Finite Group Databasebeta

← #509724

Subgroups — #509724

Order 512.

503 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2255C211yesmaximal, characteristic, complemented
2C2×#2228C211yesmaximal, characteristic
3#28812C211yesmaximal, characteristic, complemented
4#26482C211yesmaximal, characteristic
5#13528C211yesmaximal, characteristic
6#13516C211yesmaximal, characteristic, complemented
7#13507C211yesmaximal, characteristic, complemented
8#13453C211yesmaximal, characteristic, complemented
9#13453C211yesmaximal, characteristic, complemented
10#13363C211yesmaximal, characteristic
11#13149C211yesmaximal, characteristic
12#13131C211yesmaximal, characteristic
13#13109C211yesmaximal, characteristic, complemented
14#13105C211yesmaximal, characteristic, complemented
15#12935C211yesmaximal, characteristic
16C2×#500C2211yescharacteristic
17C2×#499C2211yescharacteristic
18C2×#499C2211yescharacteristic
19C2×C4×D8C2211yescharacteristic
20#2550—21no—
21#2454—21no—
22#2255—24no—
23#2228—24no—
24C2×#367C2211yescharacteristic
25C2×#367C2211yescharacteristic
26C2×#367C2211yescharacteristic
27C2×#366C2211yescharacteristic
28C2×#366C2211yescharacteristic
29C2×#366C2211yescharacteristic
30C2×C4⋊C16C2211yescharacteristic
31C2×#357C2211yescharacteristic
32C2×#357C2211yescharacteristic
33C2×#357C2211yescharacteristic
34C2×#357C2211yescharacteristic
35C2×#346C2211yescharacteristic
36#2065—21no—
37#2059—21no—
38#2058—21no—
39#2025—21no—
40#1970C2211yescharacteristic
41#1970C2211yescharacteristic
42#1949—21no—
43#1942C2211yescharacteristic
44#1942C2211yescharacteristic
45#1942C2211yescharacteristic
46#1942C2211yescharacteristic
47#1864—21no—
48#1846C2211yescharacteristic
49C4×#107C2211yescharacteristic
50C8×C4⋊C4C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59#1374C2211yescharacteristic
60#1374C2211yescharacteristic
61C22×D8C2311yescharacteristic
62C2×C4⋊Q8—21no—
63C2×#125—21no—
64C2×#125—21no—
65C2×C4×D4—21no—
66C22×C16C2311yescharacteristic
67C22×C16C2311yescharacteristic
68C22×C16C2311yescharacteristic
69C22×C16C2311yescharacteristic
70#500—22no—
71#500—41no—
72#499—22no—
73#499—22no—
74#499—41no—
75#499—41no—
76C4×D8—22no—
77C4×D8—41no—
78C2×#107—21no—
79C2×#107C2311yescharacteristic
80C2×#107C2311yescharacteristic
81C2×#107C2311yescharacteristic
82C2×#107C2311yescharacteristic
83C2×#107C2311yescharacteristic
84C2×#107C2311yescharacteristic
85C2×#107C2311yescharacteristic
86C2×#106—21no—
87C2×#106—21no—
88C2×#106—21no—
89C2×C4⋊C8D411yescharacteristic
90C2×#102—21no—
91C2×#102—21no—
92C2×#102—21no—
93C2×C4×C8C2311yescharacteristic
94C2×C4×C8C2311yescharacteristic
95C2×C4×C8C2311yescharacteristic
96#399—41no—
97#398—41no—
98#396—41no—
99#395—41no—
100#391—41no—
101#390—21no—
102#390—21no—
103#389—21no—
104#389—21no—
105#389—21no—
106#389—21no—
107#389—41no—
108#385—41no—
109#382—21no—
110#380—41no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4D411yescharacteristic
113#367—22no—
114#367—22no—
115#367—22no—
116#366—22no—
117#366—41no—
118#366D414yes—
119C4⋊C16—22no—
120#357—22no—
121#357—22no—
122#357—41no—
123#357Q814yescomplemented
124#346—22no—
125#340—21no—
126#340—21no—
127#340—21no—
128#340—21no—
129#340—21no—
130#336D411yescharacteristic
131#336D411yescharacteristic
132C22×Q8—41no—
133C22×D4—21no—
134C23×C4—41no—
135C2×D8—21no—
136C2×D8—21no—
137C2×D8—21no—
138C2×D8—21no—
139C2×D8—21no—
140C2×D8C2×Q812yes—
141C22×C8C2×D411yescharacteristic
142C22×C8C2×D411yescharacteristic
143C22×C8C2×D411yescharacteristic
144C22×C8C2411yescharacteristic
145C4⋊Q8—42no—
146C4⋊Q8—81no—
147#125—42no—
148#125—42no—
149#125—81no—
150#125—81no—
151C4×D4—42no—
152C4×D4—81no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4D811yescharacteristic
170C2×C4⋊C4D811yescharacteristic
171C2×C22⋊C4—41no—
172C2×C22⋊C4—41no—
173C2×C22⋊C4—41no—
174C2×C22⋊C4—41no—
175C2×C22⋊C4—41no—
176C2×C42—41no—
177C2×C42C2×D411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C16—21no—
181C2×C16—21no—
182C2×C16—21no—
183C2×C16—21no—
184C2×C16—21no—
185C2×C16—21no—
186C2×C16—21no—
187C2×C16—21no—
188C2×C16—21no—
189C2×C16—21no—
190C2×C16C2×D412yes—
191C2×C16C2×Q812yes—
192#107—21no—
193#107—21no—
194#107—21no—
195#107—21no—
196#107—21no—
197#107—21no—
198#107—21no—
199#107—21no—
200#107—21no—
201#107—21no—
202#107—21no—
203#107—42no—
204#107C2×D412yes—
205#107C2×D412yes—
206#107C2×Q812yes—
207#106—42no—
208#106—42no—
209#106—81no—
210C4⋊C8—22no—
211#102—42no—
212#102—42no—
213#102—81no—
214C4×C8—21no—
215C4×C8—21no—
216C4×C8—21no—
217C4×C8—21no—
218C4×C8—21no—
219C4×C8—21no—
220#95—41no—
221#95—41no—
222#95—41no—
223#95—41no—
224#95—41no—
225#95—41no—
226#95—41no—
227#95D811yescharacteristic
228#95D811yescharacteristic
229C24—41no—
230C2×Q8—82no—
231C2×Q8—82no—
232C2×Q8—161no—
233C2×D4—22no—
234C2×D4—41no—
235C2×D4—41no—
236C2×D4—41no—
237C2×D4—41no—
238C2×D4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—82no—
247C22×C4—82no—
248C22×C4—161no—
249C22×C4C2×D811yescharacteristic
250C22×C4C2×D811yescharacteristic
251C22×C4C2×D811yescharacteristic
252C22×C4C2×D811yescharacteristic
253C22×C4C2×D811yescharacteristic
254C22×C4C2×D811yescharacteristic
255C22×C4C22×D411yescharacteristic
256D8—22no—
257D8—22no—
258D8—41no—
259D8—41no—
260C2×C8—21no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8—21no—
264C2×C8—21no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8#13712yes—
269C2×C8C22×D411yescharacteristic
270C2×C8C22×Q811yescharacteristic
271C2×C8H(5,𝔽2)11yescharacteristic
272C2×C82−1+411yescharacteristic
273C2×C82−1+411yescharacteristic
274C2×C82−1+411yescharacteristic
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—82no—
292C4⋊C4—82no—
293C4⋊C4—82no—
294C4⋊C4—82no—
295C4⋊C4—82no—
296C4⋊C4—82no—
297C4⋊C4—161no—
298C4⋊C4—161no—
299C4⋊C4—161no—
300C22⋊C4—82no—
301C22⋊C4—82no—
302C22⋊C4—161no—
303C22⋊C4—161no—
304C22⋊C4—161no—
305C42—21no—
306C42—21no—
307C42—21no—
308C42—21no—
309C42—21no—
310C42—21no—
311C42—161no—
312C16—22no—
313C16—22no—
314C16—41no—
315C16—41no—
316C23—42no—
317C23—81no—
318C23—81no—
319C23—81no—
320C23—81no—
321C23—81no—
322C23—81no—
323C23C22×D811yescharacteristic
324Q8—164no—
325D4—42no—
326D4—42no—
327D4—81no—
328D4—81no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—164no—
369C2×C4#53611yescharacteristic
370C2×C4D4211yescharacteristic
371C2×C4D4×Q811yescharacteristic
372C2×C4C2×#13711yescharacteristic
373C2×C4#57611yescharacteristic
374C2×C4#57811yescharacteristic
375C8—22no—
376C8—41no—
377C8—41no—
378C8#53611yescharacteristic
379C8#54411yescharacteristic
380C8D4×Q811yescharacteristic
381C8#56311yescharacteristic
382C22—82no—
383C22—82no—
384C22—82no—
385C22—161no—
386C22—161no—
387C22—161no—
388C22—161no—
389C22#322611yescharacteristic
390C22D4×D811yescharacteristic
391C22Q8×D811yescharacteristic
392C22#341411yescharacteristic
393C22#341411yescharacteristic
394C22#341511yescharacteristic
395C22#341611yescharacteristic
396C4—41no—
397C4—41no—
398C4—41no—
399C4—41no—
400C4—41no—
401C4—41no—
402C4—82no—
403C4—82no—
404C4—82no—
405C4—161no—
406C4—161no—
407C4—161no—
408C4—161no—
409C4#267711yescharacteristic
410C4#324311yescharacteristic
411C4#327011yescharacteristic
412C4#338511yescharacteristic
413C2—162no—
414C2#3094611yescharacteristic
415C2#3380611yescharacteristic
416C2#3380711yescharacteristic
417C2#3383711yescharacteristic
418C2#3383911yescharacteristic
419C2#3395511yescharacteristic
420C2#3395611yescharacteristic
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×#2255C2semidirect
3. #28812C2semidirect
6. #13516C2semidirect
7. #13507C2semidirect
8. #13453C2semidirect
9. #13453C2semidirect
13. #13109C2semidirect
14. #13105C2semidirect
123. #357Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#22552
C2×#22282
#288122
#264822
#135282
#135162
#135072
#134532
#133632
#131492
#131312
#131092
#131052
#129352