Finite Group Databasebeta

← #128396

Subgroups — #128396

Order 512.

848 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 848 conjugacy classes of subgroups — too many to draw. Only its 126 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#385C211yesmaximal, characteristic
2C2×#1326C212yesmaximal
3#8152C211yesmaximal, characteristic
4#8152C211yesmaximal, characteristic
5#7187C212yesmaximal
6C23×C42C411yescharacteristic
7C2×#385C411yescharacteristic
8C2×#385C2211yescharacteristic
9C2×C4×C4⋊C4—21no—
10C2×C4×C22⋊C4C411yescharacteristic
11C2×C4×C22⋊C4C2211yescharacteristic
12#1749—21no—
13#1749—21no—
14C2×#323C2212yes—
15C2×#323C2212yes—
16#1436—21no—
17#1436—21no—
18C22⋊#95C411yescharacteristic
19C22⋊#95C2211yescharacteristic
20C4×#95—21no—
21#1326C418yescomplemented
22#1280—21no—
23#1280—21no—
24C24×C4C2×C411yescharacteristic
25C24×C4C2×C411yescharacteristic
26C24×C4C2×C411yescharacteristic
27C22×C4⋊C4—21no—
28C22×C4⋊C4—21no—
29C22×C22⋊C4C2×C411yescharacteristic
30C22×C22⋊C4C2×C411yescharacteristic
31C22×C22⋊C4C2×C411yescharacteristic
32C22×C22⋊C4C2311yescharacteristic
33C22×C42—21no—
34C22×C42—21no—
35C22×C42—21no—
36C22×C42—44no—
37C22×C42—41no—
38C22×C42—41no—
39C22×C42D411yescharacteristic
40C22×C42D411yescharacteristic
41C2×C22⋊C8—22no—
42C2×C22⋊C8—22no—
43C2×#97—41no—
44C2×#97—41no—
45#385—22no—
46#385—22no—
47#385—22no—
48#385C2×C414yes—
49C4×C4⋊C4—44no—
50C4×C22⋊C4—22no—
51C4×C22⋊C4—22no—
52C4×C22⋊C4—41no—
53C4×C22⋊C4—41no—
54C2×#95—21no—
55C2×#95—21no—
56C2×#95—21no—
57C2×#95—21no—
58C2×#95—21no—
59C2×#95—21no—
60C2×#95—21no—
61C2×#95D411yescharacteristic
62C2×#95D411yescharacteristic
63#336—41no—
64#336—41no—
65#323—22no—
66#323—22no—
67#323C2×C414yes—
68#323C2×C414yes—
69C25C22×C411yescharacteristic
70C23×C4—21no—
71C23×C4—21no—
72C23×C4—21no—
73C23×C4—21no—
74C23×C4—21no—
75C23×C4—21no—
76C23×C4—21no—
77C23×C4—22no—
78C23×C4—21no—
79C23×C4—21no—
80C23×C4—21no—
81C23×C4—42no—
82C23×C4—42no—
83C23×C4—42no—
84C23×C4—42no—
85C23×C4—42no—
86C23×C4—42no—
87C23×C4—41no—
88C23×C4—41no—
89C23×C4—41no—
90C23×C4—41no—
91C23×C4C4212yes—
92C23×C4C22⋊C411yescharacteristic
93C23×C4C22⋊C411yescharacteristic
94C23×C4C22⋊C411yescharacteristic
95C23×C4C22⋊C411yescharacteristic
96C23×C4M1612yes—
97C23×C4C2×D411yescharacteristic
98C23×C4C2×D411yescharacteristic
99C22×C8—42no—
100C22×C8—42no—
101C2×C4⋊C4—24no—
102C2×C4⋊C4—24no—
103C2×C4⋊C4—42no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4—42no—
106C2×C4⋊C4—42no—
107C2×C22⋊C4—21no—
108C2×C22⋊C4—21no—
109C2×C22⋊C4—21no—
110C2×C22⋊C4—21no—
111C2×C22⋊C4—21no—
112C2×C22⋊C4—21no—
113C2×C22⋊C4—41no—
114C2×C22⋊C4—41no—
115C2×C22⋊C4—41no—
116C2×C22⋊C4—41no—
117C2×C22⋊C4—41no—
118C2×C22⋊C4—41no—
119C2×C22⋊C4C4212yes—
120C2×C22⋊C4M1612yes—
121C2×C22⋊C4M1612yes—
122C2×C22⋊C4M1612yes—
123C2×C22⋊C4C22×C412yes—
124C2×C22⋊C4#4112yes—
125C2×C42—24no—
126C2×C42—24no—
127C2×C42—21no—
128C2×C42—21no—
129C2×C42—21no—
130C2×C42—21no—
131C2×C42—42no—
132C2×C42—42no—
133C2×C42—42no—
134C2×C42—42no—
135C2×C42—42no—
136C2×C42—42no—
137C2×C42—44no—
138C2×C42—44no—
139C2×C42—44no—
140C2×C42—41no—
141C2×C42—41no—
142C2×C42—44no—
143C2×C42—42no—
144C2×C42—42no—
145C22⋊C8—24no—
146C22⋊C8—24no—
147C22⋊C8—42no—
148C22⋊C8—42no—
149#97—82no—
150#97—82no—
151#95—24no—
152#95—24no—
153#95—24no—
154#95—41no—
155#95—41no—
156#95—42no—
157#95—42no—
158#95—42no—
159#95—42no—
160C24—21no—
161C24—21no—
162C24—21no—
163C24—21no—
164C24—41no—
165C24—41no—
166C24—41no—
167C24—41no—
168C24C2×C4211yescharacteristic
169C24C2×C22⋊C411yescharacteristic
170C24#11711yescharacteristic
171C24C2×M1611yescharacteristic
172C24C2×M1611yescharacteristic
173C24C2×M1611yescharacteristic
174C24C2×M1611yescharacteristic
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—21no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—22no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—42no—
206C22×C4—44no—
207C22×C4—44no—
208C22×C4—48no—
209C22×C4—42no—
210C22×C4—44no—
211C22×C4—44no—
212C22×C4—44no—
213C22×C4—44no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—48no—
227C22×C4—48no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—41no—
237C22×C4—42no—
238C22×C4—41no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4#9911yescharacteristic
259C22×C4#9911yescharacteristic
260C22×C4#10112yes—
261C22×C4C4×D412yes—
262C22×C4C4×D412yes—
263C2×C8—44no—
264C2×C8—44no—
265C2×C8—82no—
266C2×C8—82no—
267C2×C8—82no—
268C2×C8—82no—
269C4⋊C4—48no—
270C4⋊C4—48no—
271C22⋊C4—22no—
272C22⋊C4—22no—
273C22⋊C4—22no—
274C22⋊C4—42no—
275C22⋊C4—42no—
276C22⋊C4—42no—
277C22⋊C4—42no—
278C22⋊C4—42no—
279C22⋊C4—42no—
280C22⋊C4#9714yescomplemented
281C42—48no—
282C42—44no—
283C42—48no—
284C42—42no—
285C42—44no—
286C42—42no—
287C42—81no—
288C42—81no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23—41no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—42no—
318C23—42no—
319C23—42no—
320C23—42no—
321C23—42no—
322C23—42no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23C4×C22⋊C411yescharacteristic
328C23C2×#9711yescharacteristic
329C23C4×M1611yescharacteristic
330C23#40711yescharacteristic
331C23#40711yescharacteristic
332C23C2×#9911yescharacteristic
333C23#41011yescharacteristic
334C23C2×#10111yescharacteristic
335C23#41311yescharacteristic
336C23#43211yescharacteristic
337C23#43211yescharacteristic
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—22no—
347C2×C4—24no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—42no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4AΣL(1,𝔽16)12yes—
410C2×C4#35212yes—
411C8—84no—
412C8—84no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—41no—
430C22—41no—
431C22—41no—
432C22—41no—
433C22—42no—
434C22—42no—
435C22—42no—
436C22—42no—
437C22—42no—
438C22—42no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22—41no—
444C22—41no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22—41no—
449C22—41no—
450C22—41no—
451C22#146311yescharacteristic
452C22#146411yescharacteristic
453C22#146511yescharacteristic
454C22#146711yescharacteristic
455C22#174911yescharacteristic
456C22C4×#9911yescharacteristic
457C22C4×#10111yescharacteristic
458C22C2×AΣL(1,𝔽16)11yescharacteristic
459C22C2×#35211yescharacteristic
460C22#212011yescharacteristic
461C22#212011yescharacteristic
462C4—22no—
463C4—22no—
464C4—44no—
465C4—44no—
466C4—42no—
467C4—44no—
468C4—42no—
469C4—42no—
470C4—82no—
471C4—81no—
472C4—81no—
473C2—21no—
474C2—21no—
475C2—21no—
476C2—21no—
477C2—41no—
478C2—41no—
479C2—41no—
480C2—41no—
481C2#815311yescharacteristic
482C2#1151711yescharacteristic
483C2#1151811yescharacteristic
484C2#1152511yescharacteristic
485C2#1152611yescharacteristic
486C2C4×AΣL(1,𝔽16)11yescharacteristic
487C2C4×#35211yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC24C2×C42
Frattini, cosocleC22×C22⋊C4C23
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
21. #1326C4semidirect
280. C22⋊C4#97semidirect

Maximal subgroups

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

subgroupindex
C4×#3852
C2×#13262
#81522
#71872