Finite Group Databasebeta

← #164166

Subgroups — #164166

Order 512.

807 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#36287C211yesmaximal, characteristic
2#22632C211yesmaximal, characteristic
3#20430C211yesmaximal, characteristic
4#7688C211yesmaximal, characteristic
5#7683C212yesmaximal, complemented
6#7665C211yesmaximal, characteristic
7#7660C212yesmaximal, complemented
8#7617C211yesmaximal, characteristic
9#7615C212yesmaximal, complemented
10#7566C211yesmaximal, characteristic
11#7562C212yesmaximal, complemented
12C2×#424—21no—
13C2×#405—21no—
14#2859C2212yes—
15#2857C2211yescharacteristic
16#2612—22no—
17#2567C411yescharacteristic
18#2556—22no—
19#2552C411yescharacteristic
20#2506—21no—
21#2497C412yes—
22#2441—21no—
23#2423C412yes—
24#2370—21no—
25#2369C411yescharacteristic
26Q8×C4⋊C4C411yescharacteristic
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#1272C2211yescharacteristic
46#1271C2211yescharacteristic
47#1271C2212yes—
48#1271C2212yes—
49#1271C2212yes—
50#1271C2212yes—
51#1271C2212yes—
52#1271C2211yescharacteristic
53C22×M16—21no—
54C23×C8—21no—
55C2×C4⋊Q8C2×C411yescharacteristic
56C2×#125—21no—
57C2×#125—22no—
58C2×C22⋊Q8—21no—
59C2×C22⋊Q8C2×C411yescharacteristic
60C2×C4×Q8C2×C412yes—
61C2×C4×Q8C2×C411yescharacteristic
62C2×#117D411yescharacteristic
63C2×#117D411yescharacteristic
64C22×C42C2311yescharacteristic
65#424—28no—
66C2×C4⋊C8—22no—
67C2×C4⋊C8—22no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—22no—
70C2×C22⋊C8—21no—
71C2×C22⋊C8—21no—
72#405—44no—
73C2×#97—22no—
74C2×C4×C8—22no—
75C2×C4×C8—22no—
76C2×C4×C8—21no—
77#400—22no—
78#399—22no—
79#398—22no—
80#398—22no—
81#396—22no—
82#395—22no—
83#391—21no—
84#391—22no—
85#391C2×C412yes—
86#391C2311yescharacteristic
87#389—21no—
88#389—21no—
89#389—21no—
90#389—21no—
91#389—21no—
92#389—22no—
93#389—22no—
94#389—22no—
95#389C2×C412yes—
96#389C2×C412yes—
97#389C2×C412yes—
98#389C2312yes—
99#389C2311yescharacteristic
100#388—22no—
101#388—22no—
102#387—21no—
103#387—22no—
104#387—22no—
105#387—22no—
106#387—22no—
107#385—21no—
108#384—21no—
109#384C2×C412yes—
110#384C2×C411yescharacteristic
111#384C2×C412yes—
112#382—22no—
113#380—21no—
114#380C2×C412yes—
115#380C2×C411yescharacteristic
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—22no—
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—21no—
143C22×Q8C22×C411yescharacteristic
144C23×C4C22×C411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C2×M16—24no—
148C2×M16—42no—
149C2×M16—42no—
150C22×C8—24no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—22no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—22no—
157C22×C8—42no—
158C22×C8—42no—
159C4⋊Q8—42no—
160#125—44no—
161#125—82no—
162#125—82no—
163C22⋊Q8—22no—
164C22⋊Q8—41no—
165C22⋊Q8—81no—
166C22⋊Q8—81no—
167C4×Q8—24no—
168C4×Q8—42no—
169C4×Q8—42no—
170#117—24no—
171#117SD1614yes—
172#117Q1614yes—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
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×C4⋊C4C22×C411yescharacteristic
195C2×C4⋊C4C22×C412yes—
196C2×C4⋊C4C22×C411yescharacteristic
197C2×C22⋊C4—21no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4C22⋊C412yes—
201C2×C22⋊C4C22⋊C412yes—
202C2×C42—22no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—22no—
206C2×C42—42no—
207C2×C42C22×C411yescharacteristic
208C2×C42C22×C412yes—
209C2×C42C22×C412yes—
210C2×C42C2×D412yes—
211C2×C42C2×D412yes—
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D412yes—
215C2×C42C2×D412yes—
216C2×C42C2411yescharacteristic
217C4⋊C8—28no—
218C4⋊C8—28no—
219C4⋊C8—42no—
220C4⋊C8—82no—
221C22⋊C8—42no—
222C22⋊C8—81no—
223#97—44no—
224C4×C8—42no—
225C4×C8—44no—
226C4×C8—44no—
227#95—22no—
228#95—22no—
229#95—21no—
230#95—22no—
231#95—22no—
232#95—22no—
233#95—21no—
234#95—22no—
235#95—22no—
236#95—21no—
237#95—22no—
238#95—21no—
239#95—21no—
240#95C22⋊C412yes—
241#95C22⋊C412yes—
242#95C22⋊C412yes—
243#95C22⋊C412yes—
244C24C2×C22⋊C411yescharacteristic
245C2×Q8—21no—
246C2×Q8—42no—
247C2×Q8—41no—
248C2×Q8—41no—
249C2×Q8—81no—
250C2×Q8—81no—
251C2×Q8#13112yes—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4C2×C22⋊C411yescharacteristic
269C22×C4C2×C22⋊C411yescharacteristic
270C22×C4C2×C22⋊C412yes—
271C22×C4C2×C22⋊C412yes—
272C22×C4C2×C22⋊C412yes—
273C22×C4C2×C22⋊C412yes—
274C22×C4C2×C22⋊C411yescharacteristic
275C22×C4C2×SD1612yes—
276C22×C4C2×Q1612yes—
277C22×C4C23×C411yescharacteristic
278C22×C4C22×D411yescharacteristic
279C22×C4C22×D411yescharacteristic
280M16—48no—
281C2×C8—22no—
282C2×C8—22no—
283C2×C8—22no—
284C2×C8—24no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—48no—
289C2×C8—42no—
290C2×C8—41no—
291C2×C8—42no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—42no—
296C2×C8—41no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—41no—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—21no—
303C4⋊C4—22no—
304C4⋊C4—21no—
305C4⋊C4—44no—
306C4⋊C4—42no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—41no—
313C4⋊C4—42no—
314C4⋊C4—41no—
315C4⋊C4—42no—
316C4⋊C4—44no—
317C4⋊C4—44no—
318C4⋊C4—42no—
319C4⋊C4—82no—
320C4⋊C4—81no—
321C4⋊C4—82no—
322C4⋊C4—82no—
323C4⋊C4—82no—
324C4⋊C4—82no—
325C4⋊C4#10314yes—
326C4⋊C4#10314yes—
327C4⋊C4#10414yes—
328C22⋊C4—24no—
329C22⋊C4—42no—
330C22⋊C4—81no—
331C22⋊C4—82no—
332C22⋊C4#10318yes—
333C42—22no—
334C42—22no—
335C42—22no—
336C42—22no—
337C42—22no—
338C42—22no—
339C42—24no—
340C42—42no—
341C42—41no—
342C42—41no—
343C42—42no—
344C42—44no—
345C42—42no—
346C42—42no—
347C42—82no—
348C42C2×SD1614yes—
349C42C2×Q1614yes—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23C2×#10312yes—
357C23C22×C22⋊C411yescharacteristic
358Q8—42no—
359Q8—42no—
360Q8—162no—
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—42no—
391C2×C4—41no—
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#57611yescharacteristic
417C2×C4#57711yescharacteristic
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#288911yescharacteristic
434C22C22×#10311yescharacteristic
435C22#289711yescharacteristic
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#1741311yescharacteristic
459C2#1750911yescharacteristic
460C2#1752011yescharacteristic
461C2C2×#150311yescharacteristic
462C2#1761011yescharacteristic
463C2#1765311yescharacteristic
464C2#1772311yescharacteristic
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. #7683C2semidirect
7. #7660C2semidirect
9. #7615C2semidirect
11. #7562C2semidirect

Maximal subgroups

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

subgroupindex
#362872
#226322
#204302
#76882
#76832
#76652
#76602
#76172
#76152
#75662
#75622