Finite Group Databasebeta

← #502006

Subgroups — #502006

Order 512.

547 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2219C211yesmaximal, characteristic
2C2×C4×Q32C211yesmaximal, characteristic, complemented
3#25035C211yesmaximal, characteristic
4#25033C211yesmaximal, characteristic
5#13483C211yesmaximal, characteristic, complemented
6#13481C211yesmaximal, characteristic, complemented
7#13383C211yesmaximal, characteristic
8#13382C211yesmaximal, characteristic
9#13240C211yesmaximal, characteristic, complemented
10#13166C211yesmaximal, characteristic, complemented
11#12938C211yesmaximal, characteristic, complemented
12#12936C211yesmaximal, characteristic, complemented
13#12908C211yesmaximal, characteristic, complemented
14#12836C211yesmaximal, characteristic
15#12612C211yesmaximal, characteristic
16C22×Q32C2211yescharacteristic
17C22×M32C2211yescharacteristic
18C2×#467C2211yescharacteristic
19C2×#467C2211yescharacteristic
20C2×C4×Q16C2211yescharacteristic
21C2×C4×Q16C2211yescharacteristic
22#2423—21no—
23#2421—21no—
24#2219—24no—
25C4×Q32—24no—
26C2×#367C2211yescharacteristic
27C2×#367C2211yescharacteristic
28C2×#367C2211yescharacteristic
29C2×#366C2211yescharacteristic
30C2×#366C2211yescharacteristic
31C2×#358C2211yescharacteristic
32C2×#358C2211yescharacteristic
33C2×#358C2211yescharacteristic
34C2×#358C2211yescharacteristic
35C2×#358C2211yescharacteristic
36C2×#358C2211yescharacteristic
37C2×C22⋊C16C2211yescharacteristic
38C2×#346C2211yescharacteristic
39C2×C4×C16C2211yescharacteristic
40#2074—21no—
41#2072—21no—
42#2036—21no—
43#2035—21no—
44#1969C2211yescharacteristic
45#1969C2211yescharacteristic
46#1942C2211yescharacteristic
47#1942C2211yescharacteristic
48#1865—21no—
49#1863—21no—
50#1846C2211yescharacteristic
51#1845C2211yescharacteristic
52#1808C2211yescharacteristic
53#1808C2211yescharacteristic
54C8×C22⋊C4C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1374C2211yescharacteristic
60#1374C2211yescharacteristic
61C22×Q16C2311yescharacteristic
62C22×Q16C2311yescharacteristic
63C23×C8C2311yescharacteristic
64C2×C22⋊Q8—21no—
65C2×C22⋊Q8—21no—
66C2×C4×Q8—21no—
67C2×C4×Q8—21no—
68C2×Q32—22no—
69C2×Q32—22no—
70C2×Q32D414yes—
71C2×M32—22no—
72C2×M32—22no—
73C2×M32—22no—
74C22×C16C2311yescharacteristic
75C22×C16C2311yescharacteristic
76C22×C16C2311yescharacteristic
77C22×C16C2311yescharacteristic
78#467—22no—
79#467—22no—
80#467—41no—
81#467—41no—
82C4×Q16—22no—
83C4×Q16—22no—
84C4×Q16—41no—
85C4×Q16—41no—
86C2×#107—21no—
87C2×#107C2311yescharacteristic
88C2×#107C2311yescharacteristic
89C2×#107C2311yescharacteristic
90C2×#107C2311yescharacteristic
91C2×#107C2311yescharacteristic
92C2×#107C2311yescharacteristic
93C2×#106—21no—
94C2×#103—21no—
95C2×#103—21no—
96C2×#103—21no—
97C2×#103—21no—
98C2×#103—21no—
99C2×#103—21no—
100C2×C22⋊C8D411yescharacteristic
101C2×C4×C8C2311yescharacteristic
102C2×C4×C8C2311yescharacteristic
103#400—41no—
104#399—41no—
105#398—41no—
106#397—41no—
107#391—21no—
108#391—21no—
109#389—21no—
110#389—21no—
111#388—41no—
112#388—41no—
113#387—41no—
114#387—41no—
115#384—21no—
116#382—21no—
117#380—21no—
118#380—21no—
119C4×C22⋊C4D411yescharacteristic
120#367—22no—
121#367—22no—
122#367—41no—
123#366—22no—
124#366—41no—
125#358—22no—
126#358—22no—
127#358—22no—
128#358—22no—
129#358—22no—
130#358—22no—
131C22⋊C16—22no—
132#346—22no—
133C4×C16—22no—
134#340—21no—
135#340—21no—
136#340—21no—
137#340—21no—
138#336D411yescharacteristic
139#336D411yescharacteristic
140C22×Q8—21no—
141C22×Q8—21no—
142C23×C4C2×D411yescharacteristic
143C2×Q16—21no—
144C2×Q16—21no—
145C2×Q16—21no—
146C2×Q16—21no—
147C2×Q16—21no—
148C2×Q16—21no—
149C2×Q16—21no—
150C2×Q16—21no—
151C2×Q16—21no—
152C2×Q16—21no—
153C2×Q16C2×D412yes—
154C2×Q16C2×D412yes—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8C2×D411yescharacteristic
162C22×C8C2×D411yescharacteristic
163C22×C8C2×D411yescharacteristic
164C22×C8C2411yescharacteristic
165C22⋊Q8—42no—
166C22⋊Q8—42no—
167C22⋊Q8—81no—
168C22⋊Q8—81no—
169C4×Q8—42no—
170C4×Q8—42no—
171C4×Q8—81no—
172C4×Q8—81no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—41no—
187C2×C22⋊C4—41no—
188C2×C22⋊C4—41no—
189C2×C22⋊C4—41no—
190C2×C22⋊C4—41no—
191C2×C22⋊C4D811yescharacteristic
192C2×C22⋊C4D811yescharacteristic
193C2×C42—41no—
194C2×C42—41no—
195C2×C42C2×D411yescharacteristic
196C2×C42C2×D411yescharacteristic
197Q32—28no—
198M32—44no—
199C2×C16—21no—
200C2×C16—21no—
201C2×C16—21no—
202C2×C16—21no—
203C2×C16—21no—
204C2×C16—21no—
205C2×C16—21no—
206C2×C16—21no—
207C2×C16C2×D412yes—
208C2×C16#4112yes—
209C2×C16#4112yes—
210C2×C16#4112yes—
211#107—21no—
212#107—21no—
213#107—21no—
214#107—21no—
215#107—21no—
216#107—21no—
217#107—21no—
218#107—21no—
219#107—21no—
220#107—21no—
221#107—21no—
222#107—21no—
223#107—81no—
224#106—81no—
225#103—42no—
226#103—42no—
227#103—42no—
228#103—42no—
229#103—42no—
230#103—42no—
231C22⋊C8—22no—
232C4×C8—21no—
233C4×C8—21no—
234C4×C8—21no—
235C4×C8—21no—
236#95—41no—
237#95—41no—
238#95—41no—
239#95—41no—
240#95—41no—
241#95—41no—
242#95—41no—
243#95—41no—
244#95—41no—
245#95—41no—
246#95D811yescharacteristic
247#95D811yescharacteristic
248C24C2×D811yescharacteristic
249C2×Q8—22no—
250C2×Q8—22no—
251C2×Q8—41no—
252C2×Q8—41no—
253C2×Q8—41no—
254C2×Q8—41no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4C2×D811yescharacteristic
276C22×C4C2×D811yescharacteristic
277C22×C4C2×D811yescharacteristic
278C22×C4C2×D811yescharacteristic
279C22×C4C2×D811yescharacteristic
280C22×C4C22×D411yescharacteristic
281Q16—22no—
282Q16—22no—
283Q16—22no—
284Q16—22no—
285Q16—41no—
286Q16—41no—
287Q16—41no—
288Q16—41no—
289C2×C8—21no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8#13512yes—
302C2×C8C22×D411yescharacteristic
303C2×C8C2×#4111yescharacteristic
304C2×C8C2×#4111yescharacteristic
305C2×C8C2×#4111yescharacteristic
306C2×C8H(5,𝔽2)11yescharacteristic
307C2×C82−1+411yescharacteristic
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—82no—
321C4⋊C4—82no—
322C4⋊C4—82no—
323C4⋊C4—82no—
324C4⋊C4—82no—
325C4⋊C4—82no—
326C4⋊C4—161no—
327C4⋊C4—161no—
328C22⋊C4—22no—
329C22⋊C4—41no—
330C22⋊C4—82no—
331C22⋊C4—82no—
332C22⋊C4—82no—
333C22⋊C4—161no—
334C42—21no—
335C42—21no—
336C42—21no—
337C42—21no—
338C42—82no—
339C42—82no—
340C16—22no—
341C16—41no—
342C16—41no—
343C16#12314yes—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—21no—
351C23C22×D811yescharacteristic
352Q8—42no—
353Q8—42no—
354Q8—42no—
355Q8—42no—
356Q8—81no—
357Q8—81no—
358Q8—81no—
359Q8—81no—
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—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4#51012yes—
403C2×C4#53211yescharacteristic
404C2×C4D4211yescharacteristic
405C2×C4#54711yescharacteristic
406C2×C4C2×#13511yescharacteristic
407C2×C4#57611yescharacteristic
408C2×C4#57811yescharacteristic
409C8—22no—
410C8—41no—
411C8—41no—
412C8C2×#12311yescharacteristic
413C8#54211yescharacteristic
414C8#54711yescharacteristic
415C8#55311yescharacteristic
416C22—41no—
417C22—41no—
418C22—41no—
419C22—41no—
420C22—41no—
421C22—41no—
422C22—41no—
423C22#316911yescharacteristic
424C22D4×D811yescharacteristic
425C22#333411yescharacteristic
426C22C2×#51011yescharacteristic
427C22#341411yescharacteristic
428C22#341611yescharacteristic
429C22#341711yescharacteristic
430C4—22no—
431C4—41no—
432C4—41no—
433C4—41no—
434C4—41no—
435C4—82no—
436C4—82no—
437C4—161no—
438C4—161no—
439C4—161no—
440C4—161no—
441C4—161no—
442C4—161no—
443C4#242211yescharacteristic
444C4#317511yescharacteristic
445C4#328311yescharacteristic
446C4#333911yescharacteristic
447C2—81no—
448C2#2503411yescharacteristic
449C2#3373811yescharacteristic
450C2#3374511yescharacteristic
451C2#3383611yescharacteristic
452C2#3383711yescharacteristic
453C2#3391911yescharacteristic
454C2#3392311yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC24C2×D8
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ C8×C22⋊C4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×C4×Q32C2semidirect
5. #13483C2semidirect
6. #13481C2semidirect
9. #13240C2semidirect
10. #13166C2semidirect
11. #12938C2semidirect
12. #12936C2semidirect
13. #12908C2semidirect

Maximal subgroups

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

subgroupindex
C2×#22192
C2×C4×Q322
#250352
#250332
#134832
#134812
#133832
#133822
#132402
#131662
#129382
#129362
#129082
#128362
#126122