Finite Group Databasebeta

← #120092

Subgroups — #120092

Order 512.

624 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21170C211yesmaximal, characteristic
2#13955C211yesmaximal, characteristic
3C2×#1303C211yesmaximal, characteristic
4#8026C211yesmaximal, characteristic
5#8014C211yesmaximal, characteristic
6C2×#1286C211yesmaximal, characteristic
7#7892C211yesmaximal, characteristic
8C2×#431C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#428—21no—
11C2×#423C2211yescharacteristic
12#2285C2211yescharacteristic
13C2×#382C2211yescharacteristic
14C2×C4×C4⋊C4C2211yescharacteristic
15C2×C4×C4⋊C4C2211yescharacteristic
16#1949—22no—
17#1948C412yescomplemented
18#1947C412yescomplemented
19#1736—21no—
20#1442—22no—
21C4⋊#95—21no—
22C4⋊#95—21no—
23#1434—21no—
24#1434—21no—
25#1303—22no—
26#1294—21no—
27#1294—21no—
28#1294—22no—
29#1286—22no—
30#1279—22no—
31C22×M16—21no—
32C2×#117—21no—
33C22×C4⋊C4—21no—
34C22×C4⋊C4—21no—
35C22×C4⋊C4D411yescharacteristic
36C22×C4⋊C4D411yescharacteristic
37C22×C42—21no—
38C22×C42—21no—
39C22×C42C2311yescharacteristic
40#431—21no—
41#431—21no—
42#431—21no—
43#431—21no—
44#431—21no—
45#431—21no—
46#431Q812yes—
47#431Q812yes—
48#428—42no—
49#428—42no—
50C2×#107—22no—
51C2×#106—22no—
52#423—21no—
53#423—21no—
54#423D412yes—
55#423Q812yes—
56C2×C4⋊C8—21no—
57C2×C4⋊C8—21no—
58C2×C4⋊C8—21no—
59C2×C4⋊C8C2×C412yes—
60C2×C22⋊C8—21no—
61C2×C22⋊C8—21no—
62C2×C4×C8—21no—
63C2×C4×C8—21no—
64#384C2×C411yescharacteristic
65#384C2×C411yescharacteristic
66#382—21no—
67#382—21no—
68#382—21no—
69#382—21no—
70#382—21no—
71#380—21no—
72#380—21no—
73C4×C4⋊C4—21no—
74C4×C4⋊C4—21no—
75C4×C4⋊C4—21no—
76C4×C4⋊C4—21no—
77C4×C4⋊C4—21no—
78C4×C4⋊C4—21no—
79C4×C4⋊C4—21no—
80C4×C4⋊C4—21no—
81C4×C4⋊C4C2×C412yes—
82C4×C4⋊C4D412yes—
83C2×#95—21no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—21no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95D411yescharacteristic
90C2×#95D411yescharacteristic
91C2×#95D412yes—
92#340—22no—
93#340—22no—
94C23×C4—21no—
95C23×C4—21no—
96C23×C4—21no—
97C23×C4—21no—
98C23×C4—22no—
99C23×C4C2×D411yescharacteristic
100C23×C4C2×D411yescharacteristic
101C23×C4C2×D411yescharacteristic
102C2×M16—22no—
103C2×M16—22no—
104C2×M16—41no—
105C2×M16—41no—
106C2×M16—41no—
107C2×M16—41no—
108C22×C8—22no—
109C22×C8—41no—
110C22×C8—41no—
111#117—41no—
112#117—41no—
113#117—41no—
114#117—41no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4—41no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4—41no—
138C2×C4⋊C4—41no—
139C2×C4⋊C4—41no—
140C2×C4⋊C4—41no—
141C2×C4⋊C4C22⋊C412yes—
142C2×C4⋊C4C22⋊C412yes—
143C2×C22⋊C4—41no—
144C2×C42—22no—
145C2×C42—22no—
146C2×C42—41no—
147C2×C42—41no—
148C2×C42—41no—
149C2×C42—41no—
150C2×C42—41no—
151C2×C42—41no—
152C2×C42—41no—
153C2×C42—82no—
154C2×C42C22×C411yescharacteristic
155C2×C42C2×D411yescharacteristic
156C2×C42C2×D411yescharacteristic
157C2×C42C2×Q811yescharacteristic
158C2×C42C2×Q811yescharacteristic
159C2×C42C2×Q811yescharacteristic
160C2×C42#4111yescharacteristic
161C2×C42#4111yescharacteristic
162C2×C42#4111yescharacteristic
163C2×C42#4111yescharacteristic
164C2×C42#4111yescharacteristic
165C2×C42#4111yescharacteristic
166#107—44no—
167#106—44no—
168C4⋊C8—22no—
169C4⋊C8—22no—
170C4⋊C8—22no—
171C4⋊C8—22no—
172C4⋊C8—22no—
173C4⋊C8—41no—
174C4⋊C8—41no—
175C4⋊C8C4⋊C414yes—
176C22⋊C8—22no—
177C22⋊C8—22no—
178C22⋊C8—41no—
179C22⋊C8—41no—
180C4×C8—22no—
181C4×C8—22no—
182C4×C8—41no—
183C4×C8—41no—
184#95—22no—
185#95—22no—
186#95—21no—
187#95—21no—
188#95—22no—
189#95—21no—
190#95—21no—
191#95—22no—
192#95—44no—
193#95—44no—
194#95—41no—
195#95—41no—
196#95—41no—
197#95—41no—
198#95—82no—
199#95—82no—
200C24#12011yescharacteristic
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4C2×C22⋊C411yescharacteristic
239C22×C4C4×D411yescharacteristic
240C22×C4C4×D411yescharacteristic
241C22×C4#12111yescharacteristic
242C22×C4#12111yescharacteristic
243C22×C4#12111yescharacteristic
244C22×C4#12111yescharacteristic
245C22×C4#12111yescharacteristic
246C22×C4C22⋊Q811yescharacteristic
247C22×C4C22⋊Q811yescharacteristic
248C22×C4C22⋊Q811yescharacteristic
249C22×C4#12311yescharacteristic
250C22×C4#12311yescharacteristic
251C22×C4#12311yescharacteristic
252C22×C4#12411yescharacteristic
253C22×C4C4⋊Q811yescharacteristic
254C22×C4C4⋊Q811yescharacteristic
255C22×C4C4⋊Q811yescharacteristic
256C22×C4#13511yescharacteristic
257C22×C4#13511yescharacteristic
258C22×C4#13511yescharacteristic
259C22×C4#13511yescharacteristic
260C22×C4AGL(1,ℤ/8)11yescharacteristic
261C22×C4AGL(1,ℤ/8)11yescharacteristic
262C22×C4#13711yescharacteristic
263C22×C4#13711yescharacteristic
264M16—42no—
265M16—42no—
266M16—42no—
267M16—42no—
268C2×C8—24no—
269C2×C8—42no—
270C2×C8—42no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—81no—
274C2×C8—81no—
275C2×C8—81no—
276C2×C8—81no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—22no—
281C4⋊C4—44no—
282C4⋊C4—44no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—81no—
292C4⋊C4—81no—
293C4⋊C4—81no—
294C4⋊C4—81no—
295C4⋊C4—81no—
296C4⋊C4—81no—
297C4⋊C4—81no—
298C4⋊C4—81no—
299C22⋊C4—81no—
300C22⋊C4—81no—
301C42—82no—
302C42—82no—
303C42—81no—
304C42—81no—
305C42—81no—
306C42—81no—
307C42—81no—
308C42—81no—
309C42C2×C4⋊C411yescharacteristic
310C42#11711yescharacteristic
311C42C4×Q811yescharacteristic
312C42C4×Q811yescharacteristic
313C42#12111yescharacteristic
314C42C22⋊Q811yescharacteristic
315C42C22⋊Q811yescharacteristic
316C42C22⋊Q811yescharacteristic
317C42C22⋊Q811yescharacteristic
318C42C22⋊Q811yescharacteristic
319C42#12511yescharacteristic
320C42#12511yescharacteristic
321C42#12511yescharacteristic
322C42#12511yescharacteristic
323C42#12611yescharacteristic
324C42#12611yescharacteristic
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23#38611yescharacteristic
330C23#39411yescharacteristic
331C23#39511yescharacteristic
332C23#44911yescharacteristic
333C23#44911yescharacteristic
334C23#45211yescharacteristic
335C23#45211yescharacteristic
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—82no—
389C2×C4#38011yescharacteristic
390C2×C4#38711yescharacteristic
391C2×C4#38711yescharacteristic
392C2×C4#38911yescharacteristic
393C2×C4#38911yescharacteristic
394C2×C4#39111yescharacteristic
395C2×C4#39611yescharacteristic
396C2×C4#39611yescharacteristic
397C2×C4#39811yescharacteristic
398C2×C4#39811yescharacteristic
399C2×C4#39811yescharacteristic
400C2×C4#40011yescharacteristic
401C2×C4#41611yescharacteristic
402C2×C4#41611yescharacteristic
403C2×C4#41811yescharacteristic
404C2×C4#41911yescharacteristic
405C2×C4#46311yescharacteristic
406C2×C4#46311yescharacteristic
407C2×C4#46411yescharacteristic
408C2×C4#46411yescharacteristic
409C2×C4#47811yescharacteristic
410C2×C4#47811yescharacteristic
411C2×C4#47911yescharacteristic
412C2×C4#47911yescharacteristic
413C2×C4#48211yescharacteristic
414C2×C4#48211yescharacteristic
415C2×C4#48511yescharacteristic
416C2×C4#48511yescharacteristic
417C2×C4#48811yescharacteristic
418C2×C4#48811yescharacteristic
419C2×C4#48911yescharacteristic
420C2×C4#48911yescharacteristic
421C2×C4#49911yescharacteristic
422C2×C4#49911yescharacteristic
423C2×C4#50111yescharacteristic
424C2×C4#50111yescharacteristic
425C8—44no—
426C8—82no—
427C8—82no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22#144111yescharacteristic
441C22#187311yescharacteristic
442C22#187511yescharacteristic
443C22#190411yescharacteristic
444C22#190511yescharacteristic
445C22#190611yescharacteristic
446C22#190611yescharacteristic
447C22#204211yescharacteristic
448C22#204211yescharacteristic
449C22#205911yescharacteristic
450C22#205911yescharacteristic
451C4—21no—
452C4—21no—
453C4—21no—
454C4—21no—
455C4—21no—
456C4—21no—
457C4—21no—
458C4—21no—
459C4—82no—
460C4—82no—
461C4—82no—
462C4—82no—
463C4—162no—
464C4—161no—
465C4—161no—
466C4#180411yescharacteristic
467C4#180511yescharacteristic
468C4#199111yescharacteristic
469C4#199311yescharacteristic
470C4#208511yescharacteristic
471C4#208511yescharacteristic
472C4#210111yescharacteristic
473C4#210111yescharacteristic
474C2—21no—
475C2—21no—
476C2—21no—
477C2—21no—
478C2#917711yescharacteristic
479C2#917811yescharacteristic
480C2#918111yescharacteristic
481C2#918211yescharacteristic
482C2#1120811yescharacteristic
483C2#1121311yescharacteristic
484C2#1121511yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#120
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
17. #1948C4semidirect
18. #1947C4semidirect

Maximal subgroups

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

subgroupindex
#211702
#139552
C2×#13032
#80262
#80142
C2×#12862
#78922