Finite Group Databasebeta

← #106790

Subgroups — #106790

Order 512.

822 conjugacy classes of subgroups.

The subgroup lattice has 437 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60125C211yesmaximal, characteristic
2#21380C211yesmaximal, characteristic
3#17363C211yesmaximal, characteristic
4#8125C212yesmaximal, complemented
5#7937C212yesmaximal, complemented
6C2×C4×#41C2211yescharacteristic
7#2925—24no—
8#2925—24no—
9#2921—21no—
10#2921C412yes—
11#2920—21no—
12#2920C2212yes—
13C2×#432C2212yes—
14C2×#432C2211yescharacteristic
15C2×#423—21no—
16#2879C412yes—
17#2873—21no—
18C2×C4×M16C2211yescharacteristic
19#1984—21no—
20#1954—21no—
21#1870C412yes—
22#1870C412yes—
23#1749—21no—
24#1749—21no—
25#1467—21no—
26#1466—21no—
27#1464—21no—
28#1464—21no—
29#1459C412yes—
30#1459C412yes—
31#1308—24no—
32#1283—24no—
33C22×#41C2×C411yescharacteristic
34C22×M16—22no—
35C22×M16C2×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—22no—
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#432D414yes—
66#431—22no—
67#431—22no—
68#431—22no—
69#431D414yes—
70#423—22no—
71#423—22no—
72#423—22no—
73#423—22no—
74C2×C4⋊C8—22no—
75C2×C4⋊C8—22no—
76C2×C4⋊C8—22no—
77C2×C4⋊C8—21no—
78#408—28no—
79#407—22no—
80#407C2×C414yes—
81C2×C22⋊C8—22no—
82C2×C22⋊C8—22no—
83C2×C22⋊C8C2×C412yes—
84C2×C22⋊C8C2×C414yes—
85#405—28no—
86C4×M16—22no—
87C4×M16—22no—
88C4×M16—22no—
89C4×M16—22no—
90C2×#97—21no—
91C2×#97—21no—
92C2×#97—22no—
93C2×C4×C8—21no—
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×D4C4211yescharacteristic
106C22×D4C4211yescharacteristic
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×C4C22×C411yescharacteristic
116C23×C4C22×C411yescharacteristic
117C2×M16—24no—
118C2×M16—24no—
119C2×M16—24no—
120C2×M16—44no—
121C2×M16—44no—
122C2×M16—44no—
123C22×C8—24no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—24no—
127C4×Q8—22no—
128C4×Q8—22no—
129C4×Q8—22no—
130C4×Q8—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—
141C4×D4—22no—
142C4×D4—22no—
143#117—22no—
144#117—22no—
145#117—22no—
146#117—22no—
147#117—22no—
148#117—22no—
149C2×C4⋊C4C22⋊C411yescharacteristic
150C2×C4⋊C4C22⋊C411yescharacteristic
151C2×C4⋊C4C4⋊C411yescharacteristic
152C2×C4⋊C4C4⋊C411yescharacteristic
153C2×C4⋊C4C4⋊C411yescharacteristic
154C2×C4⋊C4C4⋊C411yescharacteristic
155C2×C22⋊C4C4211yescharacteristic
156C2×C22⋊C4C4211yescharacteristic
157C2×C22⋊C4C22⋊C411yescharacteristic
158C2×C22⋊C4C22⋊C411yescharacteristic
159C2×C22⋊C4C4⋊C411yescharacteristic
160C2×C22⋊C4C4⋊C411yescharacteristic
161C2×C42—21no—
162C2×C42—21no—
163C2×C42—21no—
164C2×C42—21no—
165C2×C42—21no—
166C2×C42—21no—
167C2×C42—21no—
168C2×C42—21no—
169C2×C42C22⋊C411yescharacteristic
170C2×C42C22⋊C411yescharacteristic
171C2×C42C22⋊C411yescharacteristic
172C2×C42C22⋊C411yescharacteristic
173C2×C42C4⋊C411yescharacteristic
174C2×C42C4⋊C411yescharacteristic
175C2×C42C2×D412yes—
176C2×C42C2×D412yes—
177C2×C42C2×D411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C42C2×Q811yescharacteristic
181C2×C42#4112yes—
182C2×C42#4112yes—
183C4⋊C8—24no—
184C4⋊C8—24no—
185C4⋊C8—24no—
186C4⋊C8—24no—
187C4⋊C8—42no—
188C4⋊C8—42no—
189C4⋊C8—42no—
190C22⋊C8—24no—
191C22⋊C8—24no—
192C22⋊C8—24no—
193C22⋊C8—24no—
194C22⋊C8—24no—
195C22⋊C8—24no—
196C22⋊C8—44no—
197C22⋊C8—44no—
198C22⋊C8C22⋊C418yes—
199#97—24no—
200#97—24no—
201#97—41no—
202#97—41no—
203#97—42no—
204#97—41no—
205#97—41no—
206#97—42no—
207C4×C8—24no—
208C4×C8—24no—
209C4×C8—42no—
210C24#9511yescharacteristic
211C24#9511yescharacteristic
212C24C2×C4211yescharacteristic
213#41—44no—
214#41—44no—
215#41—48no—
216C2×Q8—22no—
217C2×Q8—22no—
218C2×Q8—41no—
219C2×D4—22no—
220C2×D4—22no—
221C2×D4—22no—
222C2×D4—22no—
223C2×D4—22no—
224C2×D4—22no—
225C2×D4—41no—
226C2×D4—41no—
227C2×D4—41no—
228C22×C4—22no—
229C22×C4—22no—
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—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—42no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—81no—
250C22×C4—81no—
251C22×C4—81no—
252C22×C4#9511yescharacteristic
253C22×C4#9511yescharacteristic
254C22×C4#9511yescharacteristic
255C22×C4#9511yescharacteristic
256C22×C4#9511yescharacteristic
257C22×C4#9511yescharacteristic
258C22×C4C2×C22⋊C411yescharacteristic
259C22×C4C2×C22⋊C411yescharacteristic
260C22×C4C2×C22⋊C412yes—
261C22×C4C2×C22⋊C411yescharacteristic
262C22×C4C2×C4⋊C411yescharacteristic
263C22×C4C2×C4⋊C411yescharacteristic
264C22×C4C2×C4⋊C411yescharacteristic
265C22×C4#11712yes—
266C22×C4C4×D412yes—
267C22×C4C4×D412yes—
268C22×C4C4×D412yes—
269C22×C4C4×D412yes—
270M16—28no—
271M16—48no—
272M16—44no—
273M16—44no—
274M16—44no—
275C2×C8—24no—
276C2×C8—28no—
277C2×C8—48no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—44no—
284C2×C8—84no—
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4—22no—
291C22⋊C4—22no—
292C22⋊C4—22no—
293C22⋊C4—22no—
294C22⋊C4—22no—
295C22⋊C4—22no—
296C22⋊C4—22no—
297C42—21no—
298C42—21no—
299C42—21no—
300C42—21no—
301C42—21no—
302C42—21no—
303C42—21no—
304C42—21no—
305C42—21no—
306C42—21no—
307C42—21no—
308C42—21no—
309C42#12012yes—
310C42#12011yescharacteristic
311C42#12011yescharacteristic
312C42#12112yes—
313C42#12112yes—
314C42C22⋊Q812yes—
315C42C22⋊Q812yes—
316C42#12311yescharacteristic
317C42#12311yescharacteristic
318C42#12312yes—
319C23—21no—
320C23—21no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23#34412yes—
333C23C2×#9511yescharacteristic
334C23C4×C22⋊C412yes—
335Q8—44no—
336D4—44no—
337D4—44no—
338D4—44no—
339C2×C4—22no—
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—22no—
349C2×C4—24no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
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#38511yescharacteristic
385C2×C4#38512yes—
386C2×C4#38611yescharacteristic
387C2×C4#38612yes—
388C2×C4#38611yescharacteristic
389C8—48no—
390C8—44no—
391C8—82no—
392C8—84no—
393C22—21no—
394C22—21no—
395C22—22no—
396C22—22no—
397C22—22no—
398C22—22no—
399C22—42no—
400C22—42no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—81no—
407C22—81no—
408C22—81no—
409C22—81no—
410C22C22⋊#9511yescharacteristic
411C22C2×#34411yescharacteristic
412C22C2×#34411yescharacteristic
413C4—22no—
414C4—21no—
415C4—21no—
416C4—21no—
417C4—21no—
418C4—22no—
419C4—22no—
420C4—21no—
421C4—22no—
422C4—41no—
423C4—41no—
424C4—41no—
425C4—41no—
426C4—41no—
427C4#189112yes—
428C2—21no—
429C2—21no—
430C2—42no—
431C2—42no—
432C2—81no—
433C2#1073811yescharacteristic
434C2#1074611yescharacteristic
435C2#1075711yescharacteristic
subgroupG/H
center, socleC22C22⋊#95
commutatorC24C2×C42
Frattini, cosocleC22×C42C23
prime radicalC22×#41C2×C4
LCSG ▷ C24 ▷ C22 ▷ 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. #8125C2semidirect
5. #7937C2semidirect

Maximal subgroups

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

subgroupindex
#601252
#213802
#173632
#81252
#79372