Finite Group Databasebeta

← #128134

Subgroups — #128134

Order 512.

786 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21788C211yesmaximal, characteristic
2C2×#1799C211yesmaximal, characteristic
3#17362C211yesmaximal, characteristic
4#7322C214yesmaximal, complemented
5C22×C4×D4C2211yescharacteristic
6C2×#432C411yescharacteristic, complemented
7C2×#431C411yescharacteristic, complemented
8C22×#99C412yes—
9C2×#407—21no—
10C2×#380—21no—
11#1955C412yescomplemented
12#1955C2212yes—
13#1954—21no—
14#1954—21no—
15#1799—21no—
16#1799—21no—
17#1799C412yes—
18#1799C2212yes—
19#1779—21no—
20#1779—21no—
21C2×#342—21no—
22#1465—21no—
23#1465C2212yes—
24#1463C412yescomplemented
25#1462—21no—
26#1459—21no—
27#1459C412yescomplemented
28C23×D4C2×C411yescharacteristic
29C24×C4D411yescharacteristic
30C24×C4D411yescharacteristic
31C22×M16—41no—
32C2×C4×D4—41no—
33C2×C4×D4—41no—
34C2×C4×D4—41no—
35C2×C4×D4—41no—
36C2×C4×D4C2×C411yescharacteristic
37C2×C4×D4C2×C411yescharacteristic
38C2×C4×D4C2×C411yescharacteristic
39C2×C4×D4D411yescharacteristic
40C2×C4×D4D411yescharacteristic
41C2×C4×D4D411yescharacteristic
42C2×C4×D4Q811yescharacteristic
43C2×C4×D4C2311yescharacteristic
44C22×C4⋊C4—41no—
45C22×C4⋊C4C2×C411yescharacteristic
46C22×C22⋊C4—22no—
47C22×C22⋊C4D411yescharacteristic
48C22×C22⋊C4Q811yescharacteristic
49C22×C42C2×C411yescharacteristic
50#432—41no—
51#432—41no—
52#431—41no—
53#431—41no—
54C2×C4⋊C8—21no—
55C2×C4⋊C8C2×C412yescomplemented
56C2×#99—22no—
57C2×#99—44no—
58C2×#99C2×C414yes—
59#407—22no—
60#407—22no—
61#407—41no—
62#407—41no—
63C2×C22⋊C8—22no—
64C2×C22⋊C8—21no—
65C2×C22⋊C8—21no—
66C2×C22⋊C8—22no—
67C2×C22⋊C8—22no—
68C2×C22⋊C8—22no—
69C2×C22⋊C8—22no—
70C2×C22⋊C8—41no—
71C2×#97C2×C412yescomplemented
72C2×C4×C8—21no—
73#380—22no—
74#380—22no—
75#380—41no—
76#380—41no—
77C2×#95—41no—
78#342—22no—
79#342—22no—
80#324—22no—
81#324C2×C414yescomplemented
82#323—22no—
83#323—22no—
84C25C22⋊C411yescharacteristic
85C25C4⋊C411yescharacteristic
86C22×D4—22no—
87C22×D4—22no—
88C22×D4—41no—
89C22×D4—41no—
90C22×D4—42no—
91C22×D4C4211yescharacteristic
92C22×D4C22⋊C411yescharacteristic
93C22×D4C4⋊C411yescharacteristic
94C22×D4C22×C411yescharacteristic
95C23×C4—21no—
96C23×C4—21no—
97C23×C4—21no—
98C23×C4—21no—
99C23×C4—21no—
100C23×C4—21no—
101C23×C4—21no—
102C23×C4—21no—
103C23×C4—42no—
104C23×C4—82no—
105C23×C4—82no—
106C23×C4C4211yescharacteristic
107C23×C4C22⋊C411yescharacteristic
108C23×C4C22⋊C411yescharacteristic
109C23×C4C22⋊C411yescharacteristic
110C23×C4C22⋊C411yescharacteristic
111C23×C4C22⋊C411yescharacteristic
112C23×C4C22⋊C411yescharacteristic
113C23×C4C4⋊C411yescharacteristic
114C23×C4C4⋊C411yescharacteristic
115C23×C4C4⋊C411yescharacteristic
116C23×C4C4⋊C411yescharacteristic
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C2×M16—42no—
120C2×M16—42no—
121C2×M16—84no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—42no—
125C4×D4—41no—
126C4×D4—41no—
127C4×D4—41no—
128C4×D4—41no—
129C4×D4—41no—
130C4×D4—41no—
131C4×D4—41no—
132C4×D4—41no—
133C4×D4—41no—
134C4×D4—41no—
135C4×D4—41no—
136C4×D4—41no—
137C4×D4—41no—
138C4×D4—41no—
139C4×D4—41no—
140C4×D4—41no—
141C2×C4⋊C4—41no—
142C2×C4⋊C4—41no—
143C2×C4⋊C4—42no—
144C2×C4⋊C4—42no—
145C2×C4⋊C4—84no—
146C2×C4⋊C4C4211yescharacteristic
147C2×C4⋊C4C22⋊C411yescharacteristic
148C2×C4⋊C4C4⋊C411yescharacteristic
149C2×C4⋊C4C22×C411yescharacteristic
150C2×C22⋊C4—24no—
151C2×C22⋊C4—41no—
152C2×C22⋊C4—41no—
153C2×C22⋊C4—41no—
154C2×C22⋊C4—41no—
155C2×C22⋊C4—44no—
156C2×C22⋊C4—44no—
157C2×C22⋊C4—42no—
158C2×C22⋊C4C22⋊C411yescharacteristic
159C2×C22⋊C4C22⋊C411yescharacteristic
160C2×C22⋊C4C4⋊C411yescharacteristic
161C2×C22⋊C4C4⋊C411yescharacteristic
162C2×C22⋊C4C4⋊C411yescharacteristic
163C2×C22⋊C4C4⋊C411yescharacteristic
164C2×C22⋊C4C2×D411yescharacteristic
165C2×C22⋊C4C2×Q811yescharacteristic
166C2×C42—21no—
167C2×C42—21no—
168C2×C42—21no—
169C2×C42—21no—
170C2×C42C4211yescharacteristic
171C2×C42C22⋊C411yescharacteristic
172C2×C42C4⋊C411yescharacteristic
173C2×C42C22×C411yescharacteristic
174C4⋊C8—42no—
175C4⋊C8—42no—
176#99—44no—
177#99—44no—
178C22⋊C8—24no—
179C22⋊C8—42no—
180C22⋊C8—42no—
181C22⋊C8—42no—
182C22⋊C8—41no—
183C22⋊C8—41no—
184C22⋊C8—41no—
185C22⋊C8—41no—
186C22⋊C8—82no—
187C22⋊C8—82no—
188C22⋊C8—82no—
189C22⋊C8—82no—
190#97—42no—
191C4×C8—42no—
192#95—42no—
193#95—42no—
194C24—21no—
195C24—21no—
196C24—22no—
197C24—21no—
198C24—21no—
199C24—21no—
200C24—21no—
201C24—22no—
202C24—41no—
203C24—42no—
204C24—42no—
205C24—81no—
206C24—81no—
207C24#9511yescharacteristic
208C24#9511yescharacteristic
209C24#9511yescharacteristic
210C24C2×C22⋊C411yescharacteristic
211C24C2×C4⋊C411yescharacteristic
212C2×D4—22no—
213C2×D4—22no—
214C2×D4—22no—
215C2×D4—21no—
216C2×D4—21no—
217C2×D4—22no—
218C2×D4—41no—
219C2×D4—41no—
220C2×D4—41no—
221C2×D4—41no—
222C2×D4—41no—
223C2×D4—41no—
224C2×D4—44no—
225C2×D4—44no—
226C2×D4—44no—
227C2×D4—44no—
228C2×D4#10012yes—
229C2×D4#10112yes—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—22no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—42no—
256C22×C4—41no—
257C22×C4—44no—
258C22×C4—44no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—44no—
265C22×C4—82no—
266C22×C4—82no—
267C22×C4—81no—
268C22×C4—81no—
269C22×C4—81no—
270C22×C4—81no—
271C22×C4—81no—
272C22×C4—81no—
273C22×C4—82no—
274C22×C4—82no—
275C22×C4—82no—
276C22×C4—82no—
277C22×C4—84no—
278C22×C4—84no—
279C22×C4—84no—
280C22×C4#9511yescharacteristic
281C22×C4#9511yescharacteristic
282C22×C4#9511yescharacteristic
283C22×C4#9511yescharacteristic
284C22×C4#9511yescharacteristic
285C22×C4#9911yescharacteristic
286C22×C4#9911yescharacteristic
287C22×C4#9911yescharacteristic
288C22×C4#9911yescharacteristic
289C22×C4C2×C4211yescharacteristic
290C22×C4C2×C22⋊C411yescharacteristic
291C22×C4C2×C22⋊C411yescharacteristic
292C22×C4C2×C4⋊C411yescharacteristic
293C22×C4C2×C4⋊C411yescharacteristic
294M16—84no—
295M16—84no—
296C2×C8—44no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—84no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—84no—
309C4⋊C4—84no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C22⋊C4—44no—
319C22⋊C4—44no—
320C22⋊C4—84no—
321C42—21no—
322C42—21no—
323C42—21no—
324C42—21no—
325C42—21no—
326C42—21no—
327C42—21no—
328C42—21no—
329C23—22no—
330C23—22no—
331C23—22no—
332C23—22no—
333C23—22no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—41no—
346C23—42no—
347C23—42no—
348C23—42no—
349C23—41no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23—42no—
358C23—42no—
359C23—42no—
360C23—81no—
361C23—81no—
362C23—81no—
363C23—81no—
364C23—84no—
365C23—82no—
366C23—82no—
367C23—84no—
368C23—84no—
369C23#34211yescharacteristic
370C23#34211yescharacteristic
371C23#34312yes—
372C23#34412yes—
373C23C2×#9511yescharacteristic
374C23C2×#10011yescharacteristic
375C23C2×#10111yescharacteristic
376D4—42no—
377D4—42no—
378D4—44no—
379D4—44no—
380D4—42no—
381D4—42no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—84no—
430C2×C4—84no—
431C2×C4—84no—
432C2×C4—168no—
433C2×C4#34211yescharacteristic
434C2×C4#34211yescharacteristic
435C2×C4#34312yes—
436C2×C4C2×#9911yescharacteristic
437C2×C4C2×#9911yescharacteristic
438C8—82no—
439C8—84no—
440C8—82no—
441C22—21no—
442C22—21no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—22no—
450C22—21no—
451C22—22no—
452C22—42no—
453C22—42no—
454C22—42no—
455C22—42no—
456C22—41no—
457C22—41no—
458C22—44no—
459C22—44no—
460C22—42no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—41no—
465C22—41no—
466C22—41no—
467C22—41no—
468C22—41no—
469C22—41no—
470C22—81no—
471C22—81no—
472C22—81no—
473C22—84no—
474C22—81no—
475C22—82no—
476C22—82no—
477C22—84no—
478C22#139512yes—
479C22C2×#34211yescharacteristic
480C22C2×#34311yescharacteristic
481C22C2×#34411yescharacteristic
482C4—21no—
483C4—21no—
484C4—21no—
485C4—22no—
486C4—41no—
487C4—41no—
488C4—41no—
489C4—41no—
490C4—41no—
491C4—41no—
492C4—41no—
493C4—164no—
494C4#212712yes—
495C2—22no—
496C2—21no—
497C2—21no—
498C2—41no—
499C2—41no—
500C2—42no—
501C2—41no—
502C2—42no—
503C2—81no—
504C2#810711yescharacteristic
505C2#1258211yescharacteristic
506C2C2×#139511yescharacteristic
subgroupG/H
center, socleC22C2×#342
commutatorC22×C4C2×C42
Frattini, cosocleC2×C4×D4C23
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7322C2semidirect
6. C2×#432C4semidirect
7. C2×#431C4semidirect
11. #1955C4semidirect
24. #1463C4semidirect
27. #1459C4semidirect
55. C2×C4⋊C8C2×C4semidirect
71. C2×#97C2×C4semidirect
81. #324C2×C4semidirect

Maximal subgroups

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

subgroupindex
#217882
C2×#17992
#173622
#73222