Finite Group Databasebeta

← #99153

Subgroups — #99153

Order 512.

658 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 658 conjugacy classes of subgroups — too many to draw. Only its 106 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#13778C211yesmaximal, characteristic
2#8698C211yesmaximal, characteristic
3#8698C211yesmaximal, characteristic
4#7032C212yesmaximal
5#7032C212yesmaximal
6C2×#380C411yescharacteristic
7C2×#380C411yescharacteristic
8C2×C4×C22⋊C4C411yescharacteristic
9C2×C4×C22⋊C4C411yescharacteristic
10C22×#95C2211yescharacteristic
11#1779C2212yes—
12#1779C2212yes—
13#1436—21no—
14#1436—21no—
15#1436—21no—
16#1436—21no—
17C22⋊#95C2211yescharacteristic
18C22⋊#95C2211yescharacteristic
19#1280—21no—
20#1280—21no—
21#1280—21no—
22#1280—21no—
23C24×C4C2×C411yescharacteristic
24C24×C4C2×C411yescharacteristic
25C24×C4C2311yescharacteristic
26C22×C4⋊C4—21no—
27C22×C4⋊C4—21no—
28C22×C22⋊C4C2×C411yescharacteristic
29C22×C22⋊C4C2×C411yescharacteristic
30C22×C22⋊C4C2×C411yescharacteristic
31C22×C22⋊C4C2×C411yescharacteristic
32C22×C42—21no—
33C22×C42—21no—
34C2×C22⋊C8—22no—
35C2×C22⋊C8—22no—
36C2×C22⋊C8—22no—
37C2×C22⋊C8—22no—
38C2×C22⋊C8—22no—
39C2×C22⋊C8—22no—
40#380—22no—
41#380—22no—
42#380—22no—
43#380C814yescomplemented
44C4×C22⋊C4—42no—
45C4×C22⋊C4—42no—
46C2×#95—21no—
47C2×#95—21no—
48C2×#95—22no—
49C2×#95—22no—
50C2×#95—21no—
51C2×#95—21no—
52C2×#95—21no—
53C2×#95—21no—
54C2×#95—21no—
55C2×#95—21no—
56C2×#95D411yescharacteristic
57C2×#95D411yescharacteristic
58C2×#95D411yescharacteristic
59C2×#95D411yescharacteristic
60C25C22×C411yescharacteristic
61C23×C4—21no—
62C23×C4—21no—
63C23×C4—21no—
64C23×C4—21no—
65C23×C4—21no—
66C23×C4—21no—
67C23×C4—22no—
68C23×C4—22no—
69C23×C4—21no—
70C23×C4—21no—
71C23×C4—21no—
72C23×C4—21no—
73C23×C4—41no—
74C23×C4—41no—
75C23×C4—81no—
76C23×C4—81no—
77C23×C4—81no—
78C23×C4—81no—
79C23×C4—81no—
80C23×C4—81no—
81C23×C4C22⋊C411yescharacteristic
82C23×C4C22⋊C411yescharacteristic
83C23×C4C22⋊C411yescharacteristic
84C23×C4C22⋊C411yescharacteristic
85C23×C4C2×C812yes—
86C23×C4M1612yes—
87C23×C4C2×D411yescharacteristic
88C23×C4C2×D411yescharacteristic
89C23×C4#4112yes—
90C23×C4#4112yes—
91C22×C8—42no—
92C22×C8—42no—
93C2×C4⋊C4—24no—
94C2×C4⋊C4—24no—
95C2×C4⋊C4—81no—
96C2×C4⋊C4—81no—
97C2×C4⋊C4—81no—
98C2×C4⋊C4—81no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—21no—
102C2×C22⋊C4—21no—
103C2×C22⋊C4—41no—
104C2×C22⋊C4—41no—
105C2×C22⋊C4—41no—
106C2×C22⋊C4—41no—
107C2×C22⋊C4—41no—
108C2×C22⋊C4—41no—
109C2×C22⋊C4—41no—
110C2×C22⋊C4—41no—
111C2×C22⋊C4C2×C812yes—
112C2×C22⋊C4C2×C812yes—
113C2×C22⋊C4M1612yes—
114C2×C22⋊C4M1612yes—
115C2×C42—42no—
116C2×C42—42no—
117C2×C42—81no—
118C2×C42—81no—
119C2×C42—81no—
120C2×C42—81no—
121C22⋊C8—42no—
122C22⋊C8—42no—
123C22⋊C8—42no—
124C22⋊C8—42no—
125C22⋊C8—82no—
126C22⋊C8—82no—
127C22⋊C8—82no—
128C22⋊C8—82no—
129#95—24no—
130#95—24no—
131#95—22no—
132#95—22no—
133#95—22no—
134#95—22no—
135#95—42no—
136#95—42no—
137#95—81no—
138#95—81no—
139#95—81no—
140#95—81no—
141C24—21no—
142C24—21no—
143C24—21no—
144C24—21no—
145C24—41no—
146C24—41no—
147C24—41no—
148C24—41no—
149C24C2×C22⋊C411yescharacteristic
150C24#11711yescharacteristic
151C24#11711yescharacteristic
152C24C22×C811yescharacteristic
153C24C2×M1611yescharacteristic
154C24C2×M1611yescharacteristic
155C24C2×M1611yescharacteristic
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—24no—
165C22×C4—24no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—41no—
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—81no—
221C22×C4—81no—
222C22×C4—81no—
223C22×C4—81no—
224C22×C4—81no—
225C22×C4—81no—
226C22×C4—81no—
227C22×C4—81no—
228C22×C4—84no—
229C22×C4—82no—
230C22×C4—82no—
231C22×C4—84no—
232C22×C4—82no—
233C22×C4—82no—
234C22×C4—84no—
235C22×C4—84no—
236C22×C4C22⋊C812yes—
237C22×C4C22⋊C812yes—
238C22×C4#10012yes—
239C22×C4#10112yes—
240C22×C4#12312yes—
241C22×C4#12312yes—
242C2×C8—82no—
243C2×C8—82no—
244C2×C8—82no—
245C2×C8—82no—
246C2×C8—82no—
247C2×C8—82no—
248C4⋊C4—84no—
249C4⋊C4—84no—
250C22⋊C4—42no—
251C22⋊C4—42no—
252C22⋊C4—42no—
253C22⋊C4—42no—
254C22⋊C4—42no—
255C22⋊C4—42no—
256C22⋊C4—42no—
257C22⋊C4—42no—
258C42—84no—
259C42—84no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—21no—
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—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—21no—
283C23—21no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23—42no—
293C23—42no—
294C23—42no—
295C23—42no—
296C23—42no—
297C23—42no—
298C23—42no—
299C23—42no—
300C23#38111yescharacteristic
301C23C2×C22⋊C811yescharacteristic
302C23#40711yescharacteristic
303C23#41011yescharacteristic
304C23#41011yescharacteristic
305C23C2×#10011yescharacteristic
306C23C2×#10111yescharacteristic
307C23#43111yescharacteristic
308C23#43111yescharacteristic
309C23#43211yescharacteristic
310C23#43211yescharacteristic
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
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—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—82no—
354C2×C4—82no—
355C2×C4—82no—
356C2×C4—82no—
357C2×C4—84no—
358C2×C4—84no—
359C2×C4—84no—
360C2×C4—84no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C8—162no—
378C8—162no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—42no—
404C22—42no—
405C22—42no—
406C22—42no—
407C22—42no—
408C22—42no—
409C22—42no—
410C22—42no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22#146011yescharacteristic
420C22#146111yescharacteristic
421C22#146611yescharacteristic
422C22#146711yescharacteristic
423C22#181711yescharacteristic
424C22#182011yescharacteristic
425C22#182111yescharacteristic
426C22#182811yescharacteristic
427C22#182811yescharacteristic
428C22#182911yescharacteristic
429C22#183011yescharacteristic
430C4—42no—
431C4—42no—
432C4—42no—
433C4—42no—
434C4—42no—
435C4—42no—
436C4—82no—
437C4—82no—
438C4—82no—
439C4—82no—
440C2—21no—
441C2—21no—
442C2—21no—
443C2—21no—
444C2—41no—
445C2—41no—
446C2—41no—
447C2—41no—
448C2#870111yescharacteristic
449C2#873411yescharacteristic
450C2#873511yescharacteristic
451C2#875011yescharacteristic
452C2#875511yescharacteristic
453C2#875611yescharacteristic
454C2#875711yescharacteristic
subgroupG/H
center, socleC23#381
commutatorC24C22×C8
Frattini, cosocleC24×C4C23
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
43. #380C8semidirect

Maximal subgroups

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

subgroupindex
#137782
#86982
#70322