Finite Group Databasebeta

← #100453

Subgroups — #100453

Order 512.

474 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
1#33147C211yesmaximal, characteristic
2#22629C211yesmaximal, characteristic
3#10807C211yesmaximal, characteristic
4#9679C211yesmaximal, characteristic
5#7665C211yesmaximal, characteristic
6#7039C211yesmaximal, characteristic
7#7029C211yesmaximal, characteristic
8#2857C2211yescharacteristic
9C2×C4×C4⋊C4C2211yescharacteristic
10#1988C411yescharacteristic, complemented
11#1988C411yescharacteristic, complemented
12#1948C411yescharacteristic, complemented
13#1948C411yescharacteristic, complemented
14#1939—21no—
15#1919—21no—
16#1919—21no—
17#1917—22no—
18#1853—21no—
19#1824—21no—
20#1797—21no—
21#1797C2211yescharacteristic
22#1797C2211yescharacteristic
23#1770—21no—
24#1749—21no—
25C4×C22⋊C8C2211yescharacteristic
26C2×#340—21no—
27C2×#340—21no—
28C2×#336C2211yescharacteristic
29C2×#328—21no—
30C2×#328—21no—
31C4⋊#95C2211yescharacteristic
32#1271—22no—
33C23×C8—21no—
34C2×C4⋊Q8C2×C411yescharacteristic
35C2×C22⋊Q8—21no—
36C22×C4⋊C4—21no—
37C22×C4⋊C4D411yescharacteristic
38C22×C4⋊C4D411yescharacteristic
39C22×C42—21no—
40C22×C42C2311yescharacteristic
41C2×#106—21no—
42C2×#106—21no—
43C2×C4⋊C8—21no—
44C2×C4⋊C8—22no—
45C2×C4⋊C8C2×C411yescharacteristic
46C2×C4⋊C8C2×C411yescharacteristic
47C2×C22⋊C8—21no—
48C2×C22⋊C8—21no—
49C2×C22⋊C8D411yescharacteristic
50C2×C22⋊C8D411yescharacteristic
51C2×#97—22no—
52C2×C4×C8—21no—
53#391—21no—
54#389—21no—
55#389—21no—
56#384C2×C411yescharacteristic
57#380—21no—
58#380—21no—
59C4×C4⋊C4—21no—
60C4×C4⋊C4—21no—
61C4×C4⋊C4C2×C411yescharacteristic
62C4×C4⋊C4C2×C411yescharacteristic
63C4×C4⋊C4D411yescharacteristic
64C4×C4⋊C4D411yescharacteristic
65C2×#95—21no—
66C2×#95—21no—
67C2×#95—21no—
68#340—24no—
69#340—21no—
70#340—21no—
71#340—21no—
72#340—21no—
73#340—21no—
74#340—21no—
75#340—42no—
76#336—22no—
77#336—22no—
78#336—21no—
79#336—21no—
80#336—21no—
81#336—21no—
82#336D412yes—
83#328—42no—
84#328—42no—
85C22×Q8—21no—
86C23×C4—21no—
87C23×C4—21no—
88C23×C4—41no—
89C23×C4C2×D411yescharacteristic
90C23×C4C2×D411yescharacteristic
91C23×C4#4111yescharacteristic
92C22×C8—21no—
93C22×C8—22no—
94C22×C8—21no—
95C22×C8—21no—
96C22×C8—21no—
97C22×C8—22no—
98C22×C8—41no—
99C22×C8—41no—
100C22×C8—41no—
101C22×C8—41no—
102C22×C8—41no—
103C22×C8—41no—
104C4⋊Q8—41no—
105C4⋊Q8—41no—
106C22⋊Q8—41no—
107C22⋊Q8—41no—
108C22⋊Q8—41no—
109C22⋊Q8—41no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4—41no—
125C2×C4⋊C4—41no—
126C2×C4⋊C4—41no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4—41no—
129C2×C4⋊C4—41no—
130C2×C4⋊C4C22⋊C411yescharacteristic
131C2×C4⋊C4C22⋊C411yescharacteristic
132C2×C4⋊C4C22⋊C411yescharacteristic
133C2×C4⋊C4C22⋊C411yescharacteristic
134C2×C22⋊C4—41no—
135C2×C42—21no—
136C2×C42—21no—
137C2×C42—21no—
138C2×C42—21no—
139C2×C42—41no—
140C2×C42—41no—
141C2×C42—82no—
142C2×C42C22×C411yescharacteristic
143C2×C42C2×D411yescharacteristic
144C2×C42C2×D411yescharacteristic
145C2×C42#4111yescharacteristic
146#106—42no—
147#106—42no—
148C4⋊C8—41no—
149C4⋊C8—41no—
150C4⋊C8—41no—
151C4⋊C8—41no—
152C4⋊C8—82no—
153C22⋊C8—21no—
154C22⋊C8—21no—
155C22⋊C8—21no—
156C22⋊C8—41no—
157C22⋊C8—41no—
158C22⋊C8—41no—
159C22⋊C8—41no—
160C22⋊C8SD1612yes—
161#97—82no—
162C4×C8—41no—
163C4×C8—41no—
164#95—21no—
165#95—21no—
166#95—21no—
167#95—21no—
168#95—41no—
169#95—41no—
170#95—41no—
171#95—41no—
172#95—81no—
173C24#12111yescharacteristic
174C2×Q8—41no—
175C2×Q8—41no—
176C2×Q8—81no—
177C2×Q8—81no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—42no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—42no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—81no—
201C22×C4—81no—
202C22×C4—81no—
203C22×C4—81no—
204C22×C4—81no—
205C22×C4—81no—
206C22×C4—81no—
207C22×C4—81no—
208C22×C4—81no—
209C22×C4C2×C22⋊C411yescharacteristic
210C22×C4C4×D411yescharacteristic
211C22×C4C4×D411yescharacteristic
212C22×C4#12011yescharacteristic
213C22×C4#12111yescharacteristic
214C22×C4#12311yescharacteristic
215C22×C4C2×SD1611yescharacteristic
216C22×C4#13511yescharacteristic
217C22×C4AGL(1,ℤ/8)11yescharacteristic
218C22×C4AGL(1,ℤ/8)11yescharacteristic
219C22×C4#13711yescharacteristic
220C22×C4#13711yescharacteristic
221C22×C4#13711yescharacteristic
222C22×C4#13711yescharacteristic
223C2×C8—22no—
224C2×C8—41no—
225C2×C8—41no—
226C2×C8—41no—
227C2×C8—42no—
228C2×C8—42no—
229C2×C8—41no—
230C2×C8—42no—
231C2×C8—42no—
232C2×C8—42no—
233C2×C8—41no—
234C2×C8—41no—
235C2×C8—41no—
236C2×C8—41no—
237C2×C8—41no—
238C2×C8—41no—
239C2×C8—41no—
240C2×C8—42no—
241C2×C8—81no—
242C2×C8—81no—
243C2×C8—81no—
244C2×C8—81no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—21no—
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—42no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—81no—
265C4⋊C4—81no—
266C4⋊C4—81no—
267C4⋊C4—81no—
268C4⋊C4—81no—
269C4⋊C4#10412yes—
270C22⋊C4—81no—
271C22⋊C4—81no—
272C42—21no—
273C42—21no—
274C42—21no—
275C42—21no—
276C42—41no—
277C42—41no—
278C42—42no—
279C42—42no—
280C42—81no—
281C42—81no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23#38611yescharacteristic
287C23#46011yescharacteristic
288C23#46411yescharacteristic
289C23#46811yescharacteristic
290C23#47011yescharacteristic
291C23#47111yescharacteristic
292C23#47311yescharacteristic
293Q8—161no—
294Q8—161no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—22no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4—82no—
338C2×C4—82no—
339C2×C4—82no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—161no—
347C2×C4—161no—
348C2×C4—161no—
349C2×C4—161no—
350C2×C4#41011yescharacteristic
351C2×C4#41011yescharacteristic
352C2×C4#41711yescharacteristic
353C2×C4#41911yescharacteristic
354C2×C4C2×#10411yescharacteristic
355C2×C4#42111yescharacteristic
356C2×C4C4×SD1611yescharacteristic
357C2×C4#44011yescharacteristic
358C2×C4#44811yescharacteristic
359C2×C4#44911yescharacteristic
360C2×C4#46811yescharacteristic
361C2×C4#47011yescharacteristic
362C8—42no—
363C8—81no—
364C8—82no—
365C8—82no—
366C8—81no—
367C8—81no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22#168611yescharacteristic
381C22#169011yescharacteristic
382C22#170011yescharacteristic
383C22#170411yescharacteristic
384C22#187111yescharacteristic
385C22#188011yescharacteristic
386C22#188111yescharacteristic
387C22#189411yescharacteristic
388C22#189911yescharacteristic
389C22#190411yescharacteristic
390C22#190811yescharacteristic
391C4—21no—
392C4—21no—
393C4—21no—
394C4—21no—
395C4—21no—
396C4—21no—
397C4—41no—
398C4—41no—
399C4—42no—
400C4—81no—
401C4—81no—
402C4—81no—
403C4—162no—
404C4—161no—
405C4—161no—
406C4#150611yescharacteristic
407C4#150911yescharacteristic
408C4#151211yescharacteristic
409C4#151511yescharacteristic
410C2—21no—
411C2—21no—
412C2—21no—
413C2—21no—
414C2#899211yescharacteristic
415C2#911611yescharacteristic
416C2#916511yescharacteristic
417C2#922811yescharacteristic
418C2#926111yescharacteristic
419C2#930711yescharacteristic
420C2#934611yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
10. #1988C4semidirect
11. #1988C4semidirect
12. #1948C4semidirect
13. #1948C4semidirect

Maximal subgroups

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

subgroupindex
#331472
#226292
#108072
#96792
#76652
#70392
#70292