Finite Group Databasebeta

← #193166

Subgroups — #193166

Order 512.

815 conjugacy classes of subgroups.

The subgroup lattice has 431 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60251C211yesmaximal, characteristic
2#21329C211yesmaximal, characteristic
3#17820C211yesmaximal, characteristic
4C2×#1569C211yesmaximal, characteristic
5#17648C211yesmaximal, characteristic
6#8919C218yesmaximal, complemented
7#8071C211yesmaximal, characteristic
8#8068C211yesmaximal, characteristic
9C2×#533C2211yescharacteristic
10#2987C2214yes—
11#2985—22no—
12#2984—22no—
13#2983—22no—
14#2962—21no—
15#2962—21no—
16#2961—21no—
17#2961—21no—
18#2924—22no—
19#2920C2211yescharacteristic
20#2920C2211yescharacteristic
21C2×#432C2211yescharacteristic
22C2×#423C2211yescharacteristic
23C2×C4×M16C2211yescharacteristic
24C4×#117C2211yescharacteristic
25#1995C2214yes—
26#1994—22no—
27#1733—22no—
28#1574—22no—
29#1571—21no—
30#1570—21no—
31#1569—22no—
32#1569C2214yes—
33#1569C2214yes—
34C2×#335—21no—
35C2×#334—21no—
36#1531C412yescomplemented
37#1531C412yescomplemented
38#1528C414yescomplemented
39#1526C2214yes—
40#1300C2214yes—
41#1300C2214yes—
42C22×M16—21no—
43C22×M16—21no—
44#533—21no—
45#533—21no—
46#533—21no—
47#533—21no—
48#533C2×C412yes—
49#533D412yes—
50#533Q812yes—
51#533C2312yes—
52C2×C4⋊Q8Q811yescharacteristic
53C2×C4⋊Q8Q811yescharacteristic
54C2×#125C2×C412yes—
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8—21no—
57C2×C4×Q8—21no—
58C2×C4×Q8—21no—
59C2×#117—21no—
60C2×#117D412yes—
61C22×C42C2311yescharacteristic
62#446—24no—
63#446—24no—
64#446—22no—
65#446—22no—
66#446—24no—
67#446—24no—
68#446—24no—
69#446—24no—
70C8×Q8—22no—
71C8×Q8—22no—
72#433—24no—
73#433—24no—
74#432—24no—
75#432—21no—
76#432—21no—
77#432—21no—
78#432C2×C412yes—
79#432C2×C412yes—
80#432D412yes—
81#432C2312yes—
82#432C2312yes—
83#431—24no—
84#431—21no—
85#431—21no—
86#431—21no—
87#431D412yes—
88#424—22no—
89#424—22no—
90#423—21no—
91#423—21no—
92#423—21no—
93#423—21no—
94#423D411yescharacteristic
95#423D411yescharacteristic
96#423C2312yes—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8D411yescharacteristic
100C2×C4⋊C8D411yescharacteristic
101#407—21no—
102#407—21no—
103C2×C22⋊C8—21no—
104#405—22no—
105#405—22no—
106C4×M16—21no—
107C4×M16—21no—
108C4×M16—21no—
109C4×M16C2×C412yes—
110C4×M16C2312yes—
111C4×M16C2312yes—
112C2×#97—21no—
113C2×#97C2×C411yescharacteristic
114C2×#97C2×C411yescharacteristic
115C2×C4×C8—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C22⋊C4—21no—
119C4×C22⋊C4—21no—
120#376C2312yes—
121C43—21no—
122#339—28no—
123#335—24no—
124#335—24no—
125#334—24no—
126#334—24no—
127#330C2×C418yes—
128#330C2×C418yes—
129C22×Q8—21no—
130C22×Q8—21no—
131C23×C4C2×D411yescharacteristic
132C23×C4#4111yescharacteristic
133C23×C4#4111yescharacteristic
134C2×M16—22no—
135C2×M16—21no—
136C2×M16—21no—
137C2×M16—22no—
138C2×M16—22no—
139C2×M16—22no—
140C2×M16—44no—
141C2×M16—41no—
142C2×M16—44no—
143C22×C8—21no—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—41no—
148C4⋊Q8—21no—
149C4⋊Q8—21no—
150C4⋊Q8—21no—
151C4⋊Q8—21no—
152C4⋊Q8C4⋊C412yes—
153C4⋊Q8C4⋊C412yes—
154C4⋊Q8C2×Q812yes—
155C4⋊Q8C2×Q812yes—
156#125—22no—
157#125—22no—
158#125C4⋊C414yes—
159#125C22×C414yes—
160C22⋊Q8—22no—
161C22⋊Q8—22no—
162C22⋊Q8—22no—
163C22⋊Q8—22no—
164C22⋊Q8—22no—
165C22⋊Q8—22no—
166C22⋊Q8—22no—
167C22⋊Q8—22no—
168C4×Q8—22no—
169C4×Q8—22no—
170C4×Q8—22no—
171C4×Q8—22no—
172C4×Q8—22no—
173C4×Q8—22no—
174C4×Q8—22no—
175C4×Q8—22no—
176#117—22no—
177#117—22no—
178#117—24no—
179#117—42no—
180#117C4⋊C414yes—
181#117C2×D414yes—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4C4⋊C412yes—
188C2×C4⋊C4C4⋊C412yes—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—41no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—41no—
194C2×C42—41no—
195C2×C42C22×C411yescharacteristic
196C2×C42C22×C411yescharacteristic
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C42C2×Q811yescharacteristic
203C2×C42C2×Q811yescharacteristic
204C2×C42#4111yescharacteristic
205C2×C42#4111yescharacteristic
206C2×C42C2411yescharacteristic
207C4⋊C8—24no—
208C4⋊C8—24no—
209C4⋊C8—24no—
210C4⋊C8—24no—
211C4⋊C8—22no—
212C4⋊C8—22no—
213C4⋊C8—21no—
214C4⋊C8—21no—
215C4⋊C8—21no—
216C4⋊C8—21no—
217C4⋊C8—22no—
218C4⋊C8—22no—
219C4⋊C8—41no—
220C4⋊C8—41no—
221C4⋊C8C2×D412yes—
222C4⋊C8C2×D412yes—
223C4⋊C8C2×D412yes—
224C4⋊C8C2×D412yes—
225C22⋊C8—22no—
226C22⋊C8—22no—
227C22⋊C8—22no—
228C22⋊C8—22no—
229C22⋊C8—22no—
230C22⋊C8—22no—
231#97—24no—
232#97—24no—
233#97—21no—
234#97—21no—
235#97—22no—
236#97—22no—
237#97—22no—
238#97C22×C412yes—
239#97C22×C412yes—
240#97C22×C414yes—
241C4×C8—24no—
242C4×C8—24no—
243C4×C8—22no—
244C4×C8—22no—
245C4×C8—22no—
246C4×C8—22no—
247#95—22no—
248#95—22no—
249C24#12311yescharacteristic
250C2×Q8—22no—
251C2×Q8—22no—
252C2×Q8—22no—
253C2×Q8—22no—
254C2×Q8—82no—
255C2×Q8—82no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—81no—
263C22×C4C2×C4⋊C411yescharacteristic
264C22×C4C2×C4⋊C411yescharacteristic
265C22×C4C4×D411yescharacteristic
266C22×C4C4×D411yescharacteristic
267C22×C4C4×D411yescharacteristic
268C22×C4C4×D411yescharacteristic
269C22×C4#12011yescharacteristic
270C22×C4#12011yescharacteristic
271C22×C4C22⋊Q811yescharacteristic
272C22×C4C22⋊Q811yescharacteristic
273C22×C4C22⋊Q811yescharacteristic
274C22×C4C22⋊Q811yescharacteristic
275C22×C4#12311yescharacteristic
276C22×C4#12311yescharacteristic
277C22×C4#12311yescharacteristic
278C22×C4C22×D411yescharacteristic
279C22×C4C2×#4111yescharacteristic
280C22×C4C2×#4111yescharacteristic
281M16—44no—
282M16—44no—
283M16—42no—
284M16—42no—
285M16—44no—
286M16—82no—
287M16—82no—
288C2×C8—24no—
289C2×C8—22no—
290C2×C8—22no—
291C2×C8—24no—
292C2×C8—22no—
293C2×C8—22no—
294C2×C8—22no—
295C2×C8—22no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—82no—
301C4⋊C4—24no—
302C4⋊C4—24no—
303C4⋊C4—22no—
304C4⋊C4—22no—
305C4⋊C4—24no—
306C4⋊C4—24no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—44no—
312C4⋊C4C2×C4⋊C414yes—
313C4⋊C4C2×C4⋊C414yes—
314C22⋊C4—24no—
315C22⋊C4—24no—
316C22⋊C4—44no—
317C42—24no—
318C42—24no—
319C42—24no—
320C42—42no—
321C42—82no—
322C42—82no—
323C42C2×C4⋊C411yescharacteristic
324C42C2×C4⋊C411yescharacteristic
325C42C4×D411yescharacteristic
326C42C4×D411yescharacteristic
327C42C4×D411yescharacteristic
328C42C4×D411yescharacteristic
329C42#12011yescharacteristic
330C42#12011yescharacteristic
331C42C22⋊Q811yescharacteristic
332C42C22⋊Q811yescharacteristic
333C42C22⋊Q811yescharacteristic
334C42C22⋊Q811yescharacteristic
335C42C23×C411yescharacteristic
336C42C22×D411yescharacteristic
337C42C22×D411yescharacteristic
338C42C22×Q811yescharacteristic
339C23—22no—
340C23—41no—
341C23—41no—
342C23#38511yescharacteristic
343C23#38511yescharacteristic
344C23C2×#12311yescharacteristic
345Q8—44no—
346Q8—82no—
347Q8—82no—
348Q8—82no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—24no—
360C2×C4—24no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4#38511yescharacteristic
376C2×C4#38511yescharacteristic
377C2×C4#38511yescharacteristic
378C2×C4#38511yescharacteristic
379C2×C4#38511yescharacteristic
380C2×C4#38511yescharacteristic
381C2×C4#45412yes—
382C2×C4#45612yes—
383C2×C4C22×C4⋊C411yescharacteristic
384C2×C4C2×C4×D411yescharacteristic
385C2×C4C2×C4×D411yescharacteristic
386C2×C4C2×#12011yescharacteristic
387C2×C4C2×C22⋊Q811yescharacteristic
388C2×C4C2×C22⋊Q811yescharacteristic
389C8—24no—
390C8—42no—
391C8—44no—
392C8—42no—
393C8—42no—
394C8—42no—
395C8—42no—
396C8—82no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—82no—
406C22#185712yes—
407C22C2×#38511yescharacteristic
408C4—21no—
409C4—21no—
410C4—21no—
411C4—21no—
412C4—21no—
413C4—21no—
414C4—21no—
415C4—21no—
416C4—21no—
417C4—44no—
418C4—44no—
419C4—84no—
420C4#185712yes—
421C4#186112yes—
422C4C2×#45411yescharacteristic
423C4C2×#45611yescharacteristic
424C2—22no—
425C2—41no—
426C2—41no—
427C2C2×#185711yescharacteristic
428C2C2×#186111yescharacteristic
429C2#2086711yescharacteristic
subgroupG/H
centerC2×C4C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#385
prime radicalC24#123
LCSG ▷ C42 ▷ 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. #8919C2semidirect
36. #1531C4semidirect
37. #1531C4semidirect
38. #1528C4semidirect

Maximal subgroups

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

subgroupindex
#602512
#213292
#178202
C2×#15692
#176482
#89192
#80712
#80682