Finite Group Databasebeta

← #106388

Subgroups — #106388

Order 512.

821 conjugacy classes of subgroups.

The subgroup lattice has 447 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60124C211yesmaximal, characteristic
2#20717C211yesmaximal, characteristic
3C2×#1464C211yesmaximal, characteristic
4#7989C212yesmaximal, complemented
5#7966C212yesmaximal, complemented
6C22×#117C2211yescharacteristic
7#2924—24no—
8#2924—24no—
9#2921—21no—
10#2921C412yes—
11#2920—21no—
12#2920C2212yes—
13C2×#432C2212yes—
14C2×#432C2211yescharacteristic
15C2×#431C2211yescharacteristic
16#2905—21no—
17C2×#407—21no—
18#2873C412yes—
19#1817—21no—
20#1798—21no—
21#1798—21no—
22#1779—21no—
23#1750C412yes—
24#1750C412yes—
25#1464—21no—
26#1464—21no—
27#1464—21no—
28#1464—21no—
29C2×#324C412yes—
30C2×#323C412yes—
31#1290—24no—
32#1287—24no—
33C24×C4D411yescharacteristic
34C22×M16—22no—
35C2×#117—22no—
36C2×#117—22no—
37C2×#117—22no—
38C2×#117—22no—
39C2×#117C2×C411yescharacteristic
40C2×#117C2×C411yescharacteristic
41C2×#117C2×C411yescharacteristic
42C2×#117C2×C411yescharacteristic
43C2×#117D411yescharacteristic
44C2×#117D411yescharacteristic
45C2×#117D411yescharacteristic
46C2×#117D411yescharacteristic
47C22×C4⋊C4C2×C411yescharacteristic
48C22×C4⋊C4Q811yescharacteristic
49C22×C22⋊C4C2×C411yescharacteristic
50C22×C22⋊C4Q811yescharacteristic
51C22×C42D411yescharacteristic
52C22×C42C2311yescharacteristic
53#433—28no—
54#433—28no—
55#432—22no—
56#432—22no—
57#432—22no—
58#432—22no—
59#432—22no—
60#432—22no—
61#432—22no—
62#432—22no—
63#432D414yes—
64#432D414yes—
65#431—22no—
66#431—22no—
67#431—22no—
68#431—22no—
69#431—22no—
70#431—22no—
71#424—28no—
72#423—22no—
73#423—22no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—22no—
77#407—22no—
78#407—22no—
79#407—22no—
80#407—22no—
81C2×C22⋊C8—22no—
82C2×C22⋊C8—22no—
83C2×C22⋊C8—22no—
84C2×C22⋊C8—22no—
85C2×C22⋊C8—21no—
86C2×C22⋊C8—21no—
87C2×C22⋊C8C2×C414yes—
88#405—28no—
89C4×M16—22no—
90C4×M16C2×C414yes—
91C2×#97—22no—
92C2×#97C2×C412yes—
93C2×C4×C8C2×C412yes—
94#336—22no—
95#336—22no—
96#324—22no—
97#324—22no—
98#323—22no—
99#323—22no—
100C25C4⋊C411yescharacteristic
101C23×C4—21no—
102C23×C4—21no—
103C23×C4—21no—
104C23×C4—21no—
105C23×C4—42no—
106C23×C4—42no—
107C23×C4C22⋊C411yescharacteristic
108C23×C4C22⋊C411yescharacteristic
109C23×C4C22⋊C411yescharacteristic
110C23×C4C22⋊C411yescharacteristic
111C23×C4C4⋊C411yescharacteristic
112C23×C4C4⋊C411yescharacteristic
113C23×C4C4⋊C411yescharacteristic
114C23×C4C22×C411yescharacteristic
115C23×C4C2×D411yescharacteristic
116C23×C4C2×Q811yescharacteristic
117C2×M16—24no—
118C2×M16—24no—
119C2×M16—44no—
120C2×M16—44no—
121C22×C8—24no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—24no—
125#117—22no—
126#117—22no—
127#117—22no—
128#117—22no—
129#117—22no—
130#117—22no—
131#117—22no—
132#117—22no—
133#117—22no—
134#117—22no—
135#117—22no—
136#117—22no—
137#117—22no—
138#117—22no—
139#117—22no—
140#117—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4C4211yescharacteristic
146C2×C4⋊C4C4211yescharacteristic
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⋊C4—22no—
154C2×C22⋊C4—22no—
155C2×C22⋊C4—22no—
156C2×C22⋊C4—22no—
157C2×C22⋊C4C4211yescharacteristic
158C2×C22⋊C4C4211yescharacteristic
159C2×C22⋊C4C22⋊C411yescharacteristic
160C2×C22⋊C4C22⋊C411yescharacteristic
161C2×C22⋊C4C4⋊C411yescharacteristic
162C2×C22⋊C4C4⋊C411yescharacteristic
163C2×C22⋊C4C4⋊C411yescharacteristic
164C2×C22⋊C4C4⋊C411yescharacteristic
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×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—42no—
186C4⋊C8—42no—
187C4⋊C8—42no—
188C4⋊C8—42no—
189C22⋊C8—24no—
190C22⋊C8—24no—
191C22⋊C8—24no—
192C22⋊C8—24no—
193C22⋊C8—24no—
194C22⋊C8—24no—
195C22⋊C8—44no—
196C22⋊C8—44no—
197C22⋊C8—41no—
198C22⋊C8—41no—
199C22⋊C8—41no—
200C22⋊C8—41no—
201#97—24no—
202#97—24no—
203#97—42no—
204#97—42no—
205#97C22⋊C418yes—
206C4×C8—24no—
207C4×C8—24no—
208C4×C8—24no—
209C24—21no—
210C24—21no—
211C24—22no—
212C24—41no—
213C24—42no—
214C24—81no—
215C24#9511yescharacteristic
216C24#9511yescharacteristic
217C24C2×C4⋊C411yescharacteristic
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—22no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—42no—
245C22×C4—44no—
246C22×C4—44no—
247C22×C4—44no—
248C22×C4—44no—
249C22×C4—42no—
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×C4211yescharacteristic
259C22×C4C2×C22⋊C411yescharacteristic
260C22×C4C2×C22⋊C411yescharacteristic
261C22×C4C2×C22⋊C411yescharacteristic
262C22×C4C2×C4⋊C411yescharacteristic
263C22×C4C2×C4⋊C411yescharacteristic
264C22×C4C4×D412yes—
265C22×C4C4×D412yes—
266C22×C4#12011yescharacteristic
267C22×C4#12011yescharacteristic
268C22×C4C22⋊Q812yes—
269C22×C4C22⋊Q812yes—
270C22×C4#12311yescharacteristic
271C22×C4#12311yescharacteristic
272M16—48no—
273M16—44no—
274M16—44no—
275C2×C8—28no—
276C2×C8—48no—
277C2×C8—42no—
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—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C22⋊C4—22no—
294C22⋊C4—22no—
295C22⋊C4—22no—
296C22⋊C4—22no—
297C22⋊C4—22no—
298C22⋊C4—22no—
299C22⋊C4—22no—
300C22⋊C4—22no—
301C42—21no—
302C42—21no—
303C42—21no—
304C42—21no—
305C42—21no—
306C42—21no—
307C42—21no—
308C42—21no—
309C42C2×C22⋊C412yes—
310C42#11712yes—
311C42C4×D412yes—
312C42C4×D412yes—
313C42#12012yes—
314C42#12112yes—
315C42#12112yes—
316C42#12312yes—
317C23—22no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—41no—
326C23—42no—
327C23—42no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—42no—
333C23—42no—
334C23—81no—
335C23—81no—
336C23—84no—
337C23—84no—
338C23H(3,ℤ/4)12yes—
339C23C2×#9511yescharacteristic
340C23#38511yescharacteristic
341C23#38511yescharacteristic
342C2×C4—24no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—42no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4H(3,ℤ/4)12yes—
386C2×C4C4×C22⋊C412yes—
387C2×C4#37911yescharacteristic
388C2×C4#38012yes—
389C2×C4#38111yescharacteristic
390C2×C4#38512yes—
391C2×C4#38611yescharacteristic
392C2×C4#38611yescharacteristic
393C2×C4#38612yes—
394C8—48no—
395C8—82no—
396C8—82no—
397C8—84no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—22no—
405C22—21no—
406C22—42no—
407C22—42no—
408C22—42no—
409C22—42no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—81no—
416C22—84no—
417C22—81no—
418C22—84no—
419C22C22⋊#9511yescharacteristic
420C22C2×H(3,ℤ/4)11yescharacteristic
421C22C2×H(3,ℤ/4)11yescharacteristic
422C22#185812yes—
423C4—22no—
424C4—22no—
425C4—22no—
426C4—22no—
427C4—21no—
428C4—21no—
429C4—21no—
430C4—22no—
431C4—21no—
432C4—41no—
433C4—41no—
434C4—41no—
435C4—41no—
436C4—41no—
437C2—21no—
438C2—21no—
439C2—22no—
440C2—41no—
441C2—42no—
442C2—81no—
443C2#1071611yescharacteristic
444C2#1075311yescharacteristic
445C2#1075711yescharacteristic
subgroupG/H
center, socleC22C22⋊#95
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
prime radicalC25C4⋊C4
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. #7989C2semidirect
5. #7966C2semidirect

Maximal subgroups

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

subgroupindex
#601242
#207172
C2×#14642
#79892
#79662