Finite Group Databasebeta

← #125309

Subgroups — #125309

Order 512.

760 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 760 conjugacy classes of subgroups — too many to draw. Only its 96 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#14914C211yesmaximal, characteristic
2#7909C211yesmaximal, characteristic
3C2×#1280C211yesmaximal, characteristic
4#7131C212yesmaximal, complemented
5#7123C212yesmaximal, complemented
6C22×C22⋊Q8C411yescharacteristic
7C2×#407—21no—
8C22×C22⋊C8—21no—
9C2×#385—21no—
10C22×#95C2211yescharacteristic
11#1446C411yescharacteristic
12#1446C2211yescharacteristic
13#1444C411yescharacteristic
14#1444C2211yescharacteristic
15#1441—21no—
16#1441—21no—
17C22⋊#95—21no—
18C22⋊#95C411yescharacteristic
19#1280—21no—
20#1280—21no—
21#1280C2212yes—
22#1280C2212yes—
23C23×Q8C2×C411yescharacteristic
24C24×C4C2×C411yescharacteristic
25C24×C4D411yescharacteristic
26C24×C4D411yescharacteristic
27C22×M16—41no—
28C23×C8—41no—
29C2×C22⋊Q8—44no—
30C2×C22⋊Q8—41no—
31C2×C22⋊Q8—41no—
32C22×C4⋊C4—21no—
33C22×C4⋊C4—21no—
34C22×C4⋊C4—21no—
35C22×C4⋊C4C2×C411yescharacteristic
36C22×C22⋊C4—21no—
37C22×C22⋊C4—21no—
38#407—24no—
39#407—42no—
40C2×C22⋊C8—24no—
41C2×C22⋊C8—22no—
42C2×C22⋊C8—22no—
43C2×C22⋊C8—41no—
44C2×C22⋊C8—42no—
45C2×C22⋊C8—41no—
46#385—44no—
47C2×#95—21no—
48C2×#95—21no—
49C2×#95—21no—
50C2×#95—21no—
51C2×#95—21no—
52C2×#95—21no—
53C2×#95—21no—
54C2×#95—41no—
55C2×#95—41no—
56C2×#95C2×C411yescharacteristic
57C2×#95C2×C411yescharacteristic
58C2×#95C2×C411yescharacteristic
59C2×#95D411yescharacteristic
60C2×#95D411yescharacteristic
61C2×#95C2311yescharacteristic
62C25C22⋊C411yescharacteristic
63C22×Q8—22no—
64C22×Q8—42no—
65C22×Q8—41no—
66C22×Q8—41no—
67C22×Q8—81no—
68C23×C4—21no—
69C23×C4—21no—
70C23×C4—21no—
71C23×C4—21no—
72C23×C4—22no—
73C23×C4—22no—
74C23×C4—21no—
75C23×C4—21no—
76C23×C4—21no—
77C23×C4—21no—
78C23×C4—21no—
79C23×C4—21no—
80C23×C4—21no—
81C23×C4—21no—
82C23×C4—22no—
83C23×C4—41no—
84C23×C4—42no—
85C23×C4—42no—
86C23×C4—81no—
87C23×C4—81no—
88C23×C4—81no—
89C23×C4—81no—
90C23×C4C22⋊C411yescharacteristic
91C23×C4C22⋊C411yescharacteristic
92C23×C4C22⋊C411yescharacteristic
93C23×C4SD1612yes—
94C23×C4Q1612yes—
95C23×C4C22×C411yescharacteristic
96C23×C4C2×D411yescharacteristic
97C23×C4C2×D411yescharacteristic
98C2×M16—44no—
99C2×M16—82no—
100C2×M16—82no—
101C22×C8—44no—
102C22×C8—42no—
103C22×C8—42no—
104C22×C8—82no—
105C22×C8—82no—
106C22⋊Q8—44no—
107C22⋊Q8—44no—
108C22⋊Q8—44no—
109C22⋊Q8—44no—
110C2×C4⋊C4—41no—
111C2×C4⋊C4—42no—
112C2×C4⋊C4—42no—
113C2×C4⋊C4—42no—
114C2×C4⋊C4—42no—
115C2×C4⋊C4—42no—
116C2×C4⋊C4—42no—
117C2×C4⋊C4—42no—
118C2×C4⋊C4—41no—
119C2×C4⋊C4—41no—
120C2×C4⋊C4—81no—
121C2×C4⋊C4—81no—
122C2×C22⋊C4—24no—
123C2×C22⋊C4—24no—
124C2×C22⋊C4—42no—
125C2×C22⋊C4—42no—
126C2×C22⋊C4—41no—
127C2×C22⋊C4—41no—
128C2×C22⋊C4—42no—
129C22⋊C8—24no—
130C22⋊C8—24no—
131C22⋊C8—44no—
132C22⋊C8—44no—
133C22⋊C8—42no—
134C22⋊C8—42no—
135#95—22no—
136#95—41no—
137#95—41no—
138#95—41no—
139#95—41no—
140#95—41no—
141#95—42no—
142#95—42no—
143#95—81no—
144#95—81no—
145#95—81no—
146#95—81no—
147#95—81no—
148C24—22no—
149C24—22no—
150C24—21no—
151C24—21no—
152C24—21no—
153C24—21no—
154C24—41no—
155C24—41no—
156C24#9911yescharacteristic
157C24#9911yescharacteristic
158C24#10312yes—
159C24#10412yes—
160C24C2×C22⋊C411yescharacteristic
161C2×Q8—42no—
162C2×Q8—42no—
163C2×Q8—82no—
164C2×Q8—84no—
165C2×Q8—84no—
166C2×Q8—82no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
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—42no—
221C22×C4—42no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—82no—
237C22×C4—82no—
238C22×C4—82no—
239C22×C4—84no—
240C22×C4—82no—
241C22×C4—81no—
242C22×C4—81no—
243C22×C4—81no—
244C22×C4—81no—
245C22×C4—81no—
246C22×C4—81no—
247C22×C4—84no—
248C22×C4—84no—
249C22×C4—84no—
250C22×C4#10312yes—
251C22×C4#10412yes—
252C22×C4#13112yes—
253C22×C4C2×SD1611yescharacteristic
254C22×C4C2×Q1611yescharacteristic
255C22×C4AGL(1,ℤ/8)11yescharacteristic
256C22×C4#13711yescharacteristic
257M16—88no—
258C2×C8—44no—
259C2×C8—44no—
260C2×C8—88no—
261C2×C8—82no—
262C2×C8—82no—
263C2×C8—82no—
264C2×C8—82no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—48no—
268C4⋊C4—84no—
269C4⋊C4—82no—
270C4⋊C4—82no—
271C22⋊C4—48no—
272C22⋊C4—44no—
273C22⋊C4—44no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—22no—
283C23—21no—
284C23—21no—
285C23—24no—
286C23—24no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—22no—
291C23—22no—
292C23—21no—
293C23—22no—
294C23—42no—
295C23—42no—
296C23—42no—
297C23—42no—
298C23—42no—
299C23—42no—
300C23—42no—
301C23—42no—
302C23—42no—
303C23—42no—
304C23#32812yes—
305C23#32812yes—
306C23AΣL(1,𝔽16)12yes—
307C23#35612yes—
308C23#40811yescharacteristic
309C23C2×#9911yescharacteristic
310C23#41311yescharacteristic
311C23C2×#10311yescharacteristic
312C23#41711yescharacteristic
313C23C2×#10411yescharacteristic
314C23C2×#10411yescharacteristic
315Q8—84no—
316Q8—84no—
317C2×C4—22no—
318C2×C4—21no—
319C2×C4—22no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—22no—
323C2×C4—21no—
324C2×C4—44no—
325C2×C4—44no—
326C2×C4—44no—
327C2×C4—44no—
328C2×C4—42no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4—84no—
370C2×C4—84no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C8—84no—
390C8—84no—
391C22—22no—
392C22—21no—
393C22—21no—
394C22—22no—
395C22—21no—
396C22—21no—
397C22—22no—
398C22—21no—
399C22—21no—
400C22—22no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—22no—
406C22—21no—
407C22—21no—
408C22—22no—
409C22—22no—
410C22—22no—
411C22—22no—
412C22—44no—
413C22—44no—
414C22—44no—
415C22—44no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22—41no—
420C22#135414yes—
421C22#146811yescharacteristic
422C22#148311yescharacteristic
423C22#149311yescharacteristic
424C22C2×#32811yescharacteristic
425C22#150411yescharacteristic
426C22#152411yescharacteristic
427C22#155011yescharacteristic
428C22C2×AΣL(1,𝔽16)11yescharacteristic
429C22#212111yescharacteristic
430C22#212711yescharacteristic
431C22C2×#35611yescharacteristic
432C4—42no—
433C4—41no—
434C4—42no—
435C4—41no—
436C4—41no—
437C4—42no—
438C4—41no—
439C4—82no—
440C4—82no—
441C4—81no—
442C4—82no—
443C4—82no—
444C2—21no—
445C2—21no—
446C2—21no—
447C2—24no—
448C2—21no—
449C2—42no—
450C2#768111yescharacteristic
451C2#1154111yescharacteristic
452C2#1155511yescharacteristic
453C2#1171611yescharacteristic
454C2C2×#135411yescharacteristic
455C2#1175311yescharacteristic
456C2#1176811yescharacteristic
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. #7131C2semidirect
5. #7123C2semidirect

Maximal subgroups

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

subgroupindex
#149142
#79092
C2×#12802
#71312
#71232