Finite Group Databasebeta

← #164151

Subgroups — #164151

Order 512.

811 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36288C211yesmaximal, characteristic
2#22632C211yesmaximal, characteristic
3#20430C211yesmaximal, characteristic
4#7687C211yesmaximal, characteristic
5#7682C212yesmaximal, complemented
6#7667C211yesmaximal, characteristic
7#7662C212yesmaximal, complemented
8#7617C211yesmaximal, characteristic
9#7615C212yesmaximal, complemented
10#7566C211yesmaximal, characteristic
11#7562C212yesmaximal, complemented
12C2×#424—21no—
13C2×#405—21no—
14#2862C2212yes—
15#2857C2211yescharacteristic
16#2612—22no—
17#2574—22no—
18#2551C411yescharacteristic
19#2546C411yescharacteristic
20#2508—21no—
21#2492C412yes—
22#2439—21no—
23#2429C412yes—
24#2369—21no—
25#2365C411yescharacteristic
26#2355C411yescharacteristic
27#2346—21no—
28C4×C22⋊Q8C2211yescharacteristic
29#2295C2212yes—
30C4×#117C2211yescharacteristic
31#1939—21no—
32#1923—22no—
33#1831C2212yes—
34#1797C2211yescharacteristic
35C8×C4⋊C4—21no—
36#1768—22no—
37#1767C2212yes—
38C4×C4⋊C8C2212yes—
39C4×C22⋊C8C2211yescharacteristic
40#1744C2212yes—
41#1728C2211yescharacteristic
42C2×#336C2211yescharacteristic
43#1272C2211yescharacteristic
44#1272C2212yes—
45#1272C2212yes—
46#1272C2212yes—
47#1272C2212yes—
48#1272C2212yes—
49#1272C2211yescharacteristic
50#1271C2211yescharacteristic
51#1271C2212yes—
52#1271C2211yescharacteristic
53C22×M16—21no—
54C23×C8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8C2×C411yescharacteristic
57C2×C4⋊Q8C2×C412yes—
58C2×C4⋊Q8C2×C412yes—
59C2×C22⋊Q8C2×C411yescharacteristic
60C2×C22⋊Q8C2×C412yes—
61C2×C4×Q8C2×C412yes—
62C2×C4×Q8C2×C411yescharacteristic
63C2×#117D411yescharacteristic
64C2×#117D411yescharacteristic
65C22×C42C2311yescharacteristic
66#424—28no—
67C2×C4⋊C8—22no—
68C2×C4⋊C8—22no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—22no—
71C2×C22⋊C8—21no—
72C2×C22⋊C8—21no—
73#405—44no—
74C2×#97—22no—
75C2×C4×C8—22no—
76C2×C4×C8—22no—
77C2×C4×C8—21no—
78#400—22no—
79#398—22no—
80#398—22no—
81#396—22no—
82#395—22no—
83#393—22no—
84#391—22no—
85#391—22no—
86#391—22no—
87#391C2×C412yes—
88#391C2×C412yes—
89#391C2×C412yes—
90#391C2312yes—
91#391C2311yescharacteristic
92#389—21no—
93#389—21no—
94#389—21no—
95#389—21no—
96#389—21no—
97#389—21no—
98#389—22no—
99#389C2×C412yes—
100#389C2311yescharacteristic
101#388—22no—
102#388—22no—
103#387—22no—
104#387—21no—
105#387—22no—
106#387—22no—
107#387—22no—
108#385—21no—
109#384C2×C411yescharacteristic
110#382—21no—
111#382—22no—
112#382—22no—
113#380—21no—
114#380—21no—
115#380C2×C411yescharacteristic
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4C2×C412yes—
121C4×C4⋊C4C2×C411yescharacteristic
122C4×C4⋊C4C2×C411yescharacteristic
123C4×C4⋊C4C2×C411yescharacteristic
124C4×C4⋊C4D411yescharacteristic
125C4×C4⋊C4D411yescharacteristic
126C4×C22⋊C4—21no—
127C4×C22⋊C4C2×C412yes—
128C4×C22⋊C4D412yes—
129#376—22no—
130#376D412yes—
131C43C2312yes—
132#336—22no—
133#336—22no—
134#336—21no—
135#336—21no—
136#336C2312yes—
137#336C2311yescharacteristic
138#336C2311yescharacteristic
139#336C2311yescharacteristic
140#336C2312yes—
141#336C2311yescharacteristic
142C22×Q8—22no—
143C22×Q8C22×C412yes—
144C22×Q8C22×C411yescharacteristic
145C23×C4C22×C411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C2×M16—24no—
149C2×M16—42no—
150C2×M16—42no—
151C22×C8—24no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—22no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—22no—
158C22×C8—42no—
159C22×C8—42no—
160C4⋊Q8—42no—
161C4⋊Q8—44no—
162C4⋊Q8—82no—
163C4⋊Q8—82no—
164C22⋊Q8—22no—
165C22⋊Q8—41no—
166C22⋊Q8—82no—
167C4×Q8—24no—
168C4×Q8—42no—
169C4×Q8—42no—
170#117—24no—
171#117SD1614yes—
172#117Q1614yes—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4C22⋊C411yescharacteristic
187C2×C4⋊C4C22⋊C411yescharacteristic
188C2×C4⋊C4C22⋊C411yescharacteristic
189C2×C4⋊C4C22⋊C411yescharacteristic
190C2×C4⋊C4C22×C411yescharacteristic
191C2×C4⋊C4C22×C411yescharacteristic
192C2×C4⋊C4C22×C412yes—
193C2×C4⋊C4C22×C411yescharacteristic
194C2×C22⋊C4—21no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4C22⋊C412yes—
198C2×C22⋊C4C22⋊C412yes—
199C2×C42—22no—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—22no—
203C2×C42—42no—
204C2×C42C22×C411yescharacteristic
205C2×C42C22×C412yes—
206C2×C42C22×C412yes—
207C2×C42C2×D412yes—
208C2×C42C2×D412yes—
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D412yes—
212C2×C42C2×D412yes—
213C2×C42C2411yescharacteristic
214C4⋊C8—28no—
215C4⋊C8—28no—
216C4⋊C8—42no—
217C4⋊C8—82no—
218C22⋊C8—42no—
219C22⋊C8—81no—
220#97—44no—
221C4×C8—42no—
222C4×C8—44no—
223C4×C8—44no—
224#95—22no—
225#95—22no—
226#95—21no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95—21no—
232#95—22no—
233#95—22no—
234#95—21no—
235#95—22no—
236#95—21no—
237#95—22no—
238#95—21no—
239#95C22⋊C412yes—
240#95C22⋊C412yes—
241#95C22⋊C412yes—
242#95C22⋊C412yes—
243C24C2×C22⋊C411yescharacteristic
244C2×Q8—22no—
245C2×Q8—21no—
246C2×Q8—22no—
247C2×Q8—21no—
248C2×Q8—41no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×Q8—41no—
252C2×Q8—44no—
253C2×Q8—82no—
254C2×Q8—82no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—21no—
266C22×C4—22no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4C2×C22⋊C412yes—
272C22×C4C2×C22⋊C411yescharacteristic
273C22×C4C2×C22⋊C411yescharacteristic
274C22×C4C2×C22⋊C412yes—
275C22×C4C2×C22⋊C412yes—
276C22×C4C2×C22⋊C412yes—
277C22×C4C2×C22⋊C411yescharacteristic
278C22×C4C2×SD1612yes—
279C22×C4C2×Q1612yes—
280C22×C4C23×C411yescharacteristic
281C22×C4C22×D411yescharacteristic
282C22×C4C22×D411yescharacteristic
283M16—48no—
284C2×C8—22no—
285C2×C8—22no—
286C2×C8—22no—
287C2×C8—24no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—48no—
292C2×C8—42no—
293C2×C8—41no—
294C2×C8—42no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—42no—
299C2×C8—41no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—41no—
303C4⋊C4—21no—
304C4⋊C4—22no—
305C4⋊C4—44no—
306C4⋊C4—42no—
307C4⋊C4—41no—
308C4⋊C4—42no—
309C4⋊C4—41no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—42no—
315C4⋊C4—44no—
316C4⋊C4—42no—
317C4⋊C4—81no—
318C4⋊C4—82no—
319C4⋊C4—82no—
320C4⋊C4—82no—
321C4⋊C4—82no—
322C4⋊C4#10314yes—
323C4⋊C4#10314yes—
324C4⋊C4#10414yes—
325C4⋊C4#13112yes—
326C22⋊C4—24no—
327C22⋊C4—42no—
328C22⋊C4—81no—
329C22⋊C4—82no—
330C22⋊C4#10318yes—
331C42—22no—
332C42—22no—
333C42—22no—
334C42—22no—
335C42—22no—
336C42—22no—
337C42—24no—
338C42—42no—
339C42—41no—
340C42—41no—
341C42—42no—
342C42—44no—
343C42—42no—
344C42—42no—
345C42—82no—
346C42C2×SD1614yes—
347C42C2×Q1614yes—
348C23—21no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23C2×#10312yes—
355C23C22×C22⋊C411yescharacteristic
356Q8—82no—
357Q8—81no—
358Q8—82no—
359Q8—81no—
360Q8—164no—
361C2×C4—22no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—22no—
382C2×C4—21no—
383C2×C4—22no—
384C2×C4—21no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—42no—
399C2×C4—41no—
400C2×C4—42no—
401C2×C4—41no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4#41012yes—
407C2×C4C2×#10314yes—
408C2×C4C2×#10314yes—
409C2×C4C2×#10312yes—
410C2×C4C2×#10414yes—
411C2×C4C2×#10412yes—
412C2×C4C2×#13111yescharacteristic
413C2×C4#56811yescharacteristic
414C2×C4C22×SD1611yescharacteristic
415C2×C4C22×Q1611yescharacteristic
416C2×C4#57711yescharacteristic
417C2×C4#57811yescharacteristic
418C8—42no—
419C8—42no—
420C8—42no—
421C8—44no—
422C8—81no—
423C8—82no—
424C22—22no—
425C22—22no—
426C22—22no—
427C22—41no—
428C22—41no—
429C22—41no—
430C22—41no—
431C22#288011yescharacteristic
432C22C2×#41011yescharacteristic
433C22#288811yescharacteristic
434C22C22×#10311yescharacteristic
435C22#289811yescharacteristic
436C22C22×#10411yescharacteristic
437C22#290111yescharacteristic
438C4—21no—
439C4—21no—
440C4—24no—
441C4—24no—
442C4—21no—
443C4—21no—
444C4—21no—
445C4—21no—
446C4—24no—
447C4—41no—
448C4—42no—
449C4—42no—
450C4—41no—
451C4—82no—
452C4—81no—
453C4—81no—
454C4—82no—
455C4—81no—
456C4#150314yes—
457C2—42no—
458C2#1741811yescharacteristic
459C2#1750911yescharacteristic
460C2#1751811yescharacteristic
461C2C2×#150311yescharacteristic
462C2#1760511yescharacteristic
463C2#1765211yescharacteristic
464C2#1772211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #7682C2semidirect
7. #7662C2semidirect
9. #7615C2semidirect
11. #7562C2semidirect

Maximal subgroups

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

subgroupindex
#362882
#226322
#204302
#76872
#76822
#76672
#76622
#76172
#76152
#75662
#75622