Finite Group Databasebeta

← #97026

Subgroups — #97026

Order 512.

952 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 952 conjugacy classes of subgroups — too many to draw. Only its 130 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
1C2×#2323C211yesmaximal, characteristic
2#8300C211yesmaximal, characteristic
3#8300C211yesmaximal, characteristic
4#7595C212yesmaximal, complemented
5#7593C212yesmaximal, complemented
6C22×C22⋊Q8C411yescharacteristic
7C22×C4×Q8C411yescharacteristic
8C22×#117C2211yescharacteristic
9#2323—24no—
10#2323—24no—
11C2×#391C2211yescharacteristic
12C2×#389—21no—
13C2×#389C2211yescharacteristic
14C2×#388—21no—
15C2×#388—21no—
16C2×#387—21no—
17C2×#380C411yescharacteristic
18C2×C4×C4⋊C4C411yescharacteristic
19#1817C2212yes—
20#1817C2212yes—
21C2×#324—21no—
22C2×#324—21no—
23C2×#323—21no—
24C2×#323—21no—
25#1272—24no—
26#1271—24no—
27C23×Q8C2×C411yescharacteristic
28C24×C4C2×C411yescharacteristic
29C2×C22⋊Q8—22no—
30C2×C22⋊Q8—22no—
31C2×C22⋊Q8—22no—
32C2×C22⋊Q8—22no—
33C2×C22⋊Q8—22no—
34C2×C22⋊Q8C814yescomplemented
35C2×C4×Q8—22no—
36C2×C4×Q8—22no—
37C2×C4×Q8—42no—
38C2×C4×Q8—42no—
39C2×#117—22no—
40C2×#117—22no—
41C2×#117—42no—
42C2×#117—42no—
43C22×C4⋊C4—21no—
44C22×C4⋊C4—21no—
45C22×C4⋊C4—21no—
46C22×C4⋊C4C2×C411yescharacteristic
47C22×C4⋊C4C2×C411yescharacteristic
48C22×C4⋊C4C2×C411yescharacteristic
49C22×C4⋊C4D411yescharacteristic
50C22×C4⋊C4D411yescharacteristic
51C22×C22⋊C4—21no—
52C22×C22⋊C4—21no—
53C22×C22⋊C4D411yescharacteristic
54C22×C22⋊C4D411yescharacteristic
55C22×C42—21no—
56C22×C42—21no—
57C22×C42C2×C411yescharacteristic
58C22×C42C2311yescharacteristic
59C2×C22⋊C8—22no—
60C2×C22⋊C8—22no—
61#391—22no—
62#391D414yes—
63#389—28no—
64#389—22no—
65#389D414yes—
66#388—28no—
67#388—28no—
68#387—28no—
69#380—22no—
70#380—22no—
71C4×C4⋊C4—22no—
72C4×C4⋊C4—22no—
73C2×#95—21no—
74C2×#95—21no—
75C2×#95—21no—
76C2×#95—21no—
77C2×#95—21no—
78#336—22no—
79#336—22no—
80#336—22no—
81#336—22no—
82#324—42no—
83#324—42no—
84#323—42no—
85#323—42no—
86C25C22⋊C411yescharacteristic
87C22×Q8—21no—
88C22×Q8—21no—
89C22×Q8—21no—
90C22×Q8—41no—
91C22×Q8—41no—
92C22×Q8—41no—
93C22×Q8—41no—
94C22×Q8C22⋊C412yes—
95C22×Q8C2×C812yes—
96C22×Q8M1612yes—
97C23×C4—21no—
98C23×C4—21no—
99C23×C4—21no—
100C23×C4—21no—
101C23×C4—21no—
102C23×C4—21no—
103C23×C4—21no—
104C23×C4—21no—
105C23×C4—41no—
106C23×C4—41no—
107C23×C4—41no—
108C23×C4—41no—
109C23×C4C22⋊C411yescharacteristic
110C23×C4C22⋊C411yescharacteristic
111C23×C4C22⋊C411yescharacteristic
112C23×C4C2×C812yes—
113C23×C4M1612yes—
114C23×C4C22×C411yescharacteristic
115C23×C4C2×D411yescharacteristic
116C23×C4C2×D411yescharacteristic
117C22×C8—42no—
118C22×C8—42no—
119C22⋊Q8—28no—
120C22⋊Q8—28no—
121C22⋊Q8—28no—
122C22⋊Q8—28no—
123C4×Q8—48no—
124C4×Q8—48no—
125#117—48no—
126#117—48no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—24no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—41no—
142C2×C4⋊C4—41no—
143C2×C4⋊C4—41no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4—41no—
147C2×C4⋊C4—42no—
148C2×C4⋊C4—42no—
149C2×C4⋊C4—82no—
150C2×C4⋊C4C22⋊C412yes—
151C2×C4⋊C4C22⋊C412yes—
152C2×C4⋊C4C22⋊C412yes—
153C2×C4⋊C4C2×C812yes—
154C2×C4⋊C4M1612yes—
155C2×C22⋊C4—24no—
156C2×C22⋊C4—24no—
157C2×C22⋊C4—24no—
158C2×C22⋊C4—22no—
159C2×C22⋊C4—22no—
160C2×C22⋊C4—22no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—24no—
163C2×C22⋊C4—41no—
164C2×C22⋊C4—41no—
165C2×C22⋊C4—82no—
166C2×C42—22no—
167C2×C42—22no—
168C2×C42—21no—
169C2×C42—21no—
170C2×C42—24no—
171C2×C42—24no—
172C2×C42—41no—
173C2×C42—41no—
174C2×C42—42no—
175C2×C42—42no—
176C2×C42—82no—
177C2×C42C2×D412yes—
178C2×C42C2×D412yes—
179C22⋊C8—42no—
180C22⋊C8—42no—
181C22⋊C8—42no—
182C22⋊C8—42no—
183#95—24no—
184#95—24no—
185#95—24no—
186#95—24no—
187#95—24no—
188C24—21no—
189C24—21no—
190C24—21no—
191C24—21no—
192C24—21no—
193C24—21no—
194C24—41no—
195C24—41no—
196C24—41no—
197C24—41no—
198C24C22⋊C812yes—
199C24C2×C22⋊C411yescharacteristic
200C2×Q8—22no—
201C2×Q8—22no—
202C2×Q8—22no—
203C2×Q8—42no—
204C2×Q8—42no—
205C2×Q8—42no—
206C2×Q8—42no—
207C2×Q8—42no—
208C2×Q8—42no—
209C2×Q8—42no—
210C2×Q8—42no—
211C2×Q8—42no—
212C2×Q8—42no—
213C2×Q8—42no—
214C2×Q8—42no—
215C2×Q8C22⋊C814yes—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4C22⋊C812yes—
281C22×C4#9912yes—
282C22×C4#9912yes—
283C22×C4C2×C22⋊C411yescharacteristic
284C22×C4C2×C22⋊C411yescharacteristic
285C22×C4#12012yes—
286C22×C4#12012yes—
287C22×C4C22×C811yescharacteristic
288C22×C4C2×M1611yescharacteristic
289C22×C4C2×M1611yescharacteristic
290C22×C4C2×M1611yescharacteristic
291C2×C8—82no—
292C2×C8—82no—
293C2×C8—82no—
294C2×C8—82no—
295C2×C8—82no—
296C2×C8—82no—
297C4⋊C4—216no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—44no—
302C4⋊C4—44no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—48no—
308C4⋊C4—84no—
309C4⋊C4C22⋊C814yes—
310C22⋊C4—216no—
311C22⋊C4—44no—
312C22⋊C4—44no—
313C22⋊C4—84no—
314C42—24no—
315C42—24no—
316C42—42no—
317C42—42no—
318C42—48no—
319C42—84no—
320C23—21no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—22no—
325C23—22no—
326C23—22no—
327C23—42no—
328C23—42no—
329C23—42no—
330C23—42no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—81no—
340C23—81no—
341C23—81no—
342C23—81no—
343C23—81no—
344C23—81no—
345C23—81no—
346C23—81no—
347C23#37911yescharacteristic
348C23C2×C22⋊C811yescharacteristic
349C23#40711yescharacteristic
350C23C2×#9911yescharacteristic
351C23C2×#9911yescharacteristic
352C23#41311yescharacteristic
353C23#41311yescharacteristic
354Q8—44no—
355Q8—44no—
356Q8—44no—
357Q8—44no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—24no—
377C2×C4—24no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—41no—
392C2×C4—42no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4C2×C22⋊C811yescharacteristic
424C2×C4C2×C22⋊C811yescharacteristic
425C2×C4#40711yescharacteristic
426C2×C4#40711yescharacteristic
427C8—162no—
428C8—162no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—42no—
442C22—42no—
443C22—42no—
444C22—82no—
445C22—82no—
446C22—82no—
447C22—82no—
448C22—81no—
449C22—81no—
450C22—81no—
451C22—81no—
452C22#145911yescharacteristic
453C22#145911yescharacteristic
454C22#146511yescharacteristic
455C22#146511yescharacteristic
456C22#177911yescharacteristic
457C22#178111yescharacteristic
458C22#178511yescharacteristic
459C22#179511yescharacteristic
460C22#179611yescharacteristic
461C22#179611yescharacteristic
462C22#179611yescharacteristic
463C4—21no—
464C4—21no—
465C4—21no—
466C4—21no—
467C4—42no—
468C4—42no—
469C4—42no—
470C4—42no—
471C4—42no—
472C4—44no—
473C4—44no—
474C4—82no—
475C4—82no—
476C2—21no—
477C2—21no—
478C2—21no—
479C2—21no—
480C2—82no—
481C2#830111yescharacteristic
482C2#832311yescharacteristic
483C2#832411yescharacteristic
484C2#851711yescharacteristic
485C2#852011yescharacteristic
486C2#852311yescharacteristic
487C2#852711yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC22×C4C22×C8
Frattini, cosocleC22×C42C23
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7595C2semidirect
5. #7593C2semidirect
34. C2×C22⋊Q8C8semidirect

Maximal subgroups

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

subgroupindex
C2×#23232
#83002
#75952
#75932