Finite Group Databasebeta

← #192542

Subgroups — #192542

Order 512.

833 conjugacy classes of subgroups.

The subgroup lattice has 419 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60251C211yesmaximal, characteristic
2#21953C211yesmaximal, characteristic
3#17803C211yesmaximal, characteristic
4C2×#1503C211yesmaximal, characteristic
5#17464C211yesmaximal, characteristic
6#8881C214yesmaximal, complemented
7#8879C214yesmaximal, complemented
8#8098C211yesmaximal, characteristic
9#7921C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11#2987—22no—
12#2985—22no—
13#2984—22no—
14#2983—22no—
15#2962—22no—
16#2961C2214yes—
17#2924—22no—
18#2920C2211yescharacteristic
19#2920C2211yescharacteristic
20C2×#432C2211yescharacteristic
21C2×#431C2211yescharacteristic
22C2×#431C2211yescharacteristic
23C2×C4×C4⋊C4C2211yescharacteristic
24#1940C2214yes—
25#1939—22no—
26#1797—22no—
27#1572—22no—
28#1571C2212yes—
29#1570C2212yes—
30#1564—22no—
31#1503C412yes—
32#1503C412yes—
33#1503C2212yes—
34#1503C2212yes—
35C2×#328C414yes—
36#1485—22no—
37#1481C2214yes—
38#1479—22no—
39#1303C2214yes—
40#1281C2212yes—
41#1281C2212yes—
42#533—21no—
43#533—21no—
44#533—21no—
45#533—21no—
46#533C2×C412yes—
47#533D412yes—
48#533Q812yes—
49#533C2312yes—
50C2×C4⋊Q8Q811yescharacteristic
51C2×C4⋊Q8Q811yescharacteristic
52C2×#125—21no—
53C2×C22⋊Q8—21no—
54C2×C22⋊Q8C2×C412yes—
55C2×C4×Q8—21no—
56C2×C4×Q8D412yes—
57C2×#117—21no—
58C22×C4⋊C4—21no—
59C22×C42—21no—
60C22×C42C2311yescharacteristic
61#446—28no—
62#446—24no—
63#446—28no—
64#446—28no—
65C8×Q8—24no—
66#433—28no—
67#432—24no—
68#432—21no—
69#432—21no—
70#432—21no—
71#432—21no—
72#432—21no—
73#432—21no—
74#432—21no—
75#432D412yes—
76#431—24no—
77#431—21no—
78#431—21no—
79#431—21no—
80#431—21no—
81#431—21no—
82#431—21no—
83#431—21no—
84#431—21no—
85#431C2×C412yes—
86#431D412yes—
87#431C2312yes—
88#431C2311yescharacteristic
89#431C2311yescharacteristic
90#424—24no—
91#423—21no—
92#423C2312yes—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—24no—
95C2×C4⋊C8D412yes—
96C2×C4⋊C8C2312yes—
97#407—21no—
98#407—21no—
99C2×C22⋊C8—21no—
100C2×C22⋊C8C2×C412yes—
101C2×C22⋊C8D412yes—
102#405—24no—
103C4×M16—21no—
104C4×M16—21no—
105C2×#97C2312yes—
106C2×C4×C8—21no—
107C2×C4×C8—21no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4C2×C411yescharacteristic
111C4×C4⋊C4C2×C411yescharacteristic
112C4×C4⋊C4C2312yes—
113C2×#95C2×C412yes—
114#336—24no—
115#335—24no—
116#334—24no—
117#328C2×C414yes—
118#328C2×C414yes—
119#328C2×C414yes—
120#328C2×C414yes—
121Q8⋊C8—28no—
122C22×Q8—21no—
123C22×Q8C4⋊C412yes—
124C23×C4—22no—
125C23×C4C22×C411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C23×C4#4111yescharacteristic
128C2×M16—22no—
129C2×M16—41no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8C2×D414yes—
133C4⋊Q8—21no—
134C4⋊Q8—21no—
135C4⋊Q8—21no—
136C4⋊Q8—21no—
137C4⋊Q8C4⋊C412yes—
138C4⋊Q8C4⋊C412yes—
139C4⋊Q8C2×Q812yes—
140C4⋊Q8C2×Q812yes—
141#125—22no—
142#125—22no—
143#125—22no—
144#125—22no—
145C22⋊Q8—22no—
146C22⋊Q8—22no—
147C22⋊Q8—22no—
148C22⋊Q8—22no—
149C22⋊Q8—22no—
150C22⋊Q8—22no—
151C22⋊Q8C4⋊C414yes—
152C22⋊Q8C22×C414yes—
153C4×Q8—22no—
154C4×Q8—22no—
155C4×Q8—22no—
156C4×Q8—22no—
157C4×Q8—22no—
158C4×Q8—22no—
159C4×Q8C4⋊C414yes—
160C4×Q8C2×D414yes—
161#117—22no—
162#117—22no—
163#117—22no—
164#117—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4C4⋊C412yes—
175C2×C22⋊C4—22no—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—21no—
179C2×C42—21no—
180C2×C42—82no—
181C2×C42C22×C411yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C42C2×D411yescharacteristic
184C2×C42C2×D411yescharacteristic
185C2×C42C2×D411yescharacteristic
186C2×C42C2×D411yescharacteristic
187C2×C42C2×Q811yescharacteristic
188C2×C42C2×Q811yescharacteristic
189C2×C42#4111yescharacteristic
190C2×C42#4111yescharacteristic
191C2×C42#4111yescharacteristic
192C2×C42C2411yescharacteristic
193C4⋊C8—28no—
194C4⋊C8—28no—
195C4⋊C8—22no—
196C4⋊C8—22no—
197C4⋊C8—22no—
198C4⋊C8—22no—
199C4⋊C8—22no—
200C4⋊C8—22no—
201C4⋊C8—21no—
202C4⋊C8—21no—
203C4⋊C8—22no—
204C4⋊C8—41no—
205C4⋊C8—84no—
206C4⋊C8C2×D414yes—
207C22⋊C8—22no—
208C22⋊C8—22no—
209C22⋊C8—22no—
210C22⋊C8—22no—
211C22⋊C8—22no—
212C22⋊C8—22no—
213C22⋊C8—41no—
214C22⋊C8—41no—
215C22⋊C8—41no—
216C22⋊C8C22×C412yes—
217C22⋊C8C22×C412yes—
218#97—28no—
219#97—22no—
220#97—22no—
221#97—22no—
222#97—41no—
223C4×C8—28no—
224C4×C8—22no—
225C4×C8—22no—
226C4×C8—21no—
227C4×C8—21no—
228C4×C8—22no—
229C4×C8—41no—
230C4×C8—41no—
231#95—22no—
232#95C22×C412yes—
233#95C22×C412yes—
234C24C4×D411yescharacteristic
235C2×Q8—22no—
236C2×Q8—22no—
237C2×Q8—22no—
238C2×Q8—44no—
239C2×Q8—82no—
240C2×Q8C2×C4⋊C414yes—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—44no—
250C22×C4—44no—
251C22×C4—82no—
252C22×C4C2×C4⋊C411yescharacteristic
253C22×C4C2×C4⋊C411yescharacteristic
254C22×C4C4×D411yescharacteristic
255C22×C4C4×D411yescharacteristic
256C22×C4C4×D411yescharacteristic
257C22×C4C4×D411yescharacteristic
258C22×C4C4×D411yescharacteristic
259C22×C4#12011yescharacteristic
260C22×C4#12011yescharacteristic
261C22×C4C22⋊Q811yescharacteristic
262C22×C4C22⋊Q811yescharacteristic
263C22×C4C22⋊Q811yescharacteristic
264C22×C4C22⋊Q811yescharacteristic
265C22×C4#12311yescharacteristic
266C22×C4#12311yescharacteristic
267C22×C4C23×C411yescharacteristic
268C22×C4C22×D411yescharacteristic
269C22×C4C2×#4111yescharacteristic
270M16—42no—
271M16—42no—
272M16—82no—
273C2×C8—22no—
274C2×C8—22no—
275C2×C8—24no—
276C2×C8—24no—
277C2×C8—24no—
278C2×C8—24no—
279C2×C8—44no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—42no—
284C4⋊C4—24no—
285C4⋊C4—24no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4—24no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—24no—
294C4⋊C4—24no—
295C4⋊C4—22no—
296C4⋊C4—42no—
297C4⋊C4—44no—
298C4⋊C4C2×C4⋊C414yes—
299C22⋊C4—24no—
300C22⋊C4—24no—
301C42—24no—
302C42—24no—
303C42—44no—
304C42—81no—
305C42—81no—
306C42C2×C4⋊C411yescharacteristic
307C42C2×C4⋊C411yescharacteristic
308C42C4×D411yescharacteristic
309C42C4×D411yescharacteristic
310C42#12011yescharacteristic
311C42#12011yescharacteristic
312C42C22⋊Q811yescharacteristic
313C42C22⋊Q811yescharacteristic
314C42C22⋊Q811yescharacteristic
315C42C22⋊Q811yescharacteristic
316C42#12311yescharacteristic
317C42#12311yescharacteristic
318C42C22×D411yescharacteristic
319C42C22×D411yescharacteristic
320C42C22×Q811yescharacteristic
321C42C2×#4111yescharacteristic
322C23—22no—
323C23—21no—
324C23—21no—
325C23—41no—
326C23#38511yescharacteristic
327C23#38511yescharacteristic
328C23C2×C4×D411yescharacteristic
329Q8—44no—
330Q8—44no—
331Q8—82no—
332Q8—82no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—42no—
355C2×C4—44no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—82no—
361C2×C4—88no—
362C2×C4#38511yescharacteristic
363C2×C4#38511yescharacteristic
364C2×C4#38511yescharacteristic
365C2×C4#38511yescharacteristic
366C2×C4#38511yescharacteristic
367C2×C4#38511yescharacteristic
368C2×C4C22×C4⋊C411yescharacteristic
369C2×C4C2×C4×D411yescharacteristic
370C2×C4C2×#12011yescharacteristic
371C2×C4C2×C22⋊Q811yescharacteristic
372C2×C4C2×C22⋊Q811yescharacteristic
373C2×C4C2×#12311yescharacteristic
374C2×C4#57712yes—
375C2×C4#57812yes—
376C8—42no—
377C8—42no—
378C8—44no—
379C8—44no—
380C8—82no—
381C8—82no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—42no—
390C22—42no—
391C22#186812yes—
392C22C2×#38511yescharacteristic
393C22#294911yescharacteristic
394C22#295011yescharacteristic
395C4—21no—
396C4—21no—
397C4—21no—
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—21no—
404C4—44no—
405C4—44no—
406C4—44no—
407C4—82no—
408C4#186112yes—
409C4#301312yes—
410C4#308212yes—
411C2—21no—
412C2—22no—
413C2—21no—
414C2—41no—
415C2C2×#186111yescharacteristic
416C2#2088011yescharacteristic
417C2C2×#186811yescharacteristic
subgroupG/H
centerC2×C4C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#385
prime radicalC24C4×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #8881C2semidirect
7. #8879C2semidirect

Maximal subgroups

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

subgroupindex
#602512
#219532
#178032
C2×#15032
#174642
#88812
#88792
#80982
#79212