Finite Group Databasebeta

← #224079

Subgroups — #224079

Order 512.

810 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
1C2×#2511C211yesmaximal, characteristic
2#22627C212yesmaximal
3#9882C212yesmaximal
4#9881C212yesmaximal
5#9880C212yesmaximal
6#8562C212yesmaximal
7#8560C211yesmaximal, characteristic
8#8154C212yesmaximal
9#8153C211yesmaximal, characteristic
10#2511—22no—
11#2511—22no—
12#2511—22no—
13#2511—22no—
14C2×#400C412yes—
15C2×#400C412yes—
16C2×#391C411yescharacteristic
17C2×#391C411yescharacteristic
18C2×#391C2211yescharacteristic
19C2×#389C411yescharacteristic
20C2×#387C2212yes—
21C2×#387C2212yes—
22C2×#384C411yescharacteristic
23C2×C4×C4⋊C4C2211yescharacteristic
24C2×C4×C4⋊C4C2211yescharacteristic
25#1940—22no—
26#1917—22no—
27#1916—22no—
28#1797C2212yes—
29#1797C2212yes—
30#1797C2212yes—
31#1771—22no—
32#1749C2212yes—
33#1749C2212yes—
34C4×C22⋊C8C2212yes—
35C2×#324C2212yes—
36C2×#324C2211yescharacteristic
37C2×#324C2211yescharacteristic
38#1280C2211yescharacteristic
39#1280C2211yescharacteristic
40#1280C2212yes—
41#1280C2212yes—
42#1280C2212yes—
43#1280C2212yes—
44#1280C2212yes—
45C23×Q8C2×C411yescharacteristic
46C22×C4⋊C4C2×C411yescharacteristic
47C22×C4⋊C4C2×C412yes—
48C22×C4⋊C4C2×C411yescharacteristic
49C22×C4⋊C4C2×C412yes—
50C22×C4⋊C4C2×C412yes—
51C22×C4⋊C4C2×C411yescharacteristic
52C22×C4⋊C4C2×C411yescharacteristic
53C22×C4⋊C4C2311yescharacteristic
54C22×C42C2×C411yescharacteristic
55C22×C42C2×C411yescharacteristic
56C22×C42C2311yescharacteristic
57C22×C42C2312yes—
58C2×C4⋊C8—22no—
59C2×C4⋊C8—22no—
60C2×C4⋊C8—22no—
61C2×C22⋊C8C2312yes—
62C2×C22⋊C8C2312yes—
63C2×C22⋊C8C2312yes—
64C2×C22⋊C8C2312yes—
65C2×#97—22no—
66C2×#97—22no—
67C2×C4×C8—22no—
68#400—22no—
69#400—22no—
70#400—22no—
71#400—22no—
72#400—22no—
73#400—22no—
74#400—22no—
75#400—22no—
76#391—21no—
77#391—21no—
78#391—22no—
79#391—22no—
80#391—21no—
81#391—21no—
82#391—22no—
83#391—22no—
84#389—22no—
85#389—21no—
86#389—21no—
87#387—22no—
88#387—22no—
89#387—22no—
90#387—22no—
91#387—22no—
92#387—22no—
93#387—22no—
94#387—22no—
95#384—22no—
96#384—21no—
97#384—21no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4D412yes—
101C4×C4⋊C4D412yes—
102C4×C4⋊C4D412yes—
103C4×C4⋊C4D412yes—
104C2×#95C2×C411yescharacteristic
105C2×#95C2×C412yes—
106C2×#95C2×C412yes—
107C2×#95C2×C412yes—
108C2×#95C2×C412yes—
109C2×#95C2×C412yes—
110C2×#95C2×C412yes—
111C2×#95C2×C412yes—
112C2×#95C2311yescharacteristic
113C2×#95C2312yes—
114#336—22no—
115#336—22no—
116#336—22no—
117#336—22no—
118#336—22no—
119#336—22no—
120#324—24no—
121#324—22no—
122#324Q814yescomplemented
123C22×Q8—21no—
124C22×Q8—21no—
125C22×Q8—41no—
126C22×Q8—41no—
127C22×Q8—42no—
128C22×Q8—42no—
129C23×C4C22×C411yescharacteristic
130C23×C4C22×C411yescharacteristic
131C23×C4C22×C412yes—
132C23×C4C22×C411yescharacteristic
133C23×C4C22×C412yes—
134C23×C4C22×C412yes—
135C23×C4C22×C412yes—
136C23×C4C22×C411yescharacteristic
137C23×C4C22×C411yescharacteristic
138C23×C4C22×C411yescharacteristic
139C23×C4C2411yescharacteristic
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4—82no—
164C2×C4⋊C4—82no—
165C2×C4⋊C4C22⋊C412yes—
166C2×C4⋊C4C22⋊C411yescharacteristic
167C2×C4⋊C4C22⋊C411yescharacteristic
168C2×C4⋊C4C22⋊C412yes—
169C2×C4⋊C4C22⋊C411yescharacteristic
170C2×C4⋊C4C22⋊C411yescharacteristic
171C2×C4⋊C4C2×Q812yes—
172C2×C4⋊C4#4112yes—
173C2×C42—21no—
174C2×C42—21no—
175C2×C42—21no—
176C2×C42—21no—
177C2×C42—42no—
178C2×C42—41no—
179C2×C42—41no—
180C2×C42—42no—
181C2×C42—82no—
182C2×C42C2×D412yes—
183C2×C42C2×D412yes—
184C2×C42C2×D412yes—
185C2×C42C2×D412yes—
186C2×C42C2×D412yes—
187C2×C42C2×D412yes—
188C4⋊C8—44no—
189C4⋊C8—82no—
190C4⋊C8—82no—
191C22⋊C8—22no—
192C22⋊C8—22no—
193C22⋊C8—22no—
194C22⋊C8—22no—
195C22⋊C8—22no—
196C22⋊C8—22no—
197C22⋊C8—22no—
198C22⋊C8C2×Q814yes—
199#97—44no—
200#97—82no—
201C4×C8—82no—
202#95—22no—
203#95—22no—
204#95—22no—
205#95—22no—
206#95—22no—
207#95—22no—
208#95—22no—
209#95—22no—
210#95—22no—
211#95—22no—
212#95—22no—
213#95—22no—
214#95—22no—
215#95—22no—
216#95—22no—
217#95—22no—
218#95C22⋊C412yes—
219#95C22⋊C412yes—
220#95C22⋊C412yes—
221#95C22⋊C412yes—
222C24C23×C411yescharacteristic
223C2×Q8—42no—
224C2×Q8—42no—
225C2×Q8—42no—
226C2×Q8—42no—
227C2×Q8—42no—
228C2×Q8—42no—
229C2×Q8—84no—
230C2×Q8—84no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—21no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—21no—
251C22×C4—22no—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4C2×C22⋊C412yes—
279C22×C4C2×C22⋊C412yes—
280C22×C4C2×C22⋊C412yes—
281C22×C4C2×C22⋊C412yes—
282C22×C4C2×C22⋊C412yes—
283C22×C4C2×C22⋊C411yescharacteristic
284C22×C4C2×C22⋊C411yescharacteristic
285C22×C4C4×Q812yes—
286C22×C4C4×Q812yes—
287C22×C4#13112yes—
288C22×C4C22×D412yes—
289C22×C4C22×Q811yescharacteristic
290C22×C4C2×#4111yescharacteristic
291C22×C42−1+412yes—
292C2×C8—24no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—42no—
304C4⋊C4—24no—
305C4⋊C4—24no—
306C4⋊C4—24no—
307C4⋊C4—42no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—44no—
313C4⋊C4—44no—
314C4⋊C4—44no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—82no—
318C4⋊C4—82no—
319C4⋊C4—82no—
320C4⋊C4—82no—
321C42—24no—
322C42—42no—
323C42—42no—
324C42—42no—
325C42—42no—
326C42—42no—
327C42—42no—
328C42—42no—
329C42—42no—
330C42—42no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23C22×C22⋊C411yescharacteristic
336C23C2×C4×Q811yescharacteristic
337C23#51911yescharacteristic
338C23C2×#13111yescharacteristic
339C23#56811yescharacteristic
340C23#56811yescharacteristic
341C23#56811yescharacteristic
342Q8—84no—
343Q8—84no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—22no—
349C2×C4—21no—
350C2×C4—22no—
351C2×C4—21no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—41no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—41no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4#41012yes—
409C2×C4#41312yes—
410C2×C4#54812yes—
411C2×C4D4×Q812yes—
412C8—44no—
413C8—82no—
414C8—82no—
415C8—82no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22Q8×C22⋊C411yescharacteristic
429C22#287911yescharacteristic
430C22#288011yescharacteristic
431C22C2×#41011yescharacteristic
432C22#288311yescharacteristic
433C22C2×#41311yescharacteristic
434C22#288911yescharacteristic
435C22#296111yescharacteristic
436C22#296211yescharacteristic
437C22#298511yescharacteristic
438C22#298711yescharacteristic
439C4—22no—
440C4—22no—
441C4—22no—
442C4—22no—
443C4—42no—
444C4—42no—
445C4—42no—
446C4—42no—
447C4—42no—
448C4—42no—
449C4—42no—
450C4—42no—
451C4—82no—
452C4—82no—
453C4—81no—
454C2—21no—
455C2—21no—
456C2—21no—
457C2—21no—
458C2#1739811yescharacteristic
459C2#1740711yescharacteristic
460C2#1741311yescharacteristic
461C2#1742911yescharacteristic
462C2#2251111yescharacteristic
463C2#2251311yescharacteristic
464C2#2251711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutator, prime radicalC24C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
122. #324Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25112
#226272
#98822
#98812
#98802
#85622
#85602
#81542
#81532