Finite Group Databasebeta

← #504138

Subgroups — #504138

Order 512.

570 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
1C2×#2229C211yesmaximal, characteristic, complemented
2C2×C4×SD32C211yesmaximal, characteristic
3#26482C211yesmaximal, characteristic
4#22685C211yesmaximal, characteristic, complemented
5#13528C211yesmaximal, characteristic
6#13507C211yesmaximal, characteristic, complemented
7#13476C211yesmaximal, characteristic
8#13453C211yesmaximal, characteristic, complemented
9#13166C211yesmaximal, characteristic, complemented
10#13104C211yesmaximal, characteristic, complemented
11#13013C211yesmaximal, characteristic
12#13013C211yesmaximal, characteristic
13#12938C211yesmaximal, characteristic, complemented
14#12936C211yesmaximal, characteristic, complemented
15#12935C211yesmaximal, characteristic
16C22×SD32C2211yescharacteristic
17C2×#499C2211yescharacteristic
18C2×C4×Q16C2211yescharacteristic
19C2×C4×Q16C2211yescharacteristic
20C2×C4×Q16C2211yescharacteristic
21C2×C4×D8C2211yescharacteristic
22#2454—21no—
23#2351—21no—
24#2229—24no—
25C4×SD32—24no—
26C2×#367C2211yescharacteristic
27C2×#367C2211yescharacteristic
28C2×#367C2211yescharacteristic
29C2×#366C2211yescharacteristic
30C2×#366C2211yescharacteristic
31C2×C4⋊C16C2211yescharacteristic
32C2×#358C2211yescharacteristic
33C2×#358C2211yescharacteristic
34C2×#358C2211yescharacteristic
35C2×#357C2211yescharacteristic
36C2×#357C2211yescharacteristic
37C2×#357C2211yescharacteristic
38C2×C4×C16C2211yescharacteristic
39#2065—21no—
40#2025—21no—
41#1970C2211yescharacteristic
42#1970C2211yescharacteristic
43#1942C2211yescharacteristic
44#1942C2211yescharacteristic
45#1942C2211yescharacteristic
46#1942C2211yescharacteristic
47#1865—21no—
48#1865—21no—
49#1864—21no—
50#1846C2211yescharacteristic
51#1805—21no—
52C8×C4⋊C4C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59#1374C2211yescharacteristic
60#1374C2211yescharacteristic
61C22×Q16C2311yescharacteristic
62C22×D8C2311yescharacteristic
63C2×#125—21no—
64C2×C4×Q8—21no—
65C2×C4×Q8—21no—
66C2×C4×Q8—21no—
67C2×C4×D4—21no—
68C2×SD32—22no—
69C2×SD32—22no—
70C2×SD32Q814yes—
71C22×C16C2311yescharacteristic
72C22×C16C2311yescharacteristic
73C22×C16C2311yescharacteristic
74C22×C16C2311yescharacteristic
75#499—22no—
76#499—41no—
77C4×Q16—22no—
78C4×Q16—22no—
79C4×Q16—22no—
80C4×Q16—22no—
81C4×Q16—41no—
82C4×Q16—41no—
83C4×D8—22no—
84C4×D8—41no—
85C2×#107C2311yescharacteristic
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#107C2311yescharacteristic
89C2×#107C2311yescharacteristic
90C2×#107C2311yescharacteristic
91C2×#106—21no—
92C2×#106—21no—
93C2×C4⋊C8D411yescharacteristic
94C2×#103—21no—
95C2×#103—21no—
96C2×#103—21no—
97C2×#102—21no—
98C2×#102—21no—
99C2×#102—21no—
100C2×C4×C8C2311yescharacteristic
101C2×C4×C8C2311yescharacteristic
102C2×C4×C8C2311yescharacteristic
103#399—41no—
104#396—41no—
105#390—21no—
106#390—21no—
107#389—21no—
108#389—21no—
109#389—21no—
110#389—21no—
111#389—41no—
112#389—41no—
113#385—41no—
114#382—21no—
115#382—41no—
116#380—41no—
117C4×C4⋊C4—41no—
118C4×C4⋊C4D411yescharacteristic
119#367—22no—
120#367—22no—
121#367—22no—
122#366—22no—
123#366—41no—
124C4⋊C16—22no—
125#358—22no—
126#358—22no—
127#358D414yes—
128#357—22no—
129#357—22no—
130#357—41no—
131C4×C16—22no—
132#340—21no—
133#340—21no—
134#340—21no—
135#340—21no—
136#336D411yescharacteristic
137#336D411yescharacteristic
138C22×Q8—21no—
139C22×D4—21no—
140C23×C4—41no—
141C2×Q16—21no—
142C2×Q16—21no—
143C2×Q16—21no—
144C2×Q16C2×D412yes—
145C2×Q16C2×Q812yes—
146C2×Q16#4112yes—
147C2×D8—21no—
148C2×D8—21no—
149C2×D8—21no—
150C2×D8—21no—
151C2×D8—21no—
152C2×D8C2×Q812yes—
153C22×C8C2×D411yescharacteristic
154C22×C8C2×D411yescharacteristic
155C22×C8C2×D411yescharacteristic
156C22×C8C2411yescharacteristic
157#125—42no—
158#125—81no—
159C4×Q8—24no—
160C4×Q8—42no—
161C4×Q8—42no—
162C4×Q8—42no—
163C4×Q8—81no—
164C4×Q8—81no—
165C4×D4—42no—
166C4×D4—81no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
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—41no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4D811yescharacteristic
183C2×C4⋊C4D811yescharacteristic
184C2×C22⋊C4—41no—
185C2×C22⋊C4—41no—
186C2×C22⋊C4—41no—
187C2×C22⋊C4—41no—
188C2×C22⋊C4—41no—
189C2×C42—41no—
190C2×C42—41no—
191C2×C42—41no—
192C2×C42C2×D411yescharacteristic
193C2×C42C2×D411yescharacteristic
194C2×C42C2×D411yescharacteristic
195SD32—28no—
196C2×C16—21no—
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×C16C2×D412yes—
207C2×C16C2×Q812yes—
208#107—21no—
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#107C2×D412yes—
219#107#4112yes—
220#106—42no—
221#106—81no—
222C4⋊C8—41no—
223#103—24no—
224#103—42no—
225#103—42no—
226#102—42no—
227#102—42no—
228#102—81no—
229C4×C8—21no—
230C4×C8—21no—
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#95—41no—
241#95D811yescharacteristic
242#95D811yescharacteristic
243C24—41no—
244C2×Q8—22no—
245C2×Q8—22no—
246C2×Q8—22no—
247C2×Q8—41no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×D4—22no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—41no—
255C2×D4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—82no—
264C22×C4—82no—
265C22×C4—161no—
266C22×C4C2×D811yescharacteristic
267C22×C4C2×D811yescharacteristic
268C22×C4C2×D811yescharacteristic
269C22×C4C2×D811yescharacteristic
270C22×C4C2×D811yescharacteristic
271C22×C4C2×D811yescharacteristic
272C22×C4C22×D411yescharacteristic
273Q16—22no—
274Q16—22no—
275Q16—22no—
276Q16C22⋊Q814yes—
277D8—22no—
278D8—22no—
279D8—41no—
280D8—41no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—21no—
290C2×C8C22×D411yescharacteristic
291C2×C8C22×Q811yescharacteristic
292C2×C8C2×#4111yescharacteristic
293C2×C8C2×#4111yescharacteristic
294C2×C8H(5,𝔽2)11yescharacteristic
295C2×C82−1+411yescharacteristic
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—41no—
300C4⋊C4—44no—
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—82no—
312C4⋊C4—82no—
313C4⋊C4—82no—
314C4⋊C4—82no—
315C4⋊C4—82no—
316C4⋊C4—82no—
317C4⋊C4—161no—
318C4⋊C4—161no—
319C22⋊C4—82no—
320C22⋊C4—82no—
321C22⋊C4—161no—
322C22⋊C4—161no—
323C22⋊C4—161no—
324C42—21no—
325C42—21no—
326C42—21no—
327C42—21no—
328C42—21no—
329C42—21no—
330C42—44no—
331C42—82no—
332C42—82no—
333C16—22no—
334C16—22no—
335C16—41no—
336C16—41no—
337C23—42no—
338C23—81no—
339C23—81no—
340C23—81no—
341C23—81no—
342C23—81no—
343C23—81no—
344C23C22×D811yescharacteristic
345Q8—24no—
346Q8—42no—
347Q8—42no—
348Q8—42no—
349D4—42no—
350D4—42no—
351D4—81no—
352D4—81no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
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—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
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—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—164no—
392C2×C4#50812yes—
393C2×C4#52511yescharacteristic
394C2×C4D4211yescharacteristic
395C2×C4D4×Q811yescharacteristic
396C2×C4#57511yescharacteristic
397C2×C4#57611yescharacteristic
398C2×C4#57811yescharacteristic
399C8—41no—
400C8—41no—
401C8—41no—
402C8C2×C22⋊Q811yescharacteristic
403C8#53911yescharacteristic
404C8#54711yescharacteristic
405C8#55411yescharacteristic
406C22—82no—
407C22—82no—
408C22—82no—
409C22—161no—
410C22—161no—
411C22—161no—
412C22—161no—
413C22#310011yescharacteristic
414C22D4×D811yescharacteristic
415C22Q8×D811yescharacteristic
416C22C2×#50811yescharacteristic
417C22#341411yescharacteristic
418C22#341611yescharacteristic
419C22#341611yescharacteristic
420C4—22no—
421C4—44no—
422C4—41no—
423C4—41no—
424C4—41no—
425C4—41no—
426C4—41no—
427C4—82no—
428C4—82no—
429C4—161no—
430C4—161no—
431C4—161no—
432C4—161no—
433C4#244911yescharacteristic
434C4#312411yescharacteristic
435C4#326911yescharacteristic
436C4#338911yescharacteristic
437C2—162no—
438C2#2636811yescharacteristic
439C2#3366211yescharacteristic
440C2#3368611yescharacteristic
441C2#3383711yescharacteristic
442C2#3384811yescharacteristic
443C2#3394811yescharacteristic
444C2#3395511yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D8C23
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ C8×C4⋊C4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2229C2semidirect
4. #22685C2semidirect
6. #13507C2semidirect
8. #13453C2semidirect
9. #13166C2semidirect
10. #13104C2semidirect
13. #12938C2semidirect
14. #12936C2semidirect

Maximal subgroups

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

subgroupindex
C2×#22292
C2×C4×SD322
#264822
#226852
#135282
#135072
#134762
#134532
#131662
#131042
#130132
#129382
#129362
#129352