Finite Group Databasebeta

← #232154

Subgroups — #232154

Order 512.

433 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19748C211yesmaximal, characteristic
2#9430C211yesmaximal, characteristic, complemented
3#9429C211yesmaximal, characteristic, complemented
4#9427C211yesmaximal, characteristic, complemented
5#9421C211yesmaximal, characteristic, complemented
6#9420C211yesmaximal, characteristic, complemented
7#9419C211yesmaximal, characteristic, complemented
8#9418C211yesmaximal, characteristic, complemented
9#9418C211yesmaximal, characteristic, complemented
10#9417C211yesmaximal, characteristic
11#9417C211yesmaximal, characteristic
12#9415C211yesmaximal, characteristic, complemented
13#9415C211yesmaximal, characteristic, complemented
14#8804C211yesmaximal, characteristic, complemented
15#8803C211yesmaximal, characteristic, complemented
16#3184C2211yescharacteristic
17#3182C2211yescharacteristic
18#3182C2211yescharacteristic
19#3165C2211yescharacteristic
20#3152C2211yescharacteristic
21#3141C2211yescharacteristic
22#2987C2211yescharacteristic
23#2980C2211yescharacteristic
24#2980C2211yescharacteristic
25#2381C2211yescharacteristic
26#1723C411yescharacteristic, complemented
27#1722C411yescharacteristic, complemented
28#1720C411yescharacteristic, complemented
29#1719C411yescharacteristic, complemented
30#1715C411yescharacteristic, complemented
31#1714C411yescharacteristic, complemented
32#1707C411yescharacteristic, complemented
33#1706C411yescharacteristic, complemented
34#1558C2211yescharacteristic
35#1556C2211yescharacteristic
36#1556C2211yescharacteristic
37#1537C2211yescharacteristic
38#1536C2211yescharacteristic
39#1536C2211yescharacteristic
40#1515C2211yescharacteristic
41#1515C2211yescharacteristic
42#1514C2211yescharacteristic
43#1514C2211yescharacteristic
44#1514C2211yescharacteristic
45#1514C2211yescharacteristic
46#1455C2211yescharacteristic
47#1291C2211yescharacteristic
48#1291C2211yescharacteristic
49#1291C2211yescharacteristic
50#1291C2211yescharacteristic
51#1291C2211yescharacteristic
52#1291C2211yescharacteristic
53#1288C2211yescharacteristic
54#1288C2211yescharacteristic
55#1288C2211yescharacteristic
56#1288C2211yescharacteristic
57#1288C2211yescharacteristic
58#1288C2211yescharacteristic
59C2×#137—41no—
60#533C2311yescharacteristic
61#532C2311yescharacteristic
62#532C2311yescharacteristic
63C2×C4⋊Q8C2311yescharacteristic
64#501—41no—
65#500C2×C411yescharacteristic
66#500C2×C411yescharacteristic
67#500C2×C411yescharacteristic
68#500C2×C411yescharacteristic
69#498C2×C411yescharacteristic
70#498C2×C411yescharacteristic
71#498C2×C411yescharacteristic
72#497—41no—
73#494C2×C411yescharacteristic
74#494C2×C411yescharacteristic
75#493C2×C411yescharacteristic
76#492C2×C411yescharacteristic
77#492C2×C411yescharacteristic
78#492C2×C411yescharacteristic
79#492C2×C411yescharacteristic
80#489—41no—
81#487C2×C411yescharacteristic
82#487C2×C411yescharacteristic
83#487C2×C411yescharacteristic
84#487C2×C411yescharacteristic
85#486C2×C411yescharacteristic
86#486C2×C411yescharacteristic
87#486C2×C411yescharacteristic
88#486C2×C411yescharacteristic
89#486C2×C411yescharacteristic
90#486C2×C411yescharacteristic
91#469—41no—
92#468—41no—
93#468—41no—
94#446—41no—
95#446—41no—
96#442—41no—
97#440—41no—
98#440—41no—
99#435—41no—
100#435—41no—
101#435—41no—
102#435—41no—
103#433—41no—
104#432C2311yescharacteristic
105#432C2311yescharacteristic
106#432C2311yescharacteristic
107#432C2311yescharacteristic
108#432C2311yescharacteristic
109#432C2311yescharacteristic
110#428—41no—
111#417—41no—
112#408—41no—
113#408—41no—
114C4×M16C2311yescharacteristic
115C4×M16C2311yescharacteristic
116C4×M16C2311yescharacteristic
117#391—41no—
118#387—41no—
119#387—41no—
120#383C2311yescharacteristic
121#383C2311yescharacteristic
122#333—41no—
123#332—41no—
124#332—41no—
125#330—41no—
126#329—41no—
127#329—41no—
128#328—41no—
129#328—41no—
130#327—41no—
131#327—41no—
132#327—41no—
133#327—41no—
134#322C2×C411yescharacteristic
135#322C2×C411yescharacteristic
136#322C2×C411yescharacteristic
137C82C2×C411yescharacteristic
138C2×#41—21no—
139C2×#41—21no—
140C22×Q8—21no—
141#137—82no—
142#137—82no—
143C2×Q16—41no—
144C2×Q16—41no—
145C2×SD16—41no—
146C2×SD16—41no—
147C2×SD16—41no—
148C2×SD16—41no—
149C2×D8—41no—
150C2×M16—21no—
151C2×M16—21no—
152C2×M16—21no—
153C2×M16—41no—
154C2×M16—41no—
155C22×C8—41no—
156C22×C8—41no—
157C4⋊Q8—21no—
158C4⋊Q8—21no—
159C4⋊Q8—21no—
160C4⋊Q8C22×C411yescharacteristic
161C4⋊Q8C22×C411yescharacteristic
162C4⋊Q8C22×C411yescharacteristic
163C4⋊Q8C22×C411yescharacteristic
164C4⋊Q8C22×C411yescharacteristic
165C4⋊Q8C22×C411yescharacteristic
166#127C22×C411yescharacteristic
167#127C22×C411yescharacteristic
168#125—21no—
169#124—21no—
170#124—21no—
171C22⋊Q8—21no—
172C22⋊Q8—21no—
173#121—21no—
174#121—21no—
175#121—21no—
176#121—21no—
177C4×Q8—21no—
178C4×Q8—21no—
179C4×D4—21no—
180C4×D4—21no—
181C4×D4—21no—
182C4×D4—21no—
183#117—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—41no—
187C2×C42—41no—
188C2×C42C2411yescharacteristic
189#107—41no—
190#107—41no—
191#107—41no—
192#107—41no—
193#106—41no—
194#106—41no—
195#106—41no—
196C4⋊C8—21no—
197C4⋊C8—21no—
198C4⋊C8—21no—
199C4⋊C8—21no—
200C4⋊C8—21no—
201C4⋊C8—21no—
202C4⋊C8—42no—
203#103—41no—
204#103—41no—
205#103—41no—
206#103—41no—
207#102—41no—
208#102—41no—
209#102—41no—
210#102—41no—
211#102—41no—
212#102—41no—
213C22⋊C8—21no—
214C22⋊C8—21no—
215C22⋊C8—21no—
216C22⋊C8—21no—
217C22⋊C8—21no—
218C22⋊C8—21no—
219C22⋊C8—42no—
220C22⋊C8—42no—
221#97—21no—
222#97—21no—
223#97—21no—
224#97—21no—
225#97—21no—
226#97—21no—
227#97—21no—
228#97—21no—
229#97—21no—
230#97—41no—
231C4×C8—41no—
232C4×C8C22×C411yescharacteristic
233C4×C8C22×C411yescharacteristic
234C4×C8C22×C411yescharacteristic
235C4×C8C22×C411yescharacteristic
236C4×C8C22×C411yescharacteristic
237C4×C8C22×C411yescharacteristic
238#95—21no—
239#95—21no—
240#95—21no—
241#95—21no—
242#95—21no—
243#95—21no—
244#95—42no—
245#41—81no—
246#41—81no—
247#41—81no—
248#41—81no—
249C2×Q8—21no—
250C2×Q8—21no—
251C2×Q8—21no—
252C2×Q8—21no—
253C2×Q8—21no—
254C2×Q8—21no—
255C2×Q8—41no—
256C2×Q8—81no—
257C2×Q8—81no—
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—21no—
261C2×D4—21no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—41no—
265C2×D4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4H(5,𝔽2)11yescharacteristic
272C22×C4H(5,𝔽2)11yescharacteristic
273C22×C4H(5,𝔽2)11yescharacteristic
274Q16—82no—
275Q16—82no—
276SD16—82no—
277SD16—82no—
278SD16—82no—
279SD16—82no—
280D8—82no—
281M16—42no—
282M16—42no—
283M16—81no—
284M16—161no—
285M16—161no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—21no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—161no—
299C2×C8—161no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—161no—
317C22⋊C4—41no—
318C22⋊C4—41no—
319C22⋊C4—41no—
320C22⋊C4—41no—
321C22⋊C4—41no—
322C42—41no—
323C42—161no—
324C42C23×C411yescharacteristic
325C42H(5,𝔽2)11yescharacteristic
326C42H(5,𝔽2)11yescharacteristic
327C42H(5,𝔽2)11yescharacteristic
328C23—41no—
329C23—41no—
330C23#56011yescharacteristic
331Q8—81no—
332Q8—81no—
333Q8—81no—
334Q8—81no—
335Q8—81no—
336Q8—81no—
337Q8—161no—
338D4—81no—
339D4—81no—
340D4—81no—
341D4—81no—
342D4—81no—
343D4—81no—
344D4—161no—
345D4—161no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
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#51811yescharacteristic
368C2×C4#51811yescharacteristic
369C2×C4#51811yescharacteristic
370C2×C4#56011yescharacteristic
371C2×C4#56011yescharacteristic
372C2×C4#56011yescharacteristic
373C8—22no—
374C8—22no—
375C8—41no—
376C8—41no—
377C8—41no—
378C8—41no—
379C8—81no—
380C8—81no—
381C8—81no—
382C8—81no—
383C8—81no—
384C8—81no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—81no—
389C22—81no—
390C22—81no—
391C22—81no—
392C22—81no—
393C22—81no—
394C22#238011yescharacteristic
395C4—21no—
396C4—21no—
397C4—21no—
398C4—21no—
399C4—21no—
400C4—21no—
401C4—81no—
402C4—81no—
403C4—81no—
404C4—81no—
405C4—81no—
406C4—81no—
407C4—161no—
408C4—161no—
409C2—41no—
410C2—161no—
411C2—161no—
412C2#2336011yescharacteristic
413C2#2336111yescharacteristic
414C2#2336211yescharacteristic
subgroupG/H
center, socleC22#2380
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radical#3184C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #9430C2semidirect
3. #9429C2semidirect
4. #9427C2semidirect
5. #9421C2semidirect
6. #9420C2semidirect
7. #9419C2semidirect
8. #9418C2semidirect
9. #9418C2semidirect
12. #9415C2semidirect
13. #9415C2semidirect
14. #8804C2semidirect
15. #8803C2semidirect
26. #1723C4semidirect
27. #1722C4semidirect
28. #1720C4semidirect
29. #1719C4semidirect
30. #1715C4semidirect
31. #1714C4semidirect
32. #1707C4semidirect
33. #1706C4semidirect

Maximal subgroups

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

subgroupindex
#197482
#94302
#94292
#94272
#94212
#94202
#94192
#94182
#94172
#94152
#88042
#88032