Finite Group Databasebeta

← #109239

Subgroups — #109239

Order 512.

966 conjugacy classes of subgroups.

The subgroup lattice has 435 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1808C211yesmaximal, characteristic
2C2×#1807C211yesmaximal, characteristic
3C2×#1752C211yesmaximal, characteristic
4#10854C212yesmaximal
5#8023C212yesmaximal
6C24×C8C2211yescharacteristic
7C22×#107C412yescomplemented
8C22×#106—21no—
9C22×C22⋊C8C2211yescharacteristic
10C22×#97—21no—
11C2×#380C2212yes—
12C2×#380C2212yes—
13C2×C4×C22⋊C4C2211yescharacteristic
14#1808—21no—
15#1808—21no—
16#1808—21no—
17#1808—21no—
18#1807—22no—
19#1807C414yescomplemented
20#1752—22no—
21#1752C414yescomplemented
22C2×#342—24no—
23C2×#340—21no—
24C2×#340C412yescomplemented
25C2×#336—21no—
26#1294—24no—
27C24×C4C2311yescharacteristic
28C23×C8—21no—
29C23×C8—44no—
30C23×C8C2×C411yescharacteristic
31C23×C8C2×C412yes—
32C23×C8C2×C411yescharacteristic
33C23×C8C2×C412yes—
34C23×C8D411yescharacteristic
35C23×C8D411yescharacteristic
36C23×C8D412yes—
37C23×C8D411yescharacteristic
38C23×C8D411yescharacteristic
39C23×C8Q812yes—
40C22×C4⋊C4—22no—
41C22×C4⋊C4C2×C414yes—
42C22×C22⋊C4—22no—
43C22×C22⋊C4—22no—
44C22×C22⋊C4D412yes—
45C22×C42—21no—
46C2×#107—22no—
47C2×#107—22no—
48C2×#107—44no—
49C2×#106—24no—
50C2×#106—44no—
51C2×C22⋊C8—21no—
52C2×C22⋊C8—21no—
53C2×C22⋊C8—21no—
54C2×C22⋊C8C2×C412yes—
55C2×C22⋊C8D412yes—
56C2×C22⋊C8Q812yes—
57C2×#97—24no—
58C2×#97—44no—
59#380—22no—
60#380—22no—
61#380—22no—
62#380—22no—
63#380—22no—
64#380—22no—
65#380—22no—
66#380C2×C414yes—
67C4×C22⋊C4—21no—
68C4×C22⋊C4—21no—
69C4×C22⋊C4C2×C412yes—
70C4×C22⋊C4Q812yes—
71C2×#95—21no—
72C2×#95—22no—
73C2×#95—22no—
74#342—48no—
75#340—24no—
76#340—22no—
77#340—22no—
78#336—24no—
79C25C2×D411yescharacteristic
80C23×C4—22no—
81C23×C4—24no—
82C23×C4—42no—
83C23×C4—42no—
84C23×C4—42no—
85C23×C4C22×C411yescharacteristic
86C23×C4C22×C411yescharacteristic
87C23×C4C22×C411yescharacteristic
88C23×C4C2×D411yescharacteristic
89C23×C4C2×D411yescharacteristic
90C23×C4C2×D411yescharacteristic
91C23×C4C2×D411yescharacteristic
92C23×C4C2×Q811yescharacteristic
93C23×C4C2×Q811yescharacteristic
94C23×C4C2×Q811yescharacteristic
95C23×C4#4111yescharacteristic
96C23×C4#4111yescharacteristic
97C23×C4#4111yescharacteristic
98C23×C4#4111yescharacteristic
99C22×C8—24no—
100C22×C8—24no—
101C22×C8—24no—
102C22×C8—22no—
103C22×C8—22no—
104C22×C8—22no—
105C22×C8—44no—
106C22×C8—44no—
107C22×C8—44no—
108C22×C8—44no—
109C22×C8—42no—
110C22×C8—42no—
111C22×C8—42no—
112C22×C8—42no—
113C22×C8—42no—
114C22×C8C4212yes—
115C22×C8C4212yes—
116C22×C8C22⋊C411yescharacteristic
117C22×C8C22⋊C411yescharacteristic
118C22×C8C22⋊C412yes—
119C22×C8C22⋊C412yes—
120C22×C8C22⋊C411yescharacteristic
121C22×C8C22⋊C412yes—
122C22×C8C22⋊C412yes—
123C22×C8C22⋊C411yescharacteristic
124C22×C8C4⋊C412yes—
125C22×C8C4⋊C412yes—
126C22×C8C4⋊C412yes—
127C22×C8C4⋊C412yes—
128C22×C8C4⋊C412yes—
129C22×C8C4⋊C412yes—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—44no—
134C2×C4⋊C4—44no—
135C2×C4⋊C4—44no—
136C2×C4⋊C4—44no—
137C2×C4⋊C4—42no—
138C2×C22⋊C4—24no—
139C2×C22⋊C4—24no—
140C2×C22⋊C4—22no—
141C2×C22⋊C4—24no—
142C2×C22⋊C4—44no—
143C2×C22⋊C4—44no—
144C2×C22⋊C4—42no—
145C2×C22⋊C4—42no—
146C2×C22⋊C4—42no—
147C2×C22⋊C4—44no—
148C2×C22⋊C4—44no—
149C2×C22⋊C4C4⋊C414yes—
150C2×C42—22no—
151C2×C42—22no—
152C2×C42—42no—
153C2×C42—42no—
154#107—44no—
155#107—44no—
156#106—48no—
157C22⋊C8—22no—
158C22⋊C8—22no—
159C22⋊C8—22no—
160C22⋊C8C4⋊C414yes—
161#97—48no—
162#95—22no—
163#95—22no—
164#95—24no—
165#95—42no—
166#95—42no—
167#95—42no—
168C24—42no—
169C24—42no—
170C24C2×C4⋊C411yescharacteristic
171C24C4×D411yescharacteristic
172C24C4×D411yescharacteristic
173C24#12111yescharacteristic
174C24#12111yescharacteristic
175C24#12711yescharacteristic
176C24C4⋊Q811yescharacteristic
177C24AGL(1,ℤ/8)14yes—
178C24#13714yes—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—24no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—44no—
202C22×C4—44no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—44no—
206C22×C4—44no—
207C22×C4—48no—
208C22×C4—84no—
209C22×C4—84no—
210C22×C4—82no—
211C22×C4—82no—
212C22×C4—84no—
213C22×C4—84no—
214C22×C4C2×C4211yescharacteristic
215C22×C4C2×C22⋊C411yescharacteristic
216C22×C4C2×C22⋊C411yescharacteristic
217C22×C4C2×C22⋊C411yescharacteristic
218C22×C4C2×C4⋊C411yescharacteristic
219C22×C4C2×C4⋊C411yescharacteristic
220C22×C4C2×C4⋊C411yescharacteristic
221C22×C4C2×C4⋊C411yescharacteristic
222C22×C4#11711yescharacteristic
223C22×C4#11711yescharacteristic
224C22×C4C4×D411yescharacteristic
225C22×C4C4×D411yescharacteristic
226C22×C4C4×Q811yescharacteristic
227C22×C4C4×Q811yescharacteristic
228C22×C4C4×Q811yescharacteristic
229C22×C4C4×Q811yescharacteristic
230C22×C4#12111yescharacteristic
231C22×C4#12111yescharacteristic
232C22×C4C22⋊Q811yescharacteristic
233C22×C4C22⋊Q811yescharacteristic
234C22×C4C22⋊Q811yescharacteristic
235C22×C4C22⋊Q811yescharacteristic
236C22×C4#12411yescharacteristic
237C22×C4#12411yescharacteristic
238C22×C4#12511yescharacteristic
239C22×C4#12511yescharacteristic
240C22×C4C4⋊Q811yescharacteristic
241C22×C4C4⋊Q811yescharacteristic
242C2×C8—22no—
243C2×C8—22no—
244C2×C8—22no—
245C2×C8—24no—
246C2×C8—24no—
247C2×C8—24no—
248C2×C8—24no—
249C2×C8—24no—
250C2×C8—24no—
251C2×C8—24no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—42no—
259C2×C8—44no—
260C2×C8—42no—
261C2×C8—44no—
262C2×C8#9512yes—
263C2×C8#9512yes—
264C2×C8#9512yes—
265C2×C8#9512yes—
266C2×C8#9914yes—
267C4⋊C4—44no—
268C4⋊C4—44no—
269C4⋊C4—44no—
270C4⋊C4—44no—
271C4⋊C4—84no—
272C4⋊C4—84no—
273C22⋊C4—28no—
274C22⋊C4—44no—
275C22⋊C4—48no—
276C22⋊C4—84no—
277C22⋊C4—84no—
278C22⋊C4—82no—
279C22⋊C4—82no—
280C42—42no—
281C42—42no—
282C42—44no—
283C23—22no—
284C23—22no—
285C23—22no—
286C23—22no—
287C23—22no—
288C23—22no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23—48no—
298C23—48no—
299C23C4×C4⋊C411yescharacteristic
300C23#38011yescharacteristic
301C23#38411yescharacteristic
302C23#38911yescharacteristic
303C23#38911yescharacteristic
304C23#39011yescharacteristic
305C23#39011yescharacteristic
306C23#42814yes—
307C23#44014yes—
308C23#44112yes—
309C23#44212yes—
310C23#46814yes—
311C23#46912yes—
312C23#47012yes—
313C23#49612yes—
314C23#49712yes—
315C23#50114yes—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—84no—
353C2×C4—84no—
354C2×C4—84no—
355C2×C4—84no—
356C2×C4—84no—
357C2×C4—84no—
358C2×C4—84no—
359C2×C4—84no—
360C2×C4—164no—
361C2×C4—164no—
362C2×C4—168no—
363C2×C4C2×#9511yescharacteristic
364C2×C4C4×C4⋊C411yescharacteristic
365C2×C4#38011yescharacteristic
366C2×C4#38211yescharacteristic
367C2×C4#38911yescharacteristic
368C2×C4#38911yescharacteristic
369C2×C4#39111yescharacteristic
370C2×C4#39111yescharacteristic
371C2×C4C2×#9912yes—
372C2×C4#41012yes—
373C8—22no—
374C8—22no—
375C8—22no—
376C8—44no—
377C8#34214yescomplemented
378C22—24no—
379C22—24no—
380C22—24no—
381C22—24no—
382C22—24no—
383C22—24no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—44no—
397C22—44no—
398C22—44no—
399C22—44no—
400C22C4⋊#9511yescharacteristic
401C22#177612yes—
402C22#180912yes—
403C22#184712yes—
404C22#194312yes—
405C22#194412yes—
406C22#196412yes—
407C22#197111yescharacteristic
408C22#197211yescharacteristic
409C22#197312yes—
410C22#197612yes—
411C4—21no—
412C4—21no—
413C4—21no—
414C4—21no—
415C4—21no—
416C4—21no—
417C4—84no—
418C4—164no—
419C4—164no—
420C4C2×#34211yescharacteristic
421C4#174011yescharacteristic
422C4#199612yes—
423C2—22no—
424C2—22no—
425C2—22no—
426C2—22no—
427C2—22no—
428C2—22no—
429C2#978412yes—
430C2#978512yes—
431C2#1085411yescharacteristic
432C2#1086311yescharacteristic
433C2#1087111yescharacteristic
subgroupG/H
center, socleC23#390
commutatorC22×C4C2×C42
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
7. C22×#107C4semidirect
19. #1807C4semidirect
21. #1752C4semidirect
24. C2×#340C4semidirect
377. C8#342semidirect

Maximal subgroups

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

subgroupindex
C2×#18082
C2×#18072
C2×#17522
#108542
#80232