Finite Group Databasebeta

← #504096

Subgroups — #504096

Order 512.

588 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2228C211yesmaximal, characteristic, complemented
2C2×#2158C211yesmaximal, characteristic, complemented
3#26066C211yesmaximal, characteristic, complemented
4#23027C211yesmaximal, characteristic
5#13526C211yesmaximal, characteristic, complemented
6#13514C211yesmaximal, characteristic, complemented
7#13476C211yesmaximal, characteristic, complemented
8#13453C211yesmaximal, characteristic, complemented
9#13177C211yesmaximal, characteristic, complemented
10#13112C211yesmaximal, characteristic, complemented
11#13059C211yesmaximal, characteristic
12#12992C211yesmaximal, characteristic
13#12937C211yesmaximal, characteristic, complemented
14#12921C211yesmaximal, characteristic, complemented
15#12912C211yesmaximal, characteristic, complemented
16C23×C16C2211yescharacteristic
17C2×#499C2211yescharacteristic
18C2×#466C2211yescharacteristic
19C2×C4×D8C2211yescharacteristic
20C2×#427C2211yescharacteristic
21C2×#427C2211yescharacteristic
22#2441—21no—
23#2363—21no—
24#2228—24no—
25#2158—24no—
26C2×#367C2211yescharacteristic
27C2×#367C2211yescharacteristic
28C2×#367C2211yescharacteristic
29C2×#366C2211yescharacteristic
30C2×#366C2211yescharacteristic
31C2×#366C2211yescharacteristic
32C2×C4⋊C16C2211yescharacteristic
33C2×C4⋊C16C2211yescharacteristic
34C2×#357C2211yescharacteristic
35C2×#357C2211yescharacteristic
36C2×#357C2211yescharacteristic
37C2×#357C2211yescharacteristic
38C2×C22⋊C16C2211yescharacteristic
39#2084—21no—
40#2059—21no—
41#1970C2211yescharacteristic
42#1942C2211yescharacteristic
43#1942C2211yescharacteristic
44#1942C2211yescharacteristic
45#1926—21no—
46#1875—21no—
47#1864—21no—
48#1854—21no—
49#1845C2211yescharacteristic
50#1808C2211yescharacteristic
51C4×#107C2211yescharacteristic
52C8×C22⋊C4C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1374C2211yescharacteristic
60#1374C2211yescharacteristic
61C22×D8C2311yescharacteristic
62C23×C8C2311yescharacteristic
63C2×#125—21no—
64C2×#121—21no—
65C2×C4×D4—21no—
66C2×#117—21no—
67C2×#117—21no—
68C22×C16—22no—
69C22×C16—22no—
70C22×C16—22no—
71C22×C16C2311yescharacteristic
72C22×C16C2311yescharacteristic
73C22×C16C2311yescharacteristic
74C22×C16C2311yescharacteristic
75#499—22no—
76#499—41no—
77#466—22no—
78#466Q814yes—
79C4×D8—22no—
80C4×D8—41no—
81#427—22no—
82#427—22no—
83#427—41no—
84#427—41no—
85C2×#107—21no—
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#107C2311yescharacteristic
89C2×#107C2311yescharacteristic
90C2×#107C2311yescharacteristic
91C2×#107C2311yescharacteristic
92C2×#107C2311yescharacteristic
93C2×#106—21no—
94C2×#106—21no—
95C2×#106—21no—
96C2×#102—21no—
97C2×#102—21no—
98C2×#102—21no—
99C2×C22⋊C8D411yescharacteristic
100C2×C4×C8C2311yescharacteristic
101C2×C4×C8C2311yescharacteristic
102#399—41no—
103#396—41no—
104#390—21no—
105#390—41no—
106#389—21no—
107#389—21no—
108#389—21no—
109#389—41no—
110#388—41no—
111#387—41no—
112#386—41no—
113#384—21no—
114#380—21no—
115C4×C4⋊C4—21no—
116C4×C22⋊C4—41no—
117C4×C22⋊C4D411yescharacteristic
118#367—22no—
119#367—22no—
120#367—22no—
121#366—22no—
122#366—41no—
123#366D414yescomplemented
124C4⋊C16—22no—
125C4⋊C16—22no—
126#357—22no—
127#357—22no—
128#357—22no—
129#357—41no—
130C22⋊C16—22no—
131#340—21no—
132#340—21no—
133#340—21no—
134#340—21no—
135#340—21no—
136#336D411yescharacteristic
137#336D411yescharacteristic
138C22×D4—21no—
139C22×D4—41no—
140C23×C4—41no—
141C23×C4C2×D411yescharacteristic
142C2×D8—21no—
143C2×D8—21no—
144C2×D8—21no—
145C2×D8—21no—
146C2×D8—21no—
147C2×D8C2×Q812yes—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8C2×D411yescharacteristic
153C22×C8C2×D411yescharacteristic
154C22×C8C2×D411yescharacteristic
155C22×C8C2×Q812yes—
156C22×C8#4112yes—
157C22×C8C2411yescharacteristic
158#125—42no—
159#125—81no—
160#121—24no—
161#121—42no—
162C4×D4—42no—
163C4×D4—81no—
164#117—42no—
165#117—42no—
166#117—81no—
167#117—81no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—41no—
182C2×C22⋊C4—41no—
183C2×C22⋊C4—41no—
184C2×C22⋊C4—41no—
185C2×C22⋊C4—41no—
186C2×C22⋊C4—41no—
187C2×C22⋊C4—41no—
188C2×C22⋊C4—41no—
189C2×C22⋊C4D811yescharacteristic
190C2×C22⋊C4D811yescharacteristic
191C2×C42—41no—
192C2×C42—41no—
193C2×C42—41no—
194C2×C42C2×D411yescharacteristic
195C2×C42C2×D411yescharacteristic
196C2×C16—28no—
197C2×C16—21no—
198C2×C16—21no—
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—
209#107—21no—
210#107—21no—
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—81no—
221#107C2×D412yes—
222#107C2×D412yes—
223#107C2×Q812yes—
224#106—42no—
225#106—42no—
226#106—42no—
227#102—24no—
228#102—42no—
229#102—81no—
230C22⋊C8—22no—
231C4×C8—21no—
232C4×C8—21no—
233C4×C8—21no—
234C4×C8—21no—
235#95—41no—
236#95—41no—
237#95—41no—
238#95—41no—
239#95—41no—
240#95D811yescharacteristic
241#95D811yescharacteristic
242C24—41no—
243C24C2×D811yescharacteristic
244C2×D4—22no—
245C2×D4—44no—
246C2×D4—41no—
247C2×D4—41no—
248C2×D4—41no—
249C2×D4—41no—
250C2×D4—41no—
251C2×D4—161no—
252C2×D4—161no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—82no—
265C22×C4—82no—
266C22×C4—161no—
267C22×C4C22⋊Q812yes—
268C22×C4C2×D811yescharacteristic
269C22×C4C2×D811yescharacteristic
270C22×C4C2×D811yescharacteristic
271C22×C4C2×D811yescharacteristic
272C22×C4C2×D811yescharacteristic
273C22×C4#13712yes—
274C22×C4C22×D411yescharacteristic
275D8—22no—
276D8—22no—
277D8—41no—
278D8—41no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—22no—
282C2×C8—22no—
283C2×C8—22no—
284C2×C8—22no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—21no—
290C2×C8—21no—
291C2×C8C22⋊Q812yes—
292C2×C8C22×D411yescharacteristic
293C2×C8C22×Q811yescharacteristic
294C2×C8C2×#4111yescharacteristic
295C2×C8C2×#4111yescharacteristic
296C2×C8H(5,𝔽2)11yescharacteristic
297C2×C82−1+411yescharacteristic
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—82no—
313C4⋊C4—82no—
314C4⋊C4—82no—
315C4⋊C4—82no—
316C4⋊C4—161no—
317C4⋊C4—161no—
318C4⋊C4—161no—
319C22⋊C4—22no—
320C22⋊C4—41no—
321C22⋊C4—44no—
322C22⋊C4—82no—
323C22⋊C4—82no—
324C22⋊C4—82no—
325C22⋊C4—82no—
326C22⋊C4—161no—
327C22⋊C4—161no—
328C42—21no—
329C42—21no—
330C42—21no—
331C42—21no—
332C42—82no—
333C42—82no—
334C42—161no—
335C16—22no—
336C16—22no—
337C16—41no—
338C16—41no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—42no—
346C23—81no—
347C23—81no—
348C23—81no—
349C23—81no—
350C23—81no—
351C23—81no—
352C23#47412yes—
353C23C22×D811yescharacteristic
354D4—42no—
355D4—42no—
356D4—81no—
357D4—81no—
358D4—164no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
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—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—81no—
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—164no—
398C2×C4#47412yes—
399C2×C4#50812yes—
400C2×C4C2×C22⋊Q811yescharacteristic
401C2×C4D4211yescharacteristic
402C2×C4#54711yescharacteristic
403C2×C4C2×#13711yescharacteristic
404C2×C4#57611yescharacteristic
405C2×C4#57811yescharacteristic
406C8—22no—
407C8—41no—
408C8—41no—
409C8#52511yescharacteristic
410C8#53911yescharacteristic
411C8D4×Q811yescharacteristic
412C8#55411yescharacteristic
413C22—22no—
414C22—22no—
415C22—22no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22—41no—
420C22—82no—
421C22—82no—
422C22—82no—
423C22—161no—
424C22—161no—
425C22—161no—
426C22—161no—
427C22C2×#47411yescharacteristic
428C22D4×D811yescharacteristic
429C22#333411yescharacteristic
430C22C2×#50811yescharacteristic
431C22#341411yescharacteristic
432C22#341511yescharacteristic
433C22#341611yescharacteristic
434C4—22no—
435C4—22no—
436C4—41no—
437C4—41no—
438C4—41no—
439C4—82no—
440C4—82no—
441C4—82no—
442C4—161no—
443C4—161no—
444C4—161no—
445C4—161no—
446C4#244911yescharacteristic
447C4#308311yescharacteristic
448C4#327011yescharacteristic
449C4#334611yescharacteristic
450C2—42no—
451C2—162no—
452C2#2633611yescharacteristic
453C2#3362611yescharacteristic
454C2#3363111yescharacteristic
455C2#3383311yescharacteristic
456C2#3385311yescharacteristic
457C2#3390411yescharacteristic
458C2#3391411yescharacteristic
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×#2228C2semidirect
2. C2×#2158C2semidirect
3. #26066C2semidirect
5. #13526C2semidirect
6. #13514C2semidirect
7. #13476C2semidirect
8. #13453C2semidirect
9. #13177C2semidirect
10. #13112C2semidirect
13. #12937C2semidirect
14. #12921C2semidirect
15. #12912C2semidirect
123. #366D4semidirect

Maximal subgroups

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

subgroupindex
C2×#22282
C2×#21582
#260662
#230272
#135262
#135142
#134762
#134532
#131772
#131122
#130592
#129922
#129372
#129212
#129122