Finite Group Databasebeta

← #192548

Subgroups — #192548

Order 512.

797 conjugacy classes of subgroups.

The subgroup lattice has 401 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60251C211yesmaximal, characteristic
2#21954C211yesmaximal, characteristic
3#17803C211yesmaximal, characteristic
4#17575C211yesmaximal, characteristic
5#17464C211yesmaximal, characteristic
6#8881C214yesmaximal, complemented
7#8879C214yesmaximal, complemented
8#8094C211yesmaximal, characteristic
9#7922C211yesmaximal, 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
23#2285C2211yescharacteristic
24#1940C2214yes—
25#1939—22no—
26#1798—22no—
27#1572—22no—
28#1571C2212yes—
29#1570C2212yes—
30#1564—22no—
31#1520C414yes—
32#1518C414yes—
33#1503C2212yes—
34#1503C2212yes—
35#1485—22no—
36#1481C2214yes—
37#1479—22no—
38#1303C2214yes—
39#1281C2212yes—
40#1281C2212yes—
41#533—21no—
42#533—21no—
43#533—21no—
44#533—21no—
45#533C2×C412yes—
46#533D412yes—
47#533Q812yes—
48#533C2312yes—
49C2×C4⋊Q8Q811yescharacteristic
50C2×C4⋊Q8Q811yescharacteristic
51C2×#125—21no—
52C2×C22⋊Q8—21no—
53C2×C22⋊Q8C2×C412yes—
54C2×C4×Q8—21no—
55C2×C4×Q8D412yes—
56C2×#117—21no—
57C2×#117—21no—
58C22×C42C2311yescharacteristic
59#446—28no—
60#446—24no—
61#446—28no—
62#446—28no—
63C8×Q8—24no—
64#433—28no—
65#432—24no—
66#432—21no—
67#432—21no—
68#432—21no—
69#432—21no—
70#432—21no—
71#432—21no—
72#432—21no—
73#432D412yes—
74#431—24no—
75#431—21no—
76#431—21no—
77#431—21no—
78#431—21no—
79#431—21no—
80#431—21no—
81#431—21no—
82#431—21no—
83#431C2×C412yes—
84#431D412yes—
85#431C2312yes—
86#431C2311yescharacteristic
87#431C2311yescharacteristic
88#424—24no—
89#423—21no—
90#423C2312yes—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—24no—
93C2×C4⋊C8D412yes—
94C2×C4⋊C8C2312yes—
95#407—21no—
96#407—21no—
97C2×C22⋊C8—21no—
98C2×C22⋊C8C2×C412yes—
99C2×C22⋊C8D412yes—
100#405—24no—
101C4×M16—21no—
102C4×M16—21no—
103C2×#97C2312yes—
104C2×C4×C8—21no—
105C2×C4×C8—21no—
106#384—21no—
107#382C2×C412yes—
108#380—21no—
109#380C2×C412yes—
110C4×C4⋊C4—21no—
111C4×C4⋊C4C2312yes—
112#336—24no—
113#335—24no—
114#334—24no—
115#328C2×C418yes—
116#328C2×C418yes—
117Q8⋊C8—28no—
118C22×Q8—21no—
119C22×Q8C4⋊C412yes—
120C23×C4C22×C411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C23×C4#4111yescharacteristic
123C2×M16—22no—
124C2×M16—41no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8C2×D414yes—
128C4⋊Q8—21no—
129C4⋊Q8—21no—
130C4⋊Q8—21no—
131C4⋊Q8—21no—
132C4⋊Q8C4⋊C412yes—
133C4⋊Q8C4⋊C412yes—
134C4⋊Q8C2×Q812yes—
135C4⋊Q8C2×Q812yes—
136#125—22no—
137#125—22no—
138#125—22no—
139#125—22no—
140C22⋊Q8—22no—
141C22⋊Q8—22no—
142C22⋊Q8—22no—
143C22⋊Q8—22no—
144C22⋊Q8—22no—
145C22⋊Q8—22no—
146C22⋊Q8C4⋊C414yes—
147C22⋊Q8C22×C414yes—
148C4×Q8—22no—
149C4×Q8—22no—
150C4×Q8—22no—
151C4×Q8—22no—
152C4×Q8—22no—
153C4×Q8—22no—
154C4×Q8C4⋊C414yes—
155C4×Q8C2×D414yes—
156#117—22no—
157#117—22no—
158#117—22no—
159#117—22no—
160#117—82no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4C4⋊C412yes—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C42—22no—
173C2×C42—22no—
174C2×C42C22×C411yescharacteristic
175C2×C42C2×D411yescharacteristic
176C2×C42C2×D411yescharacteristic
177C2×C42C2×D411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C42C2×Q811yescharacteristic
181C2×C42C2×Q811yescharacteristic
182C2×C42#4111yescharacteristic
183C2×C42#4111yescharacteristic
184C2×C42#4111yescharacteristic
185C2×C42C2411yescharacteristic
186C4⋊C8—28no—
187C4⋊C8—28no—
188C4⋊C8—22no—
189C4⋊C8—22no—
190C4⋊C8—22no—
191C4⋊C8—22no—
192C4⋊C8—22no—
193C4⋊C8—22no—
194C4⋊C8—21no—
195C4⋊C8—21no—
196C4⋊C8—22no—
197C4⋊C8—41no—
198C4⋊C8—84no—
199C4⋊C8C2×D414yes—
200C22⋊C8—22no—
201C22⋊C8—22no—
202C22⋊C8—22no—
203C22⋊C8—22no—
204C22⋊C8—22no—
205C22⋊C8—22no—
206C22⋊C8—41no—
207C22⋊C8—41no—
208C22⋊C8—41no—
209C22⋊C8C22×C412yes—
210C22⋊C8C22×C412yes—
211#97—28no—
212#97—22no—
213#97—22no—
214#97—22no—
215#97—41no—
216C4×C8—28no—
217C4×C8—22no—
218C4×C8—22no—
219C4×C8—21no—
220C4×C8—21no—
221C4×C8—22no—
222C4×C8—41no—
223C4×C8—41no—
224#95—22no—
225#95C22×C414yes—
226C24C4×D411yescharacteristic
227C2×Q8—22no—
228C2×Q8—22no—
229C2×Q8—22no—
230C2×Q8—44no—
231C2×Q8—82no—
232C2×Q8C2×C4⋊C414yes—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—24no—
238C22×C4—42no—
239C22×C4—81no—
240C22×C4C2×C4⋊C411yescharacteristic
241C22×C4C2×C4⋊C411yescharacteristic
242C22×C4C4×D411yescharacteristic
243C22×C4C4×D411yescharacteristic
244C22×C4C4×D411yescharacteristic
245C22×C4C4×D411yescharacteristic
246C22×C4C4×D411yescharacteristic
247C22×C4#12011yescharacteristic
248C22×C4#12011yescharacteristic
249C22×C4C22⋊Q811yescharacteristic
250C22×C4C22⋊Q811yescharacteristic
251C22×C4C22⋊Q811yescharacteristic
252C22×C4C22⋊Q811yescharacteristic
253C22×C4#12311yescharacteristic
254C22×C4#12311yescharacteristic
255C22×C4C23×C411yescharacteristic
256C22×C4C22×D411yescharacteristic
257C22×C4C2×#4111yescharacteristic
258M16—42no—
259M16—42no—
260M16—82no—
261C2×C8—22no—
262C2×C8—22no—
263C2×C8—24no—
264C2×C8—24no—
265C2×C8—24no—
266C2×C8—24no—
267C2×C8—44no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—42no—
271C2×C8—42no—
272C4⋊C4—24no—
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—24no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—24no—
281C4⋊C4—24no—
282C4⋊C4—22no—
283C4⋊C4—44no—
284C4⋊C4—82no—
285C4⋊C4—82no—
286C4⋊C4C2×C4⋊C414yes—
287C22⋊C4—24no—
288C22⋊C4—24no—
289C22⋊C4—82no—
290C42—24no—
291C42—24no—
292C42—44no—
293C42C2×C4⋊C411yescharacteristic
294C42C2×C4⋊C411yescharacteristic
295C42C4×D411yescharacteristic
296C42C4×D411yescharacteristic
297C42#12011yescharacteristic
298C42#12011yescharacteristic
299C42C22⋊Q811yescharacteristic
300C42C22⋊Q811yescharacteristic
301C42C22⋊Q811yescharacteristic
302C42C22⋊Q811yescharacteristic
303C42#12311yescharacteristic
304C42#12311yescharacteristic
305C42C22×D411yescharacteristic
306C42C22×D411yescharacteristic
307C42C22×Q811yescharacteristic
308C42C2×#4111yescharacteristic
309C23—21no—
310C23—21no—
311C23—41no—
312C23—41no—
313C23#38511yescharacteristic
314C23#38511yescharacteristic
315C23C2×C4×D411yescharacteristic
316Q8—44no—
317Q8—44no—
318Q8—82no—
319Q8—82no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—82no—
345C2×C4#38511yescharacteristic
346C2×C4#38511yescharacteristic
347C2×C4#38511yescharacteristic
348C2×C4#38511yescharacteristic
349C2×C4#38511yescharacteristic
350C2×C4#38511yescharacteristic
351C2×C4C22×C4⋊C411yescharacteristic
352C2×C4C2×C4×D411yescharacteristic
353C2×C4C2×#12011yescharacteristic
354C2×C4C2×C22⋊Q811yescharacteristic
355C2×C4C2×C22⋊Q811yescharacteristic
356C2×C4C2×#12311yescharacteristic
357C2×C4#57712yes—
358C2×C4#57812yes—
359C8—42no—
360C8—42no—
361C8—44no—
362C8—44no—
363C8—82no—
364C8—82no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—81no—
374C22—81no—
375C22C2×#38511yescharacteristic
376C22#294911yescharacteristic
377C22#295011yescharacteristic
378C4—21no—
379C4—21no—
380C4—21no—
381C4—21no—
382C4—21no—
383C4—21no—
384C4—21no—
385C4—21no—
386C4—21no—
387C4—44no—
388C4—44no—
389C4—84no—
390C4#186812yes—
391C4#301312yes—
392C4#308212yes—
393C2—21no—
394C2—21no—
395C2—41no—
396C2—41no—
397C2#2086711yescharacteristic
398C2#2088011yescharacteristic
399C2C2×#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
#219542
#178032
#175752
#174642
#88812
#88792
#80942
#79222