Finite Group Databasebeta

← #509864

Subgroups — #509864

Order 512.

532 conjugacy classes of subgroups.

The subgroup lattice has 441 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2236C211yesmaximal, characteristic
2C2×#2227C211yesmaximal, characteristic, complemented
3#26065C211yesmaximal, characteristic
4#25035C211yesmaximal, characteristic
5#13515C211yesmaximal, characteristic, complemented
6#13505C211yesmaximal, characteristic, complemented
7#13493C211yesmaximal, characteristic
8#13481C211yesmaximal, characteristic, complemented
9#13451C211yesmaximal, characteristic, complemented
10#13383C211yesmaximal, characteristic
11#13148C211yesmaximal, characteristic, complemented
12#13120C211yesmaximal, characteristic
13#12936C211yesmaximal, characteristic, complemented
14#12919C211yesmaximal, characteristic
15#12905C211yesmaximal, characteristic, complemented
16C2×#500C2211yescharacteristic
17C2×#467C2211yescharacteristic
18C2×C4×Q16C2211yescharacteristic
19C22×#107C2211yescharacteristic
20#2440—21no—
21#2423—21no—
22#2236—24no—
23#2227—24no—
24C2×#366C2211yescharacteristic
25C2×#366C2211yescharacteristic
26C2×#366C2211yescharacteristic
27C2×#366C2211yescharacteristic
28C2×#366C2211yescharacteristic
29C2×#366C2211yescharacteristic
30C2×C4⋊C16C2211yescharacteristic
31C2×#358C2211yescharacteristic
32C2×#358C2211yescharacteristic
33C2×#358C2211yescharacteristic
34C2×#358C2211yescharacteristic
35C2×C22⋊C16C2211yescharacteristic
36C2×C22⋊C16C2211yescharacteristic
37C2×C4×C16C2211yescharacteristic
38#2083—21no—
39#2072—21no—
40#2057—21no—
41#2036—21no—
42#1969C2211yescharacteristic
43#1942C2211yescharacteristic
44#1942C2211yescharacteristic
45#1942C2211yescharacteristic
46#1863—21no—
47#1852—21no—
48#1845C2211yescharacteristic
49#1845C2211yescharacteristic
50#1808C2211yescharacteristic
51#1808C2211yescharacteristic
52C8×C22⋊C4C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1374C2211yescharacteristic
60#1374C2211yescharacteristic
61C22×Q16C2311yescharacteristic
62C23×C8C2311yescharacteristic
63C2×C4⋊Q8—21no—
64C2×C22⋊Q8—21no—
65C2×C22⋊Q8—41no—
66C2×C4×Q8—21no—
67C22×C4⋊C4—21no—
68C22×C16C2311yescharacteristic
69C22×C16C2311yescharacteristic
70C22×C16C2311yescharacteristic
71C22×C16C2311yescharacteristic
72#500—22no—
73#500—41no—
74#467—22no—
75#467—41no—
76C4×Q16—22no—
77C4×Q16—41no—
78C2×#107—22no—
79C2×#107—21no—
80C2×#107—21no—
81C2×#107—21no—
82C2×#107—21no—
83C2×#107—41no—
84C2×#107C2311yescharacteristic
85C2×#107C2311yescharacteristic
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#107C2311yescharacteristic
89C2×#107C2311yescharacteristic
90C2×#107C2311yescharacteristic
91C2×#103—21no—
92C2×#103—21no—
93C2×#103—21no—
94C2×C22⋊C8D411yescharacteristic
95C2×C4×C8C2311yescharacteristic
96C2×C4×C8C2311yescharacteristic
97#399—41no—
98#398—41no—
99#398—41no—
100#395—41no—
101#391—21no—
102#389—21no—
103#389—21no—
104#389—21no—
105#388—41no—
106#387—41no—
107#385—41no—
108#384—21no—
109#384—21no—
110#380—21no—
111#380—21no—
112C4×C22⋊C4D411yescharacteristic
113#366—22no—
114#366—22no—
115#366—22no—
116#366—22no—
117#366—41no—
118#366—41no—
119C4⋊C16—22no—
120#358—22no—
121#358—22no—
122#358—22no—
123#358D414yes—
124C22⋊C16—22no—
125C22⋊C16—22no—
126C4×C16—22no—
127#340—21no—
128#340—21no—
129#340—21no—
130#340—21no—
131#340—21no—
132#336D411yescharacteristic
133#336D411yescharacteristic
134C22×Q8—21no—
135C22×Q8—41no—
136C23×C4—41no—
137C23×C4C2×D411yescharacteristic
138C2×Q16—21no—
139C2×Q16—21no—
140C2×Q16—21no—
141C2×Q16—21no—
142C2×Q16—21no—
143C2×Q16C2×D412yes—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8C2×D411yescharacteristic
151C22×C8C2×D411yescharacteristic
152C22×C8C2×D411yescharacteristic
153C22×C8C2411yescharacteristic
154C4⋊Q8—42no—
155C4⋊Q8—81no—
156C22⋊Q8—42no—
157C22⋊Q8—84no—
158C22⋊Q8—81no—
159C4×Q8—42no—
160C4×Q8—81no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—81no—
181C2×C22⋊C4—41no—
182C2×C22⋊C4—41no—
183C2×C22⋊C4—41no—
184C2×C22⋊C4D811yescharacteristic
185C2×C22⋊C4D811yescharacteristic
186C2×C42—41no—
187C2×C42C2×D411yescharacteristic
188C2×C42C2×D411yescharacteristic
189C2×C16—21no—
190C2×C16—21no—
191C2×C16—21no—
192C2×C16—21no—
193C2×C16—21no—
194C2×C16—21no—
195C2×C16—21no—
196C2×C16—21no—
197C2×C16—21no—
198C2×C16—21no—
199C2×C16C2×D412yes—
200C2×C16#4112yes—
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—21no—
212#107—21no—
213#107—21no—
214#107—42no—
215#107—42no—
216#107—81no—
217#107—81no—
218#107C2×D412yes—
219#103—42no—
220#103—42no—
221#103—42no—
222C22⋊C8—22no—
223C4×C8—21no—
224C4×C8—21no—
225C4×C8—21no—
226C4×C8—21no—
227#95—41no—
228#95—41no—
229#95—41no—
230#95—41no—
231#95—41no—
232#95—41no—
233#95—41no—
234#95D811yescharacteristic
235#95D811yescharacteristic
236C24C2×D811yescharacteristic
237C2×Q8—22no—
238C2×Q8—41no—
239C2×Q8—41no—
240C2×Q8—41no—
241C2×Q8—41no—
242C2×Q8—41no—
243C2×Q8—82no—
244C2×Q8—82no—
245C2×Q8—161no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—82no—
261C22×C4—82no—
262C22×C4—161no—
263C22×C4C2×D811yescharacteristic
264C22×C4C2×D811yescharacteristic
265C22×C4C2×D811yescharacteristic
266C22×C4C2×D811yescharacteristic
267C22×C4C2×D811yescharacteristic
268C22×C4C22×D411yescharacteristic
269Q16—22no—
270Q16—22no—
271Q16—41no—
272Q16—41no—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8#13712yes—
286C2×C8C22×D411yescharacteristic
287C2×C8C2×#4111yescharacteristic
288C2×C8H(5,𝔽2)11yescharacteristic
289C2×C82−1+411yescharacteristic
290C2×C82−1+411yescharacteristic
291C2×C82−1+411yescharacteristic
292C4⋊C4—22no—
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—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—82no—
307C4⋊C4—82no—
308C4⋊C4—82no—
309C4⋊C4—82no—
310C4⋊C4—82no—
311C4⋊C4—82no—
312C4⋊C4—82no—
313C4⋊C4—82no—
314C4⋊C4—161no—
315C4⋊C4—161no—
316C4⋊C4—161no—
317C22⋊C4—22no—
318C22⋊C4—82no—
319C22⋊C4—82no—
320C22⋊C4—161no—
321C22⋊C4Q3214yes—
322C42—21no—
323C42—21no—
324C42—21no—
325C42—21no—
326C42—82no—
327C16—22no—
328C16—22no—
329C16—41no—
330C16—41no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23C2×Q3212yes—
338C23C22×D811yescharacteristic
339Q8—42no—
340Q8—42no—
341Q8—81no—
342Q8—81no—
343Q8—164no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—42no—
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—81no—
384C2×C4—164no—
385C2×C4C2×Q3212yes—
386C2×C4C2×Q3212yes—
387C2×C4#50912yes—
388C2×C4#53611yescharacteristic
389C2×C4D4211yescharacteristic
390C2×C4#54711yescharacteristic
391C2×C4C2×#13711yescharacteristic
392C2×C4#57611yescharacteristic
393C2×C4#57811yescharacteristic
394C8—22no—
395C8—41no—
396C8—41no—
397C8#53611yescharacteristic
398C8#53911yescharacteristic
399C8#54711yescharacteristic
400C8#56311yescharacteristic
401C22—22no—
402C22—22no—
403C22—22no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22#322611yescharacteristic
409C22D4×D811yescharacteristic
410C22#333411yescharacteristic
411C22C22×Q3211yescharacteristic
412C22C2×#50911yescharacteristic
413C22#341511yescharacteristic
414C22#341711yescharacteristic
415C4—22no—
416C4—22no—
417C4—22no—
418C4—41no—
419C4—41no—
420C4—82no—
421C4—82no—
422C4—161no—
423C4—161no—
424C4—161no—
425C4—161no—
426C4—161no—
427C4—161no—
428C4#268011yescharacteristic
429C4#324311yescharacteristic
430C4#327611yescharacteristic
431C4#334611yescharacteristic
432C2—42no—
433C2#3107811yescharacteristic
434C2#3379611yescharacteristic
435C2#3380911yescharacteristic
436C2#3384011yescharacteristic
437C2D4×Q3211yescharacteristic
438C2#3390511yescharacteristic
439C2#3391011yescharacteristic
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×#2227C2semidirect
5. #13515C2semidirect
6. #13505C2semidirect
8. #13481C2semidirect
9. #13451C2semidirect
11. #13148C2semidirect
13. #12936C2semidirect
15. #12905C2semidirect

Maximal subgroups

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

subgroupindex
C2×#22362
C2×#22272
#260652
#250352
#135152
#135052
#134932
#134812
#134512
#133832
#131482
#131202
#129362
#129192
#129052