Finite Group Databasebeta

← #106397

Subgroups — #106397

Order 512.

798 conjugacy classes of subgroups.

The subgroup lattice has 431 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60282C211yesmaximal, characteristic
2#21380C211yesmaximal, characteristic
3#17361C211yesmaximal, characteristic
4#7995C212yesmaximal, complemented
5#7994C212yesmaximal, complemented
6C2×C4×#41C2211yescharacteristic
7#2990—24no—
8#2983—21no—
9#2973—21no—
10#2973C412yes—
11#2972C412yes—
12#2970—24no—
13#2920—21no—
14#2920C2212yes—
15C2×#432—21no—
16C2×#432C2211yescharacteristic
17C2×#432C2212yes—
18C2×C4×M16C2211yescharacteristic
19#1984C412yes—
20#1954C412yes—
21#1870—21no—
22#1870—21no—
23#1749—21no—
24#1749—21no—
25#1467—21no—
26#1466—21no—
27#1465—21no—
28#1465—21no—
29#1458C412yes—
30#1458C412yes—
31#1291—24no—
32#1291—24no—
33C22×#41Q811yescharacteristic
34C22×M16—21no—
35C4×#41—22no—
36C4×#41—22no—
37C4×#41—22no—
38C4×#41—22no—
39C2×C4×Q8C2×C411yescharacteristic
40C2×C4×Q8D411yescharacteristic
41C2×C4×D4C2×C411yescharacteristic
42C2×C4×D4C2×C411yescharacteristic
43C2×C4×D4C2×C411yescharacteristic
44C2×C4×D4D411yescharacteristic
45C2×C4×D4D411yescharacteristic
46C2×C4×D4D411yescharacteristic
47C2×#117C2×C411yescharacteristic
48C2×#117C2×C411yescharacteristic
49C2×#117Q811yescharacteristic
50C22×C42D411yescharacteristic
51C22×C42D411yescharacteristic
52C22×C42C2311yescharacteristic
53#446—28no—
54#435—28no—
55#435—28no—
56C8×D4—28no—
57#432—22no—
58#432—22no—
59#432—22no—
60#432—22no—
61#432—22no—
62#432—22no—
63#432—22no—
64#432—22no—
65#432—22no—
66#432—22no—
67#432D414yes—
68#432D414yes—
69#431—22no—
70#431—22no—
71#431—22no—
72#431C2×C414yes—
73#423—22no—
74#423—22no—
75C2×C4⋊C8—22no—
76C2×C4⋊C8—22no—
77C2×C4⋊C8C2×C412yes—
78#407—22no—
79#407—22no—
80C2×C22⋊C8—22no—
81C2×C22⋊C8—22no—
82C2×C22⋊C8—22no—
83C2×C22⋊C8—21no—
84C2×C22⋊C8C2×C414yes—
85C4×M16—22no—
86C4×M16—22no—
87C4×M16—22no—
88C4×M16—22no—
89C2×#97—22no—
90C2×#97—22no—
91C2×#97—21no—
92C2×#97—21no—
93C2×C4×C8C2×C412yes—
94#336—22no—
95#336—22no—
96#324—22no—
97#324—22no—
98#323—22no—
99#323—22no—
100C2×#41—22no—
101C2×#41—22no—
102C2×#41—42no—
103C2×#41—42no—
104C22×Q8C4⋊C411yescharacteristic
105C22×D4C4⋊C411yescharacteristic
106C22×D4C4⋊C411yescharacteristic
107C22×D4C4⋊C411yescharacteristic
108C23×C4C22⋊C411yescharacteristic
109C23×C4C22⋊C411yescharacteristic
110C23×C4C22⋊C411yescharacteristic
111C23×C4C22⋊C411yescharacteristic
112C23×C4C4⋊C411yescharacteristic
113C23×C4C4⋊C411yescharacteristic
114C23×C4C22×C411yescharacteristic
115C23×C4C2×D411yescharacteristic
116C23×C4C2×Q811yescharacteristic
117C2×M16—24no—
118C2×M16—44no—
119C2×M16—42no—
120C2×M16—42no—
121C22×C8—22no—
122C22×C8—24no—
123C22×C8—24no—
124C22×C8—22no—
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⋊C4C4211yescharacteristic
148C2×C4⋊C4C4211yescharacteristic
149C2×C4⋊C4C22⋊C411yescharacteristic
150C2×C4⋊C4C22⋊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×C42C22×C411yescharacteristic
174C2×C42C22×C411yescharacteristic
175C2×C42C2×D412yes—
176C2×C42C2×D412yes—
177C2×C42C2×D411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C42#4112yes—
180C2×C42#4112yes—
181C4⋊C8—24no—
182C4⋊C8—24no—
183C4⋊C8—24no—
184C4⋊C8—24no—
185C4⋊C8—24no—
186C4⋊C8—24no—
187C4⋊C8—42no—
188C22⋊C8—24no—
189C22⋊C8—24no—
190C22⋊C8—24no—
191C22⋊C8—24no—
192C22⋊C8—44no—
193C22⋊C8—44no—
194C22⋊C8—42no—
195C22⋊C8—42no—
196C22⋊C8C22⋊C418yes—
197#97—24no—
198#97—24no—
199#97—24no—
200#97—41no—
201#97—41no—
202#97—42no—
203#97—41no—
204#97—41no—
205C4×C8—24no—
206C4×C8—24no—
207C4×C8—24no—
208C24#9511yescharacteristic
209C24#9511yescharacteristic
210C24C2×C4⋊C411yescharacteristic
211#41—48no—
212#41—48no—
213C2×Q8—22no—
214C2×Q8—22no—
215C2×Q8—41no—
216C2×D4—22no—
217C2×D4—22no—
218C2×D4—22no—
219C2×D4—22no—
220C2×D4—22no—
221C2×D4—22no—
222C2×D4—41no—
223C2×D4—41no—
224C2×D4—41no—
225C22×C4—22no—
226C22×C4—22no—
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—21no—
238C22×C4—21no—
239C22×C4—42no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—81no—
247C22×C4—81no—
248C22×C4—81no—
249C22×C4#9511yescharacteristic
250C22×C4#9511yescharacteristic
251C22×C4#9511yescharacteristic
252C22×C4#9511yescharacteristic
253C22×C4#9511yescharacteristic
254C22×C4#9511yescharacteristic
255C22×C4C2×C4211yescharacteristic
256C22×C4C2×C22⋊C411yescharacteristic
257C22×C4C2×C22⋊C412yes—
258C22×C4C2×C22⋊C411yescharacteristic
259C22×C4C2×C22⋊C411yescharacteristic
260C22×C4C2×C4⋊C411yescharacteristic
261C22×C4C2×C4⋊C411yescharacteristic
262C22×C4#11712yes—
263C22×C4#12112yes—
264C22×C4#12112yes—
265C22×C4C22⋊Q812yes—
266C22×C4C22⋊Q812yes—
267M16—48no—
268M16—84no—
269M16—84no—
270C2×C8—24no—
271C2×C8—28no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—44no—
277C2×C8—42no—
278C2×C8—84no—
279C2×C8—84no—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—22no—
284C4⋊C4—22no—
285C4⋊C4—22no—
286C22⋊C4—22no—
287C22⋊C4—22no—
288C22⋊C4—22no—
289C22⋊C4—22no—
290C22⋊C4—22no—
291C22⋊C4—22no—
292C42—21no—
293C42—21no—
294C42—21no—
295C42—21no—
296C42—21no—
297C42—21no—
298C42—21no—
299C42—21no—
300C42—21no—
301C42—21no—
302C42—21no—
303C42—21no—
304C42C4×D412yes—
305C42C4×D412yes—
306C42C4×D412yes—
307C42C4×D412yes—
308C42#12011yescharacteristic
309C42#12012yes—
310C42#12011yescharacteristic
311C42#12311yescharacteristic
312C42#12311yescharacteristic
313C42#12312yes—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23#33812yes—
328C23C2×#9511yescharacteristic
329C23#38012yes—
330Q8—44no—
331D4—28no—
332D4—44no—
333D4—44no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—24no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4#33812yes—
374C2×C4C4×C22⋊C412yes—
375C2×C4#37911yescharacteristic
376C2×C4#38111yescharacteristic
377C2×C4#38511yescharacteristic
378C2×C4#38511yescharacteristic
379C2×C4#38512yes—
380C2×C4#38611yescharacteristic
381C2×C4#38612yes—
382C2×C4#38611yescharacteristic
383C8—44no—
384C8—48no—
385C8—84no—
386C8—82no—
387C22—21no—
388C22—21no—
389C22—22no—
390C22—22no—
391C22—22no—
392C22—22no—
393C22—42no—
394C22—42no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—81no—
401C22—81no—
402C22—81no—
403C22—81no—
404C22C22⋊#9511yescharacteristic
405C22C2×#33811yescharacteristic
406C22C2×#33811yescharacteristic
407C4—21no—
408C4—21no—
409C4—22no—
410C4—21no—
411C4—21no—
412C4—22no—
413C4—22no—
414C4—21no—
415C4—22no—
416C4—41no—
417C4—41no—
418C4—41no—
419C4—41no—
420C4—41no—
421C4#188412yes—
422C2—21no—
423C2—21no—
424C2—42no—
425C2—42no—
426C2—81no—
427C2C22⋊#33811yescharacteristic
428C2#1075111yescharacteristic
429C2#1075611yescharacteristic
subgroupG/H
center, socleC22C22⋊#95
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
prime radicalC22×#41Q8
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7995C2semidirect
5. #7994C2semidirect

Maximal subgroups

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

subgroupindex
#602822
#213802
#173612
#79952
#79942