no—
30#1460—21no—
31#1305C2218yes—
32#1305C2218yes—
33C23×D4C2×C411yescharacteristic
34C24×C4C2×C412yes—
35#518—48no—
36C2×C4×D4—24no—
37C2×C4×D4—22no—
38C2×C4×D4—22no—
39C2×C4×D4—22no—
40C2×C4×D4—22no—
41C2×C4×D4—22no—
42C2×C4×D4—22no—
43C2×C4×D4D411yescharacteristic
44C2×C4×D4D411yescharacteristic
45C2×C4×D4D412yes—
46C2×C4×D4D411yescharacteristic
47C2×C4×D4D411yescharacteristic
48C2×C4×D4D412yes—
49C2×#117C2×C412yes—
50C2×#117C2×C412yes—
51C22×C4⋊C4—22no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4C2×C411yescharacteristic
54C22×C22⋊C4C2×C414yes—
55C22×C22⋊C4C2×C414yes—
56C22×C22⋊C4C2312yes—
57C22×C42C2311yescharacteristic
58#432—22no—
59#432—22no—
60#432—22no—
61#432—22no—
62#431—22no—
63#431—22no—
64#431—22no—
65#431—22no—
66#423—22no—
67#423—22no—
68#407C2314yes—
69#407C2314yes—
70C4×M16—22no—
71C4×M16—22no—
72#389—24no—
73#389—22no—
74#389—22no—
75#385—24no—
76#385—24no—
77#385—24no—
78#385—24no—
79#384—21no—
80#384—21no—
81#382—21no—
82#382—21no—
83#380—22no—
84#380—22no—
85#380—22no—
86#380—22no—
87#380C2×C414yes—
88#380C2×C414yes—
89#380C2×C411yescharacteristic
90#380C2×C411yescharacteristic
91#380C2×C412yes—
92#380C2312yes—
93#380C2312yes—
94C4×C4⋊C4—21no—
95C4×C4⋊C4—21no—
96C4×C4⋊C4—21no—
97C4×C4⋊C4—21no—
98C2×#95—22no—
99C2×#95—22no—
100#323—24no—
101#323—24no—
102#323—24no—
103#323—24no—
104C25C22×C412yes—
105C22×D4—24no—
106C22×D4—24no—
107C22×D4—22no—
108C22×D4—22no—
109C22×D4C22⋊C412yes—
110C22×D4C22⋊C411yescharacteristic
111C22×D4C22⋊C411yescharacteristic
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—24no—
117C23×C4—24no—
118C23×C4—84no—
119C23×C4C22⋊C412yes—
120C23×C4C22⋊C412yes—
121C23×C4C22⋊C412yes—
122C23×C4C22⋊C412yes—
123C23×C4C22×C411yescharacteristic
124C23×C4C22×C412yes—
125C23×C4C22×C411yescharacteristic
126C23×C4C2411yescharacteristic
127C2×M16—24no—
128C2×M16—24no—
129C4×D4—22no—
130C4×D4—22no—
131C4×D4—22no—
132C4×D4—22no—
133C4×D4—24no—
134C4×D4—24no—
135C4×D4—22no—
136C4×D4—22no—
137C4×D4—24no—
138C4×D4—22no—
139C4×D4—22no—
140C4×D4—24no—
141C4×D4—416no—
142C4×D4—48no—
143C4×D4—48no—
144#117—44no—
145#117—44no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—22no—
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⋊C4—84no—
158C2×C4⋊C4—84no—
159C2×C4⋊C4C22⋊C412yes—
160C2×C4⋊C4C22⋊C411yescharacteristic
161C2×C4⋊C4C22⋊C411yescharacteristic
162C2×C22⋊C4—24no—
163C2×C22⋊C4—24no—
164C2×C22⋊C4—24no—
165C2×C22⋊C4—24no—
166C2×C22⋊C4—48no—
167C2×C22⋊C4—48no—
168C2×C22⋊C4C22×C412yes—
169C2×C22⋊C4C22×C414yes—
170C2×C22⋊C4C22×C412yes—
171C2×C22⋊C4C2×D412yes—
172C2×C22⋊C4C2×D412yes—
173C2×C22⋊C4C2×D412yes—
174C2×C22⋊C4C2×D412yes—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42C2×D411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C42C2×D411yescharacteristic
181C2×C42#4112yes—
182C2×C42#4112yes—
183C2×C42#4112yes—
184C2×C42#4112yes—
185C4⋊C8—42no—
186C4⋊C8—42no—
187C4⋊C8—42no—
188C4⋊C8—42no—
189C22⋊C8—24no—
190C22⋊C8—24no—
191C22⋊C8—24no—
192C22⋊C8—24no—
193#97—42no—
194#97—42no—
195C4×C8—42no—
196C4×C8—42no—
197#95—21no—
198#95—21no—
199#95—22no—
200#95—24no—
201#95—24no—
202#95—22no—
203#95—22no—
204C24—22no—
205C24—22no—
206C24—24no—
207C24—24no—
208C24—82no—
209C24—82no—
210C24C2×C22⋊C412yes—
211C24C2×C22⋊C412yes—
212C24C23×C411yescharacteristic
213C2×D4—216no—
214C2×D4—216no—
215C2×D4—24no—
216C2×D4—24no—
217C2×D4—24no—
218C2×D4—22no—
219C2×D4—22no—
220C2×D4—22no—
221C2×D4—22no—
222C2×D4#10014yes—
223C2×D4#10114yes—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—24no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—42no—
240C22×C4—44no—
241C22×C4—44no—
242C22×C4—48no—
243C22×C4—48no—
244C22×C4—84no—
245C22×C4—84no—
246C22×C4—81no—
247C22×C4—84no—
248C22×C4—84no—
249C22×C4—82no—
250C22×C4—82no—
251C22×C4—82no—
252C22×C4—84no—
253C22×C4—84no—
254C22×C4—84no—
255C22×C4—84no—
256C22×C4C2×C22⋊C411yescharacteristic
257C22×C4C2×C22⋊C411yescharacteristic
258C22×C4C2×C22⋊C412yes—
259C22×C4C2×C22⋊C412yes—
260C22×C4C2×C22⋊C411yescharacteristic
261C22×C4C2×C22⋊C411yescharacteristic
262C22×C4#11712yes—
263C22×C4#11712yes—
264C22×C4#11711yescharacteristic
265C22×C4#11711yescharacteristic
266C22×C4#11711yescharacteristic
267C22×C4#11711yescharacteristic
268C22×C4C22×D411yescharacteristic
269C22×C4C22×D411yescharacteristic
270C22×C4C2×#4112yes—
271C22×C4C2×#4112yes—
272M16—84no—
273M16—84no—
274C2×C8—44no—
275C2×C8—44no—
276C4⋊C4—24no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—44no—
282C4⋊C4—44no—
283C4⋊C4—44no—
284C4⋊C4—82no—
285C4⋊C4—82no—
286C4⋊C4—84no—
287C4⋊C4—84no—
288C22⋊C4—24no—
289C22⋊C4—24no—
290C22⋊C4—24no—
291C22⋊C4—24no—
292C22⋊C4—48no—
293C22⋊C4—48no—
294C22⋊C4—42no—
295C22⋊C4—44no—
296C22⋊C4—42no—
297C42—44no—
298C42—44no—
299C42#12312yes—
300C42#12312yes—
301C42#12312yes—
302C42#12312yes—
303C42#12312yes—
304C42#12312yes—
305C42#12312yes—
306C42#12312yes—
307C23—28no—
308C23—28no—
309C23—21no—
310C23—22no—
311C23—21no—
312C23—22no—
313C23—22no—
314C23—22no—
315C23—41no—
316C23—41no—
317C23—42no—
318C23—42no—
319C23—42no—
320C23—42no—
321C23—82no—
322C23—82no—
323C23—82no—
324C23—82no—
325C23—88no—
326C23—88no—
327C23C2×#10014yes—
328C23C2×#10114yes—
329C23C22×C22⋊C411yescharacteristic
330C23C2×#11711yescharacteristic
331C23C2×#11711yescharacteristic
332D4—216no—
333D4—28no—
334D4—28no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—44no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—44no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—44no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4—168no—
373C2×C4—168no—
374C2×C4#38112yes—
375C2×C4#38111yescharacteristic
376C2×C4#38111yescharacteristic
377C2×C4#38111yescharacteristic
378C2×C4#38112yes—
379C2×C4#38111yescharacteristic
380C2×C4C2×#10012yes—
381C2×C4C2×#10112yes—
382C2×C4C2×#12312yes—
383C2×C4C2×#12312yes—
384C8—84no—
385C8—84no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—24no—
391C22—24no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—44no—
397C22—44no—
398C22—44no—
399C22—44no—
400C22—42no—
401C22—42no—
402C22—42no—
403C22—42no—
404C22—42no—
405C22—42no—
406C22—88no—
407C22—88no—
408C22C2×#38111yescharacteristic
409C22C22×#10011yescharacteristic
410C22C22×#10111yescharacteristic
411C4—22no—
412C4—22no—
413C4—22no—
414C4—21no—
415C4—21no—
416C4—21no—
417C4—42no—
418C4—42no—
419C4—84no—
420C4—82no—
421C4—82no—
422C4#182114yes—
423C4#182912yes—
424C2—21no—
425C2—21no—
426C2—41no—
427C2—41no—
428C2—44no—
429C2—44no—
430C2C2×#182111yescharacteristic
431C2#2071011yescharacteristic
432C2C2×#182911yescharacteristic
subgroupG/H
center, socleC22C2×#381
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8756C2semidirect
5. #8109C2semidirect
6. #8105C2semidirect

Maximal subgroups

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

subgroupindex
#341812
#173642
#87562
#81092
#81052