Finite Group Databasebeta

← #137992

Subgroups — #137992

Order 512.

822 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 822 conjugacy classes of subgroups — too many to draw. Only its 112 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#14821C211yesmaximal, characteristic
3C2×#1346C212yesmaximal, complemented
4#10727C211yesmaximal, characteristic
5#7251C212yesmaximal, complemented
6C23×C4⋊C4C2211yescharacteristic
7C2×#407—21no—
8C22×C22⋊C8—21no—
9C2×#398C411yescharacteristic
10C2×#398C2211yescharacteristic
11C2×#385—21no—
12C2×#380C411yescharacteristic
13C2×#343—21no—
14C2×#343C2212yes—
15C2×#324—24no—
16C2×#324—21no—
17C2×#324C2212yes—
18#1441—21no—
19#1441—21no—
20#1441C411yescharacteristic
21#1441C2211yescharacteristic
22C22⋊#95—21no—
23C22⋊#95C411yescharacteristic
24#1346—24no—
25C24×C4C2×C411yescharacteristic
26C24×C4D411yescharacteristic
27C24×C4D411yescharacteristic
28C22×M16—22no—
29C23×C8—41no—
30C22×C4⋊C4—22no—
31C22×C4⋊C4—22no—
32C22×C4⋊C4—21no—
33C22×C4⋊C4—21no—
34C22×C4⋊C4—42no—
35C22×C4⋊C4—41no—
36C22×C4⋊C4—41no—
37C22×C4⋊C4C2×C411yescharacteristic
38C22×C4⋊C4C2×C411yescharacteristic
39C22×C4⋊C4C2×C411yescharacteristic
40C22×C4⋊C4D411yescharacteristic
41C22×C4⋊C4D411yescharacteristic
42C22×C4⋊C4C2311yescharacteristic
43C22×C22⋊C4—21no—
44C22×C22⋊C4—21no—
45#407—24no—
46#407—42no—
47C2×C22⋊C8—24no—
48C2×C22⋊C8—22no—
49C2×C22⋊C8—41no—
50C2×C22⋊C8—42no—
51C2×C22⋊C8—41no—
52C2×C22⋊C8—41no—
53#398—22no—
54#398—22no—
55#398—42no—
56#385—42no—
57#385—42no—
58#380—41no—
59#380—41no—
60C2×#95—21no—
61C2×#95—21no—
62C2×#95—21no—
63C2×#95—21no—
64C2×#95—21no—
65C2×#95—21no—
66C2×#95—21no—
67C2×#95—21no—
68C2×#95C2×C411yescharacteristic
69C2×#95C2×C411yescharacteristic
70#343—22no—
71#343—22no—
72#343—42no—
73#324—24no—
74#324—22no—
75#324D414yes—
76C25C22⋊C411yescharacteristic
77C23×C4—21no—
78C23×C4—21no—
79C23×C4—21no—
80C23×C4—22no—
81C23×C4—21no—
82C23×C4—22no—
83C23×C4—22no—
84C23×C4—22no—
85C23×C4—21no—
86C23×C4—21no—
87C23×C4—21no—
88C23×C4—21no—
89C23×C4—21no—
90C23×C4—21no—
91C23×C4—22no—
92C23×C4—21no—
93C23×C4—21no—
94C23×C4—41no—
95C23×C4—41no—
96C23×C4—41no—
97C23×C4—42no—
98C23×C4—42no—
99C23×C4—81no—
100C23×C4—81no—
101C23×C4C22⋊C411yescharacteristic
102C23×C4C22⋊C411yescharacteristic
103C23×C4C22⋊C411yescharacteristic
104C23×C4D812yes—
105C23×C4SD1612yes—
106C23×C4C22×C411yescharacteristic
107C23×C4C2×D411yescharacteristic
108C23×C4C2×D411yescharacteristic
109C2×M16—24no—
110C2×M16—42no—
111C2×M16—42no—
112C2×M16—42no—
113C2×M16—42no—
114C2×M16—42no—
115C22×C8—44no—
116C22×C8—42no—
117C22×C8—42no—
118C22×C8—82no—
119C22×C8—82no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4—41no—
129C2×C4⋊C4—41no—
130C2×C4⋊C4—41no—
131C2×C4⋊C4—41no—
132C2×C4⋊C4—41no—
133C2×C4⋊C4—44no—
134C2×C4⋊C4—44no—
135C2×C4⋊C4—44no—
136C2×C4⋊C4—44no—
137C2×C4⋊C4—41no—
138C2×C4⋊C4—41no—
139C2×C4⋊C4—41no—
140C2×C4⋊C4—42no—
141C2×C4⋊C4—42no—
142C2×C4⋊C4—42no—
143C2×C4⋊C4—42no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4—42no—
147C2×C4⋊C4—81no—
148C2×C4⋊C4C2×D412yes—
149C2×C4⋊C4#4112yes—
150C2×C22⋊C4—24no—
151C2×C22⋊C4—42no—
152C2×C22⋊C4—42no—
153C2×C22⋊C4—41no—
154C2×C22⋊C4—41no—
155C2×C22⋊C4—42no—
156C2×C22⋊C4—42no—
157C22⋊C8—24no—
158C22⋊C8—44no—
159C22⋊C8—42no—
160C22⋊C8—44no—
161C22⋊C8—44no—
162#95—22no—
163#95—41no—
164#95—42no—
165#95—42no—
166#95—42no—
167#95—41no—
168#95—41no—
169#95—41no—
170#95—41no—
171#95—81no—
172#95—81no—
173#95—81no—
174C24—22no—
175C24—22no—
176C24—21no—
177C24—21no—
178C24—21no—
179C24—21no—
180C24—41no—
181C24—41no—
182C24#9911yescharacteristic
183C24#9911yescharacteristic
184C24#10212yes—
185C24#10412yes—
186C24C2×C22⋊C411yescharacteristic
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—24no—
212C22×C4—24no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—44no—
223C22×C4—44no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
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—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—82no—
267C22×C4—84no—
268C22×C4—82no—
269C22×C4—81no—
270C22×C4—81no—
271C22×C4—81no—
272C22×C4—81no—
273C22×C4—84no—
274C22×C4—84no—
275C22×C4#9912yes—
276C22×C4#10212yes—
277C22×C4C4×D412yes—
278C22×C4#12011yescharacteristic
279C22×C4#12112yes—
280C22×C4#12311yescharacteristic
281C22×C4C2×D811yescharacteristic
282C22×C4C2×SD1611yescharacteristic
283C22×C4#13512yes—
284M16—44no—
285M16—44no—
286M16—82no—
287M16—82no—
288C2×C8—44no—
289C2×C8—44no—
290C2×C8—82no—
291C2×C8—82no—
292C2×C8—88no—
293C2×C8—82no—
294C2×C8—82no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—44no—
298C4⋊C4—44no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—82no—
302C4⋊C4—84no—
303C4⋊C4—82no—
304C4⋊C4—82no—
305C4⋊C4—82no—
306C22⋊C4—48no—
307C22⋊C4—82no—
308C22⋊C4—82no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—24no—
318C23—24no—
319C23—21no—
320C23—21no—
321C23—22no—
322C23—21no—
323C23—21no—
324C23—22no—
325C23—21no—
326C23—22no—
327C23—21no—
328C23—22no—
329C23—42no—
330C23—42no—
331C23—42no—
332C23—42no—
333C23—42no—
334C23—42no—
335C23—42no—
336C23—42no—
337C23—42no—
338C23—42no—
339C23#32712yes—
340C23#32712yes—
341C23#35212yes—
342C23#35612yes—
343C23#38611yescharacteristic
344C23C2×#9911yescharacteristic
345C23C2×#9911yescharacteristic
346C23C2×#10211yescharacteristic
347C23#41611yescharacteristic
348C23C2×#10411yescharacteristic
349C23#42111yescharacteristic
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—21no—
361C2×C4—22no—
362C2×C4—21no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—42no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—84no—
400C2×C4—84no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4#46512yes—
426C2×C4#46612yes—
427C8—84no—
428C8—84no—
429C22—21no—
430C22—21no—
431C22—22no—
432C22—21no—
433C22—21no—
434C22—22no—
435C22—21no—
436C22—21no—
437C22—22no—
438C22—21no—
439C22—21no—
440C22—22no—
441C22—22no—
442C22—21no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—22no—
447C22—22no—
448C22—22no—
449C22—22no—
450C22—44no—
451C22—44no—
452C22—44no—
453C22—44no—
454C22—41no—
455C22—41no—
456C22—41no—
457C22—41no—
458C22#134614yescomplemented
459C22C2×#32711yescharacteristic
460C22#150111yescharacteristic
461C22#151611yescharacteristic
462C22#151911yescharacteristic
463C22#187111yescharacteristic
464C22#189311yescharacteristic
465C22#189811yescharacteristic
466C22C2×#35211yescharacteristic
467C22#212111yescharacteristic
468C22C2×#35611yescharacteristic
469C22#213311yescharacteristic
470C4—22no—
471C4—22no—
472C4—41no—
473C4—42no—
474C4—41no—
475C4—42no—
476C4—42no—
477C4—82no—
478C4—82no—
479C4—81no—
480C4—161no—
481C4—161no—
482C2—21no—
483C2—21no—
484C2—24no—
485C2—21no—
486C2—21no—
487C2—42no—
488C2#911311yescharacteristic
489C2C2×#134611yescharacteristic
490C2#1169811yescharacteristic
491C2#1185011yescharacteristic
492C2#1185411yescharacteristic
493C2#1308411yescharacteristic
494C2#1309411yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC22×C4⋊C4C23
prime radicalC25C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ 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×#1346C2semidirect
5. #7251C2semidirect
458. C22#1346semidirect

Maximal subgroups

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

subgroupindex
C22×#3242
#148212
C2×#13462
#107272
#72512