Finite Group Databasebeta

← #111127

Subgroups — #111127

Order 512.

818 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1826C211yesmaximal, characteristic
2#20699C211yesmaximal, characteristic
3#20685C211yesmaximal, characteristic
4#20523C211yesmaximal, characteristic
5#8008C211yesmaximal, characteristic
6#7942C211yesmaximal, characteristic
7#7927C211yesmaximal, characteristic
8C22×#117C2211yescharacteristic
9C2×#407C2211yescharacteristic
10C2×#407C2211yescharacteristic
11C22×C22⋊C8C2211yescharacteristic
12C22×C22⋊C8C2211yescharacteristic
13#2280C2211yescharacteristic
14C2×C4×C22⋊C4C2211yescharacteristic
15#1826—22no—
16#1826C414yescomplemented
17#1825—22no—
18#1824C412yescomplemented
19#1823C412yescomplemented
20#1818C414yescomplemented
21#1817—22no—
22C8×C22⋊C4—22no—
23C8×C22⋊C4—22no—
24C4×C22⋊C8—22no—
25#1735—22no—
26C2×#339—22no—
27#1729—22no—
28#1293—22no—
29#1284—22no—
30#1282—22no—
31C24×C4C2311yescharacteristic
32C22×M16—21no—
33C22×M16—21no—
34C23×C8D412yes—
35C23×C8D412yes—
36C2×#117—21no—
37C2×#117—21no—
38C2×#117—21no—
39C2×#117—21no—
40C2×#117—21no—
41C2×#117—21no—
42C2×#117—21no—
43C2×#117—21no—
44C2×#117—21no—
45C2×#117—21no—
46C2×#117—21no—
47C2×#117C2×C412yes—
48C2×#117C2×C412yes—
49C22×C4⋊C4C2×C411yescharacteristic
50C22×C4⋊C4C2×C411yescharacteristic
51C22×C22⋊C4—21no—
52C22×C22⋊C4—21no—
53C22×C42—21no—
54C22×C42D412yes—
55#407—21no—
56#407—21no—
57#407—21no—
58#407—21no—
59#407—21no—
60#407—21no—
61#407—21no—
62#407C2×C412yes—
63C2×C22⋊C8—21no—
64C2×C22⋊C8—21no—
65C2×C22⋊C8—21no—
66C2×C22⋊C8—21no—
67C2×C22⋊C8—21no—
68C2×C22⋊C8—21no—
69C2×C22⋊C8—21no—
70C2×C22⋊C8—21no—
71C2×C22⋊C8—21no—
72C2×C22⋊C8—21no—
73C2×C22⋊C8—21no—
74C2×C22⋊C8C2×C412yes—
75C2×C22⋊C8C2×C412yes—
76C2×C22⋊C8C2×C412yes—
77C2×C22⋊C8D412yes—
78C2×C22⋊C8D412yes—
79C2×C4×C8—24no—
80#381—21no—
81#380—21no—
82#380C2×C412yes—
83#379—21no—
84C4×C22⋊C4—21no—
85C4×C22⋊C4—21no—
86C4×C22⋊C4—21no—
87C4×C22⋊C4—21no—
88C4×C22⋊C4C2×C412yes—
89C4×C22⋊C4D412yes—
90C2×#95—21no—
91#340—24no—
92#340—24no—
93#339—28no—
94#336—24no—
95#336—28no—
96C25#4111yescharacteristic
97C23×C4—22no—
98C23×C4—21no—
99C23×C4—21no—
100C23×C4—44no—
101C23×C4C22×C411yescharacteristic
102C23×C4C22×C411yescharacteristic
103C23×C4C22×C411yescharacteristic
104C23×C4C2×D411yescharacteristic
105C23×C4C2×D411yescharacteristic
106C23×C4C2×D411yescharacteristic
107C23×C4C2×D411yescharacteristic
108C23×C4C2×D411yescharacteristic
109C23×C4C2×D411yescharacteristic
110C23×C4#4111yescharacteristic
111C23×C4#4111yescharacteristic
112C23×C4#4111yescharacteristic
113C23×C4#4111yescharacteristic
114C23×C4#4111yescharacteristic
115C2×M16—22no—
116C2×M16—44no—
117C2×M16—41no—
118C2×M16—41no—
119C2×M16—41no—
120C2×M16—44no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—44no—
127C22×C8—44no—
128C22×C8—41no—
129C22×C8C22⋊C414yes—
130C22×C8C22⋊C414yes—
131#117—22no—
132#117—22no—
133#117—22no—
134#117—22no—
135#117—22no—
136#117—22no—
137#117—22no—
138#117—22no—
139#117—41no—
140#117—41no—
141#117—41no—
142#117—41no—
143#117—41no—
144#117—41no—
145#117—41no—
146#117—41no—
147#117—44no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4C4212yes—
160C2×C4⋊C4C4212yes—
161C2×C22⋊C4—28no—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—22no—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—41no—
168C2×C22⋊C4—41no—
169C2×C22⋊C4—41no—
170C2×C22⋊C4—41no—
171C2×C22⋊C4—41no—
172C2×C22⋊C4—41no—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—41no—
177C2×C42—41no—
178C2×C42—41no—
179C2×C42—41no—
180C2×C42—44no—
181C2×C42—41no—
182C2×C42C22⋊C414yes—
183C22⋊C8—22no—
184C22⋊C8—22no—
185C22⋊C8—22no—
186C22⋊C8—22no—
187C22⋊C8—22no—
188C22⋊C8—22no—
189C22⋊C8—22no—
190C22⋊C8—22no—
191C22⋊C8—22no—
192C22⋊C8—22no—
193C22⋊C8—22no—
194C22⋊C8—22no—
195C22⋊C8—41no—
196C22⋊C8—41no—
197C22⋊C8—41no—
198C22⋊C8—41no—
199C4×C8—84no—
200#95—22no—
201#95—22no—
202#95—22no—
203#95—41no—
204C24—24no—
205C24—42no—
206C24—81no—
207C24#11711yescharacteristic
208C24C4×D411yescharacteristic
209C24C4×D411yescharacteristic
210C24#12111yescharacteristic
211C24#12311yescharacteristic
212C24#12411yescharacteristic
213C24#12611yescharacteristic
214C22×C4—24no—
215C22×C4—24no—
216C22×C4—24no—
217C22×C4—24no—
218C22×C4—28no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—44no—
237C22×C4—42no—
238C22×C4—82no—
239C22×C4—82no—
240C22×C4—82no—
241C22×C4C2×C4211yescharacteristic
242C22×C4C2×C22⋊C411yescharacteristic
243C22×C4C2×C22⋊C411yescharacteristic
244C22×C4C2×C22⋊C411yescharacteristic
245C22×C4#11711yescharacteristic
246C22×C4#11711yescharacteristic
247C22×C4C4×D411yescharacteristic
248C22×C4C4×D411yescharacteristic
249C22×C4C4×D411yescharacteristic
250C22×C4C4×D411yescharacteristic
251C22×C4C4×D411yescharacteristic
252C22×C4C4×D411yescharacteristic
253C22×C4C4×D411yescharacteristic
254C22×C4C4×D411yescharacteristic
255C22×C4C4×D411yescharacteristic
256C22×C4C4×D411yescharacteristic
257C22×C4#12011yescharacteristic
258C22×C4#12111yescharacteristic
259C22×C4#12111yescharacteristic
260C22×C4#12111yescharacteristic
261C22×C4#12111yescharacteristic
262C22×C4#12111yescharacteristic
263C22×C4#12311yescharacteristic
264C22×C4#12311yescharacteristic
265C22×C4#12411yescharacteristic
266C22×C4#12411yescharacteristic
267C22×C4#12611yescharacteristic
268C22×C4#12711yescharacteristic
269M16—44no—
270M16—82no—
271M16—82no—
272M16—82no—
273C2×C8—24no—
274C2×C8—24no—
275C2×C8—44no—
276C2×C8—44no—
277C2×C8—44no—
278C2×C8—44no—
279C2×C8—44no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—82no—
283C2×C8—82no—
284C2×C8—82no—
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4—44no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C22⋊C4—28no—
295C22⋊C4—28no—
296C22⋊C4—42no—
297C22⋊C4—42no—
298C22⋊C4—42no—
299C22⋊C4—42no—
300C22⋊C4—44no—
301C22⋊C4—82no—
302C22⋊C4—82no—
303C42—44no—
304C42—42no—
305C42—42no—
306C42—42no—
307C42—42no—
308C42—44no—
309C42—82no—
310C23—22no—
311C23—22no—
312C23—22no—
313C23—22no—
314C23—22no—
315C23—22no—
316C23—22no—
317C23—22no—
318C23—22no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23—88no—
331C23C4×C22⋊C411yescharacteristic
332C23#38611yescharacteristic
333C23#38811yescharacteristic
334C23#38811yescharacteristic
335C23#38811yescharacteristic
336C23#38811yescharacteristic
337C23#39011yescharacteristic
338C23#44314yes—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—28no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—44no—
373C2×C4—88no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4C4×C22⋊C411yescharacteristic
382C2×C4C4×C22⋊C411yescharacteristic
383C2×C4#38611yescharacteristic
384C2×C4#38611yescharacteristic
385C2×C4#38811yescharacteristic
386C2×C4#38811yescharacteristic
387C2×C4#39011yescharacteristic
388C2×C4#39011yescharacteristic
389C2×C4#45814yes—
390C8—44no—
391C8—44no—
392C8—82no—
393C8—82no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—44no—
412C22—44no—
413C22—44no—
414C22—44no—
415C22—82no—
416C22—82no—
417C22—82no—
418C22#143711yescharacteristic
419C22#176412yes—
420C22#189012yes—
421C22#193712yes—
422C22#197412yes—
423C4—21no—
424C4—21no—
425C4—21no—
426C4—21no—
427C4—21no—
428C4—21no—
429C4—48no—
430C4—82no—
431C4—82no—
432C4—162no—
433C4#189012yes—
434C4#189012yes—
435C2—22no—
436C2—22no—
437C2—22no—
438C2—22no—
439C2—41no—
440C2—41no—
441C2—41no—
442C2#1092911yescharacteristic
443C2#1093611yescharacteristic
444C2#1095011yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC22×C4C2×C42
Frattini, cosocleC24×C4C23
socleC22#1437
prime radicalC25#41
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
16. #1826C4semidirect
18. #1824C4semidirect
19. #1823C4semidirect
20. #1818C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18262
#206992
#206852
#205232
#80082
#79422
#79272