Finite Group Databasebeta

← #107849

Subgroups — #107849

Order 512.

762 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
1C2×#2985C211yesmaximal, characteristic
2#21394C211yesmaximal, characteristic
3C2×#1559C211yesmaximal, characteristic
4C2×#1307C211yesmaximal, characteristic
5C2×#1287C211yesmaximal, characteristic
6#7912C211yesmaximal, characteristic
7#7901C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2985—22no—
10#2985—22no—
11#2985—22no—
12#2985C414yescomplemented
13C2×#446—21no—
14C2×#446—22no—
15C2×C8×Q8—21no—
16C2×#433—22no—
17C2×#432C2211yescharacteristic
18C2×#432C2211yescharacteristic
19C2×#432C2211yescharacteristic
20C2×#431C2211yescharacteristic
21C2×#423C2211yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23#1988—21no—
24#1981—21no—
25#1872C414yes—
26#1763—22no—
27#1559—22no—
28#1559C414yescomplemented
29C2×#333—21no—
30C2×#332—21no—
31#1307—22no—
32#1287—22no—
33#1280—22no—
34#1279—21no—
35#1279—21no—
36C22×M16—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533C2×C412yes—
42#533C2×C412yes—
43#533D412yes—
44#533Q812yes—
45C2×C4⋊Q8D411yescharacteristic
46C2×C4⋊Q8D411yescharacteristic
47C2×#125—21no—
48C2×C22⋊Q8—21no—
49C2×C22⋊Q8C2×C412yes—
50C2×C4×Q8—21no—
51C2×C4×Q8D412yes—
52C2×#117—21no—
53C22×C4⋊C4C2×C412yes—
54C22×C42—21no—
55C22×C42C2311yescharacteristic
56#446—28no—
57#446—24no—
58#446—28no—
59#446—24no—
60C8×Q8—24no—
61C8×Q8—24no—
62#433—28no—
63#433—28no—
64#432—21no—
65#432—21no—
66#432—21no—
67#432—21no—
68#432—21no—
69#432—21no—
70#432—21no—
71#432—21no—
72#432—21no—
73#432C2×C412yes—
74#432C2×C412yes—
75#432D412yes—
76#431—21no—
77#431—21no—
78#431C2×C412yes—
79#431D412yes—
80#423—21no—
81#423—21no—
82#423—21no—
83#423C2×C412yes—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×#103—24no—
93C2×C22⋊C8—21no—
94C2×C22⋊C8—21no—
95C2×C22⋊C8C2×C412yes—
96C2×C22⋊C8D412yes—
97C2×#97—22no—
98C2×#97—21no—
99C2×#97—21no—
100C2×#97—21no—
101C2×C4×C8—22no—
102C2×C4×C8D411yescharacteristic
103C2×C4×C8D411yescharacteristic
104C4×C4⋊C4—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108C2×#95—21no—
109#340—24no—
110#333—24no—
111#332—24no—
112C22×Q8—21no—
113C22×Q8C22⋊C412yes—
114C23×C4—22no—
115C23×C4C22×C411yescharacteristic
116C23×C4C2×D411yescharacteristic
117C23×C4#4111yescharacteristic
118C2×M16—22no—
119C2×M16—41no—
120C2×M16—81no—
121C2×M16—81no—
122C22×C8—21no—
123C22×C8—22no—
124C22×C8—21no—
125C22×C8—22no—
126C22×C8—41no—
127C22×C8—41no—
128C4⋊Q8—21no—
129C4⋊Q8—21no—
130C4⋊Q8—21no—
131C4⋊Q8—21no—
132C4⋊Q8C22⋊C412yes—
133C4⋊Q8C22⋊C412yes—
134C4⋊Q8C4⋊C412yes—
135C4⋊Q8C4⋊C412yes—
136#125—22no—
137#125—22no—
138#125—22no—
139#125—22no—
140C22⋊Q8—22no—
141C22⋊Q8—22no—
142C22⋊Q8—22no—
143C22⋊Q8—22no—
144C22⋊Q8—22no—
145C22⋊Q8—22no—
146C22⋊Q8C4214yes—
147C22⋊Q8C4⋊C414yes—
148C4×Q8—22no—
149C4×Q8—22no—
150C4×Q8—22no—
151C4×Q8—22no—
152C4×Q8—22no—
153C4×Q8—22no—
154C4×Q8C22⋊C414yes—
155C4×Q8C4⋊C414yes—
156#117—22no—
157#117—22no—
158#117—22no—
159#117—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4C22⋊C412yes—
169C2×C22⋊C4—22no—
170C2×C42—22no—
171C2×C42—41no—
172C2×C42—41no—
173C2×C42—81no—
174C2×C42—81no—
175C2×C42C22×C411yescharacteristic
176C2×C42C22×C411yescharacteristic
177C2×C42C2×D411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C42C2×D411yescharacteristic
181C2×C42C2×D411yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C42C2×Q811yescharacteristic
184C2×C42#4111yescharacteristic
185C2×C42#4111yescharacteristic
186C2×C42#4111yescharacteristic
187C4⋊C8—28no—
188C4⋊C8—28no—
189C4⋊C8—22no—
190C4⋊C8—22no—
191C4⋊C8—22no—
192C4⋊C8—22no—
193C4⋊C8—22no—
194C4⋊C8—22no—
195C4⋊C8—22no—
196C4⋊C8—41no—
197C4⋊C8—41no—
198C4⋊C8—41no—
199C4⋊C8—41no—
200C4⋊C8—41no—
201#103—48no—
202C22⋊C8—22no—
203C22⋊C8—22no—
204C22⋊C8—22no—
205C22⋊C8—22no—
206C22⋊C8—22no—
207C22⋊C8—22no—
208C22⋊C8—41no—
209C22⋊C8C22⋊C414yes—
210#97—28no—
211#97—22no—
212#97—22no—
213#97—22no—
214#97—41no—
215#97—41no—
216#97—41no—
217C4×C8—28no—
218C4×C8—21no—
219C4×C8—21no—
220C4×C8C22⋊C412yes—
221C4×C8C22⋊C412yes—
222#95—41no—
223#95—41no—
224C24C4×D411yescharacteristic
225C2×Q8—22no—
226C2×Q8—22no—
227C2×Q8—22no—
228C2×Q8—44no—
229C2×Q8—41no—
230C2×Q8—41no—
231C2×Q8—41no—
232C2×Q8—41no—
233C2×Q8#9514yes—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—24no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—82no—
246C22×C4—82no—
247C22×C4C2×C4211yescharacteristic
248C22×C4C2×C22⋊C411yescharacteristic
249C22×C4C2×C22⋊C411yescharacteristic
250C22×C4C2×C4⋊C411yescharacteristic
251C22×C4#11711yescharacteristic
252C22×C4C4×D411yescharacteristic
253C22×C4C4×D411yescharacteristic
254C22×C4C4×D411yescharacteristic
255C22×C4C4×D411yescharacteristic
256C22×C4C4×D411yescharacteristic
257C22×C4#12011yescharacteristic
258C22×C4#12011yescharacteristic
259C22×C4#12111yescharacteristic
260C22×C4#12111yescharacteristic
261C22×C4C22⋊Q811yescharacteristic
262C22×C4C22⋊Q811yescharacteristic
263C22×C4#12311yescharacteristic
264C22×C4#12311yescharacteristic
265M16—82no—
266M16—82no—
267C2×C8—22no—
268C2×C8—24no—
269C2×C8—22no—
270C2×C8—42no—
271C2×C8—44no—
272C2×C8—42no—
273C2×C8—44no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—82no—
278C2×C8—81no—
279C2×C8—81no—
280C4⋊C4—24no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—22no—
284C4⋊C4—24no—
285C4⋊C4—24no—
286C4⋊C4—24no—
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—24no—
292C4⋊C4—24no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4#9514yes—
296C22⋊C4—24no—
297C22⋊C4—24no—
298C42—24no—
299C42—24no—
300C42—82no—
301C42—81no—
302C42—81no—
303C42C2×C22⋊C411yescharacteristic
304C42C2×C22⋊C411yescharacteristic
305C42C2×C22⋊C411yescharacteristic
306C42C2×C4⋊C411yescharacteristic
307C42C2×C4⋊C411yescharacteristic
308C42#11711yescharacteristic
309C42C4×D411yescharacteristic
310C42C4×D411yescharacteristic
311C42#12011yescharacteristic
312C42#12011yescharacteristic
313C42#12111yescharacteristic
314C42#12111yescharacteristic
315C42C22⋊Q811yescharacteristic
316C42C22⋊Q811yescharacteristic
317C42#12311yescharacteristic
318C42#12311yescharacteristic
319C23—22no—
320C23—22no—
321C23—21no—
322C23—21no—
323C23C4×C22⋊C411yescharacteristic
324C23#38511yescharacteristic
325C23#38611yescharacteristic
326Q8—44no—
327Q8—42no—
328Q8—42no—
329Q8—42no—
330Q8—42no—
331Q8—82no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—44no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—44no—
360C2×C4—84no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4—84no—
364C2×C4C2×#9511yescharacteristic
365C2×C4C4×C22⋊C411yescharacteristic
366C2×C4#37911yescharacteristic
367C2×C4#38011yescharacteristic
368C2×C4#38011yescharacteristic
369C2×C4#38111yescharacteristic
370C2×C4#38511yescharacteristic
371C2×C4#38511yescharacteristic
372C2×C4#38511yescharacteristic
373C2×C4#38611yescharacteristic
374C2×C4#38611yescharacteristic
375C2×C4#38611yescharacteristic
376C2×C4#47111yescharacteristic
377C2×C4#47111yescharacteristic
378C2×C4#47211yescharacteristic
379C2×C4#47311yescharacteristic
380C8—42no—
381C8—44no—
382C8—42no—
383C8—82no—
384C8—82no—
385C8—82no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—42no—
393C22—42no—
394C22C22⋊#9511yescharacteristic
395C22#189112yes—
396C22#190512yes—
397C22#190711yescharacteristic
398C22#190811yescharacteristic
399C4—21no—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—21no—
404C4—21no—
405C4—21no—
406C4—21no—
407C4—21no—
408C4—21no—
409C4—44no—
410C4—44no—
411C4—84no—
412C4#181511yescharacteristic
413C4#181611yescharacteristic
414C4#190711yescharacteristic
415C4#190811yescharacteristic
416C2—41no—
417C2—41no—
418C2#952812yes—
419C2#953012yes—
420C2#1074611yescharacteristic
421C2#1077411yescharacteristic
422C2#1078611yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24C4×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
12. #2985C4semidirect
28. #1559C4semidirect

Maximal subgroups

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

subgroupindex
C2×#29852
#213942
C2×#15592
C2×#13072
C2×#12872
#79122
#79012