Finite Group Databasebeta

← #165433

Subgroups — #165433

Order 512.

833 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36446C211yesmaximal, characteristic
2#22020C212yesmaximal, complemented
3#7722C212yesmaximal, complemented
4#7702C212yesmaximal, complemented
5#7662C212yesmaximal, complemented
6#7649C211yesmaximal, characteristic
7#7618C212yesmaximal
8#7601C211yesmaximal, characteristic
9#7584C212yesmaximal
10C2×#557C411yescharacteristic
11C2×D4×Q8C411yescharacteristic
12#2862C2212yes—
13#2862C2211yescharacteristic
14#2568C412yes—
15#2568C411yescharacteristic
16#2547C411yescharacteristic
17#2547C412yes—
18#2489—22no—
19#2489—22no—
20#2489—22no—
21Q8×C22⋊C4—22no—
22C4×#124C2212yes—
23C4×#124C2211yescharacteristic
24C42×Q8C2211yescharacteristic
25#1985—22no—
26#1940—22no—
27#1916—22no—
28#1916—22no—
29#1839C2212yes—
30#1839C2212yes—
31#1831C2212yes—
32#1831C2212yes—
33C4×C4⋊C8C2212yes—
34#1745C2212yes—
35#1272C2211yescharacteristic
36#1272C2212yes—
37#1272C2212yes—
38#1272C2212yes—
39#1272C2212yes—
40#1272C2212yes—
41#1272C2211yescharacteristic
42#1270C2211yescharacteristic
43#1270C2212yes—
44#1270C2211yescharacteristic
45C23×Q8—21no—
46#557—44no—
47D4×Q8—44no—
48C2×C4⋊Q8C2×C412yes—
49C2×C4⋊Q8C2×C411yescharacteristic
50C2×C4⋊Q8C2×C411yescharacteristic
51C2×C4⋊Q8C2×C412yes—
52C2×#125—22no—
53C2×#125—21no—
54C2×#124C2×C412yes—
55C2×#124C2×C411yescharacteristic
56C2×C22⋊Q8—22no—
57C2×C22⋊Q8—21no—
58C2×C4×Q8—21no—
59C2×C4×Q8C2×C412yes—
60C2×C4×Q8C2×C411yescharacteristic
61C2×C4×Q8C2×C411yescharacteristic
62C2×C4×Q8D412yes—
63C2×C4×D4C2×C411yescharacteristic
64C2×C4×D4C2×C412yes—
65C2×C4⋊C8—22no—
66C2×C4⋊C8—22no—
67C2×C4⋊C8—22no—
68C2×#97—22no—
69C2×C4×C8—22no—
70#400—21no—
71#400—21no—
72#400—22no—
73#400—22no—
74#397—21no—
75#397—21no—
76#397—22no—
77#397—22no—
78#391—22no—
79#391—22no—
80#391—22no—
81#391—22no—
82#391—22no—
83#391—22no—
84#391C2×C412yes—
85#391C2×C412yes—
86#391C2×C412yes—
87#391C2311yescharacteristic
88#391C2312yes—
89#390C2311yescharacteristic
90#388—22no—
91#388—22no—
92#388—22no—
93#382—22no—
94#382—22no—
95#382—22no—
96#382C2×C412yes—
97#382C2×C411yescharacteristic
98#380—22no—
99#380—22no—
100#380—22no—
101C4×C4⋊C4C2×C412yes—
102C4×C4⋊C4C2×C411yescharacteristic
103C4×C4⋊C4D412yes—
104C4×C4⋊C4D412yes—
105C4×C4⋊C4D412yes—
106C4×C22⋊C4—22no—
107C4×C22⋊C4—22no—
108C4×C22⋊C4—22no—
109C43C2311yescharacteristic
110C43C2312yes—
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×Q8—44no—
125C22×Q8—48no—
126C22×Q8C22⋊C411yescharacteristic
127C22×Q8C22×C412yes—
128C22×Q8C22×C411yescharacteristic
129C22×D4C22×C411yescharacteristic
130C23×C4—22no—
131C23×C4—21no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C4⋊Q8—42no—
137C4⋊Q8—44no—
138C4⋊Q8—44no—
139C4⋊Q8—42no—
140#125—48no—
141#125—44no—
142#124—24no—
143#124—22no—
144#124—42no—
145#124—41no—
146C22⋊Q8—48no—
147C22⋊Q8—44no—
148C4×Q8—24no—
149C4×Q8—44no—
150C4×Q8—44no—
151C4×Q8—42no—
152C4×Q8—44no—
153C4×D4—42no—
154C4×D4—44no—
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⋊C4C22⋊C412yes—
164C2×C4⋊C4C22⋊C412yes—
165C2×C4⋊C4C22⋊C412yes—
166C2×C4⋊C4C22⋊C412yes—
167C2×C4⋊C4C22⋊C411yescharacteristic
168C2×C4⋊C4C22×C411yescharacteristic
169C2×C4⋊C4C22×C412yes—
170C2×C4⋊C4C22×C411yescharacteristic
171C2×C22⋊C4—21no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C42—22no—
178C2×C42—21no—
179C2×C42C22×C411yescharacteristic
180C2×C42C22×C412yes—
181C2×C42C22×C411yescharacteristic
182C2×C42C22×C412yes—
183C2×C42C2×D412yes—
184C2×C42C2×D412yes—
185C2×C42C2×D412yes—
186C2×C42C2×D412yes—
187C2×C42C2×D412yes—
188C2×C42C2×D412yes—
189C2×C42C2411yescharacteristic
190C4⋊C8—44no—
191C4⋊C8—82no—
192C4⋊C8—82no—
193#97—44no—
194C4×C8—82no—
195#95—22no—
196#95—22no—
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⋊C411yescharacteristic
209#95C22⋊C412yes—
210#95C22⋊C412yes—
211C24—21no—
212C2×Q8—22no—
213C2×Q8—24no—
214C2×Q8—22no—
215C2×Q8—21no—
216C2×Q8—22no—
217C2×Q8—21no—
218C2×Q8—44no—
219C2×Q8—44no—
220C2×Q8—44no—
221C2×Q8—42no—
222C2×Q8—41no—
223C2×Q8—42no—
224C2×Q8—41no—
225C2×Q8—42no—
226C2×Q8—44no—
227C2×Q8—816no—
228C2×D4—21no—
229C2×D4—41no—
230C2×D4—41no—
231C2×D4#13112yes—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—21no—
235C22×C4—22no—
236C22×C4—21no—
237C22×C4—44no—
238C22×C4—44no—
239C22×C4—44no—
240C22×C4—42no—
241C22×C4—44no—
242C22×C4C2×C22⋊C412yes—
243C22×C4C2×C22⋊C412yes—
244C22×C4C2×C22⋊C412yes—
245C22×C4C2×C22⋊C411yescharacteristic
246C22×C4C2×C22⋊C412yes—
247C22×C4C2×C22⋊C411yescharacteristic
248C22×C4C2×C22⋊C412yes—
249C22×C4C23×C411yescharacteristic
250C22×C4C22×D412yes—
251C2×C8—42no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—42no—
259C2×C8—42no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—42no—
263C4⋊C4—24no—
264C4⋊C4—24no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—41no—
268C4⋊C4—42no—
269C4⋊C4—41no—
270C4⋊C4—42no—
271C4⋊C4—44no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—44no—
276C4⋊C4—44no—
277C4⋊C4—44no—
278C4⋊C4#10014yes—
279C22⋊C4—44no—
280C22⋊C4—44no—
281C22⋊C4—44no—
282C22⋊C4—42no—
283C22⋊C4—42no—
284C22⋊C4—42no—
285C22⋊C4—44no—
286C42—21no—
287C42—22no—
288C42—21no—
289C42—22no—
290C42—22no—
291C42—22no—
292C42—22no—
293C42—22no—
294C42—22no—
295C42—22no—
296C42—24no—
297C42—22no—
298C42—24no—
299C42—42no—
300C42—42no—
301C42—41no—
302C42—44no—
303C42—42no—
304C23—22no—
305C23—41no—
306C23—41no—
307C23—42no—
308C23—42no—
309C23C22×C22⋊C411yescharacteristic
310Q8—28no—
311Q8—82no—
312Q8—81no—
313Q8—82no—
314Q8—81no—
315Q8—84no—
316Q8—88no—
317D4—42no—
318D4—42no—
319C2×C4—22no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—21no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—42no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—84no—
353C2×C4—88no—
354C2×C4C2×#10012yes—
355C2×C4C2×#10014yes—
356C2×C4#42114yes—
357C2×C4C2×#13111yescharacteristic
358C2×C4#56811yescharacteristic
359C2×C4#57712yes—
360C2×C4#57812yes—
361C8—82no—
362C8—82no—
363C8—82no—
364C8—82no—
365C22—42no—
366C22—44no—
367C22—81no—
368C22—81no—
369C22—82no—
370C22#288011yescharacteristic
371C22#288411yescharacteristic
372C22C22×#10011yescharacteristic
373C22#289812yes—
374C22C2×#42112yes—
375C4—22no—
376C4—24no—
377C4—21no—
378C4—21no—
379C4—24no—
380C4—22no—
381C4—22no—
382C4—42no—
383C4—42no—
384C4—42no—
385C4—41no—
386C4—81no—
387C4—82no—
388C4—82no—
389C4—81no—
390C4—81no—
391C2—82no—
392C2#1742511yescharacteristic
393C2#1752312yes—
394C2#1760112yes—
395C2#1766611yescharacteristic
396C2#1777911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C22×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22020C2semidirect
3. #7722C2semidirect
4. #7702C2semidirect
5. #7662C2semidirect

Maximal subgroups

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

subgroupindex
#364462
#220202
#77222
#77022
#76622
#76492
#76182
#76012
#75842