Finite Group Databasebeta

← #142010

Subgroups — #142010

Order 512.

652 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 652 conjugacy classes of subgroups — too many to draw. Only its 110 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#15165C211yesmaximal, characteristic
2#10846C211yesmaximal, characteristic
3C4×#323C211yesmaximal, characteristic
4#7250C212yesmaximal
5#7250C212yesmaximal
6C2×#396—21no—
7C2×#389C411yescharacteristic, complemented
8C2×#380—21no—
9C2×C4×C22⋊C4C2211yescharacteristic
10C4×C22⋊C8—21no—
11C2×#342—21no—
12C2×#342C2212yes—
13C2×#323C2212yes—
14#1444C411yescharacteristic, complemented
15#1442C411yescharacteristic, complemented
16#1441—21no—
17#1439C2211yescharacteristic
18#1439C2211yescharacteristic
19#1436—21no—
20C4⋊#95C411yescharacteristic, complemented
21C22⋊#95—21no—
22#1280—21no—
23C24×C4D411yescharacteristic
24C22×C4⋊C4—21no—
25C22×C4⋊C4—21no—
26C22×C4⋊C4—21no—
27C22×C4⋊C4—41no—
28C22×C4⋊C4C2×C411yescharacteristic
29C22×C4⋊C4C2×C411yescharacteristic
30C22×C22⋊C4—22no—
31C22×C22⋊C4—21no—
32C22×C22⋊C4—21no—
33C22×C22⋊C4D411yescharacteristic
34C22×C22⋊C4C2311yescharacteristic
35C22×C42C2×C411yescharacteristic
36C22×C42D411yescharacteristic
37C2×C22⋊C8—22no—
38C2×C4×C8—41no—
39#396—42no—
40#396—42no—
41#389—41no—
42#389—41no—
43#380—24no—
44#380—42no—
45C4×C22⋊C4—41no—
46C4×C22⋊C4—41no—
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×#95C2×C411yescharacteristic
56C2×#95C2×C411yescharacteristic
57C2×#95C2×C411yescharacteristic
58C2×#95D411yescharacteristic
59#342—24no—
60#342—42no—
61#336—41no—
62#323—22no—
63#323Q814yes—
64C25C2×D411yescharacteristic
65C23×C4—21no—
66C23×C4—21no—
67C23×C4—22no—
68C23×C4—21no—
69C23×C4—21no—
70C23×C4—21no—
71C23×C4—21no—
72C23×C4—42no—
73C23×C4—81no—
74C23×C4—81no—
75C23×C4C22⋊C411yescharacteristic
76C23×C4C22⋊C411yescharacteristic
77C23×C4C22⋊C411yescharacteristic
78C23×C4C22⋊C411yescharacteristic
79C23×C4SD1612yes—
80C23×C4Q1612yes—
81C23×C4C22×C411yescharacteristic
82C23×C4C2×D411yescharacteristic
83C22×C8—42no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—42no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4—42no—
88C2×C4⋊C4—42no—
89C2×C4⋊C4—42no—
90C2×C4⋊C4—42no—
91C2×C4⋊C4—44no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4—42no—
94C2×C4⋊C4—42no—
95C2×C4⋊C4—81no—
96C2×C4⋊C4—81no—
97C2×C4⋊C4—82no—
98C2×C4⋊C4—82no—
99C2×C4⋊C4—81no—
100C2×C22⋊C4—24no—
101C2×C22⋊C4—41no—
102C2×C22⋊C4—41no—
103C2×C22⋊C4—41no—
104C2×C22⋊C4—41no—
105C2×C22⋊C4—42no—
106C2×C22⋊C4—42no—
107C2×C22⋊C4—42no—
108C2×C22⋊C4—42no—
109C2×C22⋊C4—42no—
110C2×C22⋊C4—42no—
111C2×C22⋊C4—41no—
112C2×C22⋊C4—41no—
113C2×C22⋊C4—42no—
114C2×C22⋊C4—41no—
115C2×C22⋊C4—41no—
116C2×C22⋊C4—81no—
117C2×C22⋊C4—81no—
118C2×C22⋊C4D812yes—
119C2×C22⋊C4SD1612yes—
120C2×C22⋊C4C2×Q812yes—
121C2×C22⋊C4#4112yes—
122C2×C42—21no—
123C2×C42—21no—
124C2×C42—21no—
125C2×C42—21no—
126C2×C42—81no—
127C2×C42—81no—
128C22⋊C8—24no—
129C22⋊C8—42no—
130C4×C8—82no—
131#95—41no—
132#95—41no—
133#95—41no—
134#95—41no—
135#95—42no—
136#95—44no—
137#95—41no—
138#95—41no—
139#95—42no—
140#95—42no—
141#95—81no—
142#95—81no—
143#95—81no—
144C24—21no—
145C24—21no—
146C24—21no—
147C24—21no—
148C24—41no—
149C24—41no—
150C24—41no—
151C24—41no—
152C24C2×C22⋊C411yescharacteristic
153C24#12411yescharacteristic
154C24C4⋊Q811yescharacteristic
155C24C2×D811yescharacteristic
156C24C2×SD1611yescharacteristic
157C24C2×SD1611yescharacteristic
158C24C2×Q1611yescharacteristic
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—44no—
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—81no—
217C22×C4—81no—
218C22×C4—82no—
219C22×C4—82no—
220C22×C4—84no—
221C22×C4—82no—
222C22×C4—82no—
223C22×C4—82no—
224C22×C4—82no—
225C22×C4—82no—
226C22×C4—82no—
227C22×C4#9911yescharacteristic
228C22×C4#9911yescharacteristic
229C22×C4#10212yes—
230C22×C4#10212yes—
231C22×C4#10312yes—
232C22×C4#10312yes—
233C22×C4#10412yes—
234C22×C4C4×Q812yes—
235C22×C4C22⋊Q812yes—
236C2×C8—44no—
237C2×C8—82no—
238C2×C8—82no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—84no—
242C4⋊C4—84no—
243C4⋊C4—84no—
244C4⋊C4—82no—
245C4⋊C4—88no—
246C22⋊C4—42no—
247C22⋊C4—42no—
248C22⋊C4—42no—
249C22⋊C4—42no—
250C22⋊C4—44no—
251C22⋊C4—44no—
252C22⋊C4—82no—
253C22⋊C4—82no—
254C22⋊C4—82no—
255C22⋊C4—82no—
256C22⋊C4—82no—
257C22⋊C4—82no—
258C42—42no—
259C42—81no—
260C42—81no—
261C42—81no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—81no—
295C23—81no—
296C23—81no—
297C23—81no—
298C23—81no—
299C23—81no—
300C23#39111yescharacteristic
301C23C2×#9911yescharacteristic
302C23#41011yescharacteristic
303C23C2×#10211yescharacteristic
304C23C2×#10311yescharacteristic
305C23C2×#10411yescharacteristic
306C23#42111yescharacteristic
307C23#48611yescharacteristic
308C23#48711yescharacteristic
309C23#49811yescharacteristic
310C23#50011yescharacteristic
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—44no—
323C2×C4—44no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—82no—
348C2×C4—82no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—84no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—84no—
372C2×C4—84no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—162no—
378C2×C4—162no—
379C2×C4—162no—
380C2×C4—162no—
381C2×C4#32812yes—
382C2×C4#32812yes—
383C8—84no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—81no—
417C22—81no—
418C22—81no—
419C22—81no—
420C22—81no—
421C22—81no—
422C22C2×#32811yescharacteristic
423C22#150211yescharacteristic
424C22#151811yescharacteristic
425C22#151911yescharacteristic
426C22#198611yescharacteristic
427C22#199011yescharacteristic
428C22#199611yescharacteristic
429C22#212011yescharacteristic
430C22#212111yescharacteristic
431C22#212611yescharacteristic
432C22#212711yescharacteristic
433C4—22no—
434C4—22no—
435C4—42no—
436C4—42no—
437C4—42no—
438C4—42no—
439C4—42no—
440C4—82no—
441C4—82no—
442C4—161no—
443C4—164no—
444C4—161no—
445C2—21no—
446C2—21no—
447C2—21no—
448C2—21no—
449C2—41no—
450C2—41no—
451C2—41no—
452C2—41no—
453C2#988411yescharacteristic
454C2#1171511yescharacteristic
455C2#1171611yescharacteristic
456C2#1187511yescharacteristic
457C2#1187611yescharacteristic
458C2#1329411yescharacteristic
459C2#1330011yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC22×C22⋊C4C23
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
7. C2×#389C4semidirect
14. #1444C4semidirect
15. #1442C4semidirect
20. C4⋊#95C4semidirect

Maximal subgroups

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

subgroupindex
#151652
#108462
C4×#3232
#72502