Finite Group Databasebeta

← #99916

Subgroups — #99916

Order 512.

482 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#23237C211yesmaximal, characteristic
2#22044C211yesmaximal, characteristic
3#9878C211yesmaximal, characteristic
4#7589C211yesmaximal, characteristic
5C2×#1272C211yesmaximal, characteristic
6#7029C211yesmaximal, characteristic
7#7016C211yesmaximal, characteristic
8C2×#451C411yescharacteristic
9C4×C22⋊Q8C2211yescharacteristic
10C2×#391C2211yescharacteristic
11#1984—21no—
12#1969—21no—
13#1933—21no—
14#1928—21no—
15#1916—21no—
16#1916—21no—
17#1878—21no—
18#1872C411yescharacteristic
19#1852C411yescharacteristic
20#1850—21no—
21#1797C2211yescharacteristic
22#1787C411yescharacteristic
23C8×C4⋊C4—21no—
24C8×C22⋊C4—21no—
25C4×C22⋊C8C2211yescharacteristic
26C4×C22⋊C8C2211yescharacteristic
27C2×#336C2211yescharacteristic
28C2×#336C2211yescharacteristic
29#1280—21no—
30#1280—21no—
31#1272—21no—
32#1272—21no—
33#1272—21no—
34#1271—21no—
35C23×Q8C2×C411yescharacteristic
36C22×Q16—21no—
37C23×C8—21no—
38C23×C8—21no—
39C2×C22⋊Q8C2×C411yescharacteristic
40C2×C4×Q8D411yescharacteristic
41C22×C4⋊C4C2×C411yescharacteristic
42C22×C42C2311yescharacteristic
43#451—22no—
44#451C814yescomplemented
45C2×#107—21no—
46C2×C4⋊C8—21no—
47C2×C4⋊C8—21no—
48C2×C4⋊C8—21no—
49C2×#103—21no—
50C2×#103—21no—
51C2×#103—21no—
52C2×#103—21no—
53C2×C22⋊C8—21no—
54C2×C22⋊C8C2×C411yescharacteristic
55C2×C22⋊C8C2×C411yescharacteristic
56C2×C22⋊C8D411yescharacteristic
57C2×C22⋊C8D411yescharacteristic
58C2×C4×C8—21no—
59C2×C4×C8—21no—
60C2×C4×C8—21no—
61C2×C4×C8—21no—
62#391—21no—
63#391—21no—
64#391—21no—
65#391—21no—
66#391D411yescharacteristic
67#389D411yescharacteristic
68#387—21no—
69#385—21no—
70#380C2×C411yescharacteristic
71C4×C4⋊C4—21no—
72C4×C4⋊C4D411yescharacteristic
73C4×C22⋊C4—21no—
74C2×#95—21no—
75C2×#95—21no—
76#340—21no—
77#340—21no—
78#336—21no—
79#336—21no—
80#336—21no—
81#336—21no—
82#336—21no—
83#336—21no—
84#336—21no—
85#336—21no—
86#336—21no—
87#336—21no—
88#336D411yescharacteristic
89#336D411yescharacteristic
90C22×Q8—21no—
91C22×Q8—21no—
92C22×Q8—21no—
93C22×Q8—21no—
94C22×Q8—21no—
95C22×Q8—81no—
96C22×Q8—81no—
97C22×Q8C22⋊C411yescharacteristic
98C22×Q8C2×C812yes—
99C23×C4—21no—
100C23×C4—21no—
101C23×C4C22×C411yescharacteristic
102C23×C4C2×D411yescharacteristic
103C23×C4#4111yescharacteristic
104C2×Q16—24no—
105C2×Q16—81no—
106C2×Q16—81no—
107C22×C8—21no—
108C22×C8—21no—
109C22×C8—21no—
110C22×C8—21no—
111C22×C8—21no—
112C22×C8—21no—
113C22×C8—22no—
114C22×C8—42no—
115C22×C8—41no—
116C22×C8—41no—
117C22×C8—41no—
118C22×C8—41no—
119C22×C8—41no—
120C22×C8—81no—
121C22×C8—81no—
122C22×C8—81no—
123C22⋊Q8—21no—
124C22⋊Q8—21no—
125C22⋊Q8C2×C812yes—
126C22⋊Q8M1612yes—
127C4×Q8—81no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—81no—
134C2×C4⋊C4C22⋊C411yescharacteristic
135C2×C4⋊C4C22⋊C411yescharacteristic
136C2×C4⋊C4C22⋊C411yescharacteristic
137C2×C22⋊C4—21no—
138C2×C22⋊C4—21no—
139C2×C42—21no—
140C2×C42—21no—
141C2×C42—81no—
142C2×C42C2×D411yescharacteristic
143C2×C42C2×D411yescharacteristic
144C2×C42C2×D411yescharacteristic
145C2×C42#4111yescharacteristic
146#107—81no—
147C4⋊C8—42no—
148C4⋊C8—81no—
149C4⋊C8—81no—
150#103—24no—
151#103—81no—
152#103—81no—
153#103—81no—
154C22⋊C8—21no—
155C22⋊C8—21no—
156C22⋊C8—21no—
157C22⋊C8—21no—
158C22⋊C8—21no—
159C22⋊C8—21no—
160C22⋊C8—41no—
161C22⋊C8—41no—
162C22⋊C8C2×C812yes—
163C22⋊C8Q1612yescomplemented
164C4×C8—42no—
165C4×C8—81no—
166C4×C8—81no—
167C4×C8—81no—
168#95—21no—
169#95—21no—
170#95—21no—
171#95—21no—
172#95—21no—
173#95—41no—
174#95—41no—
175#95—41no—
176#95—41no—
177C24C4×D411yescharacteristic
178C2×Q8—21no—
179C2×Q8—22no—
180C2×Q8—22no—
181C2×Q8—41no—
182C2×Q8—41no—
183C2×Q8—41no—
184C2×Q8—41no—
185C2×Q8—41no—
186C2×Q8—41no—
187C2×Q8—41no—
188C2×Q8—41no—
189C2×Q8—81no—
190C2×Q8—81no—
191C2×Q8—82no—
192C2×Q8—82no—
193C2×Q8—82no—
194C2×Q8—82no—
195C2×Q8C22⋊C812yes—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—22no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—81no—
216C22×C4—81no—
217C22×C4—81no—
218C22×C4—81no—
219C22×C4C2×C22⋊C411yescharacteristic
220C22×C4C4×D411yescharacteristic
221C22×C4#12011yescharacteristic
222C22×C4#12111yescharacteristic
223C22×C4#12111yescharacteristic
224C22×C4#12311yescharacteristic
225C22×C4C22×C811yescharacteristic
226C22×C4C2×M1611yescharacteristic
227C22×C4#13111yescharacteristic
228C22×C4#13111yescharacteristic
229C22×C4C2×Q1611yescharacteristic
230C22×C4#13511yescharacteristic
231C22×C4#13711yescharacteristic
232C22×C4#13711yescharacteristic
233Q16—84no—
234C2×C8—22no—
235C2×C8—22no—
236C2×C8—22no—
237C2×C8—41no—
238C2×C8—41no—
239C2×C8—41no—
240C2×C8—41no—
241C2×C8—41no—
242C2×C8—41no—
243C2×C8—41no—
244C2×C8—41no—
245C2×C8—41no—
246C2×C8—41no—
247C2×C8—41no—
248C2×C8—41no—
249C2×C8—42no—
250C2×C8—41no—
251C2×C8—41no—
252C2×C8—41no—
253C2×C8—82no—
254C2×C8—82no—
255C2×C8—81no—
256C2×C8—82no—
257C2×C8—81no—
258C2×C8—82no—
259C2×C8—81no—
260C2×C8—81no—
261C2×C8—81no—
262C4⋊C4—21no—
263C4⋊C4—22no—
264C4⋊C4—22no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—81no—
272C4⋊C4—81no—
273C4⋊C4C22⋊C812yes—
274C22⋊C4—22no—
275C22⋊C4—22no—
276C22⋊C4—41no—
277C22⋊C4—41no—
278C42—41no—
279C42—41no—
280C42—41no—
281C42—41no—
282C42—41no—
283C42—41no—
284C42—41no—
285C42—41no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23#38611yescharacteristic
291C23C8×D411yescharacteristic
292C23#43511yescharacteristic
293C23C4×Q1611yescharacteristic
294C23#44111yescharacteristic
295C23#44311yescharacteristic
296C23#44411yescharacteristic
297Q8—42no—
298Q8—42no—
299Q8—42no—
300Q8—81no—
301Q8—82no—
302Q8—81no—
303Q8—81no—
304Q8—82no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
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—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—42no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—82no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—82no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—161no—
351C2×C4—161no—
352C2×C4C2×C22⋊C811yescharacteristic
353C2×C4#40811yescharacteristic
354C2×C4C8×D411yescharacteristic
355C2×C4#43511yescharacteristic
356C2×C4#45111yescharacteristic
357C2×C4#45211yescharacteristic
358C2×C4#46211yescharacteristic
359C2×C4#46411yescharacteristic
360C2×C4#47111yescharacteristic
361C2×C4#47311yescharacteristic
362C8—42no—
363C8—42no—
364C8—42no—
365C8—81no—
366C8—81no—
367C8—81no—
368C8—82no—
369C8—161no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22C8×Q1611yescharacteristic
383C22#157911yescharacteristic
384C22#158411yescharacteristic
385C22#158711yescharacteristic
386C22#186911yescharacteristic
387C22#187611yescharacteristic
388C22#187811yescharacteristic
389C22#190611yescharacteristic
390C22#190811yescharacteristic
391C22#191311yescharacteristic
392C22#191411yescharacteristic
393C4—21no—
394C4—21no—
395C4—21no—
396C4—21no—
397C4—21no—
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—41no—
404C4—42no—
405C4—42no—
406C4—81no—
407C4—81no—
408C4—82no—
409C4—161no—
410C2—21no—
411C2—21no—
412C2—21no—
413C2—21no—
414C2#894611yescharacteristic
415C2#909611yescharacteristic
416C2#910711yescharacteristic
417C2#926311yescharacteristic
418C2#926511yescharacteristic
419C2#926811yescharacteristic
420C2#927611yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC22×C4C22×C8
Frattini, cosocleC22×C42C23
prime radicalC24C4×D4
LCSG ▷ C22×C4 ▷ C22 ▷ 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
44. #451C8semidirect
163. C22⋊C8Q16semidirect

Maximal subgroups

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

subgroupindex
#232372
#220442
#98782
#75892
C2×#12722
#70292
#70162