Finite Group Databasebeta

← #120810

Subgroups — #120810

Order 512.

563 conjugacy classes of subgroups.

The subgroup lattice has 427 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21944C211yesmaximal, characteristic
2#8107C211yesmaximal, characteristic
3C2×#1305C211yesmaximal, characteristic
4#8057C211yesmaximal, characteristic
5#8023C211yesmaximal, characteristic
6#7983C211yesmaximal, characteristic
7#7927C211yesmaximal, characteristic
8C2×#407C2211yescharacteristic
9C2×#407C2211yescharacteristic
10C2×#407C2211yescharacteristic
11C22×C22⋊C8C2211yescharacteristic
12#2280C2211yescharacteristic
13C2×#380C2211yescharacteristic
14C2×#380C2211yescharacteristic
15#1854—22no—
16#1853—21no—
17#1852—21no—
18#1809—22no—
19#1305—22no—
20#1305C414yescomplemented
21#1298—22no—
22#1294—22no—
23#1289—21no—
24#1289—21no—
25#1282—22no—
26C24×C4C2311yescharacteristic
27C22×M16—21no—
28C22×M16—21no—
29C22×M16Q812yes—
30C23×C8—21no—
31C2×#117—21no—
32C22×C4⋊C4—21no—
33C22×C4⋊C4—21no—
34C22×C22⋊C4—21no—
35C22×C22⋊C4—21no—
36C2×#107—22no—
37C2×#106—22no—
38#407—21no—
39#407—21no—
40#407—21no—
41#407—21no—
42#407—21no—
43#407—21no—
44#407—21no—
45#407—21no—
46#407—21no—
47#407C2×C412yes—
48#407C2×C412yes—
49#407Q812yes—
50C2×C22⋊C8—21no—
51C2×C22⋊C8—21no—
52C2×C22⋊C8—21no—
53C2×C22⋊C8—21no—
54C2×C22⋊C8—21no—
55C2×C22⋊C8—21no—
56C2×C22⋊C8—21no—
57C2×C22⋊C8—21no—
58C2×C22⋊C8—21no—
59C2×C22⋊C8—21no—
60C2×C22⋊C8D412yes—
61C2×C22⋊C8D412yes—
62#381—21no—
63#380—21no—
64#380—21no—
65#380—21no—
66#380—21no—
67#380—21no—
68#380—21no—
69#380—21no—
70#380C2×C412yes—
71#380D411yescharacteristic
72#380D411yescharacteristic
73#380D412yes—
74#379—21no—
75C4×C22⋊C4—21no—
76C4×C22⋊C4—21no—
77C2×#95—21no—
78C2×#95—21no—
79#340—22no—
80#340—22no—
81C25#4111yescharacteristic
82C23×C4—41no—
83C23×C4—41no—
84C23×C4—82no—
85C23×C4C22×C411yescharacteristic
86C23×C4C2×D411yescharacteristic
87C23×C4C2×D411yescharacteristic
88C23×C4C2×D411yescharacteristic
89C23×C4C2×D411yescharacteristic
90C23×C4C2×Q811yescharacteristic
91C23×C4C2×Q811yescharacteristic
92C23×C4#4111yescharacteristic
93C23×C4#4111yescharacteristic
94C23×C4#4111yescharacteristic
95C23×C4#4111yescharacteristic
96C23×C4#4111yescharacteristic
97C23×C4#4111yescharacteristic
98C23×C4#4111yescharacteristic
99C2×M16—22no—
100C2×M16—22no—
101C2×M16—22no—
102C2×M16—22no—
103C2×M16—22no—
104C2×M16—41no—
105C2×M16—82no—
106C2×M16—82no—
107C2×M16—82no—
108C22×C8—22no—
109C22×C8—22no—
110C22×C8—22no—
111C22×C8—41no—
112C22×C8—41no—
113C22×C8—41no—
114C22×C8—82no—
115#117—82no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4—82no—
123C2×C4⋊C4—82no—
124C2×C22⋊C4—21no—
125C2×C22⋊C4—21no—
126C2×C22⋊C4—22no—
127C2×C22⋊C4—22no—
128C2×C22⋊C4—41no—
129C2×C22⋊C4—41no—
130C2×C22⋊C4—41no—
131C2×C22⋊C4—41no—
132C2×C22⋊C4—81no—
133C2×C22⋊C4—81no—
134C2×C22⋊C4—81no—
135C2×C22⋊C4—81no—
136C2×C42—41no—
137#107—82no—
138#106—82no—
139C22⋊C8—22no—
140C22⋊C8—22no—
141C22⋊C8—22no—
142C22⋊C8—22no—
143C22⋊C8—22no—
144C22⋊C8—22no—
145C22⋊C8—22no—
146C22⋊C8—41no—
147C22⋊C8—41no—
148C22⋊C8—41no—
149C22⋊C8—41no—
150C22⋊C8—41no—
151C22⋊C8—41no—
152C22⋊C8—41no—
153C22⋊C8—41no—
154C22⋊C8C4⋊C414yes—
155#95—22no—
156#95—22no—
157#95—21no—
158#95—21no—
159#95—41no—
160#95—41no—
161#95—41no—
162C24—21no—
163C24—21no—
164C24—21no—
165C24—21no—
166C24—81no—
167C24—81no—
168C24#11711yescharacteristic
169C24#12111yescharacteristic
170C24#12311yescharacteristic
171C24#12411yescharacteristic
172C24#12511yescharacteristic
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—42no—
183C22×C4—41no—
184C22×C4—41no—
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×C4—82no—
197C22×C4—82no—
198C22×C4—81no—
199C22×C4—81no—
200C22×C4C2×C4⋊C411yescharacteristic
201C22×C4#11711yescharacteristic
202C22×C4C4×D411yescharacteristic
203C22×C4C4×D411yescharacteristic
204C22×C4C4×D411yescharacteristic
205C22×C4C4×Q811yescharacteristic
206C22×C4#12011yescharacteristic
207C22×C4#12111yescharacteristic
208C22×C4#12111yescharacteristic
209C22×C4C22⋊Q811yescharacteristic
210C22×C4C22⋊Q811yescharacteristic
211C22×C4C22⋊Q811yescharacteristic
212C22×C4C22⋊Q811yescharacteristic
213C22×C4C22⋊Q811yescharacteristic
214C22×C4#12311yescharacteristic
215C22×C4#12311yescharacteristic
216C22×C4#12311yescharacteristic
217C22×C4#12311yescharacteristic
218C22×C4#12411yescharacteristic
219C22×C4#12411yescharacteristic
220C22×C4#12411yescharacteristic
221C22×C4#12511yescharacteristic
222C22×C4#12511yescharacteristic
223C22×C4#12511yescharacteristic
224C22×C4#12611yescharacteristic
225C22×C4#12611yescharacteristic
226C22×C4#12611yescharacteristic
227C22×C4C4⋊Q811yescharacteristic
228M16—82no—
229M16—82no—
230M16—82no—
231M16—82no—
232M16—82no—
233M16—82no—
234C2×C8—24no—
235C2×C8—42no—
236C2×C8—42no—
237C2×C8—42no—
238C2×C8—42no—
239C2×C8—42no—
240C2×C8—82no—
241C2×C8—82no—
242C2×C8—82no—
243C2×C8—82no—
244C2×C8—82no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—82no—
250C4⋊C4—82no—
251C4⋊C4—82no—
252C22⋊C4—41no—
253C22⋊C4—41no—
254C22⋊C4—41no—
255C22⋊C4—41no—
256C22⋊C4—82no—
257C22⋊C4—82no—
258C22⋊C4—82no—
259C22⋊C4—81no—
260C22⋊C4—81no—
261C22⋊C4—82no—
262C22⋊C4—82no—
263C42—82no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23—82no—
291C23—82no—
292C23—82no—
293C23—82no—
294C23#38211yescharacteristic
295C23#38711yescharacteristic
296C23#38811yescharacteristic
297C23#39411yescharacteristic
298C23#39811yescharacteristic
299C23#40011yescharacteristic
300C23#40011yescharacteristic
301C23#47111yescharacteristic
302C23#47111yescharacteristic
303C23#47211yescharacteristic
304C23#47311yescharacteristic
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
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—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—82no—
338C2×C4—82no—
339C2×C4—82no—
340C2×C4—82no—
341C2×C4—81no—
342C2×C4—82no—
343C2×C4—81no—
344C2×C4—82no—
345C2×C4—162no—
346C2×C4—162no—
347C2×C4—164no—
348C2×C4#38511yescharacteristic
349C2×C4#38711yescharacteristic
350C2×C4#38811yescharacteristic
351C2×C4#38911yescharacteristic
352C2×C4#39311yescharacteristic
353C2×C4#39711yescharacteristic
354C2×C4#39811yescharacteristic
355C2×C4#40011yescharacteristic
356C8—44no—
357C8—82no—
358C8—82no—
359C8—82no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—42no—
383C22—42no—
384C22—42no—
385C22—42no—
386C22—81no—
387C22—81no—
388C22—81no—
389C22—81no—
390C22—81no—
391C22—81no—
392C22—81no—
393C22—81no—
394C22#144211yescharacteristic
395C22#193011yescharacteristic
396C22#193111yescharacteristic
397C22#205111yescharacteristic
398C22#205211yescharacteristic
399C22#208911yescharacteristic
400C22#209011yescharacteristic
401C4—21no—
402C4—21no—
403C4—21no—
404C4—21no—
405C4—21no—
406C4—21no—
407C4—21no—
408C4—161no—
409C4—161no—
410C4—162no—
411C4—162no—
412C4#203212yes—
413C2—21no—
414C2—21no—
415C2—21no—
416C2—21no—
417C2—21no—
418C2—21no—
419C2—41no—
420C2—41no—
421C2—41no—
422C2—41no—
423C2#1127911yescharacteristic
424C2#1128411yescharacteristic
425C2#1129011yescharacteristic
subgroupG/H
center, socleC22#1442
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
20. #1305C4semidirect

Maximal subgroups

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

subgroupindex
#219442
#81072
C2×#13052
#80572
#80232
#79832
#79272