Finite Group Databasebeta

← #106566

Subgroups — #106566

Order 512.

817 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60123C211yesmaximal, characteristic
2#21851C211yesmaximal, characteristic
3#17366C211yesmaximal, characteristic
4#8124C212yesmaximal, complemented
5#7935C212yesmaximal, complemented
6C2×C4×#41C2211yescharacteristic
7#2923—24no—
8#2922—24no—
9#2921—21no—
10#2921C412yes—
11#2920—21no—
12#2920C2212yes—
13#2920C2212yes—
14C2×#432C2211yescharacteristic
15#2905—21no—
16C22×C4⋊C8C2211yescharacteristic
17#2879C412yes—
18C2×C4×M16—21no—
19#1985—21no—
20#1955—21no—
21#1869C412yes—
22#1869C412yes—
23#1797—21no—
24#1797—21no—
25#1465C412yes—
26#1465C412yes—
27#1464—21no—
28#1464—21no—
29#1461—21no—
30#1460—21no—
31#1308—24no—
32#1283—24no—
33C22×#41C2×C411yescharacteristic
34C22×M16—22no—
35C23×C8C2×C412yes—
36C4×#41—22no—
37C4×#41—22no—
38C4×#41—22no—
39C4×#41—22no—
40C2×C4×Q8D411yescharacteristic
41C2×C4×Q8Q811yescharacteristic
42C2×C4×D4C2×C411yescharacteristic
43C2×C4×D4C2×C411yescharacteristic
44C2×C4×D4C2×C411yescharacteristic
45C2×C4×D4C2×C411yescharacteristic
46C2×C4×D4D411yescharacteristic
47C2×C4×D4Q811yescharacteristic
48C2×#117C2×C411yescharacteristic
49C2×#117D411yescharacteristic
50C2×#117D411yescharacteristic
51C22×C42D411yescharacteristic
52C22×C42D411yescharacteristic
53C22×C42C2311yescharacteristic
54#433—28no—
55#433—28no—
56#432—24no—
57#432—22no—
58#432—22no—
59#432—22no—
60#432—22no—
61#432—22no—
62#432D414yes—
63#431—24no—
64#431—22no—
65#431—22no—
66#431—22no—
67#431D414yes—
68#424—28no—
69#423—22no—
70#423—22no—
71#423—22no—
72C2×C4⋊C8—22no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—22no—
77#408—28no—
78#407—22no—
79#407C2×C414yes—
80#407C2×C414yes—
81C2×C22⋊C8—22no—
82C2×C22⋊C8—22no—
83C2×C22⋊C8C2×C412yes—
84C4×M16—22no—
85C4×M16—22no—
86C4×M16—22no—
87C4×M16—22no—
88C4×M16—22no—
89C2×#97—22no—
90C2×#97—21no—
91C2×C4×C8—22no—
92#336—22no—
93#336—22no—
94#324—22no—
95#324—22no—
96#323—22no—
97#323—22no—
98C2×#41—22no—
99C2×#41—22no—
100C2×#41—42no—
101C2×#41—42no—
102C22×Q8C4⋊C411yescharacteristic
103C22×D4C4211yescharacteristic
104C22×D4C4211yescharacteristic
105C22×D4C4⋊C411yescharacteristic
106C23×C4C22⋊C411yescharacteristic
107C23×C4C22⋊C411yescharacteristic
108C23×C4C22⋊C411yescharacteristic
109C23×C4C22⋊C411yescharacteristic
110C23×C4C4⋊C411yescharacteristic
111C23×C4C4⋊C411yescharacteristic
112C23×C4C22×C411yescharacteristic
113C23×C4C22×C411yescharacteristic
114C23×C4C22×C411yescharacteristic
115C2×M16—24no—
116C2×M16—24no—
117C2×M16—24no—
118C2×M16—44no—
119C2×M16—44no—
120C22×C8—24no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—24no—
124C22×C8—82no—
125C4×Q8—22no—
126C4×Q8—22no—
127C4×Q8—22no—
128C4×Q8—22no—
129C4×D4—22no—
130C4×D4—22no—
131C4×D4—22no—
132C4×D4—22no—
133C4×D4—22no—
134C4×D4—22no—
135C4×D4—22no—
136C4×D4—22no—
137C4×D4—22no—
138C4×D4—22no—
139C4×D4—22no—
140C4×D4—22no—
141#117—22no—
142#117—22no—
143#117—22no—
144#117—22no—
145#117—22no—
146#117—22no—
147C2×C4⋊C4C22⋊C411yescharacteristic
148C2×C4⋊C4C22⋊C411yescharacteristic
149C2×C4⋊C4C4⋊C411yescharacteristic
150C2×C4⋊C4C4⋊C411yescharacteristic
151C2×C4⋊C4C4⋊C411yescharacteristic
152C2×C4⋊C4C4⋊C411yescharacteristic
153C2×C22⋊C4C4211yescharacteristic
154C2×C22⋊C4C4211yescharacteristic
155C2×C22⋊C4C22⋊C411yescharacteristic
156C2×C22⋊C4C22⋊C411yescharacteristic
157C2×C22⋊C4C4⋊C411yescharacteristic
158C2×C22⋊C4C4⋊C411yescharacteristic
159C2×C42—21no—
160C2×C42—21no—
161C2×C42—21no—
162C2×C42—21no—
163C2×C42—21no—
164C2×C42—21no—
165C2×C42—21no—
166C2×C42—21no—
167C2×C42C22⋊C411yescharacteristic
168C2×C42C22⋊C411yescharacteristic
169C2×C42C22⋊C411yescharacteristic
170C2×C42C22⋊C411yescharacteristic
171C2×C42C4⋊C411yescharacteristic
172C2×C42C4⋊C411yescharacteristic
173C2×C42C2×D412yes—
174C2×C42C2×D412yes—
175C2×C42C2×D411yescharacteristic
176C2×C42C2×D411yescharacteristic
177C2×C42C2×D411yescharacteristic
178C2×C42C2×Q811yescharacteristic
179C2×C42#4112yes—
180C2×C42#4112yes—
181C4⋊C8—28no—
182C4⋊C8—24no—
183C4⋊C8—24no—
184C4⋊C8—41no—
185C4⋊C8—41no—
186C4⋊C8—42no—
187C4⋊C8—42no—
188C4⋊C8—41no—
189C4⋊C8—41no—
190C4⋊C8—42no—
191C22⋊C8—28no—
192C22⋊C8—24no—
193C22⋊C8—24no—
194C22⋊C8—24no—
195C22⋊C8—24no—
196C22⋊C8—82no—
197C22⋊C8—82no—
198C22⋊C8C22⋊C418yes—
199#97—24no—
200#97—24no—
201#97—42no—
202#97—42no—
203C4×C8—24no—
204C4×C8—24no—
205C4×C8—42no—
206C24#9511yescharacteristic
207C24#9511yescharacteristic
208C24C2×C4211yescharacteristic
209#41—48no—
210#41—48no—
211C2×Q8—22no—
212C2×Q8—22no—
213C2×Q8—41no—
214C2×D4—22no—
215C2×D4—22no—
216C2×D4—22no—
217C2×D4—22no—
218C2×D4—22no—
219C2×D4—22no—
220C2×D4—41no—
221C2×D4—41no—
222C2×D4—41no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—41no—
238C22×C4—42no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4#9511yescharacteristic
251C22×C4#9511yescharacteristic
252C22×C4#9511yescharacteristic
253C22×C4#9511yescharacteristic
254C22×C4#9511yescharacteristic
255C22×C4#9511yescharacteristic
256C22×C4C2×C22⋊C411yescharacteristic
257C22×C4C2×C22⋊C411yescharacteristic
258C22×C4C2×C22⋊C412yes—
259C22×C4C2×C22⋊C411yescharacteristic
260C22×C4C2×C4⋊C411yescharacteristic
261C22×C4C2×C4⋊C411yescharacteristic
262C22×C4C2×C4⋊C411yescharacteristic
263C22×C4#11712yes—
264C22×C4C4×D412yes—
265C22×C4C4×D412yes—
266C22×C4C4×D412yes—
267C22×C4C4×D412yes—
268M16—48no—
269M16—44no—
270M16—44no—
271M16—48no—
272C2×C8—28no—
273C2×C8—48no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—44no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—84no—
282C4⋊C4—22no—
283C4⋊C4—22no—
284C4⋊C4—22no—
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C22⋊C4—22no—
289C22⋊C4—22no—
290C22⋊C4—22no—
291C22⋊C4—22no—
292C22⋊C4—22no—
293C22⋊C4—22no—
294C42—21no—
295C42—21no—
296C42—21no—
297C42—21no—
298C42—21no—
299C42—21no—
300C42—21no—
301C42—21no—
302C42—21no—
303C42—21no—
304C42—21no—
305C42—21no—
306C42#12011yescharacteristic
307C42#12012yes—
308C42#12011yescharacteristic
309C42#12112yes—
310C42#12112yes—
311C42C22⋊Q812yes—
312C42C22⋊Q812yes—
313C42#12311yescharacteristic
314C42#12312yes—
315C42#12311yescharacteristic
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23—21no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23#34412yes—
330C23C2×#9511yescharacteristic
331C23C4×C22⋊C412yes—
332Q8—44no—
333D4—44no—
334D4—44no—
335D4—44no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—24no—
345C2×C4—22no—
346C2×C4—21no—
347C2×C4—22no—
348C2×C4—21no—
349C2×C4—22no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4#34412yes—
380C2×C4#37911yescharacteristic
381C2×C4#38012yes—
382C2×C4#38111yescharacteristic
383C2×C4#38511yescharacteristic
384C2×C4#38512yes—
385C2×C4#38511yescharacteristic
386C2×C4#38611yescharacteristic
387C2×C4#38612yes—
388C2×C4#38611yescharacteristic
389C8—48no—
390C8—82no—
391C8—84no—
392C8—82no—
393C22—21no—
394C22—21no—
395C22—22no—
396C22—22no—
397C22—22no—
398C22—22no—
399C22—42no—
400C22—42no—
401C22—42no—
402C22—42no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—81no—
409C22—81no—
410C22C22⋊#9511yescharacteristic
411C22C2×#34411yescharacteristic
412C22C2×#34411yescharacteristic
413C4—21no—
414C4—22no—
415C4—21no—
416C4—21no—
417C4—22no—
418C4—22no—
419C4—22no—
420C4—21no—
421C4—41no—
422C4—41no—
423C4—41no—
424C4—41no—
425C4—41no—
426C4#186812yes—
427C4#191411yescharacteristic
428C4#191511yescharacteristic
429C2—21no—
430C2—21no—
431C2—42no—
432C2—42no—
433C2—81no—
434C2#1073811yescharacteristic
435C2#1074811yescharacteristic
436C2#1075511yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC24C2×C42
Frattini, cosocleC22×C42C23
socleC22C22⋊#95
prime radicalC22×#41C2×C4
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8124C2semidirect
5. #7935C2semidirect

Maximal subgroups

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

subgroupindex
#601232
#218512
#173662
#81242
#79352