Finite Group Databasebeta

← #330439

Subgroups — #330439

Order 512.

597 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33289C211yesmaximal, characteristic, complemented
2#29981C211yesmaximal, characteristic
3#29798C211yesmaximal, characteristic
4#21266C211yesmaximal, characteristic
5#10602C212yesmaximal, complemented
6#10598C212yesmaximal, complemented
7#10477C212yesmaximal
8#10476C212yesmaximal
9#9903C211yesmaximal, characteristic, complemented
10#9703C211yesmaximal, characteristic, complemented
11#8807C211yesmaximal, characteristic, complemented
12C2×#461C2212yes—
13C2×#433—21no—
14#2868C2211yescharacteristic
15#2868C2211yescharacteristic
16#2293C2211yescharacteristic
17#2100—21no—
18#2094—21no—
19#2058—21no—
20#2043—21no—
21#2043—21no—
22#2023—21no—
23#2013—21no—
24#2013—21no—
25#1990—21no—
26#1990—21no—
27#1990C2212yes—
28#1990C2212yes—
29#1981C2212yes—
30#1954C2211yescharacteristic
31#1954C2211yescharacteristic
32#1948C2212yes—
33#1939C2211yescharacteristic
34#1939C2211yescharacteristic
35#1923—21no—
36#1923—21no—
37#1834—21no—
38#1834—21no—
39#1831C2211yescharacteristic
40#1831C2211yescharacteristic
41#1802C2212yes—
42#1767C2211yescharacteristic
43#1750—21no—
44C42×C8C2211yescharacteristic
45C2×#335C2212yes—
46C2×#334C2212yes—
47C2×#332C2212yes—
48C2×#332C2212yes—
49#1296C2212yes—
50#1296C2212yes—
51C22×SD16—22no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8C2312yes—
58C2×C4×D4C2311yescharacteristic
59C2×#117—21no—
60#461—42no—
61#461—42no—
62#433—44no—
63C2×#106—22no—
64C2×C4⋊C8—21no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8C2312yes—
67C2×C4⋊C8C2311yescharacteristic
68C2×C4⋊C8C2311yescharacteristic
69C2×C4⋊C8C2312yes—
70C2×#103—22no—
71C2×#103—22no—
72C2×#103—22no—
73C2×#102—22no—
74C2×C22⋊C8—21no—
75C2×C22⋊C8—21no—
76C2×C22⋊C8—21no—
77C2×C22⋊C8—21no—
78C2×#97—21no—
79C2×C4×C8D411yescharacteristic
80C2×C4×C8D411yescharacteristic
81C2×C4×C8D411yescharacteristic
82C2×C4×C8D411yescharacteristic
83C2×C4×C8C2311yescharacteristic
84C2×C4×C8C2311yescharacteristic
85#390D411yescharacteristic
86#390D411yescharacteristic
87#389D411yescharacteristic
88#389D411yescharacteristic
89#388—21no—
90#388—21no—
91#384—21no—
92#384—21no—
93#384—21no—
94#384—21no—
95#384—21no—
96#384—21no—
97#384—21no—
98#384C2312yes—
99C4×C4⋊C4C2311yescharacteristic
100C4×C22⋊C4—21no—
101C43C2311yescharacteristic
102#340—22no—
103#340—22no—
104#340—22no—
105#340—22no—
106#340—22no—
107#340—22no—
108#336—21no—
109#336—21no—
110#336—21no—
111#336—21no—
112#336—21no—
113#336—21no—
114#336—21no—
115#335—42no—
116#334—42no—
117#332—42no—
118#332—42no—
119C22×Q8—22no—
120C22×Q8—41no—
121C22×D4C2×D411yescharacteristic
122C23×C4—21no—
123C2×SD16—44no—
124C2×SD16—82no—
125C2×SD16—82no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—21no—
129C22×C8—21no—
130C22×C8C2×D411yescharacteristic
131C22×C8C2×D411yescharacteristic
132C22×C8C2×D411yescharacteristic
133C22×C8C2×D411yescharacteristic
134C4⋊Q8—22no—
135C4⋊Q8—22no—
136C4⋊Q8—22no—
137C4⋊Q8—22no—
138C4⋊Q8—42no—
139C4⋊Q8—42no—
140C4⋊Q8—82no—
141C4⋊Q8—82no—
142C4⋊Q8—82no—
143C4⋊Q8—82no—
144C4×D4—21no—
145C4×D4—21no—
146C4×D4—21no—
147C4×D4—21no—
148#117—42no—
149#117—42no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4C2×D411yescharacteristic
165C2×C4⋊C4C2×D411yescharacteristic
166C2×C4⋊C4C2×D411yescharacteristic
167C2×C22⋊C4—21no—
168C2×C22⋊C4—21no—
169C2×C22⋊C4—21no—
170C2×C22⋊C4—21no—
171C2×C22⋊C4—21no—
172C2×C42—21no—
173C2×C42C2×D411yescharacteristic
174C2×C42C2×D411yescharacteristic
175C2×C42C2×D411yescharacteristic
176C2×C42C2×D411yescharacteristic
177C2×C42#4111yescharacteristic
178C2×C42#4111yescharacteristic
179C2×C42C2411yescharacteristic
180#106—82no—
181C4⋊C8—21no—
182C4⋊C8—21no—
183C4⋊C8—22no—
184C4⋊C8—22no—
185C4⋊C8—22no—
186C4⋊C8—22no—
187C4⋊C8—21no—
188C4⋊C8—21no—
189C4⋊C8—42no—
190C4⋊C8—42no—
191#103—44no—
192#103—82no—
193#103—82no—
194#102—82no—
195C22⋊C8—42no—
196C22⋊C8—42no—
197C22⋊C8—42no—
198C22⋊C8—42no—
199#97—42no—
200C4×C8—22no—
201C4×C8—21no—
202C4×C8—21no—
203C4×C8—21no—
204C4×C8—21no—
205C4×C8—41no—
206C4×C8—41no—
207C4×C8SD1614yescomplemented
208#95—21no—
209#95—21no—
210#95—21no—
211C24—21no—
212C2×Q8—42no—
213C2×Q8—42no—
214C2×Q8—42no—
215C2×Q8—42no—
216C2×Q8—42no—
217C2×Q8—42no—
218C2×Q8—82no—
219C2×Q8—82no—
220C2×Q8—82no—
221C2×D4—21no—
222C2×D4—21no—
223C2×D4—22no—
224C2×D4—22no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4#12111yescharacteristic
237C22×C4#12111yescharacteristic
238C22×C4#12111yescharacteristic
239C22×C4#12111yescharacteristic
240C22×C4C22×D411yescharacteristic
241C22×C4C22×D411yescharacteristic
242C22×C4C2×#4111yescharacteristic
243SD16—88no—
244C2×C8—21no—
245C2×C8—21no—
246C2×C8—21no—
247C2×C8—21no—
248C2×C8—21no—
249C2×C8—21no—
250C2×C8—21no—
251C2×C8—21no—
252C2×C8—21no—
253C2×C8—21no—
254C2×C8—42no—
255C2×C8—42no—
256C2×C8—41no—
257C2×C8—41no—
258C2×C8—41no—
259C2×C8—41no—
260C2×C8—41no—
261C2×C8—41no—
262C2×C8—42no—
263C2×C8—42no—
264C2×C8—42no—
265C2×C8—42no—
266C2×C8C2×SD1612yes—
267C2×C8C2×SD1612yes—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—22no—
271C4⋊C4—22no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—82no—
283C4⋊C4—82no—
284C4⋊C4—82no—
285C4⋊C4—82no—
286C4⋊C4—82no—
287C4⋊C4—82no—
288C4⋊C4—82no—
289C4⋊C4—82no—
290C22⋊C4—41no—
291C22⋊C4—41no—
292C22⋊C4—42no—
293C22⋊C4—42no—
294C22⋊C4—42no—
295C22⋊C4—42no—
296C42—21no—
297C42—21no—
298C42—22no—
299C42—22no—
300C42—22no—
301C42—22no—
302C42—41no—
303C42—41no—
304C42C2×SD1612yes—
305C42#13712yes—
306C42H(5,𝔽2)11yescharacteristic
307C42H(5,𝔽2)11yescharacteristic
308C422−1+411yescharacteristic
309C422−1+411yescharacteristic
310C23—41no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23C2×#12111yescharacteristic
318Q8—82no—
319Q8—82no—
320Q8—82no—
321Q8—82no—
322Q8—164no—
323D4—42no—
324D4—42no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—82no—
349C2×C4—82no—
350C2×C4—82no—
351C2×C4—82no—
352C2×C4—82no—
353C2×C4—82no—
354C2×C4#46112yes—
355C2×C4#46112yes—
356C2×C4#47112yes—
357C2×C4#47312yes—
358C2×C4#53511yescharacteristic
359C2×C4#53611yescharacteristic
360C2×C4#53711yescharacteristic
361C2×C4#53711yescharacteristic
362C2×C4#54211yescharacteristic
363C2×C4#54211yescharacteristic
364C2×C4C22×SD1611yescharacteristic
365C2×C4C2×#13711yescharacteristic
366C2×C4#57511yescharacteristic
367C2×C4#57511yescharacteristic
368C2×C4#57612yes—
369C2×C4#57712yes—
370C8—22no—
371C8—22no—
372C8—41no—
373C8—41no—
374C8—82no—
375C8—82no—
376C8—81no—
377C8—81no—
378C22—81no—
379C22—81no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22#262011yescharacteristic
386C22C2×#46111yescharacteristic
387C22#304011yescharacteristic
388C22#305711yescharacteristic
389C22#305811yescharacteristic
390C22C2×#47111yescharacteristic
391C22C2×#47311yescharacteristic
392C4—22no—
393C4—21no—
394C4—21no—
395C4—22no—
396C4—22no—
397C4—22no—
398C4—81no—
399C4—81no—
400C4—81no—
401C4—162no—
402C4—162no—
403C4#322412yes—
404C4#324112yes—
405C4#324611yescharacteristic
406C4#324611yescharacteristic
407C4#325211yescharacteristic
408C4#325211yescharacteristic
409C2—161no—
410C2#1919111yescharacteristic
411C2#1919211yescharacteristic
412C2#1936311yescharacteristic
413C2#1936411yescharacteristic
414C2#2982811yescharacteristic
415C2#2984711yescharacteristic
416C2#2985911yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
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. #33289C2semidirect
5. #10602C2semidirect
6. #10598C2semidirect
9. #9903C2semidirect
10. #9703C2semidirect
11. #8807C2semidirect
207. C4×C8SD16semidirect

Maximal subgroups

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

subgroupindex
#332892
#299812
#297982
#212662
#106022
#105982
#104772
#104762
#99032
#97032
#88072