Finite Group Databasebeta

← #207598

Subgroups — #207598

Order 512.

761 conjugacy classes of subgroups.

The subgroup lattice has 415 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34629C211yesmaximal, characteristic
2#21926C211yesmaximal, characteristic
3#20472C211yesmaximal, characteristic
4#9876C211yesmaximal, characteristic, complemented
5#9875C211yesmaximal, characteristic, complemented
6#9647C211yesmaximal, characteristic, complemented
7#9643C211yesmaximal, characteristic, complemented
8#9077C212yesmaximal
9#9075C212yesmaximal
10#8178C212yesmaximal, complemented
11#8176C212yesmaximal, complemented
12C2×#431C2211yescharacteristic
13C2×C4×M16C2211yescharacteristic
14#2655—22no—
15#2652—22no—
16#2647C411yescharacteristic
17#2645C412yes—
18#2642C411yescharacteristic
19#2629C411yescharacteristic
20#2625C412yes—
21#2621C411yescharacteristic
22#2402—22no—
23#2401—22no—
24C4×C4⋊Q8C2211yescharacteristic
25C4×#125C2211yescharacteristic
26C4×C22⋊Q8C2212yes—
27#2295C2212yes—
28C4×#117C2211yescharacteristic
29#1924—22no—
30#1917—22no—
31#1842C2211yescharacteristic
32#1831C2211yescharacteristic
33#1797C2212yes—
34#1771—22no—
35#1767C2212yes—
36#1753—22no—
37#1744C2211yescharacteristic
38#1744C2211yescharacteristic
39C4×#97C2211yescharacteristic
40C42×C8C2211yescharacteristic
41C2×Q8⋊C8C2212yes—
42C2×Q8⋊C8C2212yes—
43C2×Q8⋊C8C2212yes—
44C2×Q8⋊C8C2212yes—
45#1279C2212yes—
46#1279C2212yes—
47#1279C2212yes—
48#1279C2212yes—
49C22×M16—21no—
50C2×C4⋊Q8C2×C411yescharacteristic
51C2×#125C2×C411yescharacteristic
52C2×C22⋊Q8C2×C412yes—
53C2×C4×Q8C2312yes—
54C2×#117C2×C411yescharacteristic
55C2×#117C2×C411yescharacteristic
56C22×C42C2311yescharacteristic
57#431—22no—
58#431—22no—
59C2×C4⋊C8—21no—
60C2×C4⋊C8—22no—
61C2×C4⋊C8C2312yes—
62C2×C4⋊C8C2312yes—
63C2×C22⋊C8—21no—
64C2×C22⋊C8—22no—
65C4×M16—22no—
66C4×M16—22no—
67C2×#97—21no—
68C2×#97—22no—
69C2×#97—22no—
70C2×C4×C8—22no—
71C2×C4×C8C2311yescharacteristic
72C2×C4×C8C2311yescharacteristic
73C2×C4×C8C2311yescharacteristic
74C2×C4×C8C2311yescharacteristic
75#401—22no—
76#400—22no—
77#400—22no—
78#400—22no—
79#400—22no—
80#400—22no—
81#400—22no—
82#400—22no—
83#398—22no—
84#398—22no—
85#398—22no—
86#398—22no—
87#397—22no—
88#397—22no—
89#397—22no—
90#396—22no—
91#391C2×C412yes—
92#391C2×C412yes—
93#391C2×C412yes—
94#391C2×C412yes—
95#389—22no—
96#389C2×C412yes—
97#389C2×C412yes—
98#389C2×C412yes—
99#389C2×C412yes—
100#387—22no—
101#387—22no—
102#387—22no—
103#387—22no—
104#385—22no—
105#384C2×C411yescharacteristic
106#382C2×C411yescharacteristic
107#380C2×C412yes—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4D412yes—
113C4×C4⋊C4D412yes—
114C4×C4⋊C4C2312yes—
115C4×C22⋊C4—22no—
116C4×C22⋊C4D412yes—
117C4×C22⋊C4D412yes—
118#376—22no—
119#376C2×C411yescharacteristic
120#376C2×C411yescharacteristic
121C43C2311yescharacteristic
122C43C2311yescharacteristic
123#336—22no—
124#336—22no—
125#336—22no—
126#336—22no—
127#336—22no—
128Q8⋊C8—24no—
129Q8⋊C8—24no—
130Q8⋊C8—24no—
131Q8⋊C8—24no—
132C22×Q8C22×C412yes—
133C23×C4C22×C411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C2×M16—24no—
137C2×M16—44no—
138C22×C8—21no—
139C22×C8—21no—
140C22×C8—21no—
141C22×C8—21no—
142C22×C8—22no—
143C22×C8—22no—
144C4⋊Q8—42no—
145#125—42no—
146C22⋊Q8—44no—
147C4×Q8—22no—
148C4×Q8—22no—
149C4×Q8—22no—
150C4×Q8—22no—
151#117—42no—
152#117—42no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4C22⋊C412yes—
163C2×C4⋊C4C22⋊C412yes—
164C2×C4⋊C4C22×C412yes—
165C2×C4⋊C4C22×C412yes—
166C2×C4⋊C4C22×C412yes—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4C22⋊C412yes—
170C2×C22⋊C4C22⋊C412yes—
171C2×C42—21no—
172C2×C42—21no—
173C2×C42—21no—
174C2×C42—21no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42C22×C411yescharacteristic
178C2×C42C22×C411yescharacteristic
179C2×C42C22×C411yescharacteristic
180C2×C42C22×C411yescharacteristic
181C2×C42C22×C411yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C42C2×D411yescharacteristic
184C2×C42C2×D412yes—
185C2×C42C2×D412yes—
186C2×C42C2×D412yes—
187C2×C42C2×D412yes—
188C2×C42C2411yescharacteristic
189C4⋊C8—24no—
190C4⋊C8—22no—
191C4⋊C8—22no—
192C4⋊C8—22no—
193C4⋊C8—22no—
194C4⋊C8—82no—
195C22⋊C8—24no—
196C22⋊C8—82no—
197#97—24no—
198#97—44no—
199#97—82no—
200C4×C8—21no—
201C4×C8—21no—
202C4×C8—21no—
203C4×C8—21no—
204C4×C8—44no—
205C4×C8#4112yes—
206C4×C8#4112yes—
207C4×C8#4112yes—
208C4×C8#4112yes—
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#95—22no—
219#95—22no—
220#95—22no—
221#95—22no—
222#95—22no—
223#95—22no—
224#95C22⋊C412yes—
225#95C22⋊C412yes—
226#95C22⋊C412yes—
227#95C22⋊C412yes—
228C24C2×C22⋊C411yescharacteristic
229C2×Q8—22no—
230C2×Q8—22no—
231C2×Q8—24no—
232C2×Q8—24no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—42no—
248C22×C4C2×C22⋊C411yescharacteristic
249C22×C4C2×C22⋊C412yes—
250C22×C4C2×C22⋊C412yes—
251C22×C4C2×C22⋊C412yes—
252C22×C4C2×C22⋊C411yescharacteristic
253C22×C4C2×C22⋊C411yescharacteristic
254C22×C4C2×C22⋊C412yes—
255C22×C4C23×C411yescharacteristic
256C22×C4C22×D411yescharacteristic
257C22×C4C22×D411yescharacteristic
258M16—48no—
259C2×C8—24no—
260C2×C8—22no—
261C2×C8—22no—
262C2×C8—22no—
263C2×C8—22no—
264C2×C8—42no—
265C2×C8—42no—
266C2×C8—42no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—42no—
271C2×C8—42no—
272C2×C8—42no—
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4—24no—
276C4⋊C4—24no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—44no—
280C4⋊C4—44no—
281C4⋊C4—44no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—82no—
285C4⋊C4—82no—
286C4⋊C4—82no—
287C4⋊C4—82no—
288C4⋊C4—82no—
289C22⋊C4—42no—
290C22⋊C4—42no—
291C22⋊C4—44no—
292C22⋊C4—82no—
293C42—21no—
294C42—21no—
295C42—21no—
296C42—21no—
297C42—21no—
298C42—21no—
299C42—22no—
300C42—22no—
301C42—22no—
302C42—22no—
303C42—22no—
304C42—22no—
305C42—22no—
306C42—22no—
307C42—42no—
308C42—42no—
309C42—42no—
310C42—42no—
311C42—42no—
312C42—42no—
313C42C2×#4111yescharacteristic
314C42C2×#4111yescharacteristic
315C42C2×#4111yescharacteristic
316C42C2×#4111yescharacteristic
317C23—21no—
318C23—21no—
319C23—22no—
320C23—22no—
321C23—21no—
322C23C22×C22⋊C411yescharacteristic
323Q8—44no—
324Q8—44no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—21no—
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—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4C4×#4111yescharacteristic
365C2×C4C4×#4111yescharacteristic
366C2×C4#52511yescharacteristic
367C2×C4#52511yescharacteristic
368C2×C4#53211yescharacteristic
369C2×C4#53211yescharacteristic
370C2×C4#56812yes—
371C2×C4#56812yes—
372C2×C4#57712yes—
373C2×C4#57812yes—
374C8—24no—
375C8—42no—
376C8—42no—
377C8—42no—
378C8—82no—
379C8—82no—
380C22—42no—
381C22—42no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22#229711yescharacteristic
386C22#287911yescharacteristic
387C22#287911yescharacteristic
388C22#289812yes—
389C22#290112yes—
390C4—21no—
391C4—21no—
392C4—42no—
393C4—42no—
394C4—42no—
395C4—42no—
396C4—41no—
397C4—41no—
398C4—41no—
399C4—82no—
400C4—82no—
401C4—82no—
402C4—82no—
403C4#296712yes—
404C4#296712yes—
405C4#317411yescharacteristic
406C4#317711yescharacteristic
407C4#317812yes—
408C2—81no—
409C2#1750612yes—
410C2#1750612yes—
411C2#2137611yescharacteristic
412C2#2141311yescharacteristic
413C2#2141911yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9876C2semidirect
5. #9875C2semidirect
6. #9647C2semidirect
7. #9643C2semidirect
10. #8178C2semidirect
11. #8176C2semidirect

Maximal subgroups

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

subgroupindex
#346292
#219262
#204722
#98762
#98752
#96472
#96432
#90772
#90752
#81782
#81762