Finite Group Databasebeta

← #137682

Subgroups — #137682

Order 512.

870 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 870 conjugacy classes of subgroups — too many to draw. Only its 120 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
1C22×#324C211yesmaximal, characteristic
2#14139C211yesmaximal, characteristic
3C2×#1319C212yesmaximal, complemented
4#8299C211yesmaximal, characteristic
5#7189C212yesmaximal, complemented
6C22×C22⋊Q8C411yescharacteristic
7C23×C4⋊C4C2211yescharacteristic
8C22×C22⋊C8—21no—
9C2×#391C411yescharacteristic
10C2×#391C2211yescharacteristic
11C2×C4×C22⋊C4C411yescharacteristic
12#1779—21no—
13C2×#324—24no—
14C2×#324—21no—
15C2×#324—21no—
16C2×#324C2212yes—
17C2×#324C2212yes—
18#1441—21no—
19#1441—21no—
20C4⋊#95C411yescharacteristic
21C4⋊#95C2211yescharacteristic
22C22⋊#95—21no—
23C22⋊#95—21no—
24#1319—24no—
25C23×Q8C2×C411yescharacteristic
26C24×C4C2×C411yescharacteristic
27C24×C4D411yescharacteristic
28C24×C4D411yescharacteristic
29C23×C8—41no—
30C2×C22⋊Q8—22no—
31C2×C22⋊Q8—41no—
32C2×C22⋊Q8—41no—
33C2×C22⋊Q8—42no—
34C2×C22⋊Q8C814yescomplemented
35C22×C4⋊C4—21no—
36C22×C4⋊C4—22no—
37C22×C4⋊C4—22no—
38C22×C4⋊C4—42no—
39C22×C4⋊C4—41no—
40C22×C4⋊C4—41no—
41C22×C4⋊C4C2×C411yescharacteristic
42C22×C4⋊C4C2×C411yescharacteristic
43C22×C4⋊C4D411yescharacteristic
44C22×C4⋊C4D411yescharacteristic
45C22×C4⋊C4C2311yescharacteristic
46C22×C22⋊C4—21no—
47C22×C22⋊C4—21no—
48C22×C42C2×C411yescharacteristic
49C22×C42C2×C411yescharacteristic
50C2×C22⋊C8—24no—
51C2×C22⋊C8—22no—
52C2×C22⋊C8—22no—
53C2×C22⋊C8—41no—
54C2×C22⋊C8—42no—
55C2×C22⋊C8—41no—
56C2×C22⋊C8—41no—
57#391—22no—
58#391—22no—
59#391—42no—
60C4×C22⋊C4—42no—
61C2×#95—21no—
62C2×#95—21no—
63C2×#95—21no—
64C2×#95—21no—
65C2×#95—21no—
66C2×#95—21no—
67C2×#95—21no—
68C2×#95—21no—
69C2×#95—21no—
70#324—24no—
71#324—22no—
72#324—22no—
73#324—22no—
74#324—42no—
75#324D414yes—
76C25C22⋊C411yescharacteristic
77C22×Q8—21no—
78C22×Q8—22no—
79C22×Q8—22no—
80C22×Q8—41no—
81C22×Q8—41no—
82C22×Q8—41no—
83C22×Q8—41no—
84C22×Q8C2×C812yescomplemented
85C23×C4—21no—
86C23×C4—21no—
87C23×C4—21no—
88C23×C4—22no—
89C23×C4—21no—
90C23×C4—22no—
91C23×C4—22no—
92C23×C4—22no—
93C23×C4—21no—
94C23×C4—21no—
95C23×C4—21no—
96C23×C4—21no—
97C23×C4—21no—
98C23×C4—21no—
99C23×C4—21no—
100C23×C4—21no—
101C23×C4—22no—
102C23×C4—41no—
103C23×C4—41no—
104C23×C4—41no—
105C23×C4—42no—
106C23×C4—42no—
107C23×C4—81no—
108C23×C4—81no—
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—44no—
118C22×C8—42no—
119C22×C8—42no—
120C22×C8—82no—
121C22×C8—82no—
122C22⋊Q8—44no—
123C22⋊Q8—44no—
124C22⋊Q8—44no—
125C22⋊Q8—44no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4—41no—
138C2×C4⋊C4—41no—
139C2×C4⋊C4—41no—
140C2×C4⋊C4—41no—
141C2×C4⋊C4—41no—
142C2×C4⋊C4—44no—
143C2×C4⋊C4—44no—
144C2×C4⋊C4—44no—
145C2×C4⋊C4—44no—
146C2×C4⋊C4—42no—
147C2×C4⋊C4—42no—
148C2×C4⋊C4C2×C812yescomplemented
149C2×C4⋊C4C2×D412yes—
150C2×C4⋊C4#4112yes—
151C2×C22⋊C4—24no—
152C2×C22⋊C4—24no—
153C2×C22⋊C4—42no—
154C2×C22⋊C4—41no—
155C2×C22⋊C4—41no—
156C2×C22⋊C4—42no—
157C2×C22⋊C4—42no—
158C2×C42—22no—
159C2×C42—22no—
160C2×C42—41no—
161C2×C42—41no—
162C2×C42—41no—
163C2×C42—41no—
164C22⋊C8—24no—
165C22⋊C8—44no—
166C22⋊C8—42no—
167C22⋊C8—42no—
168C22⋊C8—42no—
169C22⋊C8—82no—
170C22⋊C8—82no—
171#95—42no—
172#95—42no—
173#95—42no—
174#95—41no—
175#95—42no—
176#95—41no—
177#95—81no—
178#95—81no—
179#95—81no—
180#95—81no—
181C24—22no—
182C24—22no—
183C24—21no—
184C24—21no—
185C24—21no—
186C24—21no—
187C24—41no—
188C24—41no—
189C24C22⋊C812yes—
190C24#9911yescharacteristic
191C24#9911yescharacteristic
192C24#10012yes—
193C24C2×C22⋊C411yescharacteristic
194C2×Q8—24no—
195C2×Q8—42no—
196C2×Q8—42no—
197C2×Q8—42no—
198C2×Q8—42no—
199C2×Q8—42no—
200C2×Q8—82no—
201C2×Q8—82no—
202C2×Q8—82no—
203C2×Q8—82no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—24no—
229C22×C4—24no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—44no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—44no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—42no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—42no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—82no—
284C22×C4—84no—
285C22×C4—82no—
286C22×C4—81no—
287C22×C4—81no—
288C22×C4—81no—
289C22×C4—81no—
290C22×C4—84no—
291C22×C4—84no—
292C22×C4C22⋊C812yes—
293C22×C4#9912yes—
294C22×C4C4×D412yes—
295C22×C4#12011yescharacteristic
296C22×C4#12112yes—
297C22×C4#12311yescharacteristic
298C22×C4C22×C811yescharacteristic
299C22×C4C2×M1611yescharacteristic
300C22×C4#13112yes—
301C2×C8—44no—
302C2×C8—88no—
303C2×C8—82no—
304C2×C8—82no—
305C2×C8—82no—
306C2×C8—82no—
307C2×C8—82no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—44no—
311C4⋊C4—44no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—48no—
317C22⋊C4—48no—
318C22⋊C4—44no—
319C22⋊C4—44no—
320C42—82no—
321C42—82no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—24no—
331C23—24no—
332C23—22no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23—22no—
339C23—22no—
340C23—21no—
341C23—22no—
342C23—42no—
343C23—42no—
344C23—42no—
345C23—42no—
346C23—42no—
347C23—42no—
348C23—42no—
349C23—42no—
350C23—42no—
351C23—42no—
352C23#32312yes—
353C23#32312yes—
354C23#35412yes—
355C23#35612yes—
356C23#38611yescharacteristic
357C23C2×C22⋊C811yescharacteristic
358C23#40811yescharacteristic
359C23C2×#9911yescharacteristic
360C23C2×#9911yescharacteristic
361C23C2×#10011yescharacteristic
362C23#41311yescharacteristic
363Q8—84no—
364Q8—84no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—21no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—21no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—42no—
397C2×C4—41no—
398C2×C4—42no—
399C2×C4—41no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—84no—
422C2×C4—84no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4C8×D412yes—
441C2×C4#43512yes—
442C8—84no—
443C8—162no—
444C22—21no—
445C22—21no—
446C22—22no—
447C22—21no—
448C22—21no—
449C22—22no—
450C22—21no—
451C22—21no—
452C22—22no—
453C22—21no—
454C22—21no—
455C22—22no—
456C22—22no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—22no—
462C22—22no—
463C22—22no—
464C22—22no—
465C22—44no—
466C22—44no—
467C22—44no—
468C22—44no—
469C22—41no—
470C22—41no—
471C22—41no—
472C22—41no—
473C22#131914yescomplemented
474C22C2×#32311yescharacteristic
475C22#145911yescharacteristic
476C22#146811yescharacteristic
477C22#147111yescharacteristic
478C22#186911yescharacteristic
479C22#189811yescharacteristic
480C22#190011yescharacteristic
481C22C2×#35411yescharacteristic
482C22#212711yescharacteristic
483C22C2×#35611yescharacteristic
484C22#213311yescharacteristic
485C4—22no—
486C4—22no—
487C4—41no—
488C4—42no—
489C4—42no—
490C4—41no—
491C4—42no—
492C4—82no—
493C4—82no—
494C4—81no—
495C4—82no—
496C4—82no—
497C2—21no—
498C2—21no—
499C2—24no—
500C2—21no—
501C2—21no—
502C2—42no—
503C2#907911yescharacteristic
504C2C2×#131911yescharacteristic
505C2#1143211yescharacteristic
506C2#1155111yescharacteristic
507C2#1155511yescharacteristic
508C2#1309011yescharacteristic
509C2#1309411yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C22×C8
Frattini, cosocleC22×C4⋊C4C23
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C25 ◁ C23×C4⋊C4 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1319C2semidirect
5. #7189C2semidirect
34. C2×C22⋊Q8C8semidirect
84. C22×Q8C2×C8semidirect
148. C2×C4⋊C4C2×C8semidirect
473. C22#1319semidirect

Maximal subgroups

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

subgroupindex
C22×#3242
#141392
C2×#13192
#82992
#71892