Finite Group Databasebeta

← #134259

Subgroups — #134259

Order 512.

668 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 122 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
1#13782C211yesmaximal, characteristic
2C2×#1326C212yesmaximal
3#8557C211yesmaximal, characteristic
4#8557C211yesmaximal, characteristic
5#7187C212yesmaximal
6C2×#385C411yescharacteristic
7C2×#385C2211yescharacteristic
8C2×C4×C22⋊C4C411yescharacteristic
9C2×C4×C22⋊C4C2211yescharacteristic
10C22×#95C411yescharacteristic
11#1797—21no—
12#1797—21no—
13C2×#323C2212yes—
14C2×#323C2212yes—
15#1436—21no—
16#1436—21no—
17C4⋊#95—21no—
18#1434—21no—
19C22⋊#95C411yescharacteristic
20C22⋊#95C2211yescharacteristic
21#1326—24no—
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×C42D411yescharacteristic
35C22×C42D411yescharacteristic
36C2×C4⋊C8—41no—
37C2×C4⋊C8—41no—
38C2×C22⋊C8—22no—
39C2×C22⋊C8—22no—
40#385—22no—
41#385—22no—
42#385—21no—
43#385—21no—
44#385D412yes—
45#385Q812yes—
46C4×C22⋊C4—22no—
47C4×C22⋊C4—41no—
48C4×C22⋊C4—41no—
49C4×C22⋊C4C814yescomplemented
50C2×#95—21no—
51C2×#95—21no—
52C2×#95—21no—
53C2×#95—21no—
54C2×#95—21no—
55C2×#95—21no—
56C2×#95—21no—
57C2×#95—21no—
58C2×#95—21no—
59C2×#95—41no—
60C2×#95—41no—
61C2×#95D411yescharacteristic
62C2×#95D411yescharacteristic
63#336—41no—
64#336—41no—
65#323—22no—
66#323—22no—
67#323—22no—
68#323—22no—
69C25C22×C411yescharacteristic
70C23×C4—21no—
71C23×C4—21no—
72C23×C4—21no—
73C23×C4—22no—
74C23×C4—21no—
75C23×C4—21no—
76C23×C4—21no—
77C23×C4—21no—
78C23×C4—21no—
79C23×C4—41no—
80C23×C4—41no—
81C23×C4—41no—
82C23×C4—41no—
83C23×C4—81no—
84C23×C4—81no—
85C23×C4—81no—
86C23×C4—81no—
87C23×C4—81no—
88C23×C4—81no—
89C23×C4C22⋊C411yescharacteristic
90C23×C4C22⋊C411yescharacteristic
91C23×C4C22⋊C411yescharacteristic
92C23×C4C22⋊C411yescharacteristic
93C23×C4C4⋊C412yes—
94C23×C4C2×C812yes—
95C23×C4M1612yes—
96C23×C4M1612yes—
97C23×C4C2×D411yescharacteristic
98C23×C4C2×D411yescharacteristic
99C22×C8—42no—
100C22×C8—42no—
101C2×C4⋊C4—24no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—81no—
105C2×C4⋊C4—81no—
106C2×C4⋊C4—81no—
107C2×C4⋊C4—81no—
108C2×C22⋊C4—21no—
109C2×C22⋊C4—21no—
110C2×C22⋊C4—21no—
111C2×C22⋊C4—21no—
112C2×C22⋊C4—21no—
113C2×C22⋊C4—21no—
114C2×C22⋊C4—41no—
115C2×C22⋊C4—41no—
116C2×C22⋊C4—41no—
117C2×C22⋊C4—41no—
118C2×C22⋊C4—41no—
119C2×C22⋊C4—41no—
120C2×C22⋊C4C4⋊C412yes—
121C2×C22⋊C4C2×C812yes—
122C2×C22⋊C4C2×C812yes—
123C2×C22⋊C4M1612yes—
124C2×C22⋊C4C2×D411yescharacteristic
125C2×C22⋊C4C2×Q811yescharacteristic
126C2×C22⋊C4#4111yescharacteristic
127C2×C22⋊C4#4111yescharacteristic
128C2×C42—21no—
129C2×C42—21no—
130C2×C42—24no—
131C2×C42—21no—
132C2×C42—21no—
133C2×C42—81no—
134C2×C42—81no—
135C2×C42—81no—
136C2×C42—81no—
137C4⋊C8—82no—
138C4⋊C8—82no—
139C22⋊C8—42no—
140C22⋊C8—42no—
141C22⋊C8—42no—
142C22⋊C8—42no—
143#95—24no—
144#95—24no—
145#95—22no—
146#95—22no—
147#95—41no—
148#95—41no—
149#95—42no—
150#95—41no—
151#95—41no—
152#95—81no—
153#95—81no—
154#95—81no—
155#95—81no—
156C24—21no—
157C24—21no—
158C24—21no—
159C24—21no—
160C24—41no—
161C24—41no—
162C24—41no—
163C24—41no—
164C24C2×C22⋊C411yescharacteristic
165C24C2×C4⋊C411yescharacteristic
166C24#11711yescharacteristic
167C24C22×C811yescharacteristic
168C24C2×M1611yescharacteristic
169C24C2×M1611yescharacteristic
170C24C2×M1611yescharacteristic
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—21no—
187C22×C4—22no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—81no—
232C22×C4—81no—
233C22×C4—81no—
234C22×C4—81no—
235C22×C4—84no—
236C22×C4—82no—
237C22×C4—82no—
238C22×C4—82no—
239C22×C4—82no—
240C22×C4—82no—
241C22×C4—82no—
242C22×C4—84no—
243C22×C4—84no—
244C22×C4—82no—
245C22×C4—82no—
246C22×C4—82no—
247C22×C4—82no—
248C22×C4C22⋊C812yes—
249C22×C4C22⋊C812yes—
250C22×C4#9911yescharacteristic
251C22×C4#9911yescharacteristic
252C22×C4#10112yes—
253C22×C4#12111yescharacteristic
254C22×C4#12111yescharacteristic
255C22×C4C22⋊Q811yescharacteristic
256C22×C4C22⋊Q811yescharacteristic
257C2×C8—82no—
258C2×C8—82no—
259C2×C8—82no—
260C2×C8—82no—
261C2×C8—82no—
262C2×C8—82no—
263C4⋊C4—82no—
264C4⋊C4—82no—
265C4⋊C4—84no—
266C22⋊C4—22no—
267C22⋊C4—22no—
268C22⋊C4—22no—
269C22⋊C4—41no—
270C22⋊C4—41no—
271C22⋊C4—41no—
272C22⋊C4—41no—
273C22⋊C4—42no—
274C22⋊C4—42no—
275C22⋊C4—42no—
276C22⋊C4—42no—
277C22⋊C4C4⋊C814yescomplemented
278C42—42no—
279C42—42no—
280C42—81no—
281C42—81no—
282C42—84no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—41no—
300C23—41no—
301C23—41no—
302C23—41no—
303C23—41no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23—41no—
310C23—41no—
311C23—42no—
312C23—42no—
313C23—42no—
314C23—42no—
315C23—42no—
316C23—42no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23#38011yescharacteristic
322C23C2×C22⋊C811yescharacteristic
323C23#40711yescharacteristic
324C23C2×#9911yescharacteristic
325C23#41011yescharacteristic
326C23C2×#10111yescharacteristic
327C23#41311yescharacteristic
328C23C2×C4⋊C811yescharacteristic
329C23#42311yescharacteristic
330C23#43111yescharacteristic
331C23#43211yescharacteristic
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—24no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—22no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—42no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—84no—
394C2×C4—84no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—84no—
402C2×C4—84no—
403C2×C4#32412yes—
404C2×C4#32412yes—
405C8—162no—
406C8—162no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—41no—
424C22—41no—
425C22—41no—
426C22—41no—
427C22—42no—
428C22—42no—
429C22—42no—
430C22—42no—
431C22—42no—
432C22—42no—
433C22—41no—
434C22—41no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22—41no—
444C22—41no—
445C22C2×#32411yescharacteristic
446C22#145811yescharacteristic
447C22#146511yescharacteristic
448C22#146711yescharacteristic
449C22#179711yescharacteristic
450C22#179911yescharacteristic
451C22#180111yescharacteristic
452C22#212111yescharacteristic
453C22#212111yescharacteristic
454C22#212211yescharacteristic
455C22#212311yescharacteristic
456C4—22no—
457C4—22no—
458C4—44no—
459C4—42no—
460C4—41no—
461C4—41no—
462C4—41no—
463C4—41no—
464C4—82no—
465C4—81no—
466C4—82no—
467C4—81no—
468C4—82no—
469C2—21no—
470C2—21no—
471C2—21no—
472C2—21no—
473C2—41no—
474C2—41no—
475C2—41no—
476C2—41no—
477C2#855811yescharacteristic
478C2#1145311yescharacteristic
479C2#1145411yescharacteristic
480C2#1152711yescharacteristic
481C2#1152811yescharacteristic
482C2#1279511yescharacteristic
483C2#1279611yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC24C22×C8
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
49. C4×C22⋊C4C8semidirect
277. C22⋊C4C4⋊C8semidirect

Maximal subgroups

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

subgroupindex
#137822
C2×#13262
#85572
#71872