Finite Group Databasebeta

← #164098

Subgroups — #164098

Order 512.

932 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35307C211yesmaximal, characteristic
2#22532C211yesmaximal, characteristic
3#22530C211yesmaximal, characteristic
4#7715C212yesmaximal, complemented
5#7701C212yesmaximal, complemented
6#7675C212yesmaximal, complemented
7#7661C212yesmaximal, complemented
8#7598C211yesmaximal, characteristic, complemented
9#7575C211yesmaximal, characteristic, complemented
10#7575C211yesmaximal, characteristic, complemented
11#7564C211yesmaximal, characteristic, complemented
12C2×#536C411yescharacteristic
13C2×C4×#41C2211yescharacteristic
14C2×#446—21no—
15C2×C8×Q8—21no—
16#2855C2212yes—
17#2855C2212yes—
18#2625C412yes—
19#2625C412yes—
20#2447—21no—
21#2447—21no—
22#2446—21no—
23#2446—21no—
24#2419—22no—
25#2419—22no—
26#2316C411yescharacteristic
27#2311C411yescharacteristic
28#2310C411yescharacteristic
29#2296C2211yescharacteristic
30C2×#391C2211yescharacteristic
31#1955—22no—
32#1954—22no—
33#1954—21no—
34#1954—21no—
35#1797C2212yes—
36#1797C2211yescharacteristic
37#1797C2212yes—
38#1797C2211yescharacteristic
39#1749C2212yes—
40C4×C22⋊C8C2211yescharacteristic
41C4×C22⋊C8C2212yes—
42C4×C22⋊C8C2211yescharacteristic
43#1272C2212yes—
44#1272C2212yes—
45#1272C2212yes—
46#1272C2212yes—
47#1271C2211yescharacteristic
48#1271C2212yes—
49#1271C2211yescharacteristic
50#1271C2212yes—
51#1271C2211yescharacteristic
52#1271C2211yescharacteristic
53C22×#41C2×C411yescharacteristic
54C23×Q8C2×C411yescharacteristic
55#536—44no—
56C2×C4⋊Q8—21no—
57C2×#124—21no—
58C2×#123—22no—
59C2×C22⋊Q8C2×C412yes—
60C2×C22⋊Q8C2×C412yes—
61C4×#41—44no—
62C2×C4×Q8D411yescharacteristic
63C2×C4×Q8D411yescharacteristic
64C2×C4×D4D412yes—
65C2×C4×D4D412yes—
66C2×C4×D4D411yescharacteristic
67C2×C4×D4D411yescharacteristic
68C2×#117C2×C411yescharacteristic
69C2×#117C2×C411yescharacteristic
70C2×#117C2×C411yescharacteristic
71C2×#117C2×C412yes—
72C22×C4⋊C4C2×C411yescharacteristic
73C22×C42C2312yes—
74C22×C42C2311yescharacteristic
75#446—44no—
76C8×Q8—44no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—22no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—22no—
81C2×C22⋊C8—21no—
82C2×C22⋊C8—22no—
83C2×C22⋊C8—22no—
84C2×C22⋊C8—21no—
85C2×C22⋊C8—21no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—21no—
89C2×#97—22no—
90C2×C4×C8—21no—
91C2×C4×C8—22no—
92C2×C4×C8—21no—
93#400—22no—
94#400—22no—
95#398—22no—
96#398—22no—
97#397—22no—
98#397—22no—
99#397—22no—
100#397—22no—
101#391—21no—
102#391—21no—
103#391—21no—
104#391—21no—
105#391—21no—
106#391C2×C412yes—
107#391C2×C412yes—
108#391C2×C412yes—
109#391C2312yes—
110#390—21no—
111#389—21no—
112#389—21no—
113#389C2×C412yes—
114#389C2×C412yes—
115#389C2×C411yescharacteristic
116#389C2×C411yescharacteristic
117#389C2311yescharacteristic
118#389C2311yescharacteristic
119#387—22no—
120#387—22no—
121#387—22no—
122#387—22no—
123#386—22no—
124#385—22no—
125#385—22no—
126#385—21no—
127#385—21no—
128#381—22no—
129#381—22no—
130#381—22no—
131#380—22no—
132#380C2×C412yes—
133#380C2×C412yes—
134C4×C4⋊C4—21no—
135C4×C4⋊C4—21no—
136C2×#95—21no—
137C2×#95—21no—
138#336C2311yescharacteristic
139#336C2312yes—
140#336C2311yescharacteristic
141#336C2312yes—
142#336C2311yescharacteristic
143#336C2311yescharacteristic
144C2×#41—24no—
145C2×#41—24no—
146C2×#41—42no—
147C22×Q8—21no—
148C22×Q8—24no—
149C22×Q8—42no—
150C22×Q8—42no—
151C22×Q8C22⋊C411yescharacteristic
152C22×Q8C22×C412yes—
153C22×D4C22⋊C411yescharacteristic
154C22×D4C22⋊C412yes—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4C22×C412yes—
158C23×C4C22×C411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D412yes—
162C23×C4C2×D412yes—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—22no—
167C22×C8—21no—
168C22×C8—22no—
169C4⋊Q8—44no—
170#124—44no—
171#123—48no—
172C22⋊Q8—44no—
173C22⋊Q8—44no—
174C4×Q8—24no—
175C4×Q8—24no—
176C4×D4—44no—
177C4×D4—42no—
178C4×D4—42no—
179C4×D4—44no—
180#117—24no—
181#117—42no—
182#117—44no—
183#117—42no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4C22⋊C411yescharacteristic
195C2×C4⋊C4C22⋊C411yescharacteristic
196C2×C4⋊C4C22⋊C412yes—
197C2×C4⋊C4C22⋊C412yes—
198C2×C4⋊C4C22×C411yescharacteristic
199C2×C4⋊C4C22×C411yescharacteristic
200C2×C4⋊C4C22×C412yes—
201C2×C4⋊C4C22×C411yescharacteristic
202C2×C4⋊C4C22×C411yescharacteristic
203C2×C22⋊C4—21no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4C22⋊C412yes—
210C2×C22⋊C4C22⋊C412yes—
211C2×C22⋊C4C22⋊C412yes—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—42no—
215C2×C42—44no—
216C2×C42C22×C412yes—
217C2×C42C22×C411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D412yes—
221C2×C42C2×D412yes—
222C2×C42C2411yescharacteristic
223C4⋊C8—44no—
224C4⋊C8—42no—
225C4⋊C8—42no—
226C4⋊C8—44no—
227C22⋊C8—42no—
228C22⋊C8—44no—
229C22⋊C8—44no—
230C22⋊C8—42no—
231C22⋊C8—81no—
232C22⋊C8—82no—
233C22⋊C8—82no—
234C22⋊C8—81no—
235#97—44no—
236C4×C8—44no—
237C4×C8—42no—
238C4×C8—42no—
239#95—22no—
240#95—22no—
241#95—21no—
242#95—21no—
243#95—22no—
244#95—22no—
245#95—22no—
246#95—22no—
247#95—22no—
248#95—22no—
249#95—22no—
250#95—22no—
251#95—22no—
252#95—22no—
253#95—21no—
254#95—21no—
255C24C2×C22⋊C412yes—
256C24C2×C22⋊C411yescharacteristic
257#41—416no—
258C2×Q8—22no—
259C2×Q8—22no—
260C2×Q8—24no—
261C2×Q8—42no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8—42no—
266C2×Q8—88no—
267C2×Q8#9914yes—
268C2×D4—24no—
269C2×D4—22no—
270C2×D4—24no—
271C2×D4—22no—
272C2×D4—22no—
273C2×D4—42no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—42no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—81no—
306C22×C4—81no—
307C22×C4C2×C22⋊C411yescharacteristic
308C22×C4C2×C22⋊C411yescharacteristic
309C22×C4C2×C22⋊C412yes—
310C22×C4C2×C22⋊C412yes—
311C22×C4C2×C22⋊C412yes—
312C22×C4C2×C22⋊C411yescharacteristic
313C22×C4#13114yes—
314C22×C4C23×C411yescharacteristic
315C22×C4C22×D411yescharacteristic
316C22×C4C22×D411yescharacteristic
317C2×C8—22no—
318C2×C8—22no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—42no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—42no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—42no—
335C4⋊C4—24no—
336C4⋊C4—24no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—42no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—44no—
349C4⋊C4—44no—
350C4⋊C4—42no—
351C4⋊C4—81no—
352C4⋊C4—81no—
353C4⋊C4—82no—
354C4⋊C4—82no—
355C22⋊C4—42no—
356C22⋊C4—42no—
357C22⋊C4—42no—
358C22⋊C4—42no—
359C22⋊C4—44no—
360C22⋊C4—44no—
361C22⋊C4—42no—
362C22⋊C4—82no—
363C22⋊C4—82no—
364C42—24no—
365C42—22no—
366C42—22no—
367C42—24no—
368C42—41no—
369C42—41no—
370C42—42no—
371C42—42no—
372C42—42no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—22no—
380C23—22no—
381C23—22no—
382C23—22no—
383C23—22no—
384C23—22no—
385C23—22no—
386C23#40812yes—
387C23C22×C22⋊C411yescharacteristic
388Q8—28no—
389Q8—84no—
390Q8—84no—
391D4—48no—
392D4—44no—
393C2×C4—22no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—22no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—28no—
402C2×C4—22no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—22no—
406C2×C4—21no—
407C2×C4—22no—
408C2×C4—21no—
409C2×C4—22no—
410C2×C4—21no—
411C2×C4—22no—
412C2×C4—21no—
413C2×C4—22no—
414C2×C4—48no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—42no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—84no—
442C2×C4—162no—
443C2×C4#40812yes—
444C2×C4C2×#9914yes—
445C2×C4C2×#9912yes—
446C2×C4#41312yes—
447C2×C4C2×#13112yes—
448C2×C4#57711yescharacteristic
449C2×C4#57711yescharacteristic
450C2×C4#57811yescharacteristic
451C2×C4#57811yescharacteristic
452C8—42no—
453C8—42no—
454C8—82no—
455C8—81no—
456C8—81no—
457C8—82no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—41no—
465C22—42no—
466C22—42no—
467C22—42no—
468C22—42no—
469C22—42no—
470C22—42no—
471C22—42no—
472C22C2×#40811yescharacteristic
473C22C22×#9911yescharacteristic
474C22C2×#41311yescharacteristic
475C22#289811yescharacteristic
476C22#289811yescharacteristic
477C22#290111yescharacteristic
478C22#290111yescharacteristic
479C4—21no—
480C4—21no—
481C4—21no—
482C4—21no—
483C4—22no—
484C4—24no—
485C4—41no—
486C4—42no—
487C4—41no—
488C4—42no—
489C4—82no—
490C4—81no—
491C4—81no—
492C4—81no—
493C4—81no—
494C4—82no—
495C4#146814yes—
496C2—42no—
497C2—82no—
498C2C2×#146811yescharacteristic
499C2#1746911yescharacteristic
500C2#1748411yescharacteristic
501C2#1760611yescharacteristic
502C2#1760611yescharacteristic
503C2#1769111yescharacteristic
504C2#1777811yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7715C2semidirect
5. #7701C2semidirect
6. #7675C2semidirect
7. #7661C2semidirect
8. #7598C2semidirect
9. #7575C2semidirect
10. #7575C2semidirect
11. #7564C2semidirect

Maximal subgroups

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

subgroupindex
#353072
#225322
#225302
#77152
#77012
#76752
#76612
#75982
#75752
#75642