Finite Group Databasebeta

← #501905

Subgroups — #501905

Order 512.

522 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27662C211yesmaximal, characteristic
2#22658C211yesmaximal, characteristic
3#13537C211yesmaximal, characteristic, complemented
4#13506C211yesmaximal, characteristic, complemented
5#13494C211yesmaximal, characteristic, complemented
6#13492C211yesmaximal, characteristic, complemented
7#13112C211yesmaximal, characteristic, complemented
8#13112C211yesmaximal, characteristic, complemented
9#12977C211yesmaximal, characteristic
10#12921C211yesmaximal, characteristic, complemented
11#12921C211yesmaximal, characteristic, complemented
12#12868C211yesmaximal, characteristic, complemented
13#12866C211yesmaximal, characteristic, complemented
14#12836C211yesmaximal, characteristic, complemented
15#12557C211yesmaximal, characteristic, complemented
16C22×M32C2211yescharacteristic
17C2×C8×D4C2211yescharacteristic
18C2×#427C2211yescharacteristic
19C2×#427C2211yescharacteristic
20C22×#107C2211yescharacteristic
21#2487—21no—
22#2354—21no—
23C2×#367C2211yescharacteristic
24C2×#367C2211yescharacteristic
25C2×#366C2211yescharacteristic
26C2×#366C2211yescharacteristic
27C2×C22⋊C16C2211yescharacteristic
28C2×C22⋊C16C2211yescharacteristic
29C2×C22⋊C16C2211yescharacteristic
30#2087—21no—
31#2076—21no—
32#1948—21no—
33#1942C2211yescharacteristic
34#1942C2211yescharacteristic
35#1854—21no—
36#1854—21no—
37#1845C2211yescharacteristic
38#1824—21no—
39#1808C2211yescharacteristic
40#1808C2211yescharacteristic
41#1808C2211yescharacteristic
42#1808C2211yescharacteristic
43#1808C2211yescharacteristic
44C4×#107C2211yescharacteristic
45#1380C2211yescharacteristic
46#1380C2211yescharacteristic
47#1380C2211yescharacteristic
48#1380C2211yescharacteristic
49#1380C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C23×C8C2311yescharacteristic
60C23×C8C2311yescharacteristic
61C2×C4×D4D411yescharacteristic
62C2×#117—21no—
63C2×#117—21no—
64C22×C4⋊C4—21no—
65C2×M32—22no—
66C2×M32—22no—
67C2×M32—41no—
68C22×C16C2311yescharacteristic
69C22×C16C2311yescharacteristic
70C22×C16C2311yescharacteristic
71C22×C16C2311yescharacteristic
72C8×D4—22no—
73C8×D4—22no—
74#427—22no—
75#427—22no—
76#427—41no—
77#427—41no—
78C2×#107—22no—
79C2×#107—22no—
80C2×#107—21no—
81C2×#107—21no—
82C2×#107C2311yescharacteristic
83C2×#107C2311yescharacteristic
84C2×#107C2311yescharacteristic
85C2×#107C2311yescharacteristic
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#107C2311yescharacteristic
89C2×#107C2311yescharacteristic
90C2×#106—21no—
91C2×#106—21no—
92C2×C4⋊C8D411yescharacteristic
93C2×C22⋊C8D411yescharacteristic
94C2×C22⋊C8D411yescharacteristic
95C2×C4×C8C2311yescharacteristic
96#400—41no—
97#399—41no—
98#390—41no—
99#389—21no—
100#389—21no—
101#388—41no—
102#388—41no—
103#386—41no—
104#385—41no—
105#384—21no—
106#380—21no—
107#380—21no—
108#380—21no—
109#380—21no—
110#380—21no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—41no—
113#367—41no—
114#367—41no—
115#366—22no—
116#366—22no—
117C22⋊C16—22no—
118C22⋊C16—22no—
119C22⋊C16—41no—
120#340—21no—
121#340—21no—
122#340—21no—
123#340—21no—
124#340—21no—
125#340—21no—
126#340—21no—
127#340—21no—
128C22×D4D811yescharacteristic
129C23×C4—41no—
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C22×C8—21no—
133C22×C8—21no—
134C22×C8—21no—
135C22×C8—21no—
136C22×C8—21no—
137C22×C8—21no—
138C22×C8—21no—
139C22×C8—21no—
140C22×C8—21no—
141C22×C8—21no—
142C22×C8C2×D411yescharacteristic
143C22×C8C2×D411yescharacteristic
144C22×C8C2×D411yescharacteristic
145C22×C8#4112yes—
146C22×C8#4112yes—
147C22×C8C2411yescharacteristic
148C4×D4—22no—
149C4×D4—22no—
150#117—42no—
151#117—42no—
152#117—81no—
153#117—81no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4D811yescharacteristic
170C2×C22⋊C4—41no—
171C2×C22⋊C4—41no—
172C2×C22⋊C4—41no—
173C2×C22⋊C4—41no—
174C2×C22⋊C4—41no—
175C2×C22⋊C4—41no—
176C2×C22⋊C4—41no—
177C2×C22⋊C4D811yescharacteristic
178C2×C22⋊C4D811yescharacteristic
179C2×C42—41no—
180C2×C42—41no—
181C2×C42—41no—
182C2×C42C2×D411yescharacteristic
183M32—44no—
184C2×C16—21no—
185C2×C16—21no—
186C2×C16—21no—
187C2×C16—21no—
188C2×C16—21no—
189C2×C16—21no—
190C2×C16—21no—
191C2×C16—21no—
192C2×C16—21no—
193C2×C16—21no—
194C2×C16—21no—
195C2×C16#4112yes—
196#107—21no—
197#107—21no—
198#107—21no—
199#107—21no—
200#107—21no—
201#107—21no—
202#107—21no—
203#107—21no—
204#107—21no—
205#107—21no—
206#107—21no—
207#107—21no—
208#107—21no—
209#107—21no—
210#107—21no—
211#107—42no—
212#107—81no—
213#107#4112yes—
214#106—42no—
215#106—42no—
216C4⋊C8—22no—
217C22⋊C8—22no—
218C22⋊C8—22no—
219C4×C8—21no—
220C4×C8—21no—
221#95—41no—
222#95—41no—
223#95—41no—
224#95—41no—
225#95—41no—
226#95—41no—
227#95—41no—
228#95—41no—
229C24C2×D811yescharacteristic
230C24C2×D811yescharacteristic
231C2×D4—22no—
232C2×D4—22no—
233C2×D4—41no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—82no—
253C22×C4—161no—
254C22×C4—161no—
255C22×C4#12312yes—
256C22×C4C2×D811yescharacteristic
257C22×C4C2×D811yescharacteristic
258C22×C4C2×D811yescharacteristic
259C22×C4C2×D811yescharacteristic
260C22×C4#13712yes—
261C22×C4C22×D411yescharacteristic
262C2×C8—21no—
263C2×C8—21no—
264C2×C8—22no—
265C2×C8—22no—
266C2×C8—22no—
267C2×C8—22no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—21no—
272C2×C8—21no—
273C2×C8—21no—
274C2×C8—41no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8#12312yes—
279C2×C8C2×#4111yescharacteristic
280C2×C8C2×#4111yescharacteristic
281C2×C8C2×#4111yescharacteristic
282C2×C8C2×#4111yescharacteristic
283C2×C8H(5,𝔽2)11yescharacteristic
284C2×C82−1+411yescharacteristic
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—82no—
303C4⋊C4—82no—
304C4⋊C4—161no—
305C4⋊C4—161no—
306C4⋊C4—161no—
307C22⋊C4—22no—
308C22⋊C4—22no—
309C22⋊C4—82no—
310C22⋊C4—82no—
311C22⋊C4—82no—
312C22⋊C4—161no—
313C22⋊C4—161no—
314C22⋊C4—161no—
315C22⋊C4—161no—
316C42—21no—
317C42—21no—
318C42—82no—
319C42—82no—
320C42—82no—
321C16—22no—
322C16—41no—
323C16—41no—
324C16—41no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23#48012yes—
339C23C22×D811yescharacteristic
340D4—44no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—164no—
384C2×C4#48012yes—
385C2×C4#51012yes—
386C2×C4C2×#12311yescharacteristic
387C2×C4#54611yescharacteristic
388C2×C4#54711yescharacteristic
389C2×C4C2×#13711yescharacteristic
390C2×C4#57711yescharacteristic
391C2×C4#57811yescharacteristic
392C8—22no—
393C8—41no—
394C8—41no—
395C8#52911yescharacteristic
396C8#53911yescharacteristic
397C8#55111yescharacteristic
398C8#55211yescharacteristic
399C22—22no—
400C22—22no—
401C22—22no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22C2×#48011yescharacteristic
414C22#329411yescharacteristic
415C22#333411yescharacteristic
416C22C2×#51011yescharacteristic
417C22#341411yescharacteristic
418C22#341511yescharacteristic
419C22#341611yescharacteristic
420C4—22no—
421C4—22no—
422C4—41no—
423C4—41no—
424C4—82no—
425C4—161no—
426C4—161no—
427C4—161no—
428C4—161no—
429C4—161no—
430C4—161no—
431C4—161no—
432C4#241911yescharacteristic
433C4#309711yescharacteristic
434C4#331211yescharacteristic
435C4#334411yescharacteristic
436C2—42no—
437C2—81no—
438C2#2493311yescharacteristic
439C2#3364911yescharacteristic
440C2#3365011yescharacteristic
441C2#3387411yescharacteristic
442C2#3390011yescharacteristic
443C2#3391411yescharacteristic
444C2#3392311yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D4D8
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #13537C2semidirect
4. #13506C2semidirect
5. #13494C2semidirect
6. #13492C2semidirect
7. #13112C2semidirect
8. #13112C2semidirect
10. #12921C2semidirect
11. #12921C2semidirect
12. #12868C2semidirect
13. #12866C2semidirect
14. #12836C2semidirect
15. #12557C2semidirect

Maximal subgroups

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

subgroupindex
#276622
#226582
#135372
#135062
#134942
#134922
#131122
#129772
#129212
#128682
#128662
#128362
#125572