Finite Group Databasebeta

← #190955

Subgroups — #190955

Order 512.

987 conjugacy classes of subgroups.

The subgroup lattice has 445 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34086C211yesmaximal, characteristic
2#21926C211yesmaximal, characteristic, complemented
3#21430C211yesmaximal, characteristic, complemented
4#8711C212yesmaximal, complemented
5#8565C212yesmaximal, complemented
6#8563C212yesmaximal, complemented
7#8313C212yesmaximal
8#8179C212yesmaximal, complemented
9#8177C212yesmaximal, complemented
10C2×#532C2211yescharacteristic
11#2337—22no—
12#2333—22no—
13#2331C411yescharacteristic
14#2331C411yescharacteristic
15#2328C411yescharacteristic
16#2328C411yescharacteristic
17#2324C412yes—
18#2324C412yes—
19#2297—22no—
20#2293C2212yes—
21#2290—22no—
22#2283C2211yescharacteristic
23C42×D4C2212yes—
24C4×#117C2211yescharacteristic
25#1924—22no—
26#1923—22no—
27#1917—22no—
28#1916—22no—
29#1842C2211yescharacteristic
30#1831C2211yescharacteristic
31#1797C2212yes—
32#1767C2212yes—
33C4×C4⋊C8C2212yes—
34C4×C22⋊C8C2212yes—
35#1744C2211yescharacteristic
36C42×C8C2211yescharacteristic
37C2×D4⋊C8C2212yes—
38C2×D4⋊C8C2212yes—
39C2×D4⋊C8C2212yes—
40C2×D4⋊C8C2212yes—
41#1279C2212yes—
42#1279C2212yes—
43#1279C2212yes—
44#1279C2212yes—
45C22×#41—21no—
46#532—24no—
47#532—24no—
48C2×C4⋊Q8C2×C411yescharacteristic
49C2×#127C2×C411yescharacteristic
50C2×#124—21no—
51C2×#121—21no—
52C2×#121C2×C412yes—
53C2×C4×D4—21no—
54C2×C4×D4C2×C412yes—
55C2×C4×D4C2×C412yes—
56C2×C4×D4C2312yes—
57C2×#117—22no—
58C2×#117C2×C411yescharacteristic
59C2×#117C2×C411yescharacteristic
60C22×C42—22no—
61C22×C42C2311yescharacteristic
62C2×C4⋊C8—22no—
63C2×C4⋊C8—22no—
64C2×C4⋊C8C2312yes—
65C2×C4⋊C8C2312yes—
66C2×C22⋊C8—22no—
67C2×C22⋊C8—22no—
68C2×#97—22no—
69C2×C4×C8—22no—
70C2×C4×C8C2312yes—
71C2×C4×C8C2312yes—
72#391—22no—
73#391—22no—
74#390—22no—
75#390—22no—
76#390C2×C412yes—
77#390C2×C412yes—
78#389C2×C412yes—
79#389C2×C412yes—
80#388—22no—
81#388—22no—
82#388—22no—
83#388—22no—
84#387—22no—
85#387—22no—
86#386—22no—
87#386—22no—
88#386—22no—
89#386—22no—
90#386—22no—
91#386—22no—
92#385—22no—
93#385—22no—
94#383—21no—
95#382C2×C411yescharacteristic
96#382C2×C411yescharacteristic
97#381—21no—
98#380C2×C412yes—
99C4×C4⋊C4C2×C412yes—
100C4×C4⋊C4C2×C412yes—
101C4×C4⋊C4D412yes—
102C4×C4⋊C4D412yes—
103C4×C4⋊C4C2312yes—
104C4×C22⋊C4—21no—
105C4×C22⋊C4—22no—
106C4×C22⋊C4—22no—
107C4×C22⋊C4—22no—
108C4×C22⋊C4—22no—
109C4×C22⋊C4D412yes—
110C4×C22⋊C4D412yes—
111#376—21no—
112#376C2×C411yescharacteristic
113#376C2×C411yescharacteristic
114C43C2311yescharacteristic
115C43C2311yescharacteristic
116#336—22no—
117#336—22no—
118#336—22no—
119#336—22no—
120#336—22no—
121#336—22no—
122D4⋊C8—24no—
123D4⋊C8—24no—
124D4⋊C8—24no—
125D4⋊C8—24no—
126C2×#41—24no—
127C2×#41—24no—
128C2×#41—82no—
129C2×#41—82no—
130C22×Q8—21no—
131C22×D4—22no—
132C22×D4—21no—
133C22×D4—21no—
134C22×D4C22×C412yes—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4C22×C411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C4⋊Q8—24no—
146#127—24no—
147#124—28no—
148#121—28no—
149#121—24no—
150#121—24no—
151C4×D4—24no—
152C4×D4—24no—
153C4×D4—28no—
154C4×D4—22no—
155C4×D4—22no—
156C4×D4—44no—
157C4×D4#4114yes—
158C4×D4#4114yes—
159#117—24no—
160#117—42no—
161#117—82no—
162#117—82no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4C22⋊C412yes—
167C2×C4⋊C4C22⋊C412yes—
168C2×C4⋊C4C22×C412yes—
169C2×C4⋊C4C22×C412yes—
170C2×C4⋊C4C22×C412yes—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4C22⋊C412yes—
181C2×C22⋊C4C22⋊C412yes—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—82no—
188C2×C42—82no—
189C2×C42C22×C411yescharacteristic
190C2×C42C22×C411yescharacteristic
191C2×C42C22×C411yescharacteristic
192C2×C42C22×C411yescharacteristic
193C2×C42C22×C411yescharacteristic
194C2×C42C2×D411yescharacteristic
195C2×C42C2×D412yes—
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D412yes—
198C2×C42C2×D412yes—
199C2×C42C2×D412yes—
200C2×C42C2411yescharacteristic
201C4⋊C8—22no—
202C4⋊C8—22no—
203C4⋊C8—22no—
204C4⋊C8—22no—
205C4⋊C8—44no—
206C4⋊C8—82no—
207C22⋊C8—44no—
208C22⋊C8—82no—
209#97—82no—
210C4×C8—22no—
211C4×C8—22no—
212C4×C8—22no—
213C4×C8—22no—
214C4×C8—44no—
215#95—22no—
216#95—21no—
217#95—21no—
218#95—22no—
219#95—22no—
220#95—21no—
221#95—21no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95C22⋊C412yes—
226#95C22⋊C412yes—
227#95C22⋊C412yes—
228#95C22⋊C412yes—
229C24—22no—
230C24C2×C22⋊C411yescharacteristic
231#41—88no—
232#41—84no—
233#41—84no—
234C2×Q8—24no—
235C2×Q8—82no—
236C2×D4—24no—
237C2×D4—28no—
238C2×D4—24no—
239C2×D4—22no—
240C2×D4—24no—
241C2×D4—24no—
242C2×D4—82no—
243C2×D4—82no—
244C2×D4—82no—
245C2×D4—82no—
246C2×D4#11714yes—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—42no—
261C22×C4—44no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—44no—
267C22×C4—82no—
268C22×C4—82no—
269C22×C4—82no—
270C22×C4—82no—
271C22×C4C2×C22⋊C411yescharacteristic
272C22×C4C2×C22⋊C411yescharacteristic
273C22×C4C2×C22⋊C411yescharacteristic
274C22×C4C2×C22⋊C412yes—
275C22×C4C2×C22⋊C412yes—
276C22×C4C2×C22⋊C412yes—
277C22×C4C2×C22⋊C412yes—
278C22×C4C23×C411yescharacteristic
279C22×C4C22×D411yescharacteristic
280C22×C4C22×D411yescharacteristic
281C2×C8—24no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—42no—
293C4⋊C4—24no—
294C4⋊C4—22no—
295C4⋊C4—24no—
296C4⋊C4—24no—
297C4⋊C4—24no—
298C4⋊C4—42no—
299C4⋊C4—82no—
300C4⋊C4—82no—
301C4⋊C4#11714yes—
302C22⋊C4—28no—
303C22⋊C4—24no—
304C22⋊C4—24no—
305C22⋊C4—24no—
306C22⋊C4—42no—
307C22⋊C4—44no—
308C22⋊C4—44no—
309C22⋊C4—82no—
310C22⋊C4—82no—
311C22⋊C4—82no—
312C22⋊C4—82no—
313C22⋊C4—82no—
314C42—22no—
315C42—21no—
316C42—21no—
317C42—22no—
318C42—22no—
319C42—22no—
320C42—22no—
321C42—22no—
322C42—22no—
323C42—22no—
324C42—22no—
325C42—42no—
326C42—42no—
327C42—42no—
328C42—42no—
329C42—82no—
330C42—82no—
331C42#13114yes—
332C42C2×#4112yes—
333C42C2×#4112yes—
334C23—21no—
335C23—21no—
336C23—22no—
337C23—22no—
338C23—24no—
339C23—21no—
340C23—42no—
341C23—42no—
342C23—42no—
343C23—42no—
344C23—42no—
345C23—42no—
346C23C22×C22⋊C411yescharacteristic
347Q8—84no—
348D4—28no—
349D4—44no—
350D4—84no—
351D4—84no—
352D4—88no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—21no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—21no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—24no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—41no—
379C2×C4—44no—
380C2×C4—41no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—82no—
390C2×C4—88no—
391C2×C4—88no—
392C2×C4#40814yes—
393C2×C4C2×#11712yes—
394C2×C4#52512yes—
395C2×C4#52511yescharacteristic
396C2×C4#52511yescharacteristic
397C2×C4C2×#13112yes—
398C2×C4#56812yes—
399C2×C4#57612yes—
400C2×C4#57712yes—
401C8—44no—
402C8—82no—
403C8—82no—
404C8—82no—
405C22—41no—
406C22—44no—
407C22—41no—
408C22—44no—
409C22—42no—
410C22—42no—
411C22—41no—
412C22—44no—
413C22—82no—
414C22—82no—
415C22—82no—
416C22—82no—
417C22#228011yescharacteristic
418C22C2×#40811yescharacteristic
419C22#287911yescharacteristic
420C22#289712yes—
421C22#290112yes—
422C4—22no—
423C4—24no—
424C4—41no—
425C4—44no—
426C4—41no—
427C4—42no—
428C4—42no—
429C4—42no—
430C4—41no—
431C4—82no—
432C4—82no—
433C4#292514yes—
434C4#311812yes—
435C4#312211yescharacteristic
436C4#312311yescharacteristic
437C2—81no—
438C2—84no—
439C2#1748312yes—
440C2#1750512yes—
441C2#2072111yescharacteristic
442C2#2074411yescharacteristic
443C2#2074911yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC2×#532C22
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. #21926C2semidirect
3. #21430C2semidirect
4. #8711C2semidirect
5. #8565C2semidirect
6. #8563C2semidirect
8. #8179C2semidirect
9. #8177C2semidirect

Maximal subgroups

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

subgroupindex
#340862
#219262
#214302
#87112
#85652
#85632
#83132
#81792
#81772