Finite Group Databasebeta

← #165529

Subgroups — #165529

Order 512.

952 conjugacy classes of subgroups.

The subgroup lattice has 437 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36303C211yesmaximal, characteristic
2#21998C212yesmaximal, complemented
3#7722C212yesmaximal, complemented
4#7710C212yesmaximal, complemented
5#7670C212yesmaximal, complemented
6#7649C211yesmaximal, characteristic, complemented
7#7609C212yesmaximal, complemented
8#7601C211yesmaximal, characteristic, complemented
9#7575C212yesmaximal, complemented
10C2×D4×Q8C411yescharacteristic
11C2×#547C411yescharacteristic
12#2862C2211yescharacteristic
13#2850C2212yes—
14#2568C411yescharacteristic
15#2547C411yescharacteristic
16#2503C412yes—
17#2492—22no—
18#2488—22no—
19#2488—22no—
20#2436C412yes—
21#2339—22no—
22C4×#124C2211yescharacteristic
23C2×#391C2212yes—
24C42×D4C2211yescharacteristic
25#1923—22no—
26#1916—22no—
27#1870—22no—
28#1851—22no—
29#1839C2212yes—
30#1831C2212yes—
31#1817C2212yes—
32#1797C2212yes—
33C4×C22⋊C8C2212yes—
34#1728C2212yes—
35#1272C2212yes—
36#1272C2212yes—
37#1272C2212yes—
38#1272C2212yes—
39#1272C2211yescharacteristic
40#1272C2212yes—
41#1272C2211yescharacteristic
42#1270C2211yescharacteristic
43#1270C2212yes—
44#1270C2211yescharacteristic
45C22×#41—21no—
46C23×Q8C2×C412yes—
47C22×M16—22no—
48D4×Q8—44no—
49#547—44no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8C2×C411yescharacteristic
52C2×C4⋊Q8C2×C411yescharacteristic
53C2×#124C2×C411yescharacteristic
54C2×#124C2×C412yes—
55C2×#123—22no—
56C2×C22⋊Q8—21no—
57C2×C22⋊Q8—22no—
58C2×C22⋊Q8C2×C412yes—
59C2×#121C2×C412yes—
60C2×C4×Q8C2×C411yescharacteristic
61C2×C4×D4—21no—
62C2×C4×D4C2×C411yescharacteristic
63C2×C4×D4C2×C411yescharacteristic
64C2×C4×D4D412yes—
65C22×C4⋊C4C2×C412yes—
66C22×C42C2312yes—
67C2×C4⋊C8—22no—
68C2×C4⋊C8—22no—
69C2×C22⋊C8—22no—
70C2×C22⋊C8—22no—
71C2×C22⋊C8—22no—
72C2×C4×C8—22no—
73#400—21no—
74#400—21no—
75#400—22no—
76#400—22no—
77#397—21no—
78#397—21no—
79#393—22no—
80#393—22no—
81#391—22no—
82#391—22no—
83#391—22no—
84#391—22no—
85#391C2×C412yes—
86#391C2×C412yes—
87#391C2×C412yes—
88#391C2312yes—
89#391C2311yescharacteristic
90#390C2311yescharacteristic
91#388—22no—
92#388—22no—
93#388—22no—
94#386—22no—
95#386—22no—
96#386—22no—
97#386—22no—
98#382—22no—
99#382C2×C412yes—
100#382C2×C411yescharacteristic
101#381—22no—
102#381—22no—
103#380—22no—
104C4×C4⋊C4C2×C411yescharacteristic
105C4×C4⋊C4D412yes—
106C4×C22⋊C4—22no—
107C4×C22⋊C4C2×C412yes—
108C4×C22⋊C4D412yes—
109C4×C22⋊C4D412yes—
110C2×#95—22no—
111C2×#95—22no—
112C43C2311yescharacteristic
113#336—22no—
114#336—22no—
115#336—22no—
116#336—22no—
117#336—22no—
118#336C2312yes—
119#336C2312yes—
120#336C2312yes—
121#336C2312yes—
122C2×#41—44no—
123C2×#41—48no—
124C22×Q8—21no—
125C22×Q8—22no—
126C22×Q8—24no—
127C22×Q8—42no—
128C22×Q8—44no—
129C22×Q8—82no—
130C22×Q8C22×C412yes—
131C22×Q8C22×C411yescharacteristic
132C22×D4—22no—
133C22×D4C22⋊C411yescharacteristic
134C22×D4C22×C411yescharacteristic
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—21no—
138C23×C4C22×C412yes—
139C23×C4C2×D412yes—
140C23×C4C2×D412yes—
141C2×M16—44no—
142C2×M16—82no—
143C2×M16—82no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C4⋊Q8—42no—
149C4⋊Q8—42no—
150C4⋊Q8—44no—
151#124—22no—
152#124—41no—
153#124—82no—
154#123—48no—
155C22⋊Q8—48no—
156C22⋊Q8—44no—
157C22⋊Q8—44no—
158#121—44no—
159C4×Q8—42no—
160C4×D4—24no—
161C4×D4—42no—
162C4×D4—44no—
163C4×D4—44no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4C22⋊C412yes—
172C2×C4⋊C4C22⋊C411yescharacteristic
173C2×C4⋊C4C22×C411yescharacteristic
174C2×C4⋊C4C22×C412yes—
175C2×C4⋊C4C22×C411yescharacteristic
176C2×C22⋊C4—22no—
177C2×C22⋊C4—21no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4C22⋊C412yes—
184C2×C22⋊C4C22⋊C412yes—
185C2×C22⋊C4C22⋊C412yes—
186C2×C42—21no—
187C2×C42—82no—
188C2×C42C22×C411yescharacteristic
189C2×C42C22×C411yescharacteristic
190C2×C42C22×C412yes—
191C2×C42C2×D412yes—
192C2×C42C2×D412yes—
193C2×C42C2×D412yes—
194C2×C42C2×D412yes—
195C2×C42C2411yescharacteristic
196C4⋊C8—82no—
197C4⋊C8—82no—
198C22⋊C8—44no—
199C22⋊C8—82no—
200C22⋊C8—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#95C22⋊C412yes—
215#95C22⋊C412yes—
216#95C22⋊C411yescharacteristic
217#95C22⋊C411yescharacteristic
218C24—21no—
219C24C2×C22⋊C412yes—
220#41—816no—
221C2×Q8—21no—
222C2×Q8—22no—
223C2×Q8—22no—
224C2×Q8—44no—
225C2×Q8—44no—
226C2×Q8—41no—
227C2×Q8—41no—
228C2×Q8—42no—
229C2×Q8—42no—
230C2×Q8—44no—
231C2×Q8—42no—
232C2×Q8—82no—
233C2×Q8—88no—
234C2×Q8—88no—
235C2×Q8#13112yes—
236C2×D4—24no—
237C2×D4—21no—
238C2×D4—21no—
239C2×D4—22no—
240C2×D4—41no—
241C2×D4—41no—
242C2×D4—44no—
243C2×D4—44no—
244C2×D4—44no—
245C22×C4—21no—
246C22×C4—24no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—24no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—21no—
255C22×C4—22no—
256C22×C4—21no—
257C22×C4—22no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—44no—
263C22×C4—44no—
264C22×C4—44no—
265C22×C4—44no—
266C22×C4—44no—
267C22×C4—44no—
268C22×C4—42no—
269C22×C4—82no—
270C22×C4C2×C22⋊C412yes—
271C22×C4C2×C22⋊C412yes—
272C22×C4C2×C22⋊C412yes—
273C22×C4C2×C22⋊C411yescharacteristic
274C22×C4C2×C22⋊C412yes—
275C22×C4C2×C22⋊C411yescharacteristic
276C22×C4C23×C411yescharacteristic
277C22×C4C22×D412yes—
278M16—88no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C4⋊C4—42no—
292C4⋊C4—41no—
293C4⋊C4—42no—
294C4⋊C4—41no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—44no—
299C4⋊C4—42no—
300C4⋊C4—44no—
301C4⋊C4#10014yes—
302C22⋊C4—24no—
303C22⋊C4—24no—
304C22⋊C4—42no—
305C22⋊C4—44no—
306C22⋊C4—42no—
307C22⋊C4—44no—
308C22⋊C4—42no—
309C22⋊C4—42no—
310C22⋊C4—44no—
311C22⋊C4—82no—
312C22⋊C4—82no—
313C42—21no—
314C42—21no—
315C42—22no—
316C42—22no—
317C42—22no—
318C42—42no—
319C42—42no—
320C42—41no—
321C42—42no—
322C42—42no—
323C42—42no—
324C23—22no—
325C23—22no—
326C23—22no—
327C23—22no—
328C23—22no—
329C23—22no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—42no—
334C23—42no—
335C23#42114yes—
336C23C22×C22⋊C411yescharacteristic
337Q8—42no—
338Q8—42no—
339Q8—88no—
340Q8—82no—
341Q8—82no—
342Q8—84no—
343D4—28no—
344D4—81no—
345D4—81no—
346D4—88no—
347C2×C4—22no—
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—21no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—41no—
375C2×C4—42no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—84no—
385C2×C4—88no—
386C2×C4—88no—
387C2×C4C2×#10012yes—
388C2×C4C2×#10014yes—
389C2×C4C2×#13111yescharacteristic
390C2×C4#56811yescharacteristic
391C2×C4#57712yes—
392C2×C4#57812yes—
393C8—82no—
394C8—82no—
395C8—82no—
396C8—82no—
397C22—24no—
398C22—24no—
399C22—24no—
400C22—42no—
401C22—42no—
402C22—42no—
403C22—42no—
404C22—81no—
405C22—81no—
406C22—81no—
407C22—82no—
408C22—82no—
409C22#288011yescharacteristic
410C22#288311yescharacteristic
411C22C22×#10011yescharacteristic
412C22#289812yes—
413C22C2×#42112yes—
414C4—22no—
415C4—21no—
416C4—21no—
417C4—24no—
418C4—22no—
419C4—22no—
420C4—42no—
421C4—41no—
422C4—42no—
423C4—42no—
424C4—42no—
425C4—81no—
426C4—82no—
427C4—82no—
428C4—81no—
429C2—44no—
430C2—161no—
431C2#1742111yescharacteristic
432C2#1752412yes—
433C2#1759812yes—
434C2#1766611yescharacteristic
435C2#1777311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#547C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21998C2semidirect
3. #7722C2semidirect
4. #7710C2semidirect
5. #7670C2semidirect
6. #7649C2semidirect
7. #7609C2semidirect
8. #7601C2semidirect
9. #7575C2semidirect

Maximal subgroups

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

subgroupindex
#363032
#219982
#77222
#77102
#76702
#76492
#76092
#76012
#75752