Finite Group Databasebeta

← #137940

Subgroups — #137940

Order 512.

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

normal subgroups only

This group has 786 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#14822C211yesmaximal, characteristic
3C2×#1353C212yesmaximal, complemented
4#10727C211yesmaximal, characteristic
5#7251C212yesmaximal, complemented
6C23×C4⋊C4C2211yescharacteristic
7C2×#407—21no—
8C22×C22⋊C8—21no—
9C2×#400C411yescharacteristic
10C2×#400C2211yescharacteristic
11C2×#380C411yescharacteristic
12C2×#343—21no—
13C2×#343C2212yes—
14C2×#324—24no—
15C2×#324—21no—
16C2×#324C2212yes—
17#1442—21no—
18#1442—21no—
19#1441C411yescharacteristic
20#1441C2211yescharacteristic
21C22⋊#95—21no—
22C22⋊#95—21no—
23C22⋊#95C411yescharacteristic
24#1353—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—42no—
33C22×C4⋊C4—41no—
34C22×C4⋊C4—41no—
35C22×C4⋊C4C2×C411yescharacteristic
36C22×C4⋊C4C2×C411yescharacteristic
37C22×C4⋊C4C2×C411yescharacteristic
38C22×C4⋊C4D411yescharacteristic
39C22×C4⋊C4D411yescharacteristic
40C22×C4⋊C4C2311yescharacteristic
41C22×C22⋊C4—21no—
42C22×C22⋊C4—21no—
43#407—24no—
44#407—42no—
45C2×C22⋊C8—24no—
46C2×C22⋊C8—22no—
47C2×C22⋊C8—41no—
48C2×C22⋊C8—42no—
49C2×C22⋊C8—41no—
50C2×C22⋊C8—41no—
51#400—22no—
52#400—22no—
53#400—42no—
54#380—41no—
55#380—41no—
56C2×#95—21no—
57C2×#95—21no—
58C2×#95—21no—
59C2×#95—21no—
60C2×#95—21no—
61C2×#95—21no—
62C2×#95—21no—
63C2×#95—21no—
64C2×#95—21no—
65C2×#95—21no—
66C2×#95C2×C411yescharacteristic
67C2×#95C2×C411yescharacteristic
68#343—22no—
69#343—22no—
70#343—42no—
71#324—24no—
72#324—22no—
73#324D414yes—
74C25C22⋊C411yescharacteristic
75C23×C4—21no—
76C23×C4—21no—
77C23×C4—21no—
78C23×C4—22no—
79C23×C4—21no—
80C23×C4—22no—
81C23×C4—22no—
82C23×C4—22no—
83C23×C4—21no—
84C23×C4—21no—
85C23×C4—21no—
86C23×C4—21no—
87C23×C4—22no—
88C23×C4—21no—
89C23×C4—21no—
90C23×C4—21no—
91C23×C4—21no—
92C23×C4—41no—
93C23×C4—41no—
94C23×C4—41no—
95C23×C4—42no—
96C23×C4—42no—
97C23×C4—81no—
98C23×C4—81no—
99C23×C4C22⋊C411yescharacteristic
100C23×C4C22⋊C411yescharacteristic
101C23×C4C22⋊C411yescharacteristic
102C23×C4SD1612yes—
103C23×C4Q1612yes—
104C23×C4C22×C411yescharacteristic
105C23×C4C2×D411yescharacteristic
106C23×C4C2×D411yescharacteristic
107C2×M16—24no—
108C2×M16—42no—
109C2×M16—42no—
110C2×M16—42no—
111C2×M16—42no—
112C2×M16—42no—
113C22×C8—44no—
114C22×C8—42no—
115C22×C8—42no—
116C22×C8—82no—
117C22×C8—82no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—41no—
126C2×C4⋊C4—41no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4—41no—
129C2×C4⋊C4—41no—
130C2×C4⋊C4—41no—
131C2×C4⋊C4—44no—
132C2×C4⋊C4—44no—
133C2×C4⋊C4—44no—
134C2×C4⋊C4—44no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4—41no—
138C2×C4⋊C4—81no—
139C2×C4⋊C4C2×D412yes—
140C2×C4⋊C4#4112yes—
141C2×C22⋊C4—42no—
142C2×C22⋊C4—42no—
143C2×C22⋊C4—42no—
144C2×C22⋊C4—42no—
145C2×C22⋊C4—41no—
146C2×C22⋊C4—41no—
147C2×C22⋊C4—42no—
148C22⋊C8—24no—
149C22⋊C8—44no—
150C22⋊C8—42no—
151C22⋊C8—44no—
152C22⋊C8—44no—
153#95—22no—
154#95—41no—
155#95—42no—
156#95—42no—
157#95—42no—
158#95—42no—
159#95—41no—
160#95—42no—
161#95—41no—
162#95—42no—
163#95—81no—
164#95—81no—
165#95—81no—
166C24—22no—
167C24—22no—
168C24—21no—
169C24—21no—
170C24—21no—
171C24—21no—
172C24—41no—
173C24—41no—
174C24#9911yescharacteristic
175C24#9911yescharacteristic
176C24#10312yes—
177C24#10412yes—
178C24C2×C22⋊C411yescharacteristic
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—24no—
204C22×C4—24no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—44no—
213C22×C4—44no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—42no—
227C22×C4—42no—
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—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
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—82no—
259C22×C4—84no—
260C22×C4—82no—
261C22×C4—81no—
262C22×C4—81no—
263C22×C4—81no—
264C22×C4—81no—
265C22×C4—84no—
266C22×C4—84no—
267C22×C4#9912yes—
268C22×C4#10312yes—
269C22×C4C4×D412yes—
270C22×C4#12011yescharacteristic
271C22×C4#12112yes—
272C22×C4#12311yescharacteristic
273C22×C4C2×SD1611yescharacteristic
274C22×C4C2×Q1611yescharacteristic
275C22×C4#13512yes—
276M16—44no—
277M16—44no—
278M16—82no—
279M16—82no—
280C2×C8—44no—
281C2×C8—44no—
282C2×C8—82no—
283C2×C8—82no—
284C2×C8—88no—
285C2×C8—82no—
286C2×C8—82no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—44no—
290C4⋊C4—44no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—82no—
294C4⋊C4—82no—
295C22⋊C4—84no—
296C22⋊C4—82no—
297C22⋊C4—82no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—24no—
308C23—24no—
309C23—21no—
310C23—22no—
311C23—21no—
312C23—21no—
313C23—22no—
314C23—21no—
315C23—22no—
316C23—21no—
317C23—22no—
318C23—42no—
319C23—42no—
320C23—42no—
321C23—42no—
322C23—42no—
323C23—42no—
324C23—42no—
325C23—42no—
326C23—42no—
327C23—42no—
328C23#32812yes—
329C23#32812yes—
330C23#35212yes—
331C23#35512yes—
332C23#38611yescharacteristic
333C23C2×#9911yescharacteristic
334C23C2×#9911yescharacteristic
335C23C2×#10311yescharacteristic
336C23#41611yescharacteristic
337C23C2×#10411yescharacteristic
338C23#42111yescharacteristic
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—21no—
350C2×C4—22no—
351C2×C4—21no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—42no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—84no—
383C2×C4—84no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
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#46512yes—
415C2×C4#46712yes—
416C8—84no—
417C8—84no—
418C22—21no—
419C22—21no—
420C22—22no—
421C22—21no—
422C22—21no—
423C22—22no—
424C22—21no—
425C22—21no—
426C22—22no—
427C22—21no—
428C22—21no—
429C22—22no—
430C22—22no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—22no—
435C22—21no—
436C22—22no—
437C22—22no—
438C22—22no—
439C22—44no—
440C22—44no—
441C22—44no—
442C22—44no—
443C22—41no—
444C22—41no—
445C22—41no—
446C22—41no—
447C22#135314yescomplemented
448C22C2×#32811yescharacteristic
449C22#150111yescharacteristic
450C22#151611yescharacteristic
451C22#152011yescharacteristic
452C22#187111yescharacteristic
453C22#189211yescharacteristic
454C22#189811yescharacteristic
455C22C2×#35211yescharacteristic
456C22#212111yescharacteristic
457C22C2×#35511yescharacteristic
458C22#213311yescharacteristic
459C4—22no—
460C4—22no—
461C4—41no—
462C4—42no—
463C4—41no—
464C4—42no—
465C4—42no—
466C4—81no—
467C4—161no—
468C4—161no—
469C4—161no—
470C4—161no—
471C2—21no—
472C2—21no—
473C2—24no—
474C2—21no—
475C2—21no—
476C2—42no—
477C2#911111yescharacteristic
478C2C2×#135311yescharacteristic
479C2#1172211yescharacteristic
480C2#1185011yescharacteristic
481C2#1185211yescharacteristic
482C2#1308411yescharacteristic
483C2#1309211yescharacteristic
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×#1353C2semidirect
5. #7251C2semidirect
447. C22#1353semidirect

Maximal subgroups

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

subgroupindex
C22×#3242
#148222
C2×#13532
#107272
#72512