Finite Group Databasebeta

← #207463

Subgroups — #207463

Order 512.

884 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34621C211yesmaximal, characteristic
2#21431C211yesmaximal, characteristic, complemented
3#20543C211yesmaximal, characteristic, complemented
4#9866C212yesmaximal, complemented
5#9674C212yesmaximal, complemented
6#8940C212yesmaximal, complemented
7#8939C212yesmaximal, complemented
8#8183C212yesmaximal, complemented
9#8181C212yesmaximal, complemented
10C2×#431C2211yescharacteristic
11C22×C4⋊C8C2211yescharacteristic
12#2651—22no—
13#2648—22no—
14#2647C411yescharacteristic
15#2644C412yes—
16#2641C411yescharacteristic
17#2629C411yescharacteristic
18#2624C412yes—
19#2619C411yescharacteristic
20#2395—22no—
21#2392—22no—
22C4×#127C2211yescharacteristic
23C4×#125C2211yescharacteristic
24C4×#121C2212yes—
25#2293C2212yes—
26C4×#117C2211yescharacteristic
27#1924—22no—
28#1917—22no—
29C8×C4⋊C4—22no—
30C4×C4⋊C8C2212yes—
31C4×C4⋊C8C2212yes—
32C4×C4⋊C8C2212yes—
33C8×C22⋊C4—22no—
34C4×C22⋊C8C2212yes—
35#1744C2211yescharacteristic
36C4×#97C2211yescharacteristic
37C2×D4⋊C8C2212yes—
38C2×D4⋊C8C2212yes—
39C2×D4⋊C8C2212yes—
40C2×D4⋊C8C2212yes—
41#1279C2212yes—
42#1279C2212yes—
43#1279C2212yes—
44#1279C2212yes—
45C23×C8—21no—
46C2×#127C2×C411yescharacteristic
47C2×#125C2×C411yescharacteristic
48C2×#121C2×C412yes—
49C2×C4×D4—22no—
50C2×C4×D4—22no—
51C2×C4×D4C2312yes—
52C2×#117—22no—
53C2×#117C2×C411yescharacteristic
54C2×#117C2×C411yescharacteristic
55C22×C42C2311yescharacteristic
56#431—24no—
57C2×C4⋊C8—21no—
58C2×C4⋊C8—24no—
59C2×C4⋊C8—22no—
60C2×C4⋊C8C2312yes—
61C2×C4⋊C8C2312yes—
62C2×C22⋊C8—21no—
63C2×C22⋊C8—22no—
64C2×#97—22no—
65C2×C4×C8—21no—
66C2×C4×C8—22no—
67C2×C4×C8—22no—
68C2×C4×C8C2312yes—
69C2×C4×C8C2312yes—
70#400—22no—
71#399—22no—
72#398—22no—
73#398—22no—
74#397—22no—
75#397—22no—
76#397—22no—
77#397—22no—
78#396—22no—
79#396—22no—
80#394—22no—
81#394—22no—
82#394—22no—
83#394—22no—
84#394—22no—
85#392—22no—
86#390—22no—
87#390C2×C412yes—
88#390C2×C412yes—
89#390C2×C412yes—
90#390C2×C412yes—
91#389C2×C412yes—
92#389C2×C412yes—
93#389C2×C412yes—
94#389C2×C412yes—
95#388—22no—
96#388—22no—
97#388—22no—
98#387—22no—
99#386—22no—
100#384C2×C411yescharacteristic
101#382C2×C411yescharacteristic
102#380C2×C412yes—
103C4×C4⋊C4—22no—
104C4×C4⋊C4D412yes—
105C4×C4⋊C4D412yes—
106C4×C4⋊C4C2312yes—
107C4×C22⋊C4—22no—
108C4×C22⋊C4—22no—
109C4×C22⋊C4D412yes—
110C4×C22⋊C4D412yes—
111#376C2×C411yescharacteristic
112#376C2×C411yescharacteristic
113C43C2311yescharacteristic
114C43C2311yescharacteristic
115#336—22no—
116#336—22no—
117#336—22no—
118#336—22no—
119#336—22no—
120D4⋊C8—24no—
121D4⋊C8—24no—
122D4⋊C8—24no—
123D4⋊C8—24no—
124C22×D4—22no—
125C22×D4—22no—
126C22×D4C22×C412yes—
127C23×C4—22no—
128C23×C4C22×C411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—24no—
135C22×C8—22no—
136C22×C8—82no—
137#127—24no—
138#125—24no—
139#121—28no—
140C4×D4—22no—
141C4×D4—22no—
142C4×D4—22no—
143C4×D4—22no—
144C4×D4—82no—
145C4×D4—82no—
146C4×D4—82no—
147C4×D4—82no—
148#117—24no—
149#117—24no—
150#117—82no—
151#117—82no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4C22⋊C412yes—
159C2×C4⋊C4C22⋊C412yes—
160C2×C4⋊C4C22×C412yes—
161C2×C4⋊C4C22×C412yes—
162C2×C4⋊C4C22×C412yes—
163C2×C22⋊C4—22no—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4C22⋊C412yes—
175C2×C22⋊C4C22⋊C412yes—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42C22×C411yescharacteristic
180C2×C42C22×C411yescharacteristic
181C2×C42C22×C411yescharacteristic
182C2×C42C22×C411yescharacteristic
183C2×C42C22×C411yescharacteristic
184C2×C42C2×D411yescharacteristic
185C2×C42C2×D412yes—
186C2×C42C2×D412yes—
187C2×C42C2×D411yescharacteristic
188C2×C42C2×D412yes—
189C2×C42C2×D412yes—
190C2×C42C2411yescharacteristic
191C4⋊C8—24no—
192C4⋊C8—22no—
193C4⋊C8—22no—
194C4⋊C8—44no—
195C4⋊C8#4114yes—
196C4⋊C8#4114yes—
197C22⋊C8—24no—
198C22⋊C8—44no—
199#97—44no—
200C4×C8—24no—
201C4×C8—22no—
202C4×C8—22no—
203C4×C8—22no—
204C4×C8—22no—
205C4×C8—82no—
206C4×C8—82no—
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⋊C412yes—
217#95C22⋊C412yes—
218C24—22no—
219C24C2×C22⋊C411yescharacteristic
220C2×D4—22no—
221C2×D4—22no—
222C2×D4—24no—
223C2×D4—24no—
224C2×D4—44no—
225C2×D4—44no—
226C2×D4—82no—
227C2×D4—82no—
228C2×D4—82no—
229C2×D4—82no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4C2×C22⋊C411yescharacteristic
248C22×C4C2×C22⋊C412yes—
249C22×C4C2×C22⋊C412yes—
250C22×C4C2×C22⋊C412yes—
251C22×C4C2×C22⋊C411yescharacteristic
252C22×C4C2×C22⋊C411yescharacteristic
253C22×C4C2×C22⋊C412yes—
254C22×C4#13114yes—
255C22×C4C23×C411yescharacteristic
256C22×C4C22×D411yescharacteristic
257C22×C4C22×D411yescharacteristic
258C2×C8—24no—
259C2×C8—24no—
260C2×C8—24no—
261C2×C8—42no—
262C2×C8—42no—
263C2×C8—42no—
264C2×C8—42no—
265C2×C8—42no—
266C2×C8—42no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—84no—
271C4⋊C4—24no—
272C4⋊C4—24no—
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4—24no—
276C4⋊C4—24no—
277C4⋊C4—44no—
278C4⋊C4—82no—
279C4⋊C4—82no—
280C4⋊C4—82no—
281C4⋊C4—82no—
282C4⋊C4—82no—
283C22⋊C4—24no—
284C22⋊C4—24no—
285C22⋊C4—42no—
286C22⋊C4—42no—
287C22⋊C4—44no—
288C22⋊C4—82no—
289C22⋊C4—82no—
290C22⋊C4—82no—
291C22⋊C4—82no—
292C22⋊C4—82no—
293C22⋊C4—82no—
294C22⋊C4—82no—
295C22⋊C4—82no—
296C22⋊C4—82no—
297C42—22no—
298C42—21no—
299C42—21no—
300C42—22no—
301C42—22no—
302C42—22no—
303C42—22no—
304C42—22no—
305C42—22no—
306C42—22no—
307C42—22no—
308C42—42no—
309C42—42no—
310C42—42no—
311C42—42no—
312C42#13114yes—
313C42C2×#4112yes—
314C42C2×#4112yes—
315C23—21no—
316C23—22no—
317C23—22no—
318C23—21no—
319C23—42no—
320C23—42no—
321C23—42no—
322C23—42no—
323C23—42no—
324C23—42no—
325C23—42no—
326C23#40812yes—
327C23C22×C22⋊C411yescharacteristic
328D4—44no—
329D4—44no—
330D4—88no—
331D4—88no—
332C2×C4—21no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—21no—
337C2×C4—28no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—42no—
352C2×C4—41no—
353C2×C4—41no—
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—82no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4#40814yes—
369C2×C4#40812yes—
370C2×C4C4×#4112yes—
371C2×C4#52511yescharacteristic
372C2×C4#52511yescharacteristic
373C2×C4#53212yes—
374C2×C4C2×#13112yes—
375C2×C4C2×#13112yes—
376C2×C4#57612yes—
377C2×C4#57712yes—
378C8—44no—
379C8—44no—
380C8—44no—
381C8—82no—
382C22—24no—
383C22—22no—
384C22—42no—
385C22—41no—
386C22—41no—
387C22—82no—
388C22—82no—
389C22—82no—
390C22—82no—
391C22—82no—
392C22—82no—
393C22—82no—
394C22#229711yescharacteristic
395C22C2×#40811yescharacteristic
396C22C2×#40811yescharacteristic
397C22#289712yes—
398C22#290112yes—
399C4—24no—
400C4—22no—
401C4—22no—
402C4—42no—
403C4—42no—
404C4—42no—
405C4—41no—
406C4—41no—
407C4—82no—
408C4—82no—
409C4—82no—
410C4#296614yes—
411C4#311912yes—
412C4#312211yescharacteristic
413C4#312611yescharacteristic
414C2—42no—
415C2—162no—
416C2#1746812yes—
417C2#1746812yes—
418C2#2137511yescharacteristic
419C2#2138411yescharacteristic
420C2#2139911yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radical#2619C4
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
2. #21431C2semidirect
3. #20543C2semidirect
4. #9866C2semidirect
5. #9674C2semidirect
6. #8940C2semidirect
7. #8939C2semidirect
8. #8183C2semidirect
9. #8181C2semidirect

Maximal subgroups

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

subgroupindex
#346212
#214312
#205432
#98662
#96742
#89402
#89392
#81832
#81812