Finite Group Databasebeta

← #217531

Subgroups — #217531

Order 512.

536 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21972C211yesmaximal, characteristic
2#21926C211yesmaximal, characteristic
3#20782C211yesmaximal, characteristic
4#9606C211yesmaximal, characteristic, complemented
5#9606C211yesmaximal, characteristic, complemented
6#9605C211yesmaximal, characteristic, complemented
7#9604C211yesmaximal, characteristic, complemented
8#8831C211yesmaximal, characteristic
9#8831C211yesmaximal, characteristic
10#8830C211yesmaximal, characteristic
11#8829C211yesmaximal, characteristic
12#8812C211yesmaximal, characteristic, complemented
13#8811C211yesmaximal, characteristic, complemented
14#8789C211yesmaximal, characteristic, complemented
15#8787C211yesmaximal, characteristic, complemented
16C2×#431C2211yescharacteristic
17C22×C4⋊C8C2211yescharacteristic
18C4×#117C2211yescharacteristic
19#1939—21no—
20#1939—21no—
21#1924—21no—
22#1924—21no—
23#1917—21no—
24#1917—21no—
25#1850—21no—
26#1850—21no—
27#1842C2211yescharacteristic
28#1842C2211yescharacteristic
29#1842C2211yescharacteristic
30#1840C412yes—
31#1832C412yes—
32#1831C2211yescharacteristic
33#1831C2211yescharacteristic
34#1831C2211yescharacteristic
35#1818C412yes—
36#1798C412yes—
37#1797C2211yescharacteristic
38#1797C2211yescharacteristic
39#1767C2211yescharacteristic
40#1767C2211yescharacteristic
41C4×C4⋊C8C2211yescharacteristic
42C4×C4⋊C8C2211yescharacteristic
43C4×C22⋊C8C2211yescharacteristic
44C4×C22⋊C8C2211yescharacteristic
45#1744C2211yescharacteristic
46#1744C2211yescharacteristic
47C2×#335C2211yescharacteristic
48C2×#335C2211yescharacteristic
49C2×#335C2211yescharacteristic
50C2×#335C2211yescharacteristic
51C2×#334C2211yescharacteristic
52C2×#334C2211yescharacteristic
53C2×#334C2211yescharacteristic
54C2×#334C2211yescharacteristic
55#1277C2211yescharacteristic
56#1277C2211yescharacteristic
57#1277C2211yescharacteristic
58#1277C2211yescharacteristic
59#1277C2211yescharacteristic
60#1277C2211yescharacteristic
61#1277C2211yescharacteristic
62#1277C2211yescharacteristic
63C23×C8—21no—
64C2×#117C2×C411yescharacteristic
65C2×#117C2×C411yescharacteristic
66C22×C42C2311yescharacteristic
67#431—41no—
68#431—41no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—41no—
77C2×C4⋊C8—41no—
78C2×C4⋊C8C2×C412yes—
79C2×C4⋊C8C2×C412yes—
80C2×C4⋊C8C2311yescharacteristic
81C2×C4⋊C8C2311yescharacteristic
82C2×C4⋊C8C2311yescharacteristic
83C2×C4⋊C8C2311yescharacteristic
84C2×C4⋊C8C2311yescharacteristic
85C2×C4⋊C8C2311yescharacteristic
86C2×C4⋊C8C2311yescharacteristic
87C2×C4⋊C8C2311yescharacteristic
88C2×C22⋊C8—21no—
89C2×C22⋊C8—21no—
90C2×C22⋊C8—21no—
91C2×C22⋊C8C2×C412yes—
92C2×C22⋊C8C2×C412yes—
93C2×#97—21no—
94C2×#97—21no—
95C2×C4×C8—21no—
96C2×C4×C8C2311yescharacteristic
97C2×C4×C8C2311yescharacteristic
98C2×C4×C8C2311yescharacteristic
99C2×C4×C8C2311yescharacteristic
100C4×C4⋊C4D411yescharacteristic
101C4×C4⋊C4D411yescharacteristic
102C4×C4⋊C4Q811yescharacteristic
103C4×C4⋊C4Q811yescharacteristic
104C4×C22⋊C4D411yescharacteristic
105C4×C22⋊C4D411yescharacteristic
106C4×C22⋊C4Q811yescharacteristic
107C4×C22⋊C4Q811yescharacteristic
108#376C2×C411yescharacteristic
109#376C2×C411yescharacteristic
110C43C2311yescharacteristic
111C43C2311yescharacteristic
112#336—21no—
113#336—21no—
114#336—21no—
115#336—21no—
116#336—21no—
117#336—21no—
118#336—21no—
119#336—21no—
120#336—21no—
121#336—21no—
122#336C2×C412yes—
123#336C2×C412yes—
124#336C2×C412yes—
125#336C2×C412yes—
126#336C2×C412yes—
127#336C2×C412yes—
128#336C2×C412yes—
129#336C2×C412yes—
130#335—41no—
131#335—41no—
132#335—41no—
133#335—41no—
134#334—41no—
135#334—41no—
136#334—41no—
137#334—41no—
138C23×C4C22×C411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×Q811yescharacteristic
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—42no—
150C22×C8—81no—
151C22×C8—81no—
152C22×C8C22×C412yes—
153C22×C8C22×C412yes—
154C22×C8C22×C412yes—
155C22×C8C22×C412yes—
156#117—24no—
157#117—42no—
158C2×C4⋊C4C4⋊C411yescharacteristic
159C2×C4⋊C4C4⋊C411yescharacteristic
160C2×C4⋊C4C4⋊C411yescharacteristic
161C2×C4⋊C4C4⋊C411yescharacteristic
162C2×C22⋊C4C4⋊C411yescharacteristic
163C2×C22⋊C4C4⋊C411yescharacteristic
164C2×C22⋊C4C4⋊C411yescharacteristic
165C2×C22⋊C4C4⋊C411yescharacteristic
166C2×C42—21no—
167C2×C42—21no—
168C2×C42—21no—
169C2×C42—21no—
170C2×C42C22×C411yescharacteristic
171C2×C42C22×C411yescharacteristic
172C2×C42C22×C411yescharacteristic
173C2×C42C22×C411yescharacteristic
174C2×C42C22×C411yescharacteristic
175C2×C42C2×D411yescharacteristic
176C2×C42C2×D411yescharacteristic
177C2×C42C2×D411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C42C2×Q811yescharacteristic
181C2×C42C2×Q811yescharacteristic
182C2×C42C2×Q811yescharacteristic
183C2×C42C2×Q811yescharacteristic
184C2×C42C2×Q811yescharacteristic
185C2×C42C2411yescharacteristic
186C4⋊C8—21no—
187C4⋊C8—21no—
188C4⋊C8—21no—
189C4⋊C8—21no—
190C4⋊C8—21no—
191C4⋊C8—21no—
192C4⋊C8—21no—
193C4⋊C8—21no—
194C4⋊C8—21no—
195C4⋊C8—21no—
196C4⋊C8—21no—
197C4⋊C8—21no—
198C4⋊C8—21no—
199C4⋊C8—21no—
200C4⋊C8—21no—
201C4⋊C8—21no—
202C4⋊C8—42no—
203C4⋊C8—42no—
204C4⋊C8—42no—
205C4⋊C8—81no—
206C4⋊C8—81no—
207C4⋊C8—81no—
208C4⋊C8—81no—
209C4⋊C8—81no—
210C4⋊C8—81no—
211C22⋊C8—24no—
212C22⋊C8—24no—
213C22⋊C8—42no—
214C22⋊C8—81no—
215C22⋊C8—81no—
216#97—81no—
217#97—81no—
218C4×C8—21no—
219C4×C8—21no—
220C4×C8—21no—
221C4×C8—21no—
222C4×C8—21no—
223C4×C8—21no—
224C4×C8—21no—
225C4×C8—21no—
226C4×C8—42no—
227#95C4⋊C411yescharacteristic
228#95C4⋊C411yescharacteristic
229#95C4⋊C411yescharacteristic
230#95C4⋊C411yescharacteristic
231#95C4⋊C411yescharacteristic
232#95C4⋊C411yescharacteristic
233#95C4⋊C411yescharacteristic
234#95C4⋊C411yescharacteristic
235C24C2×C4⋊C411yescharacteristic
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—22no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4C2×C4⋊C411yescharacteristic
250C22×C4C2×C4⋊C411yescharacteristic
251C22×C4C2×C4⋊C411yescharacteristic
252C22×C4C2×C4⋊C411yescharacteristic
253C22×C4C2×C4⋊C411yescharacteristic
254C22×C4C2×C4⋊C411yescharacteristic
255C22×C4C2×C4⋊C411yescharacteristic
256C22×C4C2×C4⋊C411yescharacteristic
257C22×C4C2×C4⋊C411yescharacteristic
258C22×C4C2×C4⋊C411yescharacteristic
259C22×C4C2×C4⋊C411yescharacteristic
260C22×C4#13114yes—
261C22×C4C23×C411yescharacteristic
262C22×C4C22×D411yescharacteristic
263C22×C4C22×Q811yescharacteristic
264C2×C8—22no—
265C2×C8—22no—
266C2×C8—22no—
267C2×C8—22no—
268C2×C8—22no—
269C2×C8—22no—
270C2×C8—22no—
271C2×C8—22no—
272C2×C8—22no—
273C2×C8—22no—
274C2×C8—22no—
275C2×C8—22no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—84no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C22⋊C4—22no—
306C22⋊C4—22no—
307C22⋊C4—41no—
308C22⋊C4—41no—
309C42—21no—
310C42—21no—
311C42—21no—
312C42—21no—
313C42—21no—
314C42—21no—
315C42—22no—
316C42—22no—
317C42—22no—
318C42—22no—
319C42—21no—
320C42—21no—
321C42—21no—
322C42—21no—
323C42—21no—
324C42—21no—
325C42—21no—
326C42—21no—
327C42—41no—
328C42—41no—
329C42—41no—
330C42—41no—
331C42H(5,𝔽2)11yescharacteristic
332C42H(5,𝔽2)11yescharacteristic
333C422−1+411yescharacteristic
334C422−1+411yescharacteristic
335C23—21no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23#42412yes—
342C23C22×C4⋊C411yescharacteristic
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—24no—
348C2×C4—24no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4#42412yes—
386C2×C4#52011yescharacteristic
387C2×C4#52011yescharacteristic
388C2×C4#53511yescharacteristic
389C2×C4#53611yescharacteristic
390C2×C4#54411yescharacteristic
391C2×C4#54411yescharacteristic
392C2×C4C2×#13112yes—
393C2×C4#56812yes—
394C2×C4#57611yescharacteristic
395C2×C4#57711yescharacteristic
396C2×C4#57711yescharacteristic
397C2×C4#57811yescharacteristic
398C8—42no—
399C8—42no—
400C8—42no—
401C8—42no—
402C8—81no—
403C8—81no—
404C8—81no—
405C8—81no—
406C8—81no—
407C8—81no—
408C8—81no—
409C8—81no—
410C22—22no—
411C22—22no—
412C22—22no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22#231711yescharacteristic
418C22C2×#42411yescharacteristic
419C22#290511yescharacteristic
420C22#291211yescharacteristic
421C22#291311yescharacteristic
422C22#291611yescharacteristic
423C22#291611yescharacteristic
424C4—22no—
425C4—21no—
426C4—21no—
427C4—41no—
428C4—41no—
429C4—41no—
430C4—41no—
431C4—41no—
432C4—41no—
433C4—41no—
434C4—41no—
435C4—41no—
436C4—41no—
437C4#299212yes—
438C4#299212yes—
439C4#322411yescharacteristic
440C4#322511yescharacteristic
441C4#324111yescharacteristic
442C4#324211yescharacteristic
443C2—42no—
444C2#1781111yescharacteristic
445C2#1781211yescharacteristic
446C2#1782411yescharacteristic
447C2#1782511yescharacteristic
448C2#2197711yescharacteristic
449C2#2199211yescharacteristic
450C2#2199511yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C4⋊C4
prime radicalC24C2×C4⋊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. #9606C2semidirect
5. #9606C2semidirect
6. #9605C2semidirect
7. #9604C2semidirect
12. #8812C2semidirect
13. #8811C2semidirect
14. #8789C2semidirect
15. #8787C2semidirect

Maximal subgroups

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

subgroupindex
#219722
#219262
#207822
#96062
#96052
#96042
#88312
#88302
#88292
#88122
#88112
#87892
#87872