Finite Group Databasebeta

← #149216

Subgroups — #149216

Order 512.

546 conjugacy classes of subgroups.

The subgroup lattice has 405 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33409C211yesmaximal, characteristic
2C2×#2159C211yesmaximal, characteristic
3#20803C211yesmaximal, characteristic
4#12567C211yesmaximal, characteristic
5#12557C211yesmaximal, characteristic
6C2×#1378C211yesmaximal, characteristic
7#10871C211yesmaximal, characteristic
8C22×M32C2211yescharacteristic
9C22×M32C2211yescharacteristic
10C2×#428—21no—
11C2×#427C2211yescharacteristic
12C2×#427C2211yescharacteristic
13C2×#427C2211yescharacteristic
14C22×#107C2211yescharacteristic
15C2×#405C2211yescharacteristic
16#2317—21no—
17#2159—22no—
18#2159—22no—
19#2159—21no—
20#2159—21no—
21#2159—21no—
22#2159—21no—
23#2158—22no—
24C2×#367—21no—
25C2×#366—21no—
26#1847—22no—
27#1846—21no—
28#1845—21no—
29#1740—21no—
30#1733—21no—
31#1386—22no—
32#1380—22no—
33#1378C414yescomplemented
34C22×M16D411yescharacteristic
35C23×C8C2311yescharacteristic
36C2×#117—21no—
37C2×#117—21no—
38C2×#117—21no—
39C2×#117D411yescharacteristic
40C22×C4⋊C4—21no—
41C2×M32—21no—
42C2×M32—21no—
43C2×M32—21no—
44C2×M32—21no—
45C2×M32—21no—
46C2×M32—21no—
47C2×M32—21no—
48C2×M32C2×C412yes—
49C2×M32C2×C412yes—
50C2×M32D412yes—
51C2×M32Q812yes—
52C2×M32Q812yes—
53C22×C16—21no—
54C22×C16D412yes—
55#428—42no—
56#428—42no—
57#427—21no—
58#427—21no—
59#427—21no—
60#427—21no—
61#427—21no—
62#427—21no—
63#427—21no—
64#427—21no—
65#427—21no—
66#427—21no—
67#427—21no—
68#427Q812yes—
69C2×#107—21no—
70C2×#107—21no—
71C2×#107—21no—
72C2×#107—21no—
73C2×#107—21no—
74C2×#107—22no—
75C2×#107—21no—
76C2×#107D411yescharacteristic
77C2×#107D411yescharacteristic
78C2×#107D412yes—
79C2×#107D411yescharacteristic
80C2×#107D411yescharacteristic
81C2×#106—21no—
82C2×#106—22no—
83C2×#106—21no—
84C2×#106—21no—
85#405—21no—
86#405—21no—
87#405C2×C412yes—
88#405Q812yes—
89C2×#97—21no—
90C2×C4×C8—21no—
91#389—42no—
92#385—42no—
93#384—22no—
94#382—22no—
95#367—24no—
96#367—22no—
97#367—22no—
98#366—24no—
99#366—22no—
100#366—22no—
101#342—42no—
102#339—42no—
103C23×C4—41no—
104C23×C4C2×D411yescharacteristic
105C2×M16—21no—
106C2×M16—21no—
107C2×M16—21no—
108C2×M16C4⋊C412yes—
109C2×M16D812yes—
110C2×M16Q1612yes—
111C22×C8—21no—
112C22×C8C22×C411yescharacteristic
113C22×C8C2×D411yescharacteristic
114C22×C8C2×D411yescharacteristic
115C22×C8C2×D411yescharacteristic
116C22×C8C2×D411yescharacteristic
117C22×C8C2×D411yescharacteristic
118C22×C8C2×Q811yescharacteristic
119C22×C8C2×Q811yescharacteristic
120C22×C8C2×Q811yescharacteristic
121C22×C8C2×Q811yescharacteristic
122C22×C8#4111yescharacteristic
123C22×C8#4111yescharacteristic
124C22×C8#4111yescharacteristic
125C22×C8#4111yescharacteristic
126#117—21no—
127#117—21no—
128#117—22no—
129#117—41no—
130#117—41no—
131#117—41no—
132#117—41no—
133#117—41no—
134#117—41no—
135#117—41no—
136#117—41no—
137#117—41no—
138#117—41no—
139#117—41no—
140#117C4⋊C412yes—
141#117Q1612yes—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—44no—
155C2×C22⋊C4—41no—
156C2×C22⋊C4—41no—
157C2×C22⋊C4—41no—
158C2×C22⋊C4D812yes—
159C2×C42—21no—
160C2×C42—41no—
161C2×C42—41no—
162C2×C42—41no—
163M32—22no—
164M32—22no—
165M32—22no—
166M32—22no—
167M32—22no—
168M32—22no—
169M32—22no—
170M32C4⋊C414yescomplemented
171C2×C16—22no—
172C2×C16—22no—
173C2×C16—22no—
174C2×C16—22no—
175C2×C16—22no—
176C2×C16C4⋊C414yescomplemented
177#107—21no—
178#107—22no—
179#107—21no—
180#107—21no—
181#107—21no—
182#107—21no—
183#107—21no—
184#107—22no—
185#107—21no—
186#107—21no—
187#107—22no—
188#107—44no—
189#107—41no—
190#107—41no—
191#107—41no—
192#106—22no—
193#106—44no—
194#106—41no—
195#106—41no—
196#106—41no—
197#106—41no—
198#106—41no—
199#97—22no—
200#97—22no—
201C4×C8—22no—
202C4×C8—22no—
203#95—44no—
204C24C2×D811yescharacteristic
205C22×C4—22no—
206C22×C4—41no—
207C22×C4—42no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—81no—
212C22×C4—81no—
213C22×C4—81no—
214C22×C4—162no—
215C22×C4C2×C4⋊C411yescharacteristic
216C22×C4#12011yescharacteristic
217C22×C4#12111yescharacteristic
218C22×C4C22⋊Q811yescharacteristic
219C22×C4#12311yescharacteristic
220C22×C4#12711yescharacteristic
221C22×C4C4⋊Q811yescharacteristic
222C22×C4C2×D811yescharacteristic
223C22×C4C2×Q1611yescharacteristic
224C22×C4C2×Q1611yescharacteristic
225C22×C4AGL(1,ℤ/8)11yescharacteristic
226C22×C4AGL(1,ℤ/8)11yescharacteristic
227C22×C4#13711yescharacteristic
228C22×C4#13711yescharacteristic
229M16—22no—
230M16—22no—
231M16—22no—
232M16#10714yes—
233C2×C8—22no—
234C2×C8—22no—
235C2×C8—22no—
236C2×C8C2×C4⋊C411yescharacteristic
237C2×C8C2×C4⋊C411yescharacteristic
238C2×C8C4×D411yescharacteristic
239C2×C8C4×D411yescharacteristic
240C2×C8C4×D411yescharacteristic
241C2×C8C4×Q811yescharacteristic
242C2×C8#12011yescharacteristic
243C2×C8#12011yescharacteristic
244C2×C8#12111yescharacteristic
245C2×C8#12111yescharacteristic
246C2×C8C22⋊Q811yescharacteristic
247C2×C8C22⋊Q811yescharacteristic
248C2×C8C22⋊Q811yescharacteristic
249C2×C8C22⋊Q811yescharacteristic
250C2×C8C22⋊Q811yescharacteristic
251C2×C8C22⋊Q811yescharacteristic
252C2×C8C22⋊Q811yescharacteristic
253C2×C8C22⋊Q811yescharacteristic
254C2×C8#12311yescharacteristic
255C2×C8#12311yescharacteristic
256C2×C8#12511yescharacteristic
257C2×C8#12511yescharacteristic
258C2×C8#12511yescharacteristic
259C2×C8C4⋊Q811yescharacteristic
260C2×C8C4⋊Q811yescharacteristic
261C2×C8C4⋊Q811yescharacteristic
262C2×C8C4⋊Q811yescharacteristic
263C2×C8C4⋊Q811yescharacteristic
264C4⋊C4—22no—
265C4⋊C4—22no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—41no—
273C4⋊C4—42no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—81no—
278C4⋊C4—81no—
279C4⋊C4—81no—
280C4⋊C4—164no—
281C22⋊C4—22no—
282C22⋊C4—42no—
283C22⋊C4—81no—
284C22⋊C4—81no—
285C22⋊C4—81no—
286C22⋊C4—81no—
287C22⋊C4—81no—
288C22⋊C4#10714yes—
289C42—22no—
290C42—22no—
291C42—42no—
292C42—81no—
293C42—81no—
294C42—81no—
295C42—81no—
296C42—81no—
297C16—24no—
298C16—24no—
299C23—41no—
300C23—41no—
301C23C2×#10711yescharacteristic
302C23#44711yescharacteristic
303C23#45911yescharacteristic
304C23#47411yescharacteristic
305C23#48011yescharacteristic
306C23#49311yescharacteristic
307C23#50011yescharacteristic
308C2×C4—24no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—82no—
312C2×C4—84no—
313C2×C4—82no—
314C2×C4—82no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4—82no—
320C2×C4—161no—
321C2×C4—161no—
322C2×C4—162no—
323C2×C4—162no—
324C2×C4—161no—
325C2×C4—161no—
326C2×C4#38411yescharacteristic
327C2×C4#38511yescharacteristic
328C2×C4#38911yescharacteristic
329C2×C4#39211yescharacteristic
330C2×C4#39511yescharacteristic
331C2×C4#39811yescharacteristic
332C2×C4#39911yescharacteristic
333C2×C4C2×#10711yescharacteristic
334C2×C4#42811yescharacteristic
335C2×C4#42811yescharacteristic
336C2×C4#44711yescharacteristic
337C2×C4#45111yescharacteristic
338C2×C4#45111yescharacteristic
339C2×C4#45911yescharacteristic
340C2×C4#46211yescharacteristic
341C2×C4#46211yescharacteristic
342C2×C4#47411yescharacteristic
343C2×C4#47711yescharacteristic
344C2×C4#47711yescharacteristic
345C2×C4#48011yescharacteristic
346C2×C4#48411yescharacteristic
347C2×C4#48411yescharacteristic
348C2×C4#49411yescharacteristic
349C2×C4#49611yescharacteristic
350C2×C4#49711yescharacteristic
351C2×C4#50011yescharacteristic
352C2×C4#50111yescharacteristic
353C2×C4#50111yescharacteristic
354C8—24no—
355C8#38511yescharacteristic
356C8#38511yescharacteristic
357C8#38911yescharacteristic
358C8#38911yescharacteristic
359C8#39511yescharacteristic
360C8#39511yescharacteristic
361C8#39811yescharacteristic
362C8#39811yescharacteristic
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22#184511yescharacteristic
374C22#185211yescharacteristic
375C22#194711yescharacteristic
376C22#201111yescharacteristic
377C22#205711yescharacteristic
378C22#208311yescharacteristic
379C22#209311yescharacteristic
380C4—44no—
381C4—162no—
382C4—161no—
383C4—162no—
384C4—161no—
385C4—161no—
386C4—161no—
387C4#144411yescharacteristic
388C4#184711yescharacteristic
389C4#185211yescharacteristic
390C4#194711yescharacteristic
391C4#201611yescharacteristic
392C4#205711yescharacteristic
393C4#208311yescharacteristic
394C4#209511yescharacteristic
395C2—21no—
396C2—21no—
397C2—21no—
398C2—21no—
399C2—21no—
400C2—21no—
401C2#1132411yescharacteristic
402C2#1333411yescharacteristic
403C2#1333511yescharacteristic
subgroupG/H
center, socleC22#2011
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radicalC24C2×D8
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G
LECSG ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
33. #1378C4semidirect
170. M32C4⋊C4semidirect
176. C2×C16C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#334092
C2×#21592
#208032
#125672
#125572
C2×#13782
#108712