Finite Group Databasebeta

← #107237

Subgroups — #107237

Order 512.

920 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34359C211yesmaximal, characteristic
2#21825C211yesmaximal, characteristic
3C2×#1472C211yesmaximal, characteristic
4#8049C211yesmaximal, characteristic
5#8024C211yesmaximal, characteristic
6#7982C211yesmaximal, characteristic
7#7943C211yesmaximal, characteristic
8C22×C22⋊Q8C2211yescharacteristic
9C2×#407C2211yescharacteristic
10C22×C22⋊C8C2211yescharacteristic
11#2378—22no—
12#2368—22no—
13#2344—22no—
14#2342—22no—
15#2332—22no—
16#2329—22no—
17#2313C412yescomplemented
18#2310C412yescomplemented
19#2280C2211yescharacteristic
20#2280C2211yescharacteristic
21#2280C2211yescharacteristic
22C2×#381C2211yescharacteristic
23#1996—22no—
24#1996C414yescomplemented
25#1819—22no—
26#1472—22no—
27#1472C414yescomplemented
28C2×#324—22no—
29#1297—22no—
30#1294—22no—
31#1289—22no—
32#1284—22no—
33C23×Q8C2×C411yescharacteristic
34C24×C4C2311yescharacteristic
35C22×M16—21no—
36C23×C8—21no—
37C2×C22⋊Q8—21no—
38C2×C22⋊Q8—21no—
39C2×C22⋊Q8—21no—
40C2×C22⋊Q8—21no—
41C2×C22⋊Q8—21no—
42C2×C22⋊Q8—21no—
43C2×C22⋊Q8—21no—
44C2×C22⋊Q8—21no—
45C2×C22⋊Q8C2×C412yes—
46C2×C22⋊Q8C2×C412yes—
47C2×C22⋊Q8D412yes—
48C2×C22⋊Q8Q812yes—
49C2×#117—22no—
50C2×#117—21no—
51C2×#117—21no—
52C2×#117—21no—
53C22×C4⋊C4C2×C411yescharacteristic
54C22×C4⋊C4D412yes—
55C22×C22⋊C4—21no—
56C22×C22⋊C4D412yes—
57#407—21no—
58#407—21no—
59#407—21no—
60#407C2×C412yes—
61C2×C22⋊C8—21no—
62C2×C22⋊C8—21no—
63C2×C22⋊C8—21no—
64C2×C22⋊C8—21no—
65C2×C22⋊C8—21no—
66C2×C22⋊C8—21no—
67C2×C22⋊C8—21no—
68C2×C22⋊C8C2×C412yes—
69#391—24no—
70#391—24no—
71#389—24no—
72#388—24no—
73#388—24no—
74#387—24no—
75#387—24no—
76#387—24no—
77#387—24no—
78#385—24no—
79#381—22no—
80#381—21no—
81#381—21no—
82#381—21no—
83#381—21no—
84#381C2×C412yes—
85#381C2×C412yes—
86#381D412yes—
87#380—22no—
88#380—21no—
89#380—21no—
90#380—21no—
91#380—21no—
92#380C2×C412yes—
93#380D412yes—
94#379—21no—
95#379—21no—
96#379—21no—
97C4×C4⋊C4—24no—
98C4×C22⋊C4—24no—
99C4×C22⋊C4—22no—
100C4×C22⋊C4—21no—
101C4×C22⋊C4—21no—
102C4×C22⋊C4—21no—
103C4×C22⋊C4—21no—
104C4×C22⋊C4C2×C412yes—
105C4×C22⋊C4D412yes—
106C2×#95—21no—
107C2×#95D412yes—
108#342—28no—
109#324—28no—
110C25C2×D411yescharacteristic
111C22×Q8—21no—
112C22×Q8—21no—
113C22×Q8—21no—
114C22×Q8—21no—
115C22×Q8—82no—
116C22×Q8C4212yes—
117C22×Q8C4⋊C412yes—
118C23×C4—22no—
119C23×C4—82no—
120C23×C4C22⋊C412yes—
121C23×C4C22⋊C412yes—
122C23×C4C22×C411yescharacteristic
123C23×C4C22×C411yescharacteristic
124C23×C4C22×C411yescharacteristic
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C23×C4C2×Q811yescharacteristic
132C23×C4#4111yescharacteristic
133C23×C4#4111yescharacteristic
134C23×C4#4111yescharacteristic
135C23×C4#4111yescharacteristic
136C2×M16—22no—
137C2×M16—41no—
138C2×M16—82no—
139C22×C8—22no—
140C22×C8—41no—
141C22×C8—41no—
142C22×C8—41no—
143C22×C8—82no—
144C22⋊Q8—22no—
145C22⋊Q8—22no—
146C22⋊Q8—22no—
147C22⋊Q8—22no—
148C22⋊Q8—22no—
149C22⋊Q8—22no—
150C22⋊Q8—22no—
151C22⋊Q8—22no—
152C22⋊Q8—22no—
153C22⋊Q8—22no—
154C22⋊Q8—22no—
155C22⋊Q8—22no—
156C22⋊Q8—22no—
157C22⋊Q8—22no—
158C22⋊Q8—22no—
159C22⋊Q8—22no—
160#117—84no—
161#117—82no—
162#117—82no—
163#117—82no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4C4212yes—
178C2×C4⋊C4C22⋊C414yes—
179C2×C4⋊C4C4⋊C412yes—
180C2×C4⋊C4C4⋊C414yes—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—41no—
193C2×C22⋊C4—41no—
194C2×C22⋊C4—41no—
195C2×C22⋊C4—41no—
196C2×C22⋊C4—82no—
197C2×C22⋊C4C22⋊C414yes—
198C2×C22⋊C4C4⋊C414yes—
199C2×C42—24no—
200C2×C42—24no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—41no—
204C22⋊C8—22no—
205C22⋊C8—22no—
206C22⋊C8—22no—
207C22⋊C8—22no—
208C22⋊C8—41no—
209C22⋊C8—41no—
210C22⋊C8—41no—
211C22⋊C8—41no—
212#95—24no—
213#95—24no—
214#95—24no—
215#95—24no—
216#95—22no—
217#95—22no—
218#95—22no—
219#95—22no—
220#95—22no—
221#95—22no—
222#95—41no—
223#95—41no—
224#95C22⋊C414yes—
225#95C22⋊C414yes—
226C24—42no—
227C24—42no—
228C24—42no—
229C24C2×C22⋊C411yescharacteristic
230C24C2×C22⋊C411yescharacteristic
231C24C2×C4⋊C411yescharacteristic
232C24#12011yescharacteristic
233C24#12011yescharacteristic
234C24#12311yescharacteristic
235C24#12311yescharacteristic
236C2×Q8—22no—
237C2×Q8—22no—
238C2×Q8—22no—
239C2×Q8—22no—
240C2×Q8—82no—
241C2×Q8—82no—
242C2×Q8—82no—
243C2×Q8—82no—
244C2×Q8—81no—
245C2×Q8—82no—
246C2×Q8—81no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—44no—
260C22×C4—44no—
261C22×C4—44no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—81no—
265C22×C4—81no—
266C22×C4—81no—
267C22×C4—81no—
268C22×C4—81no—
269C22×C4—81no—
270C22×C4—81no—
271C22×C4—81no—
272C22×C4—82no—
273C22×C4—82no—
274C22×C4#9514yes—
275C22×C4#9514yes—
276C22×C4C2×C4211yescharacteristic
277C22×C4C2×C22⋊C411yescharacteristic
278C22×C4C2×C22⋊C411yescharacteristic
279C22×C4C2×C22⋊C411yescharacteristic
280C22×C4C2×C4⋊C411yescharacteristic
281C22×C4C2×C4⋊C411yescharacteristic
282C22×C4#11711yescharacteristic
283C22×C4#11711yescharacteristic
284C22×C4C4×D411yescharacteristic
285C22×C4C4×D411yescharacteristic
286C22×C4C4×D411yescharacteristic
287C22×C4C4×D411yescharacteristic
288C22×C4C4×D411yescharacteristic
289C22×C4C4×D411yescharacteristic
290C22×C4C4×D411yescharacteristic
291C22×C4C4×D411yescharacteristic
292C22×C4#12011yescharacteristic
293C22×C4#12011yescharacteristic
294C22×C4#12111yescharacteristic
295C22×C4#12111yescharacteristic
296C22×C4#12111yescharacteristic
297C22×C4#12111yescharacteristic
298C22×C4C22⋊Q811yescharacteristic
299C22×C4C22⋊Q811yescharacteristic
300C22×C4C22⋊Q811yescharacteristic
301C22×C4C22⋊Q811yescharacteristic
302C22×C4#12311yescharacteristic
303C22×C4#12311yescharacteristic
304M16—82no—
305M16—82no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—82no—
309C2×C8—82no—
310C2×C8—82no—
311C2×C8—82no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—24no—
317C4⋊C4—24no—
318C4⋊C4—24no—
319C4⋊C4—24no—
320C4⋊C4—84no—
321C4⋊C4—82no—
322C4⋊C4—84no—
323C4⋊C4—82no—
324C4⋊C4—82no—
325C22⋊C4—24no—
326C22⋊C4—24no—
327C22⋊C4—24no—
328C22⋊C4—24no—
329C22⋊C4—44no—
330C22⋊C4—44no—
331C22⋊C4—44no—
332C22⋊C4—44no—
333C22⋊C4—44no—
334C22⋊C4—82no—
335C22⋊C4—82no—
336C22⋊C4—82no—
337C22⋊C4—82no—
338C22⋊C4—82no—
339C22⋊C4—82no—
340C42—84no—
341C42—82no—
342C42—84no—
343C42—82no—
344C42—82no—
345C23—22no—
346C23—22no—
347C23—22no—
348C23—22no—
349C23—22no—
350C23—22no—
351C23—22no—
352C23—22no—
353C23—22no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23—41no—
366C23—88no—
367C23C2×#9511yescharacteristic
368C23#37911yescharacteristic
369C23#38111yescharacteristic
370C23#38511yescharacteristic
371C23#38511yescharacteristic
372C23#38611yescharacteristic
373C23#38611yescharacteristic
374Q8—82no—
375Q8—82no—
376Q8—82no—
377Q8—82no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—24no—
383C2×C4—24no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—164no—
428C2×C4C4×C22⋊C411yescharacteristic
429C2×C4C4×C22⋊C411yescharacteristic
430C2×C4#38011yescharacteristic
431C2×C4#38011yescharacteristic
432C2×C4#38511yescharacteristic
433C2×C4#38511yescharacteristic
434C2×C4#38611yescharacteristic
435C2×C4#38611yescharacteristic
436C8—82no—
437C8—82no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—21no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—21no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—44no—
457C22—44no—
458C22—44no—
459C22—82no—
460C22—82no—
461C22—82no—
462C22—82no—
463C22C22⋊#9511yescharacteristic
464C22#179412yes—
465C22#182812yes—
466C22#191212yes—
467C4—21no—
468C4—21no—
469C4—21no—
470C4—21no—
471C4—21no—
472C4—21no—
473C4—21no—
474C4—21no—
475C4—44no—
476C4—44no—
477C4—84no—
478C4—84no—
479C4—162no—
480C4—162no—
481C2—22no—
482C2—22no—
483C2—22no—
484C2—41no—
485C2—41no—
486C2—41no—
487C2—41no—
488C2#1074111yescharacteristic
489C2#1075211yescharacteristic
490C2#1076811yescharacteristic
subgroupG/H
center, socleC22C22⋊#95
commutatorC22×C4C2×C42
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
17. #2313C4semidirect
18. #2310C4semidirect
24. #1996C4semidirect
27. #1472C4semidirect

Maximal subgroups

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

subgroupindex
#343592
#218252
C2×#14722
#80492
#80242
#79822
#79432