Finite Group Databasebeta

← #189879

Subgroups — #189879

Order 512.

980 conjugacy classes of subgroups.

The subgroup lattice has 421 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34187C211yesmaximal, characteristic
2#21822C211yesmaximal, characteristic
3#13942C211yesmaximal, characteristic
4#8717C212yesmaximal
5#8716C212yesmaximal
6#8714C214yesmaximal
7#8021C212yesmaximal
8#8017C212yesmaximal
9C22×C4×Q8C2211yescharacteristic
10C2×#432C2212yes—
11#2356—22no—
12#2356—22no—
13#2355C414yescomplemented
14#2355C414yescomplemented
15#2313—21no—
16#2313—21no—
17#2310—21no—
18C2×#391—21no—
19#2285C2211yescharacteristic
20#2285C2211yescharacteristic
21#2283C2212yes—
22#2283C2212yes—
23C2×#382C2211yescharacteristic
24#2274—21no—
25C2×C4×C4⋊C4C2211yescharacteristic
26#1985—22no—
27#1985—22no—
28#1817C2212yes—
29#1817C2212yes—
30#1441—21no—
31#1441—21no—
32C4⋊#95—21no—
33#1434C2212yes—
34#1434C2212yes—
35#1294C2214yes—
36#1294C2214yes—
37#1294C2214yes—
38#1294C2214yes—
39C23×Q8D411yescharacteristic
40C2×C4×Q8—24no—
41C2×C4×Q8—24no—
42C2×C4×Q8C2×C412yes—
43C2×C4×Q8C2×C412yes—
44C2×C4×Q8D412yes—
45C2×C4×Q8D412yes—
46C2×#117—21no—
47C2×#117—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4D411yescharacteristic
51C22×C4⋊C4D411yescharacteristic
52C22×C4⋊C4D411yescharacteristic
53C22×C42—21no—
54C22×C42C2311yescharacteristic
55C22×C42C2311yescharacteristic
56C22×C42C2311yescharacteristic
57#432—24no—
58#432—24no—
59C2×C4⋊C8—22no—
60C2×C22⋊C8C2314yes—
61C2×#97—22no—
62#391—24no—
63#391—24no—
64#391—44no—
65#387—24no—
66#387—24no—
67#387—24no—
68#387—24no—
69#384C2×C412yes—
70#384C2×C412yes—
71#383—22no—
72#383—22no—
73#382—21no—
74#382—21no—
75#382—24no—
76#382C2×C414yes—
77#382C2×C414yes—
78#381—22no—
79#381—22no—
80#380—21no—
81#380—21no—
82#380C2312yes—
83#380C2312yes—
84C4×C4⋊C4—24no—
85C4×C4⋊C4—24no—
86C4×C4⋊C4—21no—
87C4×C4⋊C4—21no—
88C4×C4⋊C4—24no—
89C4×C4⋊C4C2×C414yes—
90C4×C4⋊C4C2×C414yes—
91C4×C4⋊C4C2×C412yes—
92C4×C4⋊C4C2×C412yes—
93C4×C22⋊C4—22no—
94C4×C22⋊C4—22no—
95#376—22no—
96#376—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—21no—
102C2×#95C2312yes—
103C2×#95C2312yes—
104#336—24no—
105#336—22no—
106#336—22no—
107C22×Q8—22no—
108C22×Q8—22no—
109C22×Q8—22no—
110C22×Q8—22no—
111C22×Q8—41no—
112C22×Q8—41no—
113C22×Q8C22⋊C412yes—
114C22×Q8C22⋊C412yes—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C23×C4C2411yescharacteristic
124C22×C8—24no—
125C4×Q8—28no—
126C4×Q8—28no—
127C4×Q8—24no—
128C4×Q8—24no—
129C4×Q8—24no—
130C4×Q8—24no—
131#117—42no—
132#117—42no—
133#117—42no—
134#117—42no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—44no—
146C2×C4⋊C4—42no—
147C2×C4⋊C4—42no—
148C2×C4⋊C4—42no—
149C2×C4⋊C4—42no—
150C2×C4⋊C4C22⋊C412yes—
151C2×C4⋊C4C22⋊C412yes—
152C2×C4⋊C4C22⋊C412yes—
153C2×C4⋊C4C22⋊C412yes—
154C2×C4⋊C4C22⋊C412yes—
155C2×C4⋊C4C22⋊C412yes—
156C2×C4⋊C4C22×C414yes—
157C2×C4⋊C4C22×C414yes—
158C2×C22⋊C4—22no—
159C2×C22⋊C4—22no—
160C2×C42—22no—
161C2×C42—22no—
162C2×C42—22no—
163C2×C42—22no—
164C2×C42—22no—
165C2×C42—22no—
166C2×C42—24no—
167C2×C42—24no—
168C2×C42—42no—
169C2×C42—42no—
170C2×C42—42no—
171C2×C42C22×C412yes—
172C2×C42C22×C411yescharacteristic
173C2×C42C22×C411yescharacteristic
174C2×C42C22×C412yes—
175C2×C42C2×D412yes—
176C2×C42C2×D411yescharacteristic
177C2×C42C2×D411yescharacteristic
178C2×C42C2×D412yes—
179C2×C42#4114yes—
180C2×C42#4114yes—
181C4⋊C8—44no—
182C22⋊C8—24no—
183C22⋊C8—24no—
184#97—44no—
185#95—24no—
186#95—24no—
187#95—24no—
188#95—24no—
189#95—22no—
190#95—22no—
191#95—24no—
192#95—24no—
193#95—22no—
194#95—22no—
195#95—24no—
196#95—24no—
197#95—44no—
198#95—44no—
199#95—44no—
200#95—42no—
201#95—44no—
202C24C22×D411yescharacteristic
203C2×Q8—24no—
204C2×Q8—24no—
205C2×Q8—24no—
206C2×Q8—24no—
207C2×Q8—42no—
208C2×Q8—42no—
209C2×Q8—44no—
210C2×Q8—44no—
211C2×Q8—44no—
212C2×Q8—44no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—24no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4C2×C22⋊C412yes—
252C22×C4C2×C22⋊C412yes—
253C22×C4C2×C22⋊C411yescharacteristic
254C22×C4C2×C22⋊C411yescharacteristic
255C22×C4C2×C22⋊C412yes—
256C22×C4C2×C22⋊C412yes—
257C22×C4C2×C22⋊C411yescharacteristic
258C22×C4C2×C22⋊C411yescharacteristic
259C22×C4#12312yes—
260C22×C4#12312yes—
261C22×C4#12312yes—
262C22×C4#12312yes—
263C22×C4AGL(1,ℤ/8)12yes—
264C22×C4#13712yes—
265C22×C4C23×C411yescharacteristic
266C22×C4C22×D411yescharacteristic
267C22×C4C2×#4112yes—
268C22×C4C2×#4112yes—
269C2×C8—48no—
270C2×C8—44no—
271C4⋊C4—24no—
272C4⋊C4—24no—
273C4⋊C4—24no—
274C4⋊C4—24no—
275C4⋊C4—24no—
276C4⋊C4—24no—
277C4⋊C4—44no—
278C4⋊C4—44no—
279C4⋊C4—44no—
280C4⋊C4—48no—
281C4⋊C4—44no—
282C4⋊C4—82no—
283C4⋊C4—82no—
284C4⋊C4—84no—
285C22⋊C4—42no—
286C22⋊C4—42no—
287C22⋊C4—42no—
288C22⋊C4—42no—
289C42—22no—
290C42—22no—
291C42—24no—
292C42—22no—
293C42—22no—
294C42—22no—
295C42—22no—
296C42—44no—
297C42—44no—
298C42—84no—
299C42—84no—
300C42—84no—
301C42#11718yes—
302C42#12314yes—
303C42#12314yes—
304C23—22no—
305C23—22no—
306C23C22×C22⋊C411yescharacteristic
307C23C2×#12311yescharacteristic
308C23C2×#12311yescharacteristic
309C23C2×AGL(1,ℤ/8)11yescharacteristic
310C23C2×#13711yescharacteristic
311C23#57511yescharacteristic
312C23#57511yescharacteristic
313Q8—48no—
314Q8—48no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—22no—
329C2×C4—24no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—24no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4#38114yes—
374C2×C4#38114yes—
375C2×C4#41712yes—
376C2×C4#41712yes—
377C2×C4C2×#11712yes—
378C2×C4C2×#12312yes—
379C8—84no—
380C22—21no—
381C22—21no—
382C22—22no—
383C22—22no—
384C22—22no—
385C22—22no—
386C22—21no—
387C22—21no—
388C22C2×#38111yescharacteristic
389C22#288412yes—
390C22C2×#41711yescharacteristic
391C22#289611yescharacteristic
392C22#290112yes—
393C22#309411yescharacteristic
394C22#309411yescharacteristic
395C22#309511yescharacteristic
396C22#309511yescharacteristic
397C4—22no—
398C4—22no—
399C4—24no—
400C4—22no—
401C4—22no—
402C4—24no—
403C4—42no—
404C4—42no—
405C4—41no—
406C4—41no—
407C4—84no—
408C4—84no—
409C4—84no—
410C4—162no—
411C2—21no—
412C2—21no—
413C2—21no—
414C2—21no—
415C2#1762412yes—
416C2#1762812yes—
417C2#2069111yescharacteristic
418C2#2070211yescharacteristic
419C2#2070611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
13. #2355C4semidirect
14. #2355C4semidirect

Maximal subgroups

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

subgroupindex
#341872
#218222
#139422
#87172
#87162
#87142
#80212
#80172