Finite Group Databasebeta

← #165671

Subgroups — #165671

Order 512.

907 conjugacy classes of subgroups.

The subgroup lattice has 425 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36276C211yesmaximal, characteristic
2#22788C212yesmaximal
3#7727C212yesmaximal, complemented
4#7702C212yesmaximal, complemented
5#7687C212yesmaximal, complemented
6#7639C211yesmaximal, characteristic, complemented
7#7609C212yesmaximal, complemented
8#7605C211yesmaximal, characteristic, complemented
9#7575C212yesmaximal, complemented
10C2×#554C411yescharacteristic
11C2×D4×Q8C411yescharacteristic
12#2862C2211yescharacteristic
13#2857C2212yes—
14#2567C411yescharacteristic
15#2550—22no—
16#2549—22no—
17#2549—22no—
18#2546C411yescharacteristic
19#2502C412yes—
20#2434C412yes—
21#2349—22no—
22C4×C4⋊Q8C2211yescharacteristic
23C2×#391C2212yes—
24C42×D4C2211yescharacteristic
25#1924—22no—
26#1916—22no—
27#1870—22no—
28#1839C2212yes—
29#1817C2212yes—
30#1797C2212yes—
31#1753—22no—
32C4×C22⋊C8C2212yes—
33#1745C2212yes—
34#1728C2212yes—
35#1272C2211yescharacteristic
36#1272C2212yes—
37#1272C2212yes—
38#1272C2212yes—
39#1272C2212yes—
40#1272C2212yes—
41#1272C2211yescharacteristic
42#1272C2211yescharacteristic
43#1272C2212yes—
44#1272C2211yescharacteristic
45C23×Q8C2×C412yes—
46C22×M16—22no—
47#554—44no—
48D4×Q8—44no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8C2×C411yescharacteristic
51C2×C4⋊Q8C2×C411yescharacteristic
52C2×C4⋊Q8C2×C412yes—
53C2×C4⋊Q8C2×C411yescharacteristic
54C2×#125—21no—
55C2×C22⋊Q8—22no—
56C2×C22⋊Q8—22no—
57C2×C22⋊Q8C2×C412yes—
58C2×C22⋊Q8C2×C412yes—
59C2×C4×Q8C2×C411yescharacteristic
60C2×C4×Q8C2×C411yescharacteristic
61C2×C4×D4—21no—
62C2×C4×D4—21no—
63C2×C4×D4C2×C411yescharacteristic
64C2×C4×D4D412yes—
65C22×C4⋊C4C2×C412yes—
66C22×C42C2312yes—
67C2×C4⋊C8—22no—
68C2×C22⋊C8—22no—
69C2×C22⋊C8—22no—
70C2×C22⋊C8—22no—
71C2×#97—22no—
72C2×C4×C8—22no—
73#398—22no—
74#398—22no—
75#398—21no—
76#398—21no—
77#395—21no—
78#395—21no—
79#395—22no—
80#395—22no—
81#391—22no—
82#391—22no—
83#391—22no—
84#391C2×C412yes—
85#391C2×C412yes—
86#391C2×C412yes—
87#391C2311yescharacteristic
88#391C2311yescharacteristic
89#391C2312yes—
90#389—22no—
91#389—22no—
92#389—22no—
93#389—22no—
94#389—22no—
95#385—22no—
96#385—22no—
97#385—22no—
98#385—22no—
99#384C2×C411yescharacteristic
100#384C2×C412yes—
101#382—22no—
102#380—22no—
103#380—22no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4C2×C411yescharacteristic
106C4×C4⋊C4D412yes—
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—
122C22×Q8—22no—
123C22×Q8—21no—
124C22×Q8—24no—
125C22×Q8—42no—
126C22×Q8—44no—
127C22×Q8—82no—
128C22×Q8C22×C411yescharacteristic
129C22×Q8C22×C412yes—
130C22×Q8C22×C411yescharacteristic
131C22×D4C22⋊C411yescharacteristic
132C23×C4—22no—
133C23×C4—22no—
134C23×C4C22×C412yes—
135C23×C4C2×D412yes—
136C23×C4C2×D412yes—
137C2×M16—44no—
138C2×M16—44no—
139C2×M16—82no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C4⋊Q8—22no—
145C4⋊Q8—41no—
146C4⋊Q8—42no—
147C4⋊Q8—42no—
148C4⋊Q8—44no—
149C4⋊Q8—82no—
150#125—44no—
151C22⋊Q8—48no—
152C22⋊Q8—48no—
153C22⋊Q8—44no—
154C22⋊Q8—44no—
155C4×Q8—42no—
156C4×Q8—42no—
157C4×D4—24no—
158C4×D4—24no—
159C4×D4—24no—
160C4×D4—44no—
161C4×D4—44no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4C22⋊C412yes—
178C2×C4⋊C4C22⋊C411yescharacteristic
179C2×C4⋊C4C22×C411yescharacteristic
180C2×C4⋊C4C22×C412yes—
181C2×C4⋊C4C22×C411yescharacteristic
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4C22⋊C412yes—
185C2×C22⋊C4C22⋊C412yes—
186C2×C22⋊C4C22⋊C412yes—
187C2×C42—21no—
188C2×C42—21no—
189C2×C42—82no—
190C2×C42C22×C411yescharacteristic
191C2×C42C22×C411yescharacteristic
192C2×C42C22×C412yes—
193C2×C42C2×D412yes—
194C2×C42C2×D412yes—
195C2×C42C2×D412yes—
196C2×C42C2×D412yes—
197C2×C42C2411yescharacteristic
198C4⋊C8—82no—
199C22⋊C8—44no—
200C22⋊C8—44no—
201C22⋊C8—82no—
202#97—44no—
203C4×C8—82no—
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
218C24C2×C22⋊C412yes—
219C2×Q8—21no—
220C2×Q8—21no—
221C2×Q8—22no—
222C2×Q8—21no—
223C2×Q8—22no—
224C2×Q8—44no—
225C2×Q8—42no—
226C2×Q8—41no—
227C2×Q8—41no—
228C2×Q8—41no—
229C2×Q8—42no—
230C2×Q8—41no—
231C2×Q8—42no—
232C2×Q8—44no—
233C2×Q8—82no—
234C2×Q8—82no—
235C2×Q8—88no—
236C2×Q8—88no—
237C2×Q8#13112yes—
238C2×D4—24no—
239C2×D4—22no—
240C22×C4—21no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—24no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—21no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—44no—
257C22×C4—44no—
258C22×C4—44no—
259C22×C4—82no—
260C22×C4—82no—
261C22×C4—82no—
262C22×C4C2×C22⋊C412yes—
263C22×C4C2×C22⋊C412yes—
264C22×C4C2×C22⋊C411yescharacteristic
265C22×C4C2×C22⋊C412yes—
266C22×C4C2×C22⋊C411yescharacteristic
267C22×C4C2×C22⋊C412yes—
268C22×C4#13714yes—
269C22×C4#13714yes—
270C22×C4C23×C411yescharacteristic
271C22×C4C22×D412yes—
272M16—88no—
273C2×C8—24no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—42no—
285C4⋊C4—24no—
286C4⋊C4—44no—
287C4⋊C4—41no—
288C4⋊C4—42no—
289C4⋊C4—41no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—44no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—44no—
297C4⋊C4—44no—
298C4⋊C4—82no—
299C4⋊C4—82no—
300C4⋊C4—82no—
301C4⋊C4—82no—
302C4⋊C4—82no—
303C4⋊C4#10014yes—
304C22⋊C4—24no—
305C22⋊C4—24no—
306C22⋊C4—24no—
307C22⋊C4—44no—
308C22⋊C4—44no—
309C42—21no—
310C42—21no—
311C42—22no—
312C42—22no—
313C42—22no—
314C42—42no—
315C42—42no—
316C42—42no—
317C42—41no—
318C42—42no—
319C42—42no—
320C42—42no—
321C23—22no—
322C23—22no—
323C23—22no—
324C23—22no—
325C23—22no—
326C23—22no—
327C23#41914yes—
328C23C22×C22⋊C411yescharacteristic
329Q8—42no—
330Q8—42no—
331Q8—88no—
332Q8—81no—
333Q8—81no—
334Q8—82no—
335Q8—82no—
336Q8—162no—
337D4—28no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—21no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—21no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—21no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—88no—
378C2×C4—164no—
379C2×C4C2×#10012yes—
380C2×C4C2×#10014yes—
381C2×C4#41914yes—
382C2×C4C2×#13111yescharacteristic
383C2×C4#56811yescharacteristic
384C2×C4C2×#13712yes—
385C2×C4C2×#13712yes—
386C8—44no—
387C8—82no—
388C8—82no—
389C8—82no—
390C22—24no—
391C22—24no—
392C22—24no—
393C22—42no—
394C22—42no—
395C22—42no—
396C22—42no—
397C22#288011yescharacteristic
398C22#288311yescharacteristic
399C22C22×#10011yescharacteristic
400C22C2×#41912yes—
401C22#290112yes—
402C4—21no—
403C4—21no—
404C4—24no—
405C4—22no—
406C4—22no—
407C4—22no—
408C4—24no—
409C4—42no—
410C4—41no—
411C4—42no—
412C4—42no—
413C4—82no—
414C4—81no—
415C4—81no—
416C4—81no—
417C4—82no—
418C2—44no—
419C2#1742111yescharacteristic
420C2#1751412yes—
421C2#1762112yes—
422C2#1768811yescharacteristic
423C2#1774611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C22⋊C4
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
3. #7727C2semidirect
4. #7702C2semidirect
5. #7687C2semidirect
6. #7639C2semidirect
7. #7609C2semidirect
8. #7605C2semidirect
9. #7575C2semidirect

Maximal subgroups

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

subgroupindex
#362762
#227882
#77272
#77022
#76872
#76392
#76092
#76052
#75752