Finite Group Databasebeta

← #216387

Subgroups — #216387

Order 512.

745 conjugacy classes of subgroups.

The subgroup lattice has 399 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34629C211yesmaximal, characteristic
2#21922C211yesmaximal, characteristic
3#21350C211yesmaximal, characteristic
4#9866C212yesmaximal, complemented
5#9673C212yesmaximal, complemented
6#9077C212yesmaximal
7#9075C212yesmaximal
8#8570C212yesmaximal, complemented
9#8568C212yesmaximal, complemented
10C2×#431C2211yescharacteristic
11C2×C4×M16C2211yescharacteristic
12#2655—22no—
13#2652—22no—
14#2647C411yescharacteristic
15#2645C412yes—
16#2642C411yescharacteristic
17#2629C411yescharacteristic
18#2625C412yes—
19#2621C411yescharacteristic
20#2402—22no—
21#2401—22no—
22C4×C4⋊Q8C2211yescharacteristic
23C4×#125C2211yescharacteristic
24C4×C22⋊Q8C2212yes—
25#2295C2212yes—
26C4×#117C2211yescharacteristic
27#1940—22no—
28#1923—22no—
29#1916—22no—
30#1851—22no—
31#1842C2211yescharacteristic
32#1831C2211yescharacteristic
33#1797C2212yes—
34C4×C4⋊C8C2212yes—
35C4×C4⋊C8C2212yes—
36C4×C4⋊C8C2212yes—
37C2×Q8⋊C8C2212yes—
38C2×Q8⋊C8C2212yes—
39C2×Q8⋊C8C2212yes—
40C2×Q8⋊C8C2212yes—
41#1279C2212yes—
42#1279C2212yes—
43#1279C2212yes—
44#1279C2212yes—
45C22×M16—21no—
46C2×C4⋊Q8C2×C411yescharacteristic
47C2×#125C2×C411yescharacteristic
48C2×C22⋊Q8C2×C412yes—
49C2×C4×Q8C2312yes—
50C2×#117C2×C411yescharacteristic
51C2×#117C2×C411yescharacteristic
52C22×C42C2311yescharacteristic
53#431—41no—
54#431—41no—
55C2×C4⋊C8—22no—
56C2×C4⋊C8—22no—
57C2×C4⋊C8—21no—
58C2×C4⋊C8—22no—
59C2×C4⋊C8C2312yes—
60C2×C4⋊C8C2312yes—
61C2×C22⋊C8—21no—
62C2×C22⋊C8—22no—
63C4×M16—41no—
64C4×M16—41no—
65C2×#97—21no—
66C2×C4×C8—22no—
67C2×C4×C8C2312yes—
68C2×C4×C8C2312yes—
69#401—22no—
70#400—22no—
71#400—22no—
72#400—22no—
73#400—22no—
74#400—22no—
75#400—22no—
76#400—22no—
77#398—22no—
78#398—22no—
79#398—22no—
80#398—22no—
81#397—22no—
82#397—22no—
83#397—22no—
84#396—22no—
85#391C2×C412yes—
86#391C2×C412yes—
87#391C2×C412yes—
88#391C2×C412yes—
89#389—22no—
90#389C2×C412yes—
91#389C2×C412yes—
92#389C2×C412yes—
93#389C2×C412yes—
94#387—22no—
95#387—22no—
96#387—22no—
97#387—22no—
98#385—22no—
99#384C2×C411yescharacteristic
100#382C2×C411yescharacteristic
101#380C2×C412yes—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4D412yes—
107C4×C4⋊C4D412yes—
108C4×C4⋊C4C2312yes—
109C4×C22⋊C4—22no—
110C4×C22⋊C4D412yes—
111C4×C22⋊C4D412yes—
112#376—22no—
113#376C2×C411yescharacteristic
114#376C2×C411yescharacteristic
115C43C2311yescharacteristic
116C43C2311yescharacteristic
117#336—22no—
118#336—22no—
119#336—22no—
120#336—22no—
121#336—22no—
122Q8⋊C8—42no—
123Q8⋊C8—42no—
124Q8⋊C8—42no—
125Q8⋊C8—42no—
126C22×Q8C22×C412yes—
127C23×C4C22×C411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C2×M16—44no—
131C2×M16—42no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C4⋊Q8—24no—
137#125—24no—
138C22⋊Q8—44no—
139C4×Q8—22no—
140C4×Q8—22no—
141C4×Q8—22no—
142C4×Q8—22no—
143#117—24no—
144#117—42no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4C22⋊C412yes—
155C2×C4⋊C4C22⋊C412yes—
156C2×C4⋊C4C22×C412yes—
157C2×C4⋊C4C22×C412yes—
158C2×C4⋊C4C22×C412yes—
159C2×C22⋊C4—22no—
160C2×C22⋊C4—22no—
161C2×C22⋊C4C22⋊C412yes—
162C2×C22⋊C4C22⋊C412yes—
163C2×C42—21no—
164C2×C42—21no—
165C2×C42—21no—
166C2×C42—21no—
167C2×C42—22no—
168C2×C42—22no—
169C2×C42C22×C411yescharacteristic
170C2×C42C22×C411yescharacteristic
171C2×C42C22×C411yescharacteristic
172C2×C42C22×C411yescharacteristic
173C2×C42C22×C411yescharacteristic
174C2×C42C2×D411yescharacteristic
175C2×C42C2×D411yescharacteristic
176C2×C42C2×D412yes—
177C2×C42C2×D412yes—
178C2×C42C2×D412yes—
179C2×C42C2×D412yes—
180C2×C42C2411yescharacteristic
181C4⋊C8—22no—
182C4⋊C8—22no—
183C4⋊C8—22no—
184C4⋊C8—22no—
185C4⋊C8—42no—
186C4⋊C8—44no—
187C4⋊C8—44no—
188C4⋊C8—82no—
189C22⋊C8—42no—
190C22⋊C8—82no—
191#97—42no—
192C4×C8—22no—
193C4×C8—22no—
194C4×C8—22no—
195C4×C8—22no—
196C4×C8—82no—
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#95—22no—
208#95—22no—
209#95—22no—
210#95—22no—
211#95—22no—
212#95C22⋊C412yes—
213#95C22⋊C412yes—
214#95C22⋊C412yes—
215#95C22⋊C412yes—
216C24C2×C22⋊C411yescharacteristic
217C2×Q8—22no—
218C2×Q8—22no—
219C2×Q8—24no—
220C2×Q8—24no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—42no—
236C22×C4C2×C22⋊C411yescharacteristic
237C22×C4C2×C22⋊C412yes—
238C22×C4C2×C22⋊C412yes—
239C22×C4C2×C22⋊C412yes—
240C22×C4C2×C22⋊C411yescharacteristic
241C22×C4C2×C22⋊C411yescharacteristic
242C22×C4C2×C22⋊C412yes—
243C22×C4C23×C411yescharacteristic
244C22×C4C22×D411yescharacteristic
245C22×C4C22×D411yescharacteristic
246M16—48no—
247C2×C8—24no—
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—
257C2×C8—42no—
258C2×C8—42no—
259C4⋊C4—24no—
260C4⋊C4—22no—
261C4⋊C4—22no—
262C4⋊C4—24no—
263C4⋊C4—24no—
264C4⋊C4—42no—
265C4⋊C4—44no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—44no—
269C4⋊C4—44no—
270C4⋊C4—82no—
271C4⋊C4—82no—
272C4⋊C4—82no—
273C4⋊C4—82no—
274C4⋊C4—82no—
275C22⋊C4—24no—
276C22⋊C4—42no—
277C22⋊C4—44no—
278C22⋊C4—82no—
279C42—21no—
280C42—21no—
281C42—21no—
282C42—21no—
283C42—21no—
284C42—21no—
285C42—22no—
286C42—22no—
287C42—22no—
288C42—22no—
289C42—22no—
290C42—22no—
291C42—22no—
292C42—42no—
293C42—42no—
294C42—42no—
295C42—42no—
296C42—42no—
297C42—42no—
298C42#13114yes—
299C42H(5,𝔽2)11yescharacteristic
300C42H(5,𝔽2)11yescharacteristic
301C422−1+411yescharacteristic
302C422−1+411yescharacteristic
303C23—21no—
304C23—21no—
305C23—21no—
306C23—22no—
307C23—22no—
308C23C22×C22⋊C411yescharacteristic
309Q8—44no—
310Q8—44no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—22no—
318C2×C4—21no—
319C2×C4—22no—
320C2×C4—22no—
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—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4#40814yes—
350C2×C4#52011yescharacteristic
351C2×C4#52011yescharacteristic
352C2×C4#53511yescharacteristic
353C2×C4#53611yescharacteristic
354C2×C4#53711yescharacteristic
355C2×C4#53711yescharacteristic
356C2×C4C2×#13112yes—
357C2×C4#56812yes—
358C2×C4#57712yes—
359C2×C4#57812yes—
360C8—44no—
361C8—82no—
362C8—82no—
363C8—82no—
364C22—42no—
365C22—41no—
366C22—42no—
367C22—41no—
368C22—41no—
369C22#231111yescharacteristic
370C22C2×#40811yescharacteristic
371C22#287911yescharacteristic
372C22#289812yes—
373C22#290112yes—
374C4—24no—
375C4—21no—
376C4—21no—
377C4—42no—
378C4—41no—
379C4—42no—
380C4—42no—
381C4—41no—
382C4—41no—
383C4—82no—
384C4—82no—
385C4—82no—
386C4—82no—
387C4#299312yes—
388C4#299312yes—
389C4#325812yes—
390C4#326011yescharacteristic
391C4#326111yescharacteristic
392C2—81no—
393C2#1746912yes—
394C2#1750612yes—
395C2#2185011yescharacteristic
396C2#2189811yescharacteristic
397C2#2190111yescharacteristic
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. #9866C2semidirect
5. #9673C2semidirect
8. #8570C2semidirect
9. #8568C2semidirect

Maximal subgroups

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

subgroupindex
#346292
#219222
#213502
#98662
#96732
#90772
#90752
#85702
#85682