Finite Group Databasebeta

← #119969

Subgroups — #119969

Order 512.

710 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21820C211yesmaximal, characteristic
2#20740C211yesmaximal, characteristic
3C2×#1763C211yesmaximal, characteristic
4C2×#1302C211yesmaximal, characteristic
5#8034C211yesmaximal, characteristic
6#8014C211yesmaximal, characteristic
7C2×#1285C211yesmaximal, characteristic
8C22×C4×Q8C2211yescharacteristic
9C2×#432C2211yescharacteristic
10C2×#432C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C22×#103—21no—
13C22×#97C2211yescharacteristic
14#2285C2211yescharacteristic
15C2×C4×C4⋊C4C2211yescharacteristic
16#1985—22no—
17#1984—22no—
18#1823—21no—
19#1817—22no—
20#1803C412yescomplemented
21#1803C412yescomplemented
22#1787—21no—
23#1763—22no—
24#1763—22no—
25#1763—22no—
26C2×#340—21no—
27#1302—22no—
28#1295—22no—
29#1294—22no—
30#1285—22no—
31C23×Q8D411yescharacteristic
32C23×C8—21no—
33C2×C4×Q8—21no—
34C2×C4×Q8—21no—
35C2×C4×Q8—21no—
36C2×C4×Q8—21no—
37C2×C4×Q8—21no—
38C2×C4×Q8—21no—
39C2×C4×Q8—21no—
40C2×C4×Q8—21no—
41C2×C4×Q8—21no—
42C2×C4×Q8—21no—
43C2×C4×Q8C2×C412yes—
44C2×C4×Q8D412yes—
45C2×#117—21no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4D411yescharacteristic
49C22×C42—21no—
50C22×C42—21no—
51C22×C42C2311yescharacteristic
52#432—21no—
53#432—21no—
54#432—21no—
55#432—21no—
56#432—21no—
57#432—21no—
58#432—21no—
59#432—21no—
60#431—21no—
61#431—21no—
62#431—21no—
63#431—21no—
64C2×C4⋊C8—21no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8C2×C411yescharacteristic
67C2×C4⋊C8C2×C411yescharacteristic
68C2×#103—28no—
69C2×#103—24no—
70C2×#103—22no—
71C2×#103—22no—
72C2×C22⋊C8—21no—
73C2×C22⋊C8—21no—
74C2×C22⋊C8D412yes—
75C2×#97—21no—
76C2×#97—21no—
77C2×#97—21no—
78C2×#97—21no—
79C2×#97—21no—
80C2×#97—21no—
81C2×#97D412yes—
82C2×#97D412yes—
83C2×C4×C8—21no—
84#384C2×C412yes—
85#382—21no—
86#380—21no—
87#380—21no—
88C4×C4⋊C4—21no—
89C4×C4⋊C4—21no—
90C4×C4⋊C4—21no—
91C4×C4⋊C4—21no—
92C4×C4⋊C4—21no—
93C4×C4⋊C4D412yes—
94C2×#95—21no—
95#340—24no—
96#340—22no—
97#340—22no—
98#336—28no—
99C22×Q8—21no—
100C22×Q8—24no—
101C22×Q8—21no—
102C22×Q8—21no—
103C22×Q8—21no—
104C22×Q8—21no—
105C22×Q8—42no—
106C22×Q8C22⋊C412yes—
107C23×C4—21no—
108C23×C4—21no—
109C23×C4—41no—
110C23×C4C2×D411yescharacteristic
111C23×C4C2×D411yescharacteristic
112C23×C4#4111yescharacteristic
113C22×C8—24no—
114C22×C8—22no—
115C22×C8—21no—
116C22×C8—21no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—42no—
120C22×C8—41no—
121C22×C8—41no—
122C4×Q8—22no—
123C4×Q8—22no—
124C4×Q8—22no—
125C4×Q8—22no—
126C4×Q8—22no—
127C4×Q8—22no—
128C4×Q8—22no—
129C4×Q8—22no—
130C4×Q8—41no—
131C4×Q8—41no—
132C4×Q8—41no—
133C4×Q8—41no—
134C4×Q8—41no—
135C4×Q8—41no—
136C4×Q8—41no—
137C4×Q8—41no—
138#117—81no—
139#117—81no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—42no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4C22⋊C412yes—
159C2×C22⋊C4—41no—
160C2×C42—22no—
161C2×C42—22no—
162C2×C42—22no—
163C2×C42—41no—
164C2×C42—41no—
165C2×C42—41no—
166C2×C42—41no—
167C2×C42—41no—
168C2×C42—41no—
169C2×C42—81no—
170C2×C42—81no—
171C2×C42C22×C411yescharacteristic
172C2×C42C2×D411yescharacteristic
173C2×C42C2×D411yescharacteristic
174C2×C42C2×D411yescharacteristic
175C2×C42C2×D411yescharacteristic
176C2×C42#4111yescharacteristic
177C2×C42#4111yescharacteristic
178C2×C42#4111yescharacteristic
179C2×C42#4111yescharacteristic
180C2×C42#4111yescharacteristic
181C2×C42#4111yescharacteristic
182C2×C42#4111yescharacteristic
183C4⋊C8—21no—
184C4⋊C8—21no—
185C4⋊C8—21no—
186C4⋊C8—21no—
187C4⋊C8—41no—
188C4⋊C8—41no—
189C4⋊C8—41no—
190C4⋊C8—41no—
191#103—216no—
192#103—44no—
193#103—44no—
194C22⋊C8—22no—
195C22⋊C8—22no—
196C22⋊C8—41no—
197C22⋊C8—41no—
198C22⋊C8—41no—
199C22⋊C8—41no—
200#97—22no—
201#97—22no—
202#97—22no—
203#97—22no—
204#97—41no—
205#97—41no—
206#97—41no—
207#97—41no—
208C4×C8—41no—
209C4×C8—41no—
210#95—22no—
211#95—41no—
212#95—41no—
213#95—41no—
214C24#12111yescharacteristic
215C2×Q8—22no—
216C2×Q8—24no—
217C2×Q8—24no—
218C2×Q8—22no—
219C2×Q8—22no—
220C2×Q8—22no—
221C2×Q8—22no—
222C2×Q8—44no—
223C2×Q8—44no—
224C2×Q8—44no—
225C2×Q8—42no—
226C2×Q8—42no—
227C2×Q8—41no—
228C2×Q8—41no—
229C2×Q8#10414yes—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—42no—
239C22×C4—44no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—44no—
243C22×C4—42no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—42no—
251C22×C4—82no—
252C22×C4—81no—
253C22×C4—81no—
254C22×C4C2×C22⋊C411yescharacteristic
255C22×C4C4×D411yescharacteristic
256C22×C4C4×D411yescharacteristic
257C22×C4#12011yescharacteristic
258C22×C4#12011yescharacteristic
259C22×C4#12111yescharacteristic
260C22×C4#12111yescharacteristic
261C22×C4#12111yescharacteristic
262C22×C4#12311yescharacteristic
263C22×C4#12311yescharacteristic
264C22×C4#12311yescharacteristic
265C22×C4#12311yescharacteristic
266C22×C4#12311yescharacteristic
267C22×C4#12311yescharacteristic
268C22×C4#12411yescharacteristic
269C22×C4#12411yescharacteristic
270C22×C4#12611yescharacteristic
271C22×C4#12611yescharacteristic
272C2×C8—24no—
273C2×C8—44no—
274C2×C8—44no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—44no—
278C2×C8—44no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—42no—
285C4⋊C4—24no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—44no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—82no—
299C4⋊C4—82no—
300C22⋊C4—81no—
301C22⋊C4—81no—
302C42—42no—
303C42—42no—
304C42—42no—
305C42—42no—
306C42—82no—
307C42—82no—
308C42—82no—
309C42#11711yescharacteristic
310C42#11711yescharacteristic
311C42C4×D411yescharacteristic
312C42C4×D411yescharacteristic
313C42#12111yescharacteristic
314C42#12111yescharacteristic
315C42#12111yescharacteristic
316C42#12111yescharacteristic
317C42#12311yescharacteristic
318C42#12311yescharacteristic
319C42#12311yescharacteristic
320C42#12311yescharacteristic
321C42#12411yescharacteristic
322C42#12411yescharacteristic
323C42#12611yescharacteristic
324C42#12611yescharacteristic
325C23—22no—
326C23—22no—
327C23#38611yescharacteristic
328C23#39411yescharacteristic
329C23#39711yescharacteristic
330C23#47111yescharacteristic
331C23#47111yescharacteristic
332C23#47211yescharacteristic
333C23#47311yescharacteristic
334Q8—24no—
335Q8—24no—
336Q8—44no—
337Q8—44no—
338Q8—42no—
339Q8—42no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—24no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—44no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—82no—
375C2×C4—84no—
376C2×C4—84no—
377C2×C4—82no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4#38111yescharacteristic
386C2×C4#38611yescharacteristic
387C2×C4#38811yescharacteristic
388C2×C4#38811yescharacteristic
389C2×C4#38811yescharacteristic
390C2×C4#38811yescharacteristic
391C2×C4#39411yescharacteristic
392C2×C4#39411yescharacteristic
393C2×C4#39411yescharacteristic
394C2×C4#39711yescharacteristic
395C2×C4#39711yescharacteristic
396C2×C4#39711yescharacteristic
397C2×C4C2×#10412yes—
398C2×C4#42112yes—
399C8—44no—
400C8—82no—
401C8—82no—
402C8—81no—
403C8—81no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—22no—
409C22—22no—
410C22—22no—
411C22—22no—
412C22#144011yescharacteristic
413C22#187112yes—
414C22#190512yes—
415C22#190711yescharacteristic
416C22#190811yescharacteristic
417C22#205111yescharacteristic
418C22#205211yescharacteristic
419C22#207811yescharacteristic
420C22#207911yescharacteristic
421C4—21no—
422C4—21no—
423C4—21no—
424C4—21no—
425C4—21no—
426C4—21no—
427C4—21no—
428C4—21no—
429C4—21no—
430C4—21no—
431C4—44no—
432C4—84no—
433C4—82no—
434C4—162no—
435C4#182612yes—
436C4#189611yescharacteristic
437C4#189711yescharacteristic
438C2—21no—
439C2—21no—
440C2—21no—
441C2—21no—
442C2#916412yes—
443C2#916612yes—
444C2#1118411yescharacteristic
445C2#1119211yescharacteristic
446C2#1120111yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#121
LCSG ▷ C2×C42 ▷ 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
20. #1803C4semidirect
21. #1803C4semidirect

Maximal subgroups

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

subgroupindex
#218202
#207402
C2×#17632
C2×#13022
#80342
#80142
C2×#12852