Finite Group Databasebeta

← #165676

Subgroups — #165676

Order 512.

785 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36474C211yesmaximal, characteristic
2#22830C212yesmaximal
3#7722C212yesmaximal
4#7702C212yesmaximal
5#7682C212yesmaximal
6#7639C211yesmaximal, characteristic
7#7624C212yesmaximal
8#7605C211yesmaximal, characteristic
9#7590C212yesmaximal
10C2×Q82C411yescharacteristic
11C2×#557C411yescharacteristic
12#2862C2212yes—
13#2862C2211yescharacteristic
14#2567C412yes—
15#2567C411yescharacteristic
16#2550—22no—
17#2550—22no—
18#2550—22no—
19#2546C412yes—
20#2546C411yescharacteristic
21#2351—22no—
22C4×C4⋊Q8C2212yes—
23C4×C4⋊Q8C2211yescharacteristic
24C42×Q8C2211yescharacteristic
25#1985—22no—
26#1917—22no—
27#1916—22no—
28#1839C2212yes—
29#1839C2212yes—
30#1831C2212yes—
31#1771—22no—
32#1767C2212yes—
33#1745C2212yes—
34#1744C2212yes—
35#1272C2211yescharacteristic
36#1272C2212yes—
37#1272C2212yes—
38#1272C2212yes—
39#1272C2212yes—
40#1272C2212yes—
41#1272C2211yescharacteristic
42#1272C2211yescharacteristic
43#1272C2212yes—
44#1272C2211yescharacteristic
45Q82—44no—
46#557—44no—
47C2×C4⋊Q8—22no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8C2×C412yes—
50C2×C4⋊Q8C2×C411yescharacteristic
51C2×C4⋊Q8C2×C412yes—
52C2×C4⋊Q8C2×C411yescharacteristic
53C2×C4⋊Q8C2×C412yes—
54C2×C4⋊Q8C2×C411yescharacteristic
55C2×#125—22no—
56C2×#125—21no—
57C2×C4×Q8—21no—
58C2×C4×Q8—21no—
59C2×C4×Q8C2×C412yes—
60C2×C4×Q8C2×C411yescharacteristic
61C2×C4×Q8C2×C412yes—
62C2×C4×Q8C2×C411yescharacteristic
63C2×C4×Q8C2×C411yescharacteristic
64C2×C4×Q8D412yes—
65C2×C4⋊C8—22no—
66C2×C4⋊C8—22no—
67C2×#97—22no—
68C2×#97—22no—
69C2×C4×C8—22no—
70#398—21no—
71#398—22no—
72#398—22no—
73#398—21no—
74#395—21no—
75#395—22no—
76#395—22no—
77#395—21no—
78#391—22no—
79#391—22no—
80#391—22no—
81#391C2×C412yes—
82#391C2×C412yes—
83#391C2×C412yes—
84#391C2311yescharacteristic
85#391C2312yes—
86#391C2311yescharacteristic
87#389—22no—
88#389—22no—
89#389—22no—
90#389—22no—
91#389—22no—
92#389—22no—
93#389—22no—
94#389—22no—
95#389—22no—
96#384C2×C412yes—
97#384C2×C411yescharacteristic
98#382—22no—
99#382—22no—
100#382—22no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4C2×C411yescharacteristic
105C4×C4⋊C4C2×C412yes—
106C4×C4⋊C4D412yes—
107C4×C4⋊C4D412yes—
108C4×C4⋊C4D412yes—
109C43C2312yes—
110C43C2311yescharacteristic
111#336—22no—
112#336—22no—
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—21no—
123C22×Q8—22no—
124C22×Q8C22⋊C411yescharacteristic
125C22×Q8C22×C412yes—
126C22×Q8C22×C411yescharacteristic
127C22×Q8C22×C411yescharacteristic
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C4⋊Q8—24no—
133C4⋊Q8—22no—
134C4⋊Q8—42no—
135C4⋊Q8—41no—
136C4⋊Q8—44no—
137C4⋊Q8—44no—
138C4⋊Q8—42no—
139C4⋊Q8—42no—
140C4⋊Q8—48no—
141C4⋊Q8—44no—
142#125—48no—
143#125—44no—
144C4×Q8—24no—
145C4×Q8—24no—
146C4×Q8—24no—
147C4×Q8—44no—
148C4×Q8—44no—
149C4×Q8—42no—
150C4×Q8—42no—
151C4×Q8—44no—
152C4×Q8—44no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4C22⋊C412yes—
170C2×C4⋊C4C22⋊C412yes—
171C2×C4⋊C4C22⋊C412yes—
172C2×C4⋊C4C22⋊C412yes—
173C2×C4⋊C4C22⋊C411yescharacteristic
174C2×C4⋊C4C22×C411yescharacteristic
175C2×C4⋊C4C22×C412yes—
176C2×C4⋊C4C22×C411yescharacteristic
177C2×C42—22no—
178C2×C42—21no—
179C2×C42—22no—
180C2×C42—21no—
181C2×C42C22×C411yescharacteristic
182C2×C42C22×C412yes—
183C2×C42C22×C411yescharacteristic
184C2×C42C22×C412yes—
185C2×C42C2×D412yes—
186C2×C42C2×D412yes—
187C2×C42C2×D412yes—
188C2×C42C2×D412yes—
189C2×C42C2×D412yes—
190C2×C42C2×D412yes—
191C2×C42C2411yescharacteristic
192C4⋊C8—44no—
193C4⋊C8—44no—
194#97—44no—
195#97—44no—
196C4×C8—44no—
197#95—22no—
198#95—22no—
199#95—22no—
200#95—22no—
201#95—22no—
202#95—22no—
203#95—22no—
204#95—22no—
205#95—22no—
206#95—22no—
207#95C22⋊C411yescharacteristic
208#95C22⋊C412yes—
209#95C22⋊C412yes—
210#95C22⋊C411yescharacteristic
211C2×Q8—21no—
212C2×Q8—22no—
213C2×Q8—24no—
214C2×Q8—22no—
215C2×Q8—21no—
216C2×Q8—22no—
217C2×Q8—21no—
218C2×Q8—42no—
219C2×Q8—41no—
220C2×Q8—41no—
221C2×Q8—42no—
222C2×Q8—41no—
223C2×Q8—42no—
224C2×Q8—41no—
225C2×Q8—44no—
226C2×Q8—82no—
227C2×Q8—82no—
228C2×Q8—82no—
229C2×Q8#13112yes—
230C22×C4—21no—
231C22×C4—22no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4C2×C22⋊C412yes—
237C22×C4C2×C22⋊C412yes—
238C22×C4C2×C22⋊C411yescharacteristic
239C22×C4C2×C22⋊C412yes—
240C22×C4C2×C22⋊C411yescharacteristic
241C22×C4C2×C22⋊C412yes—
242C22×C4C2×C22⋊C412yes—
243C22×C4C23×C411yescharacteristic
244C22×C4C22×D412yes—
245C2×C8—24no—
246C2×C8—42no—
247C2×C8—42no—
248C2×C8—42no—
249C2×C8—42no—
250C2×C8—42no—
251C2×C8—42no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8—42no—
256C2×C8—42no—
257C4⋊C4—24no—
258C4⋊C4—24no—
259C4⋊C4—24no—
260C4⋊C4—24no—
261C4⋊C4—44no—
262C4⋊C4—44no—
263C4⋊C4—44no—
264C4⋊C4—41no—
265C4⋊C4—42no—
266C4⋊C4—41no—
267C4⋊C4—42no—
268C4⋊C4—44no—
269C4⋊C4—44no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—44no—
275C4⋊C4—44no—
276C4⋊C4—44no—
277C4⋊C4—82no—
278C4⋊C4—82no—
279C4⋊C4—82no—
280C4⋊C4#10014yes—
281C42—22no—
282C42—21no—
283C42—22no—
284C42—21no—
285C42—22no—
286C42—22no—
287C42—22no—
288C42—22no—
289C42—24no—
290C42—22no—
291C42—42no—
292C42—42no—
293C42—42no—
294C42—41no—
295C42—44no—
296C42—44no—
297C42—42no—
298C42—42no—
299C42#13714yes—
300C42#13714yes—
301C23C22×C22⋊C411yescharacteristic
302Q8—28no—
303Q8—42no—
304Q8—42no—
305Q8—82no—
306Q8—81no—
307Q8—82no—
308Q8—81no—
309Q8—164no—
310Q8—162no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—21no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—41no—
340C2×C4—42no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4C2×#10012yes—
347C2×C4C2×#10014yes—
348C2×C4#41914yes—
349C2×C4#41914yes—
350C2×C4C2×#13111yescharacteristic
351C2×C4#56811yescharacteristic
352C2×C4C2×#13712yes—
353C2×C4C2×#13712yes—
354C8—44no—
355C8—82no—
356C8—82no—
357C8—82no—
358C22#288011yescharacteristic
359C22#288411yescharacteristic
360C22C22×#10011yescharacteristic
361C22C2×#41912yes—
362C22#290112yes—
363C4—21no—
364C4—21no—
365C4—24no—
366C4—22no—
367C4—24no—
368C4—24no—
369C4—22no—
370C4—22no—
371C4—42no—
372C4—41no—
373C4—42no—
374C4—42no—
375C4—82no—
376C4—81no—
377C4—81no—
378C4—81no—
379C4—82no—
380C2#1742511yescharacteristic
381C2#1751412yes—
382C2#1763212yes—
383C2#1768511yescharacteristic
384C2#1774611yescharacteristic
subgroupG/H
center, socle, prime radicalC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Maximal subgroups

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

subgroupindex
#364742
#228302
#77222
#77022
#76822
#76392
#76242
#76052
#75902