Finite Group Databasebeta

← #120004

Subgroups — #120004

Order 512.

634 conjugacy classes of subgroups.

The subgroup lattice has 511 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21349C211yesmaximal, characteristic
2C4×#428C211yesmaximal, characteristic
3C2×#1283C211yesmaximal, characteristic
4#7908C211yesmaximal, characteristic
5#7891C211yesmaximal, characteristic
6C2×#1279C211yesmaximal, characteristic
7C2×#1278C211yesmaximal, characteristic
8C2×#431C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#428C411yescharacteristic
11C2×#428C411yescharacteristic
12C22×C4⋊C8C2211yescharacteristic
13C2×C4×M16C2211yescharacteristic
14C22×C4×C8C2211yescharacteristic
15C4×#117C2211yescharacteristic
16C2×C4×C4⋊C4C2211yescharacteristic
17#1939—21no—
18#1939—21no—
19#1850—22no—
20#1776—21no—
21#1776—21no—
22C4×#107—21no—
23C4×#106—21no—
24C4×C4⋊C8—21no—
25C4×C4⋊C8—21no—
26#1736C411yescharacteristic
27#1736C411yescharacteristic
28#1283—22no—
29#1280—22no—
30#1279—21no—
31#1279—21no—
32#1279—21no—
33#1279—21no—
34#1278—22no—
35C22×M16C2×C411yescharacteristic
36C22×M16C2×C411yescharacteristic
37C23×C8—21no—
38C23×C8—21no—
39C2×#117C2×C411yescharacteristic
40C2×#117C2×C411yescharacteristic
41C22×C4⋊C4C2×C411yescharacteristic
42C22×C4⋊C4C2×C411yescharacteristic
43C22×C42—21no—
44C22×C42C2311yescharacteristic
45#431—21no—
46#431—21no—
47#431—21no—
48#431—21no—
49#431—21no—
50#431—21no—
51#431—21no—
52#431Q812yes—
53#428—22no—
54#428—22no—
55#428—22no—
56#428C814yescomplemented
57C2×#107—21no—
58C2×#107—21no—
59C2×#106—21no—
60C2×#106—21no—
61C2×C4⋊C8—21no—
62C2×C4⋊C8—21no—
63C2×C4⋊C8—21no—
64C2×C4⋊C8—21no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8—22no—
67C2×C4⋊C8—22no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8Q812yes—
71C2×C22⋊C8—21no—
72C2×C22⋊C8D412yes—
73C4×M16—21no—
74C4×M16—21no—
75C4×M16D412yes—
76C4×M16Q812yes—
77C2×#97—21no—
78C2×C4×C8—21no—
79C2×C4×C8—21no—
80C2×C4×C8—21no—
81C2×C4×C8—21no—
82C2×C4×C8—21no—
83C2×C4×C8—21no—
84C2×C4×C8—21no—
85C2×C4×C8—21no—
86C2×C4×C8D411yescharacteristic
87C2×C4×C8D411yescharacteristic
88C4×C4⋊C4—21no—
89C4×C4⋊C4—21no—
90C4×C4⋊C4—21no—
91C4×C4⋊C4—21no—
92C4×C4⋊C4D411yescharacteristic
93C4×C4⋊C4D411yescharacteristic
94C4×C4⋊C4D411yescharacteristic
95C4×C4⋊C4D411yescharacteristic
96C4×C22⋊C4—21no—
97C4×C22⋊C4—21no—
98#376—21no—
99C2×#95—21no—
100C43—21no—
101#340—21no—
102#340—21no—
103#340—21no—
104#340—21no—
105#336—22no—
106#336—22no—
107C23×C4—21no—
108C23×C4—21no—
109C23×C4C22×C411yescharacteristic
110C23×C4C2×D411yescharacteristic
111C23×C4#4111yescharacteristic
112C2×M16—21no—
113C2×M16—21no—
114C2×M16—21no—
115C2×M16—21no—
116C2×M16—21no—
117C2×M16—21no—
118C2×M16—21no—
119C2×M16—21no—
120C2×M16C4⋊C412yes—
121C2×M16C4⋊C412yes—
122C2×M16C2×C812yes—
123C2×M16M1612yes—
124C22×C8—21no—
125C22×C8—21no—
126C22×C8—21no—
127C22×C8—21no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—42no—
131C22×C8—42no—
132C22×C8—41no—
133C22×C8—41no—
134C22×C8—41no—
135C22×C8—41no—
136C22×C8—41no—
137C22×C8—81no—
138C22×C8—81no—
139#117—21no—
140#117—21no—
141#117—21no—
142#117—21no—
143#117—21no—
144#117—21no—
145#117C2×C812yes—
146#117M1612yes—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4C22⋊C411yescharacteristic
158C2×C4⋊C4C22⋊C411yescharacteristic
159C2×C4⋊C4C22⋊C411yescharacteristic
160C2×C4⋊C4C22⋊C411yescharacteristic
161C2×C4⋊C4C2×C812yes—
162C2×C4⋊C4M1612yes—
163C2×C22⋊C4—21no—
164C2×C22⋊C4—21no—
165C2×C42—21no—
166C2×C42—21no—
167C2×C42—21no—
168C2×C42—21no—
169C2×C42—21no—
170C2×C42—21no—
171C2×C42—41no—
172C2×C42—41no—
173C2×C42—81no—
174C2×C42—81no—
175C2×C42C2×D411yescharacteristic
176C2×C42C2×D411yescharacteristic
177C2×C42C2×D411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C42C2×Q811yescharacteristic
180C2×C42C2×Q811yescharacteristic
181C2×C42C2×Q811yescharacteristic
182C2×C42#4111yescharacteristic
183C2×C42#4111yescharacteristic
184C2×C42#4111yescharacteristic
185C2×C42#4111yescharacteristic
186C2×C42#4111yescharacteristic
187#107—24no—
188#107—24no—
189#106—24no—
190#106—24no—
191C4⋊C8—22no—
192C4⋊C8—22no—
193C4⋊C8—22no—
194C4⋊C8—44no—
195C4⋊C8—44no—
196C4⋊C8—41no—
197C4⋊C8—41no—
198C4⋊C8—41no—
199C4⋊C8—41no—
200C4⋊C8—41no—
201C22⋊C8—22no—
202C22⋊C8—22no—
203C22⋊C8—22no—
204C22⋊C8—41no—
205#97—22no—
206#97—22no—
207C4×C8—22no—
208C4×C8—21no—
209C4×C8—21no—
210C4×C8—22no—
211C4×C8—22no—
212C4×C8—21no—
213C4×C8—21no—
214C4×C8—41no—
215C4×C8—41no—
216C4×C8—41no—
217C4×C8—41no—
218C4×C8—41no—
219#95—21no—
220#95—21no—
221#95—21no—
222#95—21no—
223#95—41no—
224#95—41no—
225C24C4×D411yescharacteristic
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—81no—
243C22×C4—81no—
244C22×C4—81no—
245C22×C4—81no—
246C22×C4C2×C22⋊C411yescharacteristic
247C22×C4C2×C4⋊C411yescharacteristic
248C22×C4#11711yescharacteristic
249C22×C4C4×D411yescharacteristic
250C22×C4C4×Q811yescharacteristic
251C22×C4C4×Q811yescharacteristic
252C22×C4#12011yescharacteristic
253C22×C4#12111yescharacteristic
254C22×C4#12111yescharacteristic
255C22×C4#12111yescharacteristic
256C22×C4C22⋊Q811yescharacteristic
257C22×C4C22⋊Q811yescharacteristic
258C22×C4#12311yescharacteristic
259C22×C4#12311yescharacteristic
260C22×C4#12511yescharacteristic
261C22×C4#12511yescharacteristic
262C22×C4#12611yescharacteristic
263C22×C4C4⋊Q811yescharacteristic
264C22×C4C22×C811yescharacteristic
265C22×C4C2×M1611yescharacteristic
266C22×C4C2×M1611yescharacteristic
267C22×C4C2×M1611yescharacteristic
268C22×C4#13111yescharacteristic
269C22×C4#13111yescharacteristic
270C22×C4#13111yescharacteristic
271C22×C4#13111yescharacteristic
272M16—22no—
273M16—22no—
274M16—22no—
275M16—22no—
276M16—22no—
277M16—41no—
278M16—41no—
279M16C4⋊C814yescomplemented
280C2×C8—22no—
281C2×C8—22no—
282C2×C8—22no—
283C2×C8—22no—
284C2×C8—22no—
285C2×C8—22no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—44no—
289C2×C8—44no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—81no—
300C2×C8—81no—
301C2×C8—82no—
302C2×C8—81no—
303C2×C8—81no—
304C2×C8—82no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—22no—
308C4⋊C4—21no—
309C4⋊C4—22no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4C22⋊C812yes—
320C4⋊C4C22⋊C812yes—
321C22⋊C4—22no—
322C22⋊C4—22no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C42—22no—
326C42—22no—
327C42—42no—
328C42—42no—
329C42—41no—
330C42—41no—
331C42—41no—
332C42—41no—
333C42—41no—
334C42—41no—
335C42—82no—
336C42—81no—
337C42—81no—
338C42#12111yescharacteristic
339C42#12111yescharacteristic
340C42#12111yescharacteristic
341C42C22⋊Q811yescharacteristic
342C42C22⋊Q811yescharacteristic
343C42C22⋊Q811yescharacteristic
344C42C22⋊Q811yescharacteristic
345C42C22⋊Q811yescharacteristic
346C42C22⋊Q811yescharacteristic
347C42#12311yescharacteristic
348C42#12411yescharacteristic
349C42#12511yescharacteristic
350C42#12511yescharacteristic
351C42#12611yescharacteristic
352C42C4⋊Q811yescharacteristic
353C42C4⋊Q811yescharacteristic
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23#38611yescharacteristic
359C23#38711yescharacteristic
360C23#38911yescharacteristic
361C23C8×D411yescharacteristic
362C23#43511yescharacteristic
363C23#43511yescharacteristic
364C23#43611yescharacteristic
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—42no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4#38011yescharacteristic
406C2×C4#38711yescharacteristic
407C2×C4#38911yescharacteristic
408C2×C4#39111yescharacteristic
409C2×C4#39411yescharacteristic
410C2×C4#39511yescharacteristic
411C2×C4#39611yescharacteristic
412C2×C4#39611yescharacteristic
413C2×C4#39811yescharacteristic
414C2×C4#39811yescharacteristic
415C2×C4#39811yescharacteristic
416C2×C4#40011yescharacteristic
417C2×C4C2×C22⋊C811yescharacteristic
418C2×C4#40711yescharacteristic
419C2×C4#40811yescharacteristic
420C2×C4#40811yescharacteristic
421C2×C4C2×C4⋊C811yescharacteristic
422C2×C4#42311yescharacteristic
423C2×C4#42411yescharacteristic
424C2×C4#42411yescharacteristic
425C2×C4#43111yescharacteristic
426C2×C4#43211yescharacteristic
427C2×C4#43311yescharacteristic
428C2×C4#43311yescharacteristic
429C2×C4C8×D411yescharacteristic
430C2×C4#43511yescharacteristic
431C2×C4#43511yescharacteristic
432C2×C4#43611yescharacteristic
433C2×C4C8×Q811yescharacteristic
434C2×C4C8×Q811yescharacteristic
435C2×C4#44611yescharacteristic
436C2×C4#44611yescharacteristic
437C2×C4#44611yescharacteristic
438C2×C4#44611yescharacteristic
439C2×C4#44611yescharacteristic
440C2×C4#44611yescharacteristic
441C2×C4#47111yescharacteristic
442C2×C4#47111yescharacteristic
443C2×C4#47211yescharacteristic
444C2×C4#47311yescharacteristic
445C8—24no—
446C8—42no—
447C8—42no—
448C8—42no—
449C8—82no—
450C8—82no—
451C8—82no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22#144111yescharacteristic
465C22#186911yescharacteristic
466C22#187011yescharacteristic
467C22#190611yescharacteristic
468C22#190611yescharacteristic
469C22#190711yescharacteristic
470C22#190811yescharacteristic
471C22#191611yescharacteristic
472C22#191711yescharacteristic
473C22#193911yescharacteristic
474C22#194011yescharacteristic
475C4—21no—
476C4—21no—
477C4—21no—
478C4—21no—
479C4—21no—
480C4—21no—
481C4—42no—
482C4—42no—
483C4—82no—
484C4—81no—
485C4—82no—
486C4—81no—
487C4#179711yescharacteristic
488C4#179811yescharacteristic
489C4#191611yescharacteristic
490C4#191711yescharacteristic
491C4#193911yescharacteristic
492C4#194011yescharacteristic
493C4#198411yescharacteristic
494C4#198511yescharacteristic
495C4#206311yescharacteristic
496C4#206411yescharacteristic
497C4#208911yescharacteristic
498C4#209011yescharacteristic
499C2—21no—
500C2—21no—
501C2—21no—
502C2—21no—
503C2#909511yescharacteristic
504C2#909611yescharacteristic
505C2#909711yescharacteristic
506C2#909811yescharacteristic
507C2#1120211yescharacteristic
508C2#1121511yescharacteristic
509C2#1121611yescharacteristic
subgroupG/H
centerC22×C4#121
commutatorC22×C4C22×C8
Frattini, cosocleC22×C42C23
socleC23#386
prime radicalC24C4×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
56. #428C8semidirect
279. M16C4⋊C8semidirect

Maximal subgroups

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

subgroupindex
#213492
C4×#4282
C2×#12832
#79082
#78912
C2×#12792
C2×#12782