Finite Group Databasebeta

← #127502

Subgroups — #127502

Order 512.

832 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20627C211yesmaximal, characteristic
2C2×#1742C211yesmaximal, characteristic
3C2×#1458C211yesmaximal, characteristic
4#7253C214yesmaximal, complemented
5C23×M16—21no—
6C22×#117C2211yescharacteristic
7C2×#431—21no—
8C2×#423C411yescharacteristic
9C2×#423C411yescharacteristic
10C2×#407—21no—
11#1798—21no—
12#1798—21no—
13#1798C412yes—
14#1798C2212yes—
15#1742—21no—
16#1742—21no—
17#1742C412yes—
18#1742C2212yes—
19C2×#343—21no—
20C2×#343C412yes—
21#1728—21no—
22#1458—21no—
23#1458—21no—
24#1458C412yes—
25#1458C2212yes—
26C2×#324—21no—
27C2×#323C412yes—
28C24×C4D411yescharacteristic
29C22×M16—21no—
30C22×M16—21no—
31C22×M16—24no—
32C22×M16—24no—
33C22×M16—22no—
34C22×M16—22no—
35C22×M16—22no—
36C22×M16—41no—
37C22×M16—41no—
38C22×M16—41no—
39C22×M16—42no—
40C22×M16—42no—
41C23×C8—41no—
42C2×#117—41no—
43C2×#117—41no—
44C2×#117—41no—
45C2×#117—41no—
46C2×#117C2×C411yescharacteristic
47C2×#117C2×C411yescharacteristic
48C2×#117C2×C411yescharacteristic
49C2×#117D411yescharacteristic
50C2×#117D411yescharacteristic
51C2×#117D411yescharacteristic
52C2×#117Q811yescharacteristic
53C2×#117C2311yescharacteristic
54C22×C4⋊C4C2×C411yescharacteristic
55C22×C4⋊C4D411yescharacteristic
56C22×C22⋊C4C2×C411yescharacteristic
57C22×C22⋊C4Q811yescharacteristic
58C22×C42C2×C411yescharacteristic
59C22×C42D411yescharacteristic
60#431—22no—
61#431—22no—
62#431—41no—
63#431—41no—
64#423—41no—
65#423—41no—
66#423—41no—
67#423—41no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—41no—
71C2×C4⋊C8C2×C412yes—
72C2×C4⋊C8C2×C412yes—
73#407—42no—
74#407—42no—
75C2×C22⋊C8—22no—
76C2×C22⋊C8—22no—
77C2×C22⋊C8—22no—
78C2×C4×C8—41no—
79#343—22no—
80#343—22no—
81#343—22no—
82#343C2×C414yes—
83#336—22no—
84#336—22no—
85#324—22no—
86#324—22no—
87#323—22no—
88#323C2×C414yes—
89C25C4⋊C411yescharacteristic
90C23×C4—21no—
91C23×C4—21no—
92C23×C4—21no—
93C23×C4—21no—
94C23×C4—22no—
95C23×C4—22no—
96C23×C4C4211yescharacteristic
97C23×C4C22⋊C411yescharacteristic
98C23×C4C22⋊C411yescharacteristic
99C23×C4C22⋊C411yescharacteristic
100C23×C4C22⋊C411yescharacteristic
101C23×C4C22⋊C411yescharacteristic
102C23×C4C4⋊C411yescharacteristic
103C23×C4C4⋊C411yescharacteristic
104C23×C4C4⋊C411yescharacteristic
105C23×C4D812yes—
106C23×C4SD1612yes—
107C23×C4SD1612yes—
108C23×C4Q1612yes—
109C23×C4C2×D411yescharacteristic
110C2×M16—28no—
111C2×M16—28no—
112C2×M16—24no—
113C2×M16—24no—
114C2×M16—24no—
115C2×M16—42no—
116C2×M16—44no—
117C2×M16—44no—
118C2×M16—44no—
119C2×M16—44no—
120C2×M16—44no—
121C2×M16—44no—
122C2×M16—42no—
123C2×M16—42no—
124C2×M16—44no—
125C2×M16—44no—
126C2×M16—42no—
127C2×M16—42no—
128C2×M16—41no—
129C2×M16—41no—
130C2×M16—41no—
131C2×M16—41no—
132C2×M16—42no—
133C2×M16—42no—
134C2×M16—42no—
135C2×M16—42no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—42no—
139C22×C8—44no—
140C22×C8—44no—
141C22×C8—42no—
142C22×C8—42no—
143C22×C8—42no—
144#117—41no—
145#117—41no—
146#117—41no—
147#117—41no—
148#117—41no—
149#117—41no—
150#117—41no—
151#117—41no—
152#117—41no—
153#117—41no—
154#117—41no—
155#117—41no—
156#117—41no—
157#117—41no—
158#117—41no—
159#117—41no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4C4211yescharacteristic
165C2×C4⋊C4C22⋊C411yescharacteristic
166C2×C4⋊C4C22⋊C411yescharacteristic
167C2×C4⋊C4C22⋊C411yescharacteristic
168C2×C4⋊C4C4⋊C411yescharacteristic
169C2×C4⋊C4C4⋊C411yescharacteristic
170C2×C4⋊C4C22×C411yescharacteristic
171C2×C4⋊C4C2×D411yescharacteristic
172C2×C22⋊C4—41no—
173C2×C22⋊C4—41no—
174C2×C22⋊C4—41no—
175C2×C22⋊C4—41no—
176C2×C22⋊C4C4211yescharacteristic
177C2×C22⋊C4C22⋊C411yescharacteristic
178C2×C22⋊C4C4⋊C411yescharacteristic
179C2×C22⋊C4C4⋊C411yescharacteristic
180C2×C22⋊C4C4⋊C411yescharacteristic
181C2×C22⋊C4C4⋊C411yescharacteristic
182C2×C22⋊C4C22×C411yescharacteristic
183C2×C22⋊C4C2×Q811yescharacteristic
184C2×C42—21no—
185C2×C42—21no—
186C2×C42—21no—
187C2×C42—21no—
188C2×C42—21no—
189C2×C42—21no—
190C2×C42—21no—
191C2×C42—21no—
192C2×C42C4211yescharacteristic
193C2×C42C22⋊C411yescharacteristic
194C2×C42C22⋊C411yescharacteristic
195C2×C42C22⋊C411yescharacteristic
196C2×C42C4⋊C411yescharacteristic
197C2×C42C4⋊C411yescharacteristic
198C2×C42C22×C411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C4⋊C8—42no—
201C4⋊C8—42no—
202C4⋊C8—42no—
203C4⋊C8—42no—
204C4⋊C8—42no—
205C4⋊C8—42no—
206C22⋊C8—24no—
207C22⋊C8—42no—
208C22⋊C8—44no—
209C22⋊C8—42no—
210C22⋊C8—42no—
211C4×C8—42no—
212C4×C8—42no—
213C24—21no—
214C24—22no—
215C24—21no—
216C24—81no—
217C24—81no—
218C24#9511yescharacteristic
219C24#9511yescharacteristic
220C24#10612yes—
221C24#10712yes—
222C24C2×C4⋊C411yescharacteristic
223C22×C4—22no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—81no—
265C22×C4—81no—
266C22×C4—81no—
267C22×C4—81no—
268C22×C4#9511yescharacteristic
269C22×C4#9511yescharacteristic
270C22×C4#9511yescharacteristic
271C22×C4#9511yescharacteristic
272C22×C4#9511yescharacteristic
273C22×C4#9511yescharacteristic
274C22×C4#9911yescharacteristic
275C22×C4#9911yescharacteristic
276C22×C4#9911yescharacteristic
277C22×C4#9911yescharacteristic
278C22×C4#10212yes—
279C22×C4#10212yes—
280C22×C4#10212yes—
281C22×C4#10212yes—
282C22×C4#10312yes—
283C22×C4#10312yes—
284C22×C4#10312yes—
285C22×C4#10312yes—
286C22×C4#10412yes—
287C22×C4#10412yes—
288C22×C4#10612yes—
289C22×C4#10712yes—
290C22×C4C2×C4211yescharacteristic
291C22×C4C2×C22⋊C411yescharacteristic
292C22×C4C2×C22⋊C411yescharacteristic
293C22×C4C2×C22⋊C411yescharacteristic
294C22×C4C2×C4⋊C411yescharacteristic
295C22×C4C2×C4⋊C411yescharacteristic
296C22×C4C2×D811yescharacteristic
297C22×C4C2×SD1611yescharacteristic
298C22×C4C2×SD1611yescharacteristic
299C22×C4C2×Q1611yescharacteristic
300M16—28no—
301M16—48no—
302M16—48no—
303M16—44no—
304M16—44no—
305M16—44no—
306M16—82no—
307M16—82no—
308M16—82no—
309M16—82no—
310C2×C8—48no—
311C2×C8—44no—
312C2×C8—48no—
313C2×C8—44no—
314C2×C8—44no—
315C2×C8—44no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—84no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—41no—
334C22⋊C4—41no—
335C22⋊C4—41no—
336C22⋊C4—41no—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C42—21no—
340C42—21no—
341C42—21no—
342C42—21no—
343C42—21no—
344C42—21no—
345C42—21no—
346C42—21no—
347C42—21no—
348C42—21no—
349C42—21no—
350C42—21no—
351C42—21no—
352C42—21no—
353C42—21no—
354C42—21no—
355C23—22no—
356C23—22no—
357C23—21no—
358C23—21no—
359C23—22no—
360C23—22no—
361C23—21no—
362C23—21no—
363C23—41no—
364C23—41no—
365C23—42no—
366C23—42no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23—41no—
371C23—81no—
372C23—81no—
373C23—82no—
374C23—82no—
375C23—84no—
376C23#33912yes—
377C23#34012yes—
378C23#34012yes—
379C23#34211yescharacteristic
380C23#34211yescharacteristic
381C23C2×#9511yescharacteristic
382C23C2×#10611yescharacteristic
383C23C2×#10711yescharacteristic
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4#32712yes—
437C2×C4#32712yes—
438C2×C4#32712yes—
439C2×C4#32712yes—
440C2×C4#32812yes—
441C2×C4#32812yes—
442C2×C4#32812yes—
443C2×C4#32812yes—
444C2×C4#33912yes—
445C2×C4#34012yes—
446C2×C4#34012yes—
447C2×C4#34211yescharacteristic
448C2×C4#34211yescharacteristic
449C2×C4C2×#9911yescharacteristic
450C2×C4C2×#9911yescharacteristic
451C2×C4C2×#10211yescharacteristic
452C2×C4C2×#10211yescharacteristic
453C2×C4C2×#10311yescharacteristic
454C2×C4C2×#10311yescharacteristic
455C2×C4C2×#10411yescharacteristic
456C2×C4C2×#10411yescharacteristic
457C8—48no—
458C8—84no—
459C8—82no—
460C8—82no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22—21no—
467C22—22no—
468C22—21no—
469C22—21no—
470C22—42no—
471C22—42no—
472C22—42no—
473C22—41no—
474C22—41no—
475C22—44no—
476C22—44no—
477C22—41no—
478C22—41no—
479C22—41no—
480C22—41no—
481C22—81no—
482C22—81no—
483C22—82no—
484C22—82no—
485C22#129412yes—
486C22#129412yes—
487C22C2×#33911yescharacteristic
488C22C2×#34011yescharacteristic
489C22C2×#34211yescharacteristic
490C4—21no—
491C4—22no—
492C4—21no—
493C4—41no—
494C4—41no—
495C4—41no—
496C4—41no—
497C4—41no—
498C4—41no—
499C4—41no—
500C4—41no—
501C4#129412yes—
502C4#129412yes—
503C4C2×#32711yescharacteristic
504C4C2×#32711yescharacteristic
505C4C2×#32811yescharacteristic
506C4C2×#32811yescharacteristic
507C2—22no—
508C2—21no—
509C2—22no—
510C2—21no—
511C2—42no—
512C2—41no—
513C2—41no—
514C2C2×#129411yescharacteristic
515C2#1257911yescharacteristic
516C2#1258911yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C2×C42
Frattini, cosocleC2×#117C23
socleC22C2×#342
prime radicalC25C4⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#117 ◁ G
LECSG ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7253C2semidirect

Maximal subgroups

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

subgroupindex
#206272
C2×#17422
C2×#14582
#72532