Finite Group Databasebeta

← #120231

Subgroups — #120231

Order 512.

569 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21949C211yesmaximal, characteristic
2#21944C211yesmaximal, characteristic
3#8108C211yesmaximal, characteristic
4#8107C211yesmaximal, characteristic
5#8104C211yesmaximal, characteristic
6#7983C211yesmaximal, characteristic
7#7946C211yesmaximal, characteristic
8C2×#407C2211yescharacteristic
9C2×#407C2211yescharacteristic
10C2×#407C2211yescharacteristic
11C2×#407C2211yescharacteristic
12#2280C2211yescharacteristic
13#2280C2211yescharacteristic
14C2×#380C2211yescharacteristic
15#1855—22no—
16#1855C414yescomplemented
17#1854—22no—
18#1853—21no—
19#1852—21no—
20#1811—22no—
21#1809—22no—
22#1305—22no—
23#1305—22no—
24#1305—21no—
25#1305—21no—
26#1289—22no—
27#1284—22no—
28C24×C4C2311yescharacteristic
29C22×M16—21no—
30C22×M16—21no—
31C22×M16C2×C412yes—
32C22×M16D412yes—
33C2×#117—21no—
34C2×#117—21no—
35C22×C4⋊C4—21no—
36C22×C22⋊C4—21no—
37C2×#108—24no—
38C2×#107—22no—
39C2×#106—22no—
40#407—21no—
41#407—21no—
42#407—21no—
43#407—21no—
44#407—21no—
45#407—21no—
46#407—21no—
47#407—21no—
48#407—21no—
49#407—21no—
50#407—21no—
51#407C2×C412yes—
52#407D412yes—
53#407D412yes—
54#407Q812yes—
55#407Q812yes—
56C2×C22⋊C8—21no—
57C2×C22⋊C8—21no—
58C2×C22⋊C8—21no—
59C2×C22⋊C8—21no—
60C2×C22⋊C8—21no—
61C2×C22⋊C8—21no—
62C2×C22⋊C8—21no—
63C2×C22⋊C8C2×C412yes—
64#381—21no—
65#381—21no—
66#380—21no—
67#380—21no—
68#380—21no—
69#380—21no—
70#380—21no—
71#380—21no—
72#380D412yes—
73#380D411yescharacteristic
74#380D411yescharacteristic
75#379—21no—
76#379—21no—
77C4×C22⋊C4—21no—
78C4×C22⋊C4—21no—
79C4×C22⋊C4—21no—
80C4×C22⋊C4Q812yes—
81C2×#95—21no—
82#341—24no—
83#340—22no—
84#340—22no—
85C25#4111yescharacteristic
86C23×C4—41no—
87C23×C4—82no—
88C23×C4C22×C411yescharacteristic
89C23×C4C2×D411yescharacteristic
90C23×C4C2×D411yescharacteristic
91C23×C4C2×D411yescharacteristic
92C23×C4C2×D411yescharacteristic
93C23×C4C2×D411yescharacteristic
94C23×C4C2×Q811yescharacteristic
95C23×C4C2×Q811yescharacteristic
96C23×C4C2×Q811yescharacteristic
97C23×C4#4111yescharacteristic
98C23×C4#4111yescharacteristic
99C23×C4#4111yescharacteristic
100C23×C4#4111yescharacteristic
101C23×C4#4111yescharacteristic
102C2×M16—22no—
103C2×M16—22no—
104C2×M16—22no—
105C2×M16—22no—
106C2×M16—22no—
107C2×M16—22no—
108C2×M16—41no—
109C2×M16—82no—
110C2×M16—82no—
111C2×M16—82no—
112C2×M16—82no—
113C2×M16C22⋊C414yes—
114C22×C8—22no—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—41no—
118#117—82no—
119#117—82no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4—41no—
125C2×C4⋊C4—82no—
126C2×C22⋊C4—22no—
127C2×C22⋊C4—22no—
128C2×C22⋊C4—21no—
129C2×C22⋊C4—21no—
130C2×C22⋊C4—41no—
131C2×C22⋊C4—41no—
132C2×C22⋊C4—41no—
133C2×C22⋊C4—41no—
134C2×C22⋊C4—41no—
135C2×C22⋊C4—82no—
136C2×C42—22no—
137C2×C42—41no—
138#108—84no—
139#107—82no—
140#106—82no—
141C22⋊C8—22no—
142C22⋊C8—22no—
143C22⋊C8—22no—
144C22⋊C8—22no—
145C22⋊C8—22no—
146C22⋊C8—22no—
147C22⋊C8—22no—
148C22⋊C8—22no—
149C22⋊C8—22no—
150C22⋊C8—41no—
151C22⋊C8—41no—
152C22⋊C8—41no—
153C22⋊C8—41no—
154C22⋊C8—41no—
155C22⋊C8—41no—
156C22⋊C8C4⋊C414yes—
157#95—21no—
158#95—21no—
159#95—22no—
160#95—22no—
161#95—41no—
162#95—41no—
163#95—41no—
164C24—41no—
165C24—41no—
166C24—81no—
167C24—81no—
168C24C4×Q811yescharacteristic
169C24#12111yescharacteristic
170C24C22⋊Q811yescharacteristic
171C24#12311yescharacteristic
172C24#12411yescharacteristic
173C24#12511yescharacteristic
174C24#12611yescharacteristic
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—81no—
187C22×C4—81no—
188C22×C4—81no—
189C22×C4—81no—
190C22×C4—81no—
191C22×C4—81no—
192C22×C4—81no—
193C22×C4—82no—
194C22×C4—82no—
195C22×C4—82no—
196C22×C4C2×C22⋊C411yescharacteristic
197C22×C4C2×C4⋊C411yescharacteristic
198C22×C4#11711yescharacteristic
199C22×C4C4×D411yescharacteristic
200C22×C4C4×D411yescharacteristic
201C22×C4C4×Q811yescharacteristic
202C22×C4#12011yescharacteristic
203C22×C4#12111yescharacteristic
204C22×C4#12111yescharacteristic
205C22×C4#12111yescharacteristic
206C22×C4#12111yescharacteristic
207C22×C4#12111yescharacteristic
208C22×C4C22⋊Q811yescharacteristic
209C22×C4C22⋊Q811yescharacteristic
210C22×C4C22⋊Q811yescharacteristic
211C22×C4C22⋊Q811yescharacteristic
212C22×C4C22⋊Q811yescharacteristic
213C22×C4C22⋊Q811yescharacteristic
214C22×C4C22⋊Q811yescharacteristic
215C22×C4#12311yescharacteristic
216C22×C4#12311yescharacteristic
217C22×C4#12511yescharacteristic
218C22×C4#12511yescharacteristic
219C22×C4#12511yescharacteristic
220C22×C4#12611yescharacteristic
221C22×C4C4⋊Q811yescharacteristic
222C22×C4C4⋊Q811yescharacteristic
223C22×C4C4⋊Q811yescharacteristic
224M16—44no—
225M16—44no—
226M16—82no—
227M16—82no—
228M16—82no—
229M16—82no—
230M16—82no—
231M16—82no—
232C2×C8—24no—
233C2×C8—24no—
234C2×C8—42no—
235C2×C8—42no—
236C2×C8—44no—
237C2×C8—42no—
238C2×C8—42no—
239C2×C8—82no—
240C2×C8—82no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—82no—
246C4⋊C4—82no—
247C22⋊C4—81no—
248C22⋊C4—81no—
249C22⋊C4—82no—
250C22⋊C4—82no—
251C22⋊C4—82no—
252C22⋊C4—82no—
253C22⋊C4—81no—
254C22⋊C4—81no—
255C22⋊C4—82no—
256C22⋊C4—82no—
257C42—82no—
258C42—82no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—84no—
284C23—84no—
285C23#38711yescharacteristic
286C23#38911yescharacteristic
287C23#39111yescharacteristic
288C23#39411yescharacteristic
289C23#39611yescharacteristic
290C23#39811yescharacteristic
291C23#40011yescharacteristic
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—82no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—164no—
333C2×C4#38011yescharacteristic
334C2×C4#38611yescharacteristic
335C2×C4#38711yescharacteristic
336C2×C4#38911yescharacteristic
337C2×C4#39511yescharacteristic
338C2×C4#39611yescharacteristic
339C2×C4#39811yescharacteristic
340C2×C4#39811yescharacteristic
341C8—44no—
342C8—44no—
343C8—82no—
344C8—82no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—42no—
366C22—42no—
367C22—81no—
368C22—81no—
369C22—81no—
370C22—81no—
371C22—82no—
372C22—82no—
373C22—82no—
374C22—82no—
375C22#144111yescharacteristic
376C22#204811yescharacteristic
377C22#204911yescharacteristic
378C4—21no—
379C4—21no—
380C4—21no—
381C4—21no—
382C4—21no—
383C4—21no—
384C4—21no—
385C4—161no—
386C4—161no—
387C4—162no—
388C4—162no—
389C4#191312yes—
390C2—21no—
391C2—21no—
392C2—41no—
393C2—41no—
394C2—41no—
395C2—41no—
396C2—41no—
397C2—41no—
398C2#1120511yescharacteristic
399C2#1124711yescharacteristic
400C2#1125311yescharacteristic
subgroupG/H
center, socleC22#1441
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25#41
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
16. #1855C4semidirect

Maximal subgroups

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

subgroupindex
#219492
#219442
#81082
#81072
#81042
#79832
#79462