Finite Group Databasebeta

← #504662

Subgroups — #504662

Order 512.

584 conjugacy classes of subgroups.

The subgroup lattice has 479 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2237C211yesmaximal, characteristic, complemented
2C2×#2226C211yesmaximal, characteristic, complemented
3#27792C211yesmaximal, characteristic, complemented
4#23027C211yesmaximal, characteristic
5#13475C211yesmaximal, characteristic, complemented
6#13451C211yesmaximal, characteristic, complemented
7#13380C211yesmaximal, characteristic
8#13305C211yesmaximal, characteristic
9#13238C211yesmaximal, characteristic, complemented
10#13132C211yesmaximal, characteristic, complemented
11#13130C211yesmaximal, characteristic, complemented
12#13121C211yesmaximal, characteristic, complemented
13#13014C211yesmaximal, characteristic, complemented
14#12921C211yesmaximal, characteristic, complemented
15#12674C211yesmaximal, characteristic, complemented
16C22×SD32C2211yescharacteristic
17C2×#500C2211yescharacteristic
18C2×#467C2211yescharacteristic
19C2×#466C2211yescharacteristic
20C2×C4×D8C2211yescharacteristic
21C2×#427C2211yescharacteristic
22#2492—21no—
23#2363—21no—
24#2237—24no—
25#2226—24no—
26C2×#367C2211yescharacteristic
27C2×#367C2211yescharacteristic
28C2×#366C2211yescharacteristic
29C2×#366C2211yescharacteristic
30C2×#366C2211yescharacteristic
31C2×C4⋊C16C2211yescharacteristic
32C2×#358C2211yescharacteristic
33C2×#358C2211yescharacteristic
34C2×#358C2211yescharacteristic
35C2×#357C2211yescharacteristic
36C2×#357C2211yescharacteristic
37C2×#357C2211yescharacteristic
38C2×C22⋊C16C2211yescharacteristic
39C2×C22⋊C16C2211yescharacteristic
40C2×#346C2211yescharacteristic
41#2036—21no—
42#2024—21no—
43#1970C2211yescharacteristic
44#1969C2211yescharacteristic
45#1969C2211yescharacteristic
46#1942C2211yescharacteristic
47#1928—21no—
48#1926—21no—
49#1875—21no—
50#1864—21no—
51#1808C2211yescharacteristic
52C4×#107C2211yescharacteristic
53C4×#107C2211yescharacteristic
54C8×C22⋊C4C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1374C2211yescharacteristic
60#1374C2211yescharacteristic
61C22×Q16C2311yescharacteristic
62C22×D8C2311yescharacteristic
63C23×C8C2311yescharacteristic
64C2×C4⋊Q8—21no—
65C2×C22⋊Q8—21no—
66C2×#121—21no—
67C2×C4×D4—21no—
68C2×#117—21no—
69C2×SD32—22no—
70C2×SD32—22no—
71C2×SD32—41no—
72C22×C16C2311yescharacteristic
73C22×C16C2311yescharacteristic
74C22×C16C2311yescharacteristic
75C22×C16C2311yescharacteristic
76#500—22no—
77#500—41no—
78#467—22no—
79#467—41no—
80#466—22no—
81#466—22no—
82C4×D8—22no—
83C4×D8—41no—
84#427—22no—
85#427—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×#102—21no—
98C2×#102—21no—
99C2×#102—21no—
100C2×C22⋊C8D411yescharacteristic
101C2×C4×C8C2311yescharacteristic
102C2×C4×C8C2311yescharacteristic
103#400—41no—
104#393—41no—
105#391—21no—
106#391—21no—
107#391—41no—
108#390—21no—
109#390—41no—
110#389—21no—
111#388—41no—
112#388—41no—
113#386—41no—
114#380—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—21no—
117C4×C22⋊C4—41no—
118C4×C22⋊C4D411yescharacteristic
119#367—22no—
120#367D414yescomplemented
121#366—22no—
122#366—22no—
123#366—41no—
124C4⋊C16—22no—
125#358—22no—
126#358—22no—
127#358—22no—
128#357—22no—
129#357—22no—
130#357—22no—
131C22⋊C16—22no—
132C22⋊C16—22no—
133#346—22no—
134#340—21no—
135#340—21no—
136#340—21no—
137#340—21no—
138#336D411yescharacteristic
139#336D411yescharacteristic
140C22×Q8—21no—
141C22×Q8—41no—
142C22×D4—21no—
143C22×D4—41no—
144C23×C4—41no—
145C23×C4C2×D411yescharacteristic
146C2×Q16—21no—
147C2×Q16—21no—
148C2×Q16—21no—
149C2×Q16—21no—
150C2×Q16—21no—
151C2×Q16—21no—
152C2×D8—21no—
153C2×D8—21no—
154C2×D8—21no—
155C2×D8—21no—
156C2×D8—21no—
157C2×D8#4112yes—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8C2×D411yescharacteristic
165C22×C8C2×D411yescharacteristic
166C22×C8C2×D411yescharacteristic
167C22×C8C2411yescharacteristic
168C4⋊Q8—42no—
169C4⋊Q8—81no—
170C22⋊Q8—42no—
171C22⋊Q8—81no—
172#121—24no—
173#121—42no—
174C4×D4—42no—
175C4×D4—81no—
176#117—42no—
177#117—81no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
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⋊C4—41no—
192C2×C22⋊C4—41no—
193C2×C22⋊C4—41no—
194C2×C22⋊C4D811yescharacteristic
195C2×C22⋊C4D811yescharacteristic
196C2×C42—41no—
197C2×C42—41no—
198C2×C42—41no—
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201SD32—44no—
202C2×C16—21no—
203C2×C16—21no—
204C2×C16—21no—
205C2×C16—21no—
206C2×C16—21no—
207C2×C16—21no—
208C2×C16—21no—
209C2×C16—21no—
210C2×C16—21no—
211C2×C16—21no—
212C2×C16C2×D412yes—
213C2×C16#4112yes—
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—42no—
224#107C2×D412yes—
225#107C2×D412yes—
226#107#4112yes—
227#106—42no—
228#103—42no—
229#103—42no—
230#103—81no—
231#102—24no—
232#102—42no—
233#102—81no—
234C22⋊C8—41no—
235C4×C8—21no—
236C4×C8—21no—
237C4×C8—21no—
238C4×C8—21no—
239#95—41no—
240#95—41no—
241#95—41no—
242#95—41no—
243#95—41no—
244#95—41no—
245#95—41no—
246#95—41no—
247#95D811yescharacteristic
248#95D811yescharacteristic
249C24—41no—
250C24C2×D811yescharacteristic
251C2×Q8—41no—
252C2×Q8—41no—
253C2×Q8—41no—
254C2×Q8—41no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×Q8—82no—
258C2×Q8—82no—
259C2×Q8—161no—
260C2×D4—22no—
261C2×D4—44no—
262C2×D4—41no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—161no—
268C2×D4—161no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—82no—
283C22×C4—82no—
284C22×C4—161no—
285C22×C4C2×D811yescharacteristic
286C22×C4C2×D811yescharacteristic
287C22×C4C2×D811yescharacteristic
288C22×C4C2×D811yescharacteristic
289C22×C4C2×D811yescharacteristic
290C22×C4C22×D411yescharacteristic
291Q16—41no—
292Q16—41no—
293Q16—41no—
294Q16—41no—
295D8—22no—
296D8—22no—
297D8—41no—
298D8—41no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8C22×D411yescharacteristic
313C2×C8C2×#4111yescharacteristic
314C2×C8C2×#4111yescharacteristic
315C2×C8C2×#4111yescharacteristic
316C2×C8H(5,𝔽2)11yescharacteristic
317C2×C82−1+411yescharacteristic
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—82no—
331C4⋊C4—82no—
332C4⋊C4—161no—
333C22⋊C4—22no—
334C22⋊C4—41no—
335C22⋊C4—44no—
336C22⋊C4—82no—
337C22⋊C4—82no—
338C22⋊C4—82no—
339C22⋊C4—82no—
340C22⋊C4—161no—
341C22⋊C4—161no—
342C42—21no—
343C42—21no—
344C42—21no—
345C42—21no—
346C42—82no—
347C42—161no—
348C42—161no—
349C16—22no—
350C16—22no—
351C16—41no—
352C16—41no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—42no—
361C23—81no—
362C23—81no—
363C23—81no—
364C23—81no—
365C23—81no—
366C23—81no—
367C23C22×D811yescharacteristic
368Q8—81no—
369Q8—81no—
370Q8—81no—
371Q8—81no—
372Q8—164no—
373D4—42no—
374D4—42no—
375D4—81no—
376D4—81no—
377D4—164no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—164no—
418C2×C4#51012yes—
419C2×C4#52511yescharacteristic
420C2×C4D4211yescharacteristic
421C2×C4#54711yescharacteristic
422C2×C4#57511yescharacteristic
423C2×C4#57611yescharacteristic
424C2×C4#57811yescharacteristic
425C8—41no—
426C8—41no—
427C8—41no—
428C8#53211yescharacteristic
429C8#54211yescharacteristic
430C8#54811yescharacteristic
431C8#55111yescharacteristic
432C22—41no—
433C22—41no—
434C22—41no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22—41no—
439C22—82no—
440C22—82no—
441C22—82no—
442C22—161no—
443C22—161no—
444C22—161no—
445C22—161no—
446C22#310011yescharacteristic
447C22D4×D811yescharacteristic
448C22#333411yescharacteristic
449C22C2×#51011yescharacteristic
450C22#341411yescharacteristic
451C22#341511yescharacteristic
452C22#341611yescharacteristic
453C4—22no—
454C4—41no—
455C4—41no—
456C4—41no—
457C4—41no—
458C4—82no—
459C4—82no—
460C4—82no—
461C4—161no—
462C4—161no—
463C4—161no—
464C4—161no—
465C4#246111yescharacteristic
466C4#312411yescharacteristic
467C4#327711yescharacteristic
468C4#334711yescharacteristic
469C2—81no—
470C2—162no—
471C2#2678411yescharacteristic
472C2#3367611yescharacteristic
473C2#3367811yescharacteristic
474C2#3383211yescharacteristic
475C2#3383411yescharacteristic
476C2#3391711yescharacteristic
477C2#3392411yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC2×#466C22
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
1. C2×#2237C2semidirect
2. C2×#2226C2semidirect
3. #27792C2semidirect
5. #13475C2semidirect
6. #13451C2semidirect
9. #13238C2semidirect
10. #13132C2semidirect
11. #13130C2semidirect
12. #13121C2semidirect
13. #13014C2semidirect
14. #12921C2semidirect
15. #12674C2semidirect
120. #367D4semidirect

Maximal subgroups

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

subgroupindex
C2×#22372
C2×#22262
#277922
#230272
#134752
#134512
#133802
#133052
#132382
#131322
#131302
#131212
#130142
#129212
#126742