Finite Group Databasebeta

← #124673

Subgroups — #124673

Order 512.

646 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 646 conjugacy classes of subgroups — too many to draw. Only its 92 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#14275C211yesmaximal, characteristic
2#8698C211yesmaximal, characteristic
3#7909C211yesmaximal, characteristic
4#7123C212yesmaximal, complemented
5#7122C212yesmaximal, complemented
6C2×#407—21no—
7C2×#385—21no—
8C2×#380C411yescharacteristic
9C2×#380C411yescharacteristic
10C22×#95C2211yescharacteristic
11#1779—21no—
12#1444C411yescharacteristic
13#1444C2211yescharacteristic
14#1442C411yescharacteristic
15#1442C2211yescharacteristic
16#1441—21no—
17#1441—21no—
18C22⋊#95—21no—
19#1280—21no—
20#1280—21no—
21#1280C2212yes—
22#1280C2212yes—
23C24×C4C2×C411yescharacteristic
24C24×C4D411yescharacteristic
25C24×C4D411yescharacteristic
26C22×M16—41no—
27C22×C4⋊C4—21no—
28C22×C4⋊C4—21no—
29C22×C4⋊C4—21no—
30C22×C4⋊C4C2×C411yescharacteristic
31C22×C4⋊C4C2×C411yescharacteristic
32C22×C22⋊C4—21no—
33C22×C22⋊C4—21no—
34#407—24no—
35#407—42no—
36C2×C22⋊C8—22no—
37C2×C22⋊C8—22no—
38C2×C22⋊C8—41no—
39C2×C22⋊C8—41no—
40C2×C22⋊C8—41no—
41#385—44no—
42#380—22no—
43#380—22no—
44#380—41no—
45#380—41no—
46C2×#95—21no—
47C2×#95—21no—
48C2×#95—21no—
49C2×#95—21no—
50C2×#95—21no—
51C2×#95—21no—
52C2×#95—21no—
53C2×#95—41no—
54C2×#95—41no—
55C2×#95C2×C411yescharacteristic
56C2×#95C2×C411yescharacteristic
57C2×#95C2×C411yescharacteristic
58C2×#95D411yescharacteristic
59C2×#95D411yescharacteristic
60C2×#95C2311yescharacteristic
61C25C22⋊C411yescharacteristic
62C23×C4—21no—
63C23×C4—21no—
64C23×C4—21no—
65C23×C4—21no—
66C23×C4—22no—
67C23×C4—22no—
68C23×C4—21no—
69C23×C4—21no—
70C23×C4—21no—
71C23×C4—21no—
72C23×C4—21no—
73C23×C4—21no—
74C23×C4—22no—
75C23×C4—21no—
76C23×C4—21no—
77C23×C4—41no—
78C23×C4—81no—
79C23×C4—81no—
80C23×C4—81no—
81C23×C4—81no—
82C23×C4—81no—
83C23×C4—81no—
84C23×C4C22⋊C411yescharacteristic
85C23×C4C22⋊C411yescharacteristic
86C23×C4C22⋊C411yescharacteristic
87C23×C4SD1612yes—
88C23×C4Q1612yes—
89C23×C4C22×C411yescharacteristic
90C23×C4C2×D411yescharacteristic
91C23×C4C2×D411yescharacteristic
92C2×M16—44no—
93C2×M16—82no—
94C2×M16—82no—
95C22×C8—42no—
96C22×C8—42no—
97C2×C4⋊C4—24no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4—42no—
102C2×C4⋊C4—81no—
103C2×C4⋊C4—81no—
104C2×C4⋊C4—81no—
105C2×C4⋊C4—81no—
106C2×C4⋊C4—81no—
107C2×C4⋊C4—81no—
108C2×C4⋊C4—81no—
109C2×C4⋊C4—81no—
110C2×C22⋊C4—24no—
111C2×C22⋊C4—42no—
112C2×C22⋊C4—42no—
113C2×C22⋊C4—42no—
114C2×C22⋊C4—42no—
115C2×C22⋊C4—41no—
116C2×C22⋊C4—41no—
117C22⋊C8—24no—
118C22⋊C8—44no—
119C22⋊C8—42no—
120C22⋊C8—42no—
121C22⋊C8—42no—
122C22⋊C8—82no—
123C22⋊C8—82no—
124#95—21no—
125#95—21no—
126#95—41no—
127#95—41no—
128#95—41no—
129#95—41no—
130#95—42no—
131#95—42no—
132#95—42no—
133#95—81no—
134#95—81no—
135#95—81no—
136#95—81no—
137#95M1612yescomplemented
138#95M1612yescomplemented
139C24—21no—
140C24—21no—
141C24—21no—
142C24—21no—
143C24—41no—
144C24—41no—
145C24—41no—
146C24—41no—
147C24#9911yescharacteristic
148C24#9911yescharacteristic
149C24#10312yes—
150C24#10412yes—
151C24C2×C22⋊C411yescharacteristic
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—42no—
196C22×C4—42no—
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—82no—
220C22×C4—82no—
221C22×C4—81no—
222C22×C4—81no—
223C22×C4—81no—
224C22×C4—81no—
225C22×C4—81no—
226C22×C4—81no—
227C22×C4—81no—
228C22×C4—81no—
229C22×C4—82no—
230C22×C4—82no—
231C22×C4—84no—
232C22×C4—84no—
233C22×C4—84no—
234C22×C4—84no—
235C22×C4#10112yes—
236C22×C4#10312yes—
237C22×C4C2×M1611yescharacteristic
238C22×C4C2×M1611yescharacteristic
239C22×C4C2×SD1611yescharacteristic
240C22×C4C2×Q1611yescharacteristic
241C22×C4AGL(1,ℤ/8)12yes—
242M16—88no—
243C2×C8—44no—
244C2×C8—82no—
245C2×C8—82no—
246C2×C8—82no—
247C2×C8—82no—
248C2×C8—82no—
249C4⋊C4—84no—
250C4⋊C4—82no—
251C4⋊C4—82no—
252C4⋊C4—84no—
253C4⋊C4—84no—
254C22⋊C4—48no—
255C22⋊C4—82no—
256C22⋊C4—82no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23—22no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—22no—
273C23—22no—
274C23—21no—
275C23—21no—
276C23—22no—
277C23—42no—
278C23—42no—
279C23—42no—
280C23—42no—
281C23—42no—
282C23—42no—
283C23—42no—
284C23—42no—
285C23—42no—
286C23—42no—
287C23—42no—
288C23—42no—
289C23#32812yes—
290C23#32812yes—
291C23#40711yescharacteristic
292C23C2×#9911yescharacteristic
293C23C2×#10111yescharacteristic
294C23C2×#10311yescharacteristic
295C23#41811yescharacteristic
296C23C2×#10411yescharacteristic
297C23#42111yescharacteristic
298C2×C4—22no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—22no—
303C2×C4—21no—
304C2×C4—22no—
305C2×C4—42no—
306C2×C4—44no—
307C2×C4—44no—
308C2×C4—42no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—41no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—82no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4—82no—
344C2×C4—82no—
345C2×C4—84no—
346C2×C4—84no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
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—
370C8—84no—
371C8—162no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—22no—
385C22—22no—
386C22—22no—
387C22—22no—
388C22—22no—
389C22—22no—
390C22—44no—
391C22—44no—
392C22—44no—
393C22—44no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22#146311yescharacteristic
403C22#148711yescharacteristic
404C22#149011yescharacteristic
405C22C2×#32811yescharacteristic
406C22#151011yescharacteristic
407C22#152811yescharacteristic
408C22#154911yescharacteristic
409C22#212111yescharacteristic
410C22#212111yescharacteristic
411C22#212711yescharacteristic
412C22#213311yescharacteristic
413C4—42no—
414C4—41no—
415C4—42no—
416C4—41no—
417C4—42no—
418C4—41no—
419C4—41no—
420C4—82no—
421C4—82no—
422C4—81no—
423C4—161no—
424C4—161no—
425C2—21no—
426C2—21no—
427C2—21no—
428C2—21no—
429C2—42no—
430C2—42no—
431C2#760711yescharacteristic
432C2#1151411yescharacteristic
433C2#1151911yescharacteristic
434C2#1171611yescharacteristic
435C2#1172211yescharacteristic
436C2#1179911yescharacteristic
437C2#1180511yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC2×#95C23
prime radicalC25C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C25 ◁ C22×#95 ◁ G
LECSG ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7123C2semidirect
5. #7122C2semidirect
137. #95M16semidirect
138. #95M16semidirect

Maximal subgroups

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

subgroupindex
#142752
#86982
#79092
#71232
#71222