Finite Group Databasebeta

← #323678

Subgroups — #323678

Order 512.

670 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30305C211yesmaximal, characteristic, complemented
2#29219C211yesmaximal, characteristic, complemented
3#29179C211yesmaximal, characteristic, complemented
4#29143C211yesmaximal, characteristic
5#29139C211yesmaximal, characteristic
6#22268C211yesmaximal, characteristic
7#10686C211yesmaximal, characteristic, complemented
8#10682C211yesmaximal, characteristic, complemented
9#10678C211yesmaximal, characteristic
10#10672C211yesmaximal, characteristic
11#9829C211yesmaximal, characteristic
12#9820C211yesmaximal, characteristic
13#8821C211yesmaximal, characteristic, complemented
14#8817C211yesmaximal, characteristic, complemented
15#8264C211yesmaximal, characteristic, complemented
16C2×#492C2211yescharacteristic
17#2868C2211yescharacteristic
18#2859C2211yescharacteristic
19C4×C4⋊Q8C2211yescharacteristic
20C4×#127C2211yescharacteristic
21#2099—22no—
22#2087—22no—
23#2084—22no—
24#2058—22no—
25#2025—22no—
26#2023—22no—
27#1990C2211yescharacteristic
28#1990C2211yescharacteristic
29#1990C2211yescharacteristic
30#1990C2211yescharacteristic
31#1988—22no—
32#1957C2211yescharacteristic
33#1957C2211yescharacteristic
34#1957C2211yescharacteristic
35#1957C2211yescharacteristic
36#1957C2211yescharacteristic
37#1956C2211yescharacteristic
38#1921—22no—
39#1880—22no—
40#1877—22no—
41#1845C2211yescharacteristic
42#1844C2211yescharacteristic
43#1844C2211yescharacteristic
44#1834C2211yescharacteristic
45#1834C2211yescharacteristic
46#1834C2211yescharacteristic
47#1834C2211yescharacteristic
48#1834C2211yescharacteristic
49#1834C2211yescharacteristic
50#1834C2211yescharacteristic
51#1834C2211yescharacteristic
52#1834C2211yescharacteristic
53#1834C2211yescharacteristic
54#1804—22no—
55#1802—22no—
56C4×C4⋊C8C2211yescharacteristic
57C4×C4⋊C8C2211yescharacteristic
58#1744C2211yescharacteristic
59C2×#322C2211yescharacteristic
60C2×C82C2211yescharacteristic
61#1273C2211yescharacteristic
62#1273C2211yescharacteristic
63C22×SD16—22no—
64C2×C4⋊Q8—21no—
65C2×C4⋊Q8—21no—
66C2×C4⋊Q8—21no—
67C2×C4⋊Q8C2311yescharacteristic
68C2×#127C2311yescharacteristic
69C2×#125—21no—
70C2×C4×Q8—21no—
71C2×C4×D4—21no—
72C2×C4×D4—21no—
73C2×C4×D4—21no—
74#492—22no—
75#492Q814yescomplemented
76C2×#107—22no—
77C2×#106—22no—
78C2×#106—22no—
79C2×C4⋊C8D412yes—
80C2×C4⋊C8D412yes—
81C2×#103—22no—
82C2×#102—22no—
83C2×#102—22no—
84C2×#102—22no—
85C2×#102—22no—
86C2×#102—22no—
87C2×C4×C8C2311yescharacteristic
88C2×C4×C8C2311yescharacteristic
89C2×C4×C8C2311yescharacteristic
90C2×C4×C8C2311yescharacteristic
91C2×C4×C8C2311yescharacteristic
92C2×C4×C8C2311yescharacteristic
93#391—21no—
94#390—21no—
95#390—21no—
96#390—21no—
97#389—21no—
98#389—21no—
99#389—21no—
100#389—21no—
101#389—21no—
102#389—21no—
103#384—21no—
104#384—21no—
105#384—21no—
106#384C2311yescharacteristic
107#384C2311yescharacteristic
108#384C2311yescharacteristic
109#384C2311yescharacteristic
110#384C2311yescharacteristic
111#384C2311yescharacteristic
112#382—21no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—21no—
115C43C2311yescharacteristic
116#340—22no—
117#340—22no—
118#340—22no—
119#340—22no—
120#340—22no—
121#340—22no—
122#340—22no—
123#340—22no—
124#340—22no—
125#340—22no—
126#340—22no—
127#340—22no—
128#340—22no—
129#340—22no—
130#336D412yes—
131#336D412yes—
132#336D412yes—
133#336D412yes—
134#322—22no—
135C82—22no—
136C22×Q8—21no—
137C22×Q8—41no—
138C22×D4—21no—
139C22×D4—21no—
140C22×D4—21no—
141C23×C4—41no—
142C2×SD16—28no—
143C2×SD16—82no—
144C2×SD16—82no—
145C22×C8C2×D412yes—
146C22×C8C2×D412yes—
147C22×C8C2×D412yes—
148C22×C8C2×D412yes—
149C22×C8C2×D412yes—
150C22×C8C2×D412yes—
151C4⋊Q8—21no—
152C4⋊Q8—21no—
153C4⋊Q8—21no—
154C4⋊Q8—44no—
155C4⋊Q8—82no—
156C4⋊Q8—82no—
157C4⋊Q8C2×Q812yes—
158#127—21no—
159#127—21no—
160#127—21no—
161#127C2×Q812yes—
162#125—82no—
163C4×Q8—82no—
164C4×D4—44no—
165C4×D4—44no—
166C4×D4—82no—
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—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
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—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C22⋊C4—41no—
194C2×C22⋊C4—41no—
195C2×C22⋊C4—41no—
196C2×C42—41no—
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2411yescharacteristic
204#107—44no—
205#106—82no—
206#106—82no—
207C4⋊C8—24no—
208C4⋊C8—42no—
209#103—82no—
210#102—82no—
211#102—82no—
212#102—82no—
213#102—82no—
214#102—82no—
215C4×C8—21no—
216C4×C8—21no—
217C4×C8—21no—
218C4×C8—21no—
219C4×C8—21no—
220C4×C8—21no—
221C4×C8—21no—
222C4×C8—21no—
223C4×C8—21no—
224C4×C8—21no—
225C4×C8C2×Q812yes—
226C4×C8#4112yes—
227#95—41no—
228#95—41no—
229#95—41no—
230#95—41no—
231C24—41no—
232C2×Q8—22no—
233C2×Q8—41no—
234C2×Q8—42no—
235C2×Q8—42no—
236C2×Q8—82no—
237C2×Q8—82no—
238C2×Q8—82no—
239C2×D4—22no—
240C2×D4—22no—
241C2×D4—22no—
242C2×D4—41no—
243C2×D4—41no—
244C2×D4—41no—
245C2×D4—42no—
246C2×D4—42no—
247C2×D4—42no—
248C2×D4—42no—
249C2×D4—42no—
250C2×D4—42no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—82no—
259C22×C4—82no—
260C22×C4—82no—
261C22×C4#12711yescharacteristic
262C22×C4#12711yescharacteristic
263C22×C4#12711yescharacteristic
264C22×C4#12711yescharacteristic
265C22×C4C22×D411yescharacteristic
266C22×C4C22×D411yescharacteristic
267C22×C4C22×D411yescharacteristic
268SD16—88no—
269C2×C8—22no—
270C2×C8—22no—
271C2×C8—22no—
272C2×C8—22no—
273C2×C8—22no—
274C2×C8—22no—
275C2×C8—22no—
276C2×C8—22no—
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—22no—
280C2×C8—22no—
281C2×C8—22no—
282C2×C8—22no—
283C2×C8—22no—
284C2×C8—22no—
285C2×C8—22no—
286C2×C8C22⋊Q814yes—
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—82no—
310C4⋊C4—82no—
311C4⋊C4—82no—
312C4⋊C4—82no—
313C4⋊C4—82no—
314C4⋊C4—82no—
315C22⋊C4—82no—
316C22⋊C4—82no—
317C22⋊C4—82no—
318C42—22no—
319C42—22no—
320C42—22no—
321C42—22no—
322C42—22no—
323C42—82no—
324C42#13514yes—
325C42C22×Q811yescharacteristic
326C42C2×#4111yescharacteristic
327C422−1+411yescharacteristic
328C422−1+411yescharacteristic
329C23—42no—
330C23—81no—
331C23—81no—
332C23—81no—
333C23—81no—
334C23—81no—
335C23—81no—
336C23C2×#12711yescharacteristic
337Q8—44no—
338Q8—82no—
339Q8—164no—
340D4—44no—
341D4—44no—
342D4—44no—
343D4—82no—
344D4—82no—
345D4—82no—
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—42no—
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—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—164no—
378C2×C4#49512yes—
379C2×C4#49512yes—
380C2×C4C2×C22⋊Q811yescharacteristic
381C2×C4#53611yescharacteristic
382C2×C4#54711yescharacteristic
383C2×C4#54711yescharacteristic
384C2×C4D4×Q811yescharacteristic
385C2×C4D4×Q811yescharacteristic
386C2×C4C2×#13512yes—
387C2×C4#57512yes—
388C2×C4#57612yes—
389C2×C4#57712yes—
390C2×C4#57712yes—
391C2×C4#57712yes—
392C8—24no—
393C8—42no—
394C8—42no—
395C8—42no—
396C8—42no—
397C8—42no—
398C22—82no—
399C22—82no—
400C22—82no—
401C22—161no—
402C22—161no—
403C22—161no—
404C22—161no—
405C22#254211yescharacteristic
406C22C2×#49511yescharacteristic
407C22#315311yescharacteristic
408C22#315411yescharacteristic
409C22#315411yescharacteristic
410C22#315411yescharacteristic
411C22#315511yescharacteristic
412C4—22no—
413C4—22no—
414C4—41no—
415C4—41no—
416C4—82no—
417C4—161no—
418C4—161no—
419C4—161no—
420C4—161no—
421C4—161no—
422C4—161no—
423C4#308112yes—
424C4#324112yes—
425C4#334012yes—
426C4#334812yes—
427C4#338112yes—
428C4#338812yes—
429C2—162no—
430C2#1963711yescharacteristic
431C2#1965811yescharacteristic
432C2#1972211yescharacteristic
433C2#1973411yescharacteristic
434C2#2852511yescharacteristic
435C2#2852911yescharacteristic
436C2#2854911yescharacteristic
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #30305C2semidirect
2. #29219C2semidirect
3. #29179C2semidirect
7. #10686C2semidirect
8. #10682C2semidirect
13. #8821C2semidirect
14. #8817C2semidirect
15. #8264C2semidirect
75. #492Q8semidirect

Maximal subgroups

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

subgroupindex
#303052
#292192
#291792
#291432
#291392
#222682
#106862
#106822
#106782
#106722
#98292
#98202
#88212
#88172
#82642