Finite Group Databasebeta

← #215591

Subgroups — #215591

Order 512.

982 conjugacy classes of subgroups.

The subgroup lattice has 465 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35289C211yesmaximal, characteristic
2#21926C211yesmaximal, characteristic, complemented
3#20812C211yesmaximal, characteristic, complemented
4#9710C211yesmaximal, characteristic, complemented
5#9708C212yesmaximal, complemented
6#9708C211yesmaximal, characteristic, complemented
7#9704C212yesmaximal, complemented
8#9700C211yesmaximal, characteristic, complemented
9#9698C211yesmaximal, characteristic, complemented
10#8341C212yesmaximal
11#8340C212yesmaximal, complemented
12C2×#532C2211yescharacteristic
13C22×C4⋊Q8C2211yescharacteristic
14C2×#433—21no—
15#2862C2211yescharacteristic
16#2862C2211yescharacteristic
17#2858C2211yescharacteristic
18#2858C2211yescharacteristic
19#2640—21no—
20#2640—21no—
21#2563C414yes—
22#2547—22no—
23#2504C414yes—
24#2436—22no—
25#2311—21no—
26#2311—21no—
27C4×#117C2211yescharacteristic
28#1924—22no—
29#1917—22no—
30#1842C2212yes—
31#1842C2211yescharacteristic
32#1832—21no—
33#1831C2211yescharacteristic
34#1798—21no—
35#1797C2212yes—
36#1797C2212yes—
37#1768—21no—
38#1767C2211yescharacteristic
39#1767C2212yes—
40C4×C4⋊C8C2211yescharacteristic
41C2×#332C2212yes—
42C2×#332C2212yes—
43C2×#332C2212yes—
44C2×#332C2212yes—
45C2×#332C2212yes—
46C2×#332C2212yes—
47C2×#331C2212yes—
48C2×#331C2212yes—
49C22×#41—21no—
50C23×Q8—21no—
51#532—42no—
52#532—42no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—42no—
56C2×C4⋊Q8—42no—
57C2×C4⋊Q8C2×C412yes—
58C2×C4⋊Q8C2311yescharacteristic
59C2×C4⋊Q8C2311yescharacteristic
60C2×C4⋊Q8C2311yescharacteristic
61C2×#127C2311yescharacteristic
62C2×#124—21no—
63C2×#124—22no—
64C2×#124—22no—
65C2×#121—21no—
66C2×#121C2×C412yes—
67C2×C4×D4—21no—
68C2×C4×D4C2×C412yes—
69C2×#117D411yescharacteristic
70C2×#117D411yescharacteristic
71C22×C4⋊C4—21no—
72C22×C4⋊C4C2×C412yes—
73C22×C42C2311yescharacteristic
74#433—44no—
75C2×C4⋊C8—22no—
76C2×C4⋊C8—22no—
77C2×C4⋊C8—22no—
78C2×C4⋊C8C2312yes—
79C2×C4⋊C8C2312yes—
80C2×C4⋊C8C2312yes—
81C2×C4⋊C8C2312yes—
82C2×C22⋊C8—22no—
83C2×C22⋊C8—22no—
84C2×#97—21no—
85C2×#97—22no—
86C2×C4×C8—21no—
87#397—24no—
88#397—24no—
89#393—24no—
90#393—24no—
91#391—21no—
92#391—21no—
93#391—21no—
94#391—21no—
95#391—22no—
96#391—22no—
97#390—21no—
98#390—21no—
99#390—21no—
100#390—21no—
101#390C2×C414yes—
102#390C2×C414yes—
103#386—24no—
104#386—24no—
105#382C2×C414yes—
106#382C2×C414yes—
107#380—24no—
108#380—24no—
109C4×C4⋊C4C2×C412yes—
110C4×C4⋊C4D412yes—
111C4×C22⋊C4C2×C412yes—
112C4×C22⋊C4D412yes—
113#376D411yescharacteristic
114#376D411yescharacteristic
115C43C2311yescharacteristic
116C43C2311yescharacteristic
117#336—22no—
118#336—22no—
119#336—21no—
120#336—21no—
121#336—22no—
122#336—22no—
123#332—42no—
124#332—42no—
125#332—42no—
126#332—42no—
127#332—42no—
128#332—42no—
129#331—42no—
130#331—42no—
131C2×#41—44no—
132C2×#41—44no—
133C2×#41—42no—
134C2×#41—42no—
135C22×Q8—22no—
136C22×Q8—21no—
137C22×Q8—21no—
138C22×Q8—21no—
139C22×Q8—42no—
140C22×Q8—42no—
141C22×Q8—82no—
142C22×Q8—82no—
143C22×D4—22no—
144C22×D4—21no—
145C22×D4—21no—
146C22×D4C22×C412yes—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4C22×C411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C4⋊Q8—22no—
157C4⋊Q8—22no—
158C4⋊Q8—22no—
159C4⋊Q8—22no—
160C4⋊Q8—22no—
161C4⋊Q8—22no—
162C4⋊Q8—44no—
163C4⋊Q8—42no—
164C4⋊Q8—42no—
165C4⋊Q8—44no—
166#127—22no—
167#127—22no—
168#124—44no—
169#124—82no—
170#124—82no—
171#124—82no—
172#124—82no—
173#121—44no—
174#121—42no—
175#121—42no—
176C4×D4—44no—
177C4×D4—42no—
178C4×D4—42no—
179#117—22no—
180#117—22no—
181#117—42no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4C22⋊C412yes—
190C2×C4⋊C4C22⋊C412yes—
191C2×C4⋊C4C22×C412yes—
192C2×C4⋊C4C22×C412yes—
193C2×C4⋊C4C22×C412yes—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4C22⋊C412yes—
201C2×C22⋊C4C22⋊C412yes—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42C22×C411yescharacteristic
205C2×C42C22×C412yes—
206C2×C42C22×C412yes—
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D412yes—
214C2×C42C2×D412yes—
215C2×C42C2411yescharacteristic
216C4⋊C8—22no—
217C4⋊C8—22no—
218C4⋊C8—22no—
219C4⋊C8—22no—
220C4⋊C8—22no—
221C4⋊C8—22no—
222C4⋊C8—22no—
223C4⋊C8—22no—
224C4⋊C8—44no—
225C4⋊C8—44no—
226C4⋊C8—82no—
227C22⋊C8—44no—
228C22⋊C8—82no—
229#97—42no—
230#97—82no—
231C4×C8—42no—
232#95—24no—
233#95—24no—
234#95—24no—
235#95—24no—
236#95C22⋊C412yes—
237#95C22⋊C412yes—
238#95C22⋊C412yes—
239#95C22⋊C412yes—
240C24—22no—
241C24C2×C22⋊C411yescharacteristic
242#41—88no—
243#41—88no—
244C2×Q8—44no—
245C2×Q8—42no—
246C2×Q8—41no—
247C2×Q8—41no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×Q8—41no—
251C2×Q8—41no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×Q8—42no—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×Q8—84no—
258C2×Q8—84no—
259C2×Q8—84no—
260C2×Q8—88no—
261C2×D4—24no—
262C2×D4—44no—
263C2×D4—44no—
264C2×D4—44no—
265C2×D4—44no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—42no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—42no—
272C2×D4—42no—
273C2×D4—42no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—42no—
285C22×C4—44no—
286C22×C4—44no—
287C22×C4—44no—
288C22×C4—44no—
289C22×C4—84no—
290C22×C4C2×C22⋊C412yes—
291C22×C4C2×C22⋊C412yes—
292C22×C4C2×C22⋊C412yes—
293C22×C4C2×C22⋊C412yes—
294C22×C4C2×C22⋊C411yescharacteristic
295C22×C4C2×C22⋊C411yescharacteristic
296C22×C4C2×C22⋊C411yescharacteristic
297C22×C4AGL(1,ℤ/8)12yes—
298C22×C4#13712yes—
299C22×C4C23×C411yescharacteristic
300C22×C4C22×D411yescharacteristic
301C22×C4C22×D411yescharacteristic
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—42no—
314C4⋊C4—24no—
315C4⋊C4—24no—
316C4⋊C4—24no—
317C4⋊C4—24no—
318C4⋊C4—42no—
319C4⋊C4—44no—
320C4⋊C4—44no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C22⋊C4—24no—
326C22⋊C4—42no—
327C22⋊C4—44no—
328C22⋊C4—44no—
329C22⋊C4—84no—
330C22⋊C4—84no—
331C22⋊C4—84no—
332C22⋊C4—84no—
333C42—22no—
334C42—21no—
335C42—21no—
336C42—22no—
337C42—21no—
338C42—21no—
339C42—21no—
340C42—21no—
341C42—22no—
342C42—22no—
343C42—22no—
344C42—22no—
345C42—22no—
346C42—22no—
347C42—42no—
348C42—42no—
349C42H(5,𝔽2)12yes—
350C42H(5,𝔽2)12yes—
351C23—21no—
352C23—22no—
353C23—21no—
354C23—22no—
355C23—42no—
356C23—44no—
357C23—42no—
358C23—42no—
359C23—44no—
360C23#41712yes—
361C23C22×C22⋊C411yescharacteristic
362Q8—88no—
363Q8—82no—
364Q8—82no—
365Q8—81no—
366Q8—81no—
367Q8—81no—
368Q8—81no—
369Q8—82no—
370Q8—82no—
371D4—84no—
372D4—88no—
373D4—82no—
374D4—82no—
375D4—81no—
376D4—81no—
377D4—82no—
378C2×C4—21no—
379C2×C4—22no—
380C2×C4—21no—
381C2×C4—24no—
382C2×C4—24no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—44no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—88no—
411C2×C4—88no—
412C2×C4#41712yes—
413C2×C4#51812yes—
414C2×C4#53512yes—
415C2×C4#53511yescharacteristic
416C2×C4#53511yescharacteristic
417C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
418C2×C4C2×#13711yescharacteristic
419C2×C4#57512yes—
420C2×C4#57612yes—
421C2×C4#57712yes—
422C8—82no—
423C8—82no—
424C8—82no—
425C8—82no—
426C22—22no—
427C22—24no—
428C22—41no—
429C22—41no—
430C22—42no—
431C22—82no—
432C22—82no—
433C22—82no—
434C22—84no—
435C22—84no—
436C22#230911yescharacteristic
437C22#288912yes—
438C22C2×#41711yescharacteristic
439C22#289611yescharacteristic
440C22#289712yes—
441C4—22no—
442C4—22no—
443C4—41no—
444C4—41no—
445C4—42no—
446C4—42no—
447C4—42no—
448C4—42no—
449C4—82no—
450C4—82no—
451C4—82no—
452C4#321312yes—
453C4#321512yes—
454C4#322011yescharacteristic
455C4#322211yescharacteristic
456C4#322412yes—
457C2—42no—
458C2—162no—
459C2#1776512yes—
460C2#1776912yes—
461C2#2180211yescharacteristic
462C2#2180811yescharacteristic
463C2#2181311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21926C2semidirect
3. #20812C2semidirect
4. #9710C2semidirect
5. #9708C2semidirect
6. #9708C2semidirect
7. #9704C2semidirect
8. #9700C2semidirect
9. #9698C2semidirect
11. #8340C2semidirect

Maximal subgroups

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

subgroupindex
#352892
#219262
#208122
#97102
#97082
#97042
#97002
#96982
#83412
#83402