Finite Group Databasebeta

← #274195

Subgroups — #274195

Order 512.

702 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33153C211yesmaximal, characteristic
2#25856C211yesmaximal, characteristic
3#22639C211yesmaximal, characteristic
4#21313C211yesmaximal, characteristic
5#10600C212yesmaximal, complemented
6#10541C212yesmaximal, complemented
7#9737C212yesmaximal, complemented
8#9561C212yesmaximal, complemented
9#9135C211yesmaximal, characteristic
10#8832C211yesmaximal, characteristic
11#8340C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C2×#487C2212yes—
14C2×#423C2211yescharacteristic
15#2857C2211yescharacteristic
16C4×#117C2211yescharacteristic
17#2043—21no—
18#2043—21no—
19#1988C2212yes—
20#1988C2212yes—
21#1987—21no—
22#1987—21no—
23#1987C2211yescharacteristic
24#1987C2211yescharacteristic
25#1956C2212yes—
26#1948C2211yescharacteristic
27#1947C2211yescharacteristic
28#1939C2212yes—
29#1923—22no—
30#1919—21no—
31#1918—21no—
32#1854—21no—
33#1839—21no—
34#1825—21no—
35#1797C2211yescharacteristic
36#1797C2211yescharacteristic
37#1787—22no—
38C8×C4⋊C4C2212yes—
39C4×C4⋊C8—21no—
40C4×C22⋊C8C2211yescharacteristic
41#1736—21no—
42#1736—21no—
43C2×#335C2212yes—
44C2×#333C2212yes—
45C2×#332C2212yes—
46C2×#332C2212yes—
47#1296C2212yes—
48#1277C2212yes—
49C23×Q8—21no—
50C22×M16—21no—
51C2×C4⋊Q8—24no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—42no—
56C2×C4⋊Q8C2311yescharacteristic
57C2×C4⋊Q8C2312yes—
58C2×C22⋊Q8—21no—
59C2×#117D411yescharacteristic
60C2×#117D411yescharacteristic
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C42C2311yescharacteristic
64#487—24no—
65#487—42no—
66C2×#107—21no—
67C2×#106—21no—
68#423—42no—
69C2×C4⋊C8—22no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8C2312yes—
72C2×C4⋊C8C2311yescharacteristic
73C2×C4⋊C8C2311yescharacteristic
74C2×C4⋊C8C2312yes—
75C2×#103—22no—
76C2×#103—22no—
77C2×#103—22no—
78C2×#103—22no—
79C2×C22⋊C8—21no—
80C2×C22⋊C8—21no—
81C2×C22⋊C8D412yes—
82C2×C4×C8—22no—
83C2×C4×C8C2312yes—
84#391—21no—
85#389—21no—
86#389—21no—
87#384C2311yescharacteristic
88#380—21no—
89#380—21no—
90C4×C4⋊C4D411yescharacteristic
91C4×C4⋊C4D411yescharacteristic
92C4×C4⋊C4C2311yescharacteristic
93C4×C4⋊C4C2311yescharacteristic
94C4×C22⋊C4—21no—
95C4×C22⋊C4—21no—
96#376—21no—
97C43—21no—
98#340—21no—
99#340—21no—
100#340—21no—
101#340—21no—
102#340—21no—
103#340—21no—
104#340—22no—
105#340—22no—
106#340—22no—
107#340—22no—
108#336—22no—
109#336—22no—
110#336—21no—
111#336—21no—
112#336D411yescharacteristic
113#336D411yescharacteristic
114#335—42no—
115#333—42no—
116#332—24no—
117#332—42no—
118C22×Q8—22no—
119C22×Q8—21no—
120C22×Q8—44no—
121C22×Q8—44no—
122C22×Q8—41no—
123C22×Q8—81no—
124C22×Q8—81no—
125C23×C4—41no—
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4#4111yescharacteristic
129C2×M16—42no—
130C2×M16—42no—
131C2×M16—42no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—21no—
135C22×C8—21no—
136C22×C8C2×D412yes—
137C22×C8C2×D412yes—
138C4⋊Q8—24no—
139C4⋊Q8—22no—
140C4⋊Q8—22no—
141C4⋊Q8—22no—
142C4⋊Q8—22no—
143C4⋊Q8—22no—
144C4⋊Q8—44no—
145C4⋊Q8—44no—
146C4⋊Q8—42no—
147C4⋊Q8#4114yes—
148C22⋊Q8—82no—
149#117—22no—
150#117—22no—
151#117—41no—
152#117Q1614yes—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—44no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4C2×D411yescharacteristic
167C2×C4⋊C4C2×D411yescharacteristic
168C2×C4⋊C4C2×D411yescharacteristic
169C2×C4⋊C4C2×D411yescharacteristic
170C2×C22⋊C4—21no—
171C2×C22⋊C4—21no—
172C2×C22⋊C4—41no—
173C2×C42—21no—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—21no—
177C2×C42—21no—
178C2×C42—21no—
179C2×C42—21no—
180C2×C42—21no—
181C2×C42C2×D411yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C42#4111yescharacteristic
184C2×C42C2411yescharacteristic
185#107—42no—
186#106—42no—
187C4⋊C8—22no—
188C4⋊C8—22no—
189C4⋊C8—22no—
190C4⋊C8—22no—
191C4⋊C8—22no—
192C4⋊C8—22no—
193C4⋊C8—42no—
194C4⋊C8—44no—
195#103—44no—
196#103—44no—
197#103—82no—
198#103—82no—
199C22⋊C8—42no—
200C22⋊C8—42no—
201C22⋊C8—81no—
202C4×C8—22no—
203C4×C8—22no—
204C4×C8—82no—
205#95—21no—
206#95—21no—
207#95—21no—
208#95—21no—
209#95—41no—
210#95—41no—
211C24#12111yescharacteristic
212C2×Q8—24no—
213C2×Q8—44no—
214C2×Q8—42no—
215C2×Q8—42no—
216C2×Q8—42no—
217C2×Q8—42no—
218C2×Q8—42no—
219C2×Q8—42no—
220C2×Q8—82no—
221C2×Q8—82no—
222C2×Q8—81no—
223C2×Q8—81no—
224C2×Q8—88no—
225C2×Q8—88no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—22no—
239C22×C4—21no—
240C22×C4—41no—
241C22×C4—44no—
242C22×C4—42no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—82no—
246C22×C4—82no—
247C22×C4#12111yescharacteristic
248C22×C4#12111yescharacteristic
249C22×C4#12111yescharacteristic
250C22×C4C2×Q1612yes—
251C22×C4#13512yes—
252C22×C4C22×D411yescharacteristic
253C22×C4C22×D411yescharacteristic
254C22×C4C2×#4111yescharacteristic
255M16—84no—
256C2×C8—22no—
257C2×C8—22no—
258C2×C8—22no—
259C2×C8—22no—
260C2×C8—22no—
261C2×C8—22no—
262C2×C8—41no—
263C2×C8—42no—
264C2×C8—41no—
265C2×C8—42no—
266C2×C8—41no—
267C2×C8—41no—
268C2×C8—42no—
269C2×C8—41no—
270C2×C8—42no—
271C2×C8—41no—
272C2×C8—42no—
273C2×C8—42no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—24no—
278C4⋊C4—24no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—44no—
282C4⋊C4—44no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—42no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—81no—
293C4⋊C4—82no—
294C4⋊C4C2×Q1612yes—
295C4⋊C4C2×Q1612yes—
296C4⋊C4#13712yes—
297C22⋊C4—24no—
298C22⋊C4—42no—
299C22⋊C4—82no—
300C42—21no—
301C42—21no—
302C42—22no—
303C42—22no—
304C42—24no—
305C42—42no—
306C42—42no—
307C42—41no—
308C42—41no—
309C42—42no—
310C42—81no—
311C42C2×#4112yes—
312C42H(5,𝔽2)12yes—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23#46712yes—
320C23C2×#12111yescharacteristic
321Q8—44no—
322Q8—44no—
323Q8—88no—
324Q8—82no—
325Q8—82no—
326Q8—162no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—44no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—44no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—88no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4#46712yes—
366C2×C4#47112yes—
367C2×C4#52511yescharacteristic
368C2×C4#53511yescharacteristic
369C2×C4#54612yes—
370C2×C4#55112yes—
371C2×C4C22×Q1611yescharacteristic
372C2×C4C2×#13511yescharacteristic
373C2×C4C2×#13711yescharacteristic
374C2×C4#57511yescharacteristic
375C2×C4#57512yes—
376C2×C4#57611yescharacteristic
377C2×C4#57711yescharacteristic
378C8—42no—
379C8—42no—
380C8—81no—
381C8—81no—
382C8—82no—
383C8—82no—
384C22—22no—
385C22—22no—
386C22—22no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22#241111yescharacteristic
392C22#304011yescharacteristic
393C22#304511yescharacteristic
394C22C2×#46711yescharacteristic
395C22#305611yescharacteristic
396C22C2×#47111yescharacteristic
397C22#306911yescharacteristic
398C4—22no—
399C4—22no—
400C4—22no—
401C4—42no—
402C4—42no—
403C4—41no—
404C4—42no—
405C4—81no—
406C4—84no—
407C4—161no—
408C4—161no—
409C4#310411yescharacteristic
410C4#310811yescharacteristic
411C4#320911yescharacteristic
412C4#321111yescharacteristic
413C4#331112yes—
414C4#332212yes—
415C2—42no—
416C2#1927711yescharacteristic
417C2#1932511yescharacteristic
418C2#1941111yescharacteristic
419C2#1948211yescharacteristic
420C2#2449311yescharacteristic
421C2#2450211yescharacteristic
422C2#2452311yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #10600C2semidirect
6. #10541C2semidirect
7. #9737C2semidirect
8. #9561C2semidirect

Maximal subgroups

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

subgroupindex
#331532
#258562
#226392
#213132
#106002
#105412
#97372
#95612
#91352
#88322
#83402