Finite Group Databasebeta

← #101392

Subgroups — #101392

Order 512.

570 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 570 conjugacy classes of subgroups — too many to draw. Only its 104 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#24302C211yesmaximal, characteristic
2#22295C211yesmaximal, characteristic
3#11384C211yesmaximal, characteristic
4#8572C211yesmaximal, characteristic
5#8342C211yesmaximal, characteristic
6#7039C211yesmaximal, characteristic
7#7020C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9C2×#477—21no—
10C4×C22⋊Q8C2211yescharacteristic
11#2057C412yescomplemented
12#2024C412yescomplemented
13#1987—21no—
14#1985—21no—
15#1940—21no—
16#1917—21no—
17#1917—21no—
18#1872—21no—
19#1852—21no—
20#1850—21no—
21#1797C2211yescharacteristic
22#1797C2211yescharacteristic
23#1797C2211yescharacteristic
24#1752—21no—
25C2×#340—21no—
26C2×#340—21no—
27C2×#336C2211yescharacteristic
28C2×#333—21no—
29C2×#333—21no—
30C2×#328—21no—
31C4⋊#95C2211yescharacteristic
32#1294—21no—
33C23×Q8—21no—
34C23×C8D411yescharacteristic
35C23×C8D411yescharacteristic
36C2×C4⋊Q8—21no—
37C2×C4⋊Q8—21no—
38C2×C4⋊Q8—21no—
39C2×C4⋊Q8—42no—
40C2×C4⋊Q8—42no—
41C2×C4⋊Q8C2×C412yes—
42C2×C22⋊Q8D411yescharacteristic
43C2×C4×Q8D411yescharacteristic
44C22×C4⋊C4—21no—
45C22×C4⋊C4—21no—
46C22×C4⋊C4—21no—
47C22×C42C2311yescharacteristic
48#477—42no—
49#477—42no—
50C2×#107—22no—
51C2×C4⋊C8—21no—
52C2×C4⋊C8—21no—
53C2×C4⋊C8—21no—
54C2×C4⋊C8—21no—
55C2×C4⋊C8D411yescharacteristic
56C2×C4⋊C8D411yescharacteristic
57C2×#103—22no—
58C2×C22⋊C8—21no—
59C2×C22⋊C8—21no—
60C2×C22⋊C8—21no—
61C2×#97—21no—
62C2×#97—21no—
63#391C2×C411yescharacteristic
64#389C2×C411yescharacteristic
65#387—21no—
66#385—21no—
67#380D411yescharacteristic
68C4×C4⋊C4—21no—
69C4×C4⋊C4D411yescharacteristic
70C4×C22⋊C4—21no—
71C2×#95—21no—
72C2×#95—21no—
73#340—22no—
74#340—22no—
75#340—42no—
76#340—42no—
77#336—21no—
78#336—21no—
79#336—21no—
80#336—21no—
81#336—21no—
82#336—21no—
83#336—21no—
84#336—21no—
85#336C2×C412yes—
86#333—42no—
87#333—42no—
88#328—42no—
89C22×Q8—22no—
90C22×Q8—44no—
91C22×Q8—41no—
92C22×Q8—81no—
93C22×Q8—81no—
94C22×Q8—82no—
95C22×Q8C22⋊C411yescharacteristic
96C23×C4—41no—
97C23×C4—41no—
98C23×C4C2×D411yescharacteristic
99C23×C4C2×D411yescharacteristic
100C23×C4#4111yescharacteristic
101C22×C8—21no—
102C22×C8—21no—
103C22×C8—22no—
104C22×C8—21no—
105C22×C8—21no—
106C22×C8—21no—
107C22×C8—21no—
108C22×C8—21no—
109C22×C8—41no—
110C22×C8—41no—
111C22×C8—41no—
112C22×C8—41no—
113C22×C8—41no—
114C22×C8Q1612yes—
115C4⋊Q8—22no—
116C4⋊Q8—22no—
117C4⋊Q8—22no—
118C4⋊Q8—22no—
119C4⋊Q8—41no—
120C4⋊Q8—41no—
121C4⋊Q8—41no—
122C4⋊Q8—41no—
123C4⋊Q8—44no—
124C4⋊Q8—44no—
125C22⋊Q8—21no—
126C22⋊Q8—21no—
127C22⋊Q8—21no—
128C22⋊Q8—21no—
129C4×Q8—41no—
130C4×Q8—41no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—44no—
140C2×C4⋊C4—41no—
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—41no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4C22⋊C411yescharacteristic
150C2×C4⋊C4C22⋊C411yescharacteristic
151C2×C4⋊C4C22⋊C411yescharacteristic
152C2×C22⋊C4—21no—
153C2×C22⋊C4—21no—
154C2×C42—21no—
155C2×C42—21no—
156C2×C42—41no—
157C2×C42—41no—
158C2×C42C22×C411yescharacteristic
159C2×C42C2×D411yescharacteristic
160C2×C42C2×D411yescharacteristic
161C2×C42#4111yescharacteristic
162#107—44no—
163C4⋊C8—41no—
164C4⋊C8—41no—
165C4⋊C8—41no—
166C4⋊C8—41no—
167C4⋊C8—41no—
168C4⋊C8—41no—
169C4⋊C8—81no—
170C4⋊C8—81no—
171#103—44no—
172C22⋊C8—41no—
173C22⋊C8—41no—
174C22⋊C8—41no—
175C22⋊C8—41no—
176C22⋊C8—41no—
177C22⋊C8—41no—
178#97—81no—
179#97—81no—
180#95—21no—
181#95—21no—
182#95—21no—
183#95—21no—
184#95—21no—
185#95—41no—
186#95—41no—
187#95—81no—
188C24#12111yescharacteristic
189C2×Q8—24no—
190C2×Q8—21no—
191C2×Q8—21no—
192C2×Q8—42no—
193C2×Q8—42no—
194C2×Q8—42no—
195C2×Q8—41no—
196C2×Q8—41no—
197C2×Q8—82no—
198C2×Q8—82no—
199C2×Q8—82no—
200C2×Q8—82no—
201C2×Q8—88no—
202C2×Q8—88no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—81no—
222C22×C4—81no—
223C22×C4—81no—
224C22×C4—81no—
225C22×C4—81no—
226C22×C4—81no—
227C22×C4—81no—
228C22×C4—81no—
229C22×C4—81no—
230C22×C4—81no—
231C22×C4C2×C22⋊C411yescharacteristic
232C22×C4C4×D411yescharacteristic
233C22×C4C4×D411yescharacteristic
234C22×C4#12011yescharacteristic
235C22×C4#12111yescharacteristic
236C22×C4#12311yescharacteristic
237C22×C4C2×Q1611yescharacteristic
238C22×C4#13511yescharacteristic
239C22×C4AGL(1,ℤ/8)11yescharacteristic
240C22×C4#13711yescharacteristic
241C22×C4#13711yescharacteristic
242C22×C4#13711yescharacteristic
243C22×C4#13711yescharacteristic
244C22×C4#13711yescharacteristic
245C2×C8—22no—
246C2×C8—41no—
247C2×C8—41no—
248C2×C8—42no—
249C2×C8—42no—
250C2×C8—42no—
251C2×C8—42no—
252C2×C8—42no—
253C2×C8—41no—
254C2×C8—41no—
255C2×C8—41no—
256C2×C8—41no—
257C2×C8—41no—
258C2×C8—41no—
259C2×C8—41no—
260C2×C8—41no—
261C2×C8—41no—
262C2×C8—81no—
263C2×C8—81no—
264C2×C8—81no—
265C2×C8—81no—
266C2×C8—81no—
267C4⋊C4—24no—
268C4⋊C4—24no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—81no—
284C4⋊C4—81no—
285C4⋊C4—81no—
286C4⋊C4—81no—
287C22⋊C4—41no—
288C22⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—41no—
291C42—21no—
292C42—21no—
293C42—21no—
294C42—21no—
295C42—41no—
296C42—41no—
297C42—41no—
298C42—41no—
299C42—41no—
300C42—41no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23#38611yescharacteristic
306C23#46211yescharacteristic
307C23#46411yescharacteristic
308C23#46811yescharacteristic
309C23#47011yescharacteristic
310C23#47111yescharacteristic
311C23#47211yescharacteristic
312Q8—42no—
313Q8—81no—
314Q8—84no—
315Q8—84no—
316Q8—84no—
317Q8—84no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—44no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—161no—
367C2×C4—161no—
368C2×C4—161no—
369C2×C4—161no—
370C2×C4#41711yescharacteristic
371C2×C4#41911yescharacteristic
372C2×C4C4×Q1611yescharacteristic
373C2×C4#44111yescharacteristic
374C2×C4#45111yescharacteristic
375C2×C4#45211yescharacteristic
376C2×C4#45511yescharacteristic
377C2×C4#45611yescharacteristic
378C2×C4B(4,𝔽2)11yescharacteristic
379C2×C4B(4,𝔽2)11yescharacteristic
380C2×C4#46811yescharacteristic
381C2×C4#47011yescharacteristic
382C8—42no—
383C8—82no—
384C8—81no—
385C8—81no—
386C8—81no—
387C8—161no—
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#169211yescharacteristic
401C22#169611yescharacteristic
402C22#169811yescharacteristic
403C22#170411yescharacteristic
404C22#187811yescharacteristic
405C22#188211yescharacteristic
406C22#188911yescharacteristic
407C22#189411yescharacteristic
408C22#190311yescharacteristic
409C22#190711yescharacteristic
410C22#190911yescharacteristic
411C4—21no—
412C4—21no—
413C4—21no—
414C4—21no—
415C4—21no—
416C4—21no—
417C4—41no—
418C4—41no—
419C4—81no—
420C4—84no—
421C4—81no—
422C4—81no—
423C4—81no—
424C4—162no—
425C4#160211yescharacteristic
426C4#160311yescharacteristic
427C4#160511yescharacteristic
428C4#160611yescharacteristic
429C2—21no—
430C2—21no—
431C2—21no—
432C2—21no—
433C2#904011yescharacteristic
434C2#922411yescharacteristic
435C2#928811yescharacteristic
436C2#931911yescharacteristic
437C2#937511yescharacteristic
438C2#947111yescharacteristic
439C2#952011yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C23 ▷ 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
11. #2057C4semidirect
12. #2024C4semidirect

Maximal subgroups

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

subgroupindex
#243022
#222952
#113842
#85722
#83422
#70392
#70202