Finite Group Databasebeta

← #368595

Subgroups — #368595

Order 512.

932 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1997C211yesmaximal, characteristic
2#21313C211yesmaximal, characteristic
3C2×#1988C211yesmaximal, characteristic
4C4×#428C211yesmaximal, characteristic
5#20431C211yesmaximal, characteristic
6#11303C212yesmaximal
7#11290C212yesmaximal
8#10896C212yesmaximal
9#10853C211yesmaximal, characteristic
10#10847C211yesmaximal, characteristic
11#10827C212yesmaximal
12C22×C4⋊Q8C2211yescharacteristic
13C2×#428C2211yescharacteristic
14C2×#428C2211yescharacteristic
15C2×#423C2211yescharacteristic
16C2×#423C2211yescharacteristic
17C22×C4⋊C8C2211yescharacteristic
18C2×#405C2212yes—
19C2×C4×M16C2211yescharacteristic
20C4×#117C2211yescharacteristic
21C2×C4×C4⋊C4C2211yescharacteristic
22#1997C418yescomplemented
23#1988—21no—
24#1988—21no—
25#1988—22no—
26#1988—21no—
27#1988—21no—
28#1987—21no—
29#1987—21no—
30#1776—21no—
31#1776—21no—
32C4×#107—21no—
33C4×#106—21no—
34#1768—22no—
35#1767—21no—
36C4×C4⋊C8—21no—
37C2×#343C2212yes—
38C2×#343C2212yes—
39C2×#343C2212yes—
40C2×#343C2212yes—
41C2×#341C2212yes—
42C2×#341C2212yes—
43#1736C2212yes—
44#1736C2211yescharacteristic
45#1736C2212yes—
46#1736C2211yescharacteristic
47C2×#340C2212yes—
48C2×#340C2212yes—
49#1729C2212yes—
50C23×Q8—21no—
51C22×M16C2312yes—
52C22×M16C2311yescharacteristic
53C22×M16C2312yes—
54C22×M16C2311yescharacteristic
55C23×C8C2312yes—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8—22no—
61C2×C4⋊Q8—22no—
62C2×C4⋊Q8—22no—
63C2×C4⋊Q8C2×C414yes—
64C2×#117C2311yescharacteristic
65C2×#117C2311yescharacteristic
66C22×C4⋊C4—21no—
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C42—21no—
71C22×C42C2311yescharacteristic
72#428—22no—
73#428—22no—
74#428—22no—
75#428Q814yescomplemented
76C2×#107—21no—
77C2×#107—21no—
78C2×#106—21no—
79C2×#106—21no—
80#423—22no—
81#423—22no—
82#423—22no—
83#423C2×C414yes—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8D411yescharacteristic
91C2×C4⋊C8D411yescharacteristic
92C2×C4⋊C8D412yes—
93#405—24no—
94#405—42no—
95C4×M16—22no—
96C4×M16C2×C414yes—
97C2×#97—21no—
98C2×#97—22no—
99C2×C4×C8—21no—
100C2×C4×C8—22no—
101C4×C4⋊C4—21no—
102C4×C4⋊C4—21no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4D411yescharacteristic
106C4×C4⋊C4D411yescharacteristic
107C4×C4⋊C4D411yescharacteristic
108C4×C4⋊C4D411yescharacteristic
109C4×C22⋊C4—21no—
110C4×C22⋊C4—21no—
111#376—21no—
112C2×#95—21no—
113C43—21no—
114#343—24no—
115#343—42no—
116#343C2×C418yes—
117#343C2×C418yes—
118#341—24no—
119#341—42no—
120#340—22no—
121#340—22no—
122#340—22no—
123#340—22no—
124#340—22no—
125#340—22no—
126#340—22no—
127#340—22no—
128#340—21no—
129#340—21no—
130#340—21no—
131#340—21no—
132#336—22no—
133#336—22no—
134C22×Q8—24no—
135C22×Q8—41no—
136C22×Q8—41no—
137C22×Q8—42no—
138C22×Q8—82no—
139C22×Q8—82no—
140C23×C4—22no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4#4111yescharacteristic
144C23×C4#4111yescharacteristic
145C23×C4C2411yescharacteristic
146C2×M16—21no—
147C2×M16—21no—
148C2×M16—21no—
149C2×M16—21no—
150C2×M16—22no—
151C2×M16—22no—
152C2×M16—22no—
153C2×M16—21no—
154C2×M16—22no—
155C2×M16—21no—
156C2×M16—21no—
157C2×M16—21no—
158C2×M16—22no—
159C2×M16—22no—
160C2×M16—22no—
161C2×M16—22no—
162C2×M16—22no—
163C2×M16—22no—
164C2×M16—22no—
165C2×M16C22×C412yes—
166C2×M16C22×C414yes—
167C2×M16C22×C412yes—
168C2×M16C2×Q812yes—
169C2×M16#4112yes—
170C22×C8—21no—
171C22×C8—22no—
172C22×C8—21no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8C2×D412yes—
181C22×C8C2×D412yes—
182C4⋊Q8—24no—
183C4⋊Q8—24no—
184C4⋊Q8—24no—
185C4⋊Q8—24no—
186C4⋊Q8—42no—
187C4⋊Q8—42no—
188C4⋊Q8—42no—
189C4⋊Q8—42no—
190#117—21no—
191#117—21no—
192#117—21no—
193#117—21no—
194#117—21no—
195#117—21no—
196#117C2×Q812yes—
197#117#4112yes—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4C22×C414yes—
218C2×C4⋊C4C2×D411yescharacteristic
219C2×C4⋊C4C2×D411yescharacteristic
220C2×C4⋊C4C2×D411yescharacteristic
221C2×C4⋊C4C2×D411yescharacteristic
222C2×C4⋊C4C2×Q812yes—
223C2×C4⋊C4#4112yes—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—41no—
233C2×C42—41no—
234C2×C42—82no—
235C2×C42C22×C412yes—
236C2×C42C2×D411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D411yescharacteristic
240C2×C42#4112yes—
241C2×C42#4112yes—
242C2×C42#4112yes—
243#107—24no—
244#107—24no—
245#106—24no—
246#106—24no—
247C4⋊C8—24no—
248C4⋊C8—24no—
249C4⋊C8—22no—
250C4⋊C8—22no—
251C4⋊C8—24no—
252C4⋊C8—42no—
253C4⋊C8—42no—
254#97—24no—
255#97—44no—
256C4×C8—24no—
257C4×C8—44no—
258#95—21no—
259#95—21no—
260#95—21no—
261#95—21no—
262#95—41no—
263#95—41no—
264C24C2×#4111yescharacteristic
265C2×Q8—44no—
266C2×Q8—44no—
267C2×Q8—88no—
268C2×Q8—82no—
269C2×Q8—82no—
270C2×Q8—88no—
271C2×Q8—82no—
272C2×Q8—82no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—24no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—81no—
291C22×C4—81no—
292C22×C4—81no—
293C22×C4—81no—
294C22×C4—82no—
295C22×C4—82no—
296C22×C4#11712yes—
297C22×C4#11712yes—
298C22×C4#12111yescharacteristic
299C22×C4#12111yescharacteristic
300C22×C4#12111yescharacteristic
301C22×C4#12111yescharacteristic
302C22×C4#12612yes—
303C22×C4#12612yes—
304C22×C4C23×C411yescharacteristic
305C22×C4C22×D411yescharacteristic
306C22×C4C22×D411yescharacteristic
307C22×C4C22×Q811yescharacteristic
308C22×C4C2×#4111yescharacteristic
309C22×C4C2×#4111yescharacteristic
310C22×C4C2×#4111yescharacteristic
311C22×C4C2×#4111yescharacteristic
312C22×C4C2×#4111yescharacteristic
313C22×C4C2×#4111yescharacteristic
314C22×C4H(5,𝔽2)11yescharacteristic
315C22×C42−1+411yescharacteristic
316C22×C42−1+411yescharacteristic
317C22×C42−1+411yescharacteristic
318M16—22no—
319M16—22no—
320M16—22no—
321M16—24no—
322M16—22no—
323M16—22no—
324M16—24no—
325M16—41no—
326M16—41no—
327M16—42no—
328M16—42no—
329M16—42no—
330M16—42no—
331M16—42no—
332M16—42no—
333M16C4×Q814yes—
334C2×C8—22no—
335C2×C8—22no—
336C2×C8—22no—
337C2×C8—22no—
338C2×C8—24no—
339C2×C8—22no—
340C2×C8—22no—
341C2×C8—22no—
342C2×C8—22no—
343C2×C8—22no—
344C2×C8—22no—
345C2×C8—42no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—42no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—24no—
359C4⋊C4—24no—
360C4⋊C4—22no—
361C4⋊C4—21no—
362C4⋊C4—22no—
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—21no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—44no—
375C4⋊C4—44no—
376C4⋊C4C22⋊Q812yes—
377C4⋊C4C22⋊Q812yes—
378C22⋊C4—22no—
379C22⋊C4—22no—
380C22⋊C4—41no—
381C22⋊C4—41no—
382C42—22no—
383C42—22no—
384C42—41no—
385C42—41no—
386C42—42no—
387C42—42no—
388C42—81no—
389C42—81no—
390C42—82no—
391C42—82no—
392C42C4×D412yes—
393C42C4×D412yes—
394C42C4×D412yes—
395C42C4×D412yes—
396C42#12312yes—
397C42#12312yes—
398C42#12412yes—
399C42#12412yes—
400C23—21no—
401C23—21no—
402C23—21no—
403C23—21no—
404C23C2×#11711yescharacteristic
405C23C2×#12111yescharacteristic
406C23C2×#12611yescharacteristic
407C23#55211yescharacteristic
408C23#55211yescharacteristic
409C23#55411yescharacteristic
410C23#55611yescharacteristic
411Q8—88no—
412Q8—88no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—44no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—84no—
441C2×C4—82no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—82no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—84no—
450C2×C4#38812yes—
451C2×C4#38812yes—
452C2×C4#38812yes—
453C2×C4#38812yes—
454C2×C4#47112yes—
455C2×C4#47212yes—
456C2×C4C2×C4×D411yescharacteristic
457C2×C4C2×C4×D411yescharacteristic
458C2×C4C2×C4×Q811yescharacteristic
459C2×C4C4×#4111yescharacteristic
460C2×C4#51911yescharacteristic
461C2×C4#52011yescharacteristic
462C2×C4C2×C22⋊Q811yescharacteristic
463C2×C4C2×#12311yescharacteristic
464C2×C4#52511yescharacteristic
465C2×C4C2×#12411yescharacteristic
466C2×C4#52911yescharacteristic
467C2×C4#53311yescharacteristic
468C2×C4#53611yescharacteristic
469C2×C4#53711yescharacteristic
470C2×C4#54111yescharacteristic
471C2×C4#54211yescharacteristic
472C2×C4#54611yescharacteristic
473C2×C4#54711yescharacteristic
474C2×C4#54811yescharacteristic
475C2×C4D4×Q811yescharacteristic
476C2×C4#55611yescharacteristic
477C2×C4#55611yescharacteristic
478C2×C4#55611yescharacteristic
479C2×C4#55611yescharacteristic
480C2×C4#55711yescharacteristic
481C2×C4#55711yescharacteristic
482C2×C4#55911yescharacteristic
483C2×C4#55911yescharacteristic
484C8—24no—
485C8—42no—
486C8—44no—
487C8—42no—
488C8—42no—
489C8—82no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—21no—
494C22—21no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22C2×#38811yescharacteristic
503C22#235511yescharacteristic
504C22#235611yescharacteristic
505C22#241511yescharacteristic
506C22#241711yescharacteristic
507C22#251111yescharacteristic
508C22#251411yescharacteristic
509C22C2×#47111yescharacteristic
510C22C2×#47211yescharacteristic
511C22#306911yescharacteristic
512C22#306911yescharacteristic
513C4—22no—
514C4—22no—
515C4—22no—
516C4—42no—
517C4—42no—
518C4—82no—
519C4—84no—
520C4—81no—
521C4—82no—
522C4—81no—
523C4#193114yes—
524C4C4×C22⋊Q811yescharacteristic
525C4#232311yescharacteristic
526C4#236511yescharacteristic
527C4#236711yescharacteristic
528C4#244611yescharacteristic
529C4#244711yescharacteristic
530C4#256811yescharacteristic
531C4#259211yescharacteristic
532C2—21no—
533C2—21no—
534C2—21no—
535C2—21no—
536C2#1402111yescharacteristic
537C2C2×#193111yescharacteristic
538C2#2107711yescharacteristic
539C2#2472011yescharacteristic
540C2#2472311yescharacteristic
541C2#2485511yescharacteristic
542C2#2485811yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C2×#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
22. #1997C4semidirect
75. #428Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#19972
#213132
C2×#19882
C4×#4282
#204312
#113032
#112902
#108962
#108532
#108472
#108272