Finite Group Databasebeta

← #108617

Subgroups — #108617

Order 512.

770 conjugacy classes of subgroups.

The subgroup lattice has 473 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20385C211yesmaximal, characteristic
2#10723C211yesmaximal, characteristic
3#10721C211yesmaximal, characteristic
4C2×#1305C212yesmaximal
5#8016C212yesmaximal
6C24×C8C2211yescharacteristic
7C2×#407C2212yes—
8C2×#407C2211yescharacteristic
9C2×#405C411yescharacteristic, complemented
10C2×#405C411yescharacteristic, complemented
11#2280C2211yescharacteristic
12C2×#380C2212yes—
13#1752—21no—
14#1752—21no—
15C8×C22⋊C4—21no—
16C8×C22⋊C4—21no—
17C2×#341—21no—
18C2×#341—21no—
19C2×#340—21no—
20C2×#340—21no—
21#1729C411yescharacteristic, complemented
22#1729C411yescharacteristic, complemented
23#1305C418yescomplemented
24#1294—22no—
25#1294—22no—
26C24×C4C2311yescharacteristic
27C22×M16—22no—
28C22×M16C2×C411yescharacteristic
29C22×M16C2×C411yescharacteristic
30C23×C8—22no—
31C23×C8—22no—
32C23×C8—42no—
33C23×C8—42no—
34C23×C8C2×C411yescharacteristic
35C23×C8C2×C411yescharacteristic
36C23×C8D411yescharacteristic
37C23×C8D411yescharacteristic
38C23×C8D411yescharacteristic
39C23×C8D411yescharacteristic
40C2×#117C2×C411yescharacteristic
41C2×#117C2×C411yescharacteristic
42C22×C4⋊C4—22no—
43C22×C22⋊C4—22no—
44#407—22no—
45#407—22no—
46#407—22no—
47#407—22no—
48#407C2×C414yes—
49#407C2×C414yes—
50C2×C22⋊C8—21no—
51C2×C22⋊C8—21no—
52C2×C22⋊C8—22no—
53C2×C22⋊C8—22no—
54#405—42no—
55#405—42no—
56C2×#97—21no—
57C2×#97—21no—
58C2×C4×C8—21no—
59C2×C4×C8—21no—
60#381—21no—
61#380—22no—
62#380—22no—
63#380—22no—
64#380C2×C414yes—
65#380D411yescharacteristic
66#380D411yescharacteristic
67#380Q811yescharacteristic
68#380Q811yescharacteristic
69#379—21no—
70C4×C22⋊C4—21no—
71C4×C22⋊C4—21no—
72C2×#95—22no—
73#341—42no—
74#341—42no—
75#340—42no—
76#340—42no—
77#336—21no—
78#336—21no—
79#336—21no—
80#336—21no—
81C25#4111yescharacteristic
82C23×C4—41no—
83C23×C4—41no—
84C23×C4—41no—
85C23×C4—41no—
86C23×C4—42no—
87C23×C4C22×C411yescharacteristic
88C23×C4C22×C412yes—
89C23×C4C2×D411yescharacteristic
90C23×C4C2×D411yescharacteristic
91C23×C4C2×D411yescharacteristic
92C23×C4C2×Q811yescharacteristic
93C23×C4#4111yescharacteristic
94C23×C4#4112yes—
95C23×C4#4112yes—
96C23×C4#4112yes—
97C2×M16—21no—
98C2×M16—21no—
99C2×M16—24no—
100C2×M16—42no—
101C2×M16—41no—
102C2×M16—41no—
103C2×M16—41no—
104C2×M16—41no—
105C2×M16—42no—
106C2×M16—42no—
107C2×M16—42no—
108C2×M16—42no—
109C2×M16C4212yes—
110C2×M16C4212yes—
111C22×C8—21no—
112C22×C8—21no—
113C22×C8—22no—
114C22×C8—22no—
115C22×C8—21no—
116C22×C8—21no—
117C22×C8—21no—
118C22×C8—21no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—41no—
128C22×C8—41no—
129C22×C8—41no—
130C22×C8—41no—
131C22×C8—41no—
132C22×C8—41no—
133C22×C8—48no—
134C22×C8—48no—
135C22×C8—42no—
136C22×C8C22⋊C411yescharacteristic
137C22×C8C22⋊C411yescharacteristic
138C22×C8C22⋊C411yescharacteristic
139C22×C8C22⋊C411yescharacteristic
140#117—41no—
141#117—41no—
142#117—41no—
143#117—41no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—42no—
146C2×C4⋊C4—42no—
147C2×C4⋊C4—42no—
148C2×C4⋊C4—42no—
149C2×C4⋊C4—42no—
150C2×C4⋊C4C4⋊C411yescharacteristic
151C2×C4⋊C4C4⋊C411yescharacteristic
152C2×C4⋊C4C4⋊C411yescharacteristic
153C2×C4⋊C4C4⋊C411yescharacteristic
154C2×C22⋊C4—21no—
155C2×C22⋊C4—21no—
156C2×C22⋊C4—21no—
157C2×C22⋊C4—21no—
158C2×C22⋊C4—21no—
159C2×C22⋊C4—21no—
160C2×C22⋊C4—24no—
161C2×C22⋊C4—42no—
162C2×C22⋊C4—82no—
163C2×C22⋊C4—82no—
164C2×C42—21no—
165C2×C42—21no—
166C22⋊C8—24no—
167C22⋊C8—24no—
168C22⋊C8—24no—
169C22⋊C8—24no—
170C22⋊C8—42no—
171C22⋊C8—42no—
172#97—42no—
173#97—42no—
174C4×C8—42no—
175C4×C8—42no—
176#95—21no—
177#95—21no—
178#95—21no—
179#95—21no—
180#95—24no—
181#95—42no—
182C24—22no—
183C24—21no—
184C24—21no—
185C24—41no—
186C24—41no—
187C24—41no—
188C24—41no—
189C24#11711yescharacteristic
190C24#11712yes—
191C24#12111yescharacteristic
192C24C22⋊Q811yescharacteristic
193C24#12612yes—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—44no—
225C22×C4—82no—
226C22×C4—82no—
227C22×C4—82no—
228C22×C4—82no—
229C22×C4—82no—
230C22×C4—82no—
231C22×C4C2×C4211yescharacteristic
232C22×C4C2×C22⋊C411yescharacteristic
233C22×C4C2×C4⋊C411yescharacteristic
234C22×C4#11711yescharacteristic
235C22×C4#11711yescharacteristic
236C22×C4#11711yescharacteristic
237C22×C4#11712yes—
238C22×C4C4×D412yes—
239C22×C4C4×D412yes—
240C22×C4C4×D412yes—
241C22×C4C4×Q812yes—
242C22×C4#12111yescharacteristic
243C22×C4C22⋊Q811yescharacteristic
244C22×C4#12312yes—
245C22×C4#12312yes—
246C22×C4#12412yes—
247C22×C4#12512yes—
248C22×C4#12612yes—
249M16—42no—
250M16—42no—
251M16—42no—
252M16—42no—
253M16—44no—
254M16—44no—
255M16—82no—
256M16—82no—
257C2×C8—24no—
258C2×C8—24no—
259C2×C8—21no—
260C2×C8—21no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8—22no—
264C2×C8—22no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—44no—
272C2×C8—44no—
273C2×C8—44no—
274C2×C8—44no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—41no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—44no—
282C2×C8—82no—
283C2×C8—82no—
284C2×C8#9912yes—
285C2×C8#9912yes—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—44no—
291C4⋊C4—44no—
292C4⋊C4—82no—
293C4⋊C4—82no—
294C22⋊C4—42no—
295C22⋊C4—42no—
296C22⋊C4—42no—
297C22⋊C4—42no—
298C22⋊C4—42no—
299C22⋊C4—41no—
300C22⋊C4—41no—
301C22⋊C4—84no—
302C22⋊C4—84no—
303C42—41no—
304C42—41no—
305C42—41no—
306C42—41no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23—21no—
323C23—22no—
324C23—22no—
325C23—22no—
326C23—22no—
327C23—42no—
328C23—42no—
329C23—42no—
330C23—42no—
331C23—44no—
332C23—42no—
333C23—42no—
334C23—42no—
335C23—42no—
336C23—44no—
337C23#37611yescharacteristic
338C23#38011yescharacteristic
339C23#38311yescharacteristic
340C23#38712yes—
341C23#38812yes—
342C23#47112yes—
343C23#47211yescharacteristic
344C23#47311yescharacteristic
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—22no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—84no—
386C2×C4—84no—
387C2×C4—84no—
388C2×C4—162no—
389C2×C4—162no—
390C2×C4—162no—
391C2×C4—162no—
392C2×C4C4×C22⋊C411yescharacteristic
393C2×C4C4×C4⋊C411yescharacteristic
394C2×C4#38111yescharacteristic
395C2×C4#38211yescharacteristic
396C2×C4#38712yes—
397C2×C4#38812yes—
398C2×C4C2×#9911yescharacteristic
399C2×C4C2×#9911yescharacteristic
400C2×C4#41011yescharacteristic
401C2×C4#41011yescharacteristic
402C8—22no—
403C8—22no—
404C8—41no—
405C8—41no—
406C8—42no—
407C8—42no—
408C8—84no—
409C22—22no—
410C22—22no—
411C22—22no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—22no—
417C22—21no—
418C22—21no—
419C22—22no—
420C22—21no—
421C22—21no—
422C22—22no—
423C22—21no—
424C22—21no—
425C22—22no—
426C22—44no—
427C22—44no—
428C22—44no—
429C22—44no—
430C22—41no—
431C22—41no—
432C22—41no—
433C22—41no—
434C22—41no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22#143411yescharacteristic
439C22#179911yescharacteristic
440C22#179911yescharacteristic
441C22#181411yescharacteristic
442C22#181411yescharacteristic
443C22#181511yescharacteristic
444C22#181611yescharacteristic
445C22#193012yes—
446C22#193112yes—
447C4—22no—
448C4—21no—
449C4—21no—
450C4—22no—
451C4—81no—
452C4—81no—
453C4—81no—
454C4—81no—
455C4—164no—
456C4C4×#9911yescharacteristic
457C4C4×#9911yescharacteristic
458C4#181911yescharacteristic
459C4#182011yescharacteristic
460C2—22no—
461C2—22no—
462C2—41no—
463C2—41no—
464C2—42no—
465C2#862611yescharacteristic
466C2#862711yescharacteristic
467C2#862811yescharacteristic
468C2#862911yescharacteristic
469C2#1080111yescharacteristic
470C2#1082711yescharacteristic
471C2#1082811yescharacteristic
subgroupG/H
center, socleC23#380
commutatorC22×C4C2×C42
Frattini, cosocleC24×C4C23
prime radicalC25#41
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
9. C2×#405C4semidirect
10. C2×#405C4semidirect
21. #1729C4semidirect
22. #1729C4semidirect
23. #1305C4semidirect

Maximal subgroups

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

subgroupindex
#203852
#107232
#107212
C2×#13052
#80162