Finite Group Databasebeta

← #382962

Subgroups — #382962

Order 512.

794 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20383C211yesmaximal, characteristic
2#11300C211yesmaximal, characteristic, complemented
3#11299C211yesmaximal, characteristic, complemented
4#11260C211yesmaximal, characteristic, complemented
5#11259C211yesmaximal, characteristic, complemented
6#11149C211yesmaximal, characteristic
7#11148C211yesmaximal, characteristic
8#11099C211yesmaximal, characteristic
9#11098C211yesmaximal, characteristic
10#10859C211yesmaximal, characteristic, complemented
11#10818C211yesmaximal, characteristic, complemented
12#10800C211yesmaximal, characteristic, complemented
13#10799C211yesmaximal, characteristic, complemented
14#10764C211yesmaximal, characteristic
15#10764C211yesmaximal, characteristic
16C22×#131C2211yescharacteristic
17C22×#108C2211yescharacteristic
18C2×#427C2211yescharacteristic
19C2×#413C2211yescharacteristic
20C2×#413C2211yescharacteristic
21C2×#410C2211yescharacteristic
22C2×#410C2211yescharacteristic
23C2×#408C2211yescharacteristic
24C2×#405C2211yescharacteristic
25C22×C4×C8C2211yescharacteristic
26#2297C2211yescharacteristic
27#1753C414yescomplemented
28C8×C22⋊C4C414yescomplemented
29C2×#344C2211yescharacteristic
30C2×#344C2211yescharacteristic
31C2×#344C2211yescharacteristic
32C2×#344C2211yescharacteristic
33C2×#341C2211yescharacteristic
34#1735C2211yescharacteristic
35C2×#339C2211yescharacteristic
36C2×#339C2211yescharacteristic
37C2×#339C2211yescharacteristic
38C2×#339C2211yescharacteristic
39C2×#338C2211yescharacteristic
40C2×#338C2211yescharacteristic
41C2×H(3,ℤ/4)C2211yescharacteristic
42C2×H(3,ℤ/4)C2211yescharacteristic
43C2×H(3,ℤ/4)C2211yescharacteristic
44C2×H(3,ℤ/4)C2211yescharacteristic
45C2×H(3,ℤ/4)C2211yescharacteristic
46C2×H(3,ℤ/4)C2211yescharacteristic
47#1729C2211yescharacteristic
48#1729C2211yescharacteristic
49#1729C2211yescharacteristic
50#1729C2211yescharacteristic
51#1729C2211yescharacteristic
52#1728C2211yescharacteristic
53C22×#41C2311yescharacteristic
54C2×#131—22no—
55C2×#131—22no—
56C2×#131—41no—
57C2×#131—41no—
58C2×#131—41no—
59C2×#131—41no—
60C22×M16C2×C412yes—
61C22×M16C2311yescharacteristic
62C22×M16C2311yescharacteristic
63C22×M16C2311yescharacteristic
64C22×M16C2311yescharacteristic
65C22×M16C2311yescharacteristic
66C22×M16C2311yescharacteristic
67C23×C8C2×C412yes—
68C23×C8C2311yescharacteristic
69C23×C8C2311yescharacteristic
70C2×#117C2311yescharacteristic
71C2×#117C2311yescharacteristic
72C2×#117C2311yescharacteristic
73C2×#117C2311yescharacteristic
74C2×#117C2311yescharacteristic
75C22×C42C2311yescharacteristic
76C2×#108—22no—
77C2×#108—41no—
78C2×#108—41no—
79#427—41no—
80#427—41no—
81C2×#107—21no—
82C2×#106—21no—
83#413—41no—
84#413—41no—
85#413—41no—
86#413—41no—
87C2×#101—21no—
88C2×#101—21no—
89C2×#100—21no—
90C2×#100—21no—
91#410—41no—
92#410—41no—
93#410—41no—
94#410—41no—
95C2×#99—22no—
96C2×#99—22no—
97#408—22no—
98#408—21no—
99#408—21no—
100C2×C22⋊C8C2×C414yes—
101#405—41no—
102#405—41no—
103C2×#97C2×C412yes—
104C2×C4×C8—41no—
105C2×C4×C8—41no—
106C2×C4×C8C2×C412yes—
107C2×C4×C8C2×C412yes—
108C2×C4×C8C2×C412yes—
109#391—21no—
110#390—21no—
111#386—22no—
112C4×C22⋊C4C2×C414yes—
113#344—41no—
114#344—41no—
115#344—41no—
116#344—41no—
117#341—21no—
118#341—21no—
119#340—22no—
120#339—41no—
121#339—41no—
122#339—41no—
123#339—41no—
124#338—41no—
125#338—41no—
126H(3,ℤ/4)—41no—
127H(3,ℤ/4)—41no—
128H(3,ℤ/4)—41no—
129H(3,ℤ/4)—41no—
130H(3,ℤ/4)—41no—
131H(3,ℤ/4)—41no—
132#336—22no—
133#336—22no—
134#336—22no—
135#336—22no—
136#336C2×C414yes—
137#336C2×C414yes—
138C2×#41—21no—
139C2×#41—21no—
140C2×#41—21no—
141C2×#41—21no—
142C2×#41—21no—
143C2×#41—21no—
144C2×#41—21no—
145C2×#41—21no—
146C2×#41—21no—
147C2×#41—21no—
148C2×#41—21no—
149C2×#41—21no—
150C22×Q8#4111yescharacteristic
151C22×D4#4112yes—
152C22×D4#4111yescharacteristic
153C23×C4—21no—
154C23×C4C22×C412yes—
155C23×C4C2411yescharacteristic
156#131—44no—
157#131—44no—
158#131—44no—
159#131—44no—
160C2×M16—21no—
161C2×M16—21no—
162C2×M16—21no—
163C2×M16—21no—
164C2×M16—21no—
165C2×M16—21no—
166C2×M16—21no—
167C2×M16—21no—
168C2×M16—21no—
169C2×M16—21no—
170C2×M16—21no—
171C2×M16—21no—
172C2×M16—24no—
173C2×M16—21no—
174C2×M16—21no—
175C2×M16—21no—
176C2×M16—21no—
177C2×M16—21no—
178C2×M16—21no—
179C2×M16—21no—
180C2×M16—21no—
181C2×M16—21no—
182C2×M16—21no—
183C2×M16—21no—
184C2×M16—21no—
185C2×M16—21no—
186C2×M16—21no—
187C2×M16—21no—
188C2×M16—21no—
189C2×M16—21no—
190C2×M16—21no—
191C2×M16—21no—
192C2×M16—21no—
193C2×M16—21no—
194C2×M16—21no—
195C2×M16—21no—
196C2×M16—21no—
197C2×M16—42no—
198C2×M16—42no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—24no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—42no—
213C22×C8—42no—
214C22×C8C22×C412yes—
215C22×C8C22×C412yes—
216C22×C8C22×C411yescharacteristic
217C22×C8C22×C411yescharacteristic
218C22×C8C22×C412yes—
219C22×C8#4112yes—
220C22×C8#4112yes—
221C22×C8#4112yes—
222C22×C8#4111yescharacteristic
223C22×C8#4111yescharacteristic
224#117—21no—
225#117—21no—
226#117—21no—
227#117—21no—
228#117—21no—
229#117—21no—
230#117—21no—
231#117—21no—
232#117—21no—
233#117—21no—
234#117—21no—
235#117—21no—
236#117—21no—
237#117—21no—
238#117—21no—
239#117—21no—
240#117—21no—
241#117—21no—
242#117—21no—
243#117—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42C22×C412yes—
265C2×C42C22×C412yes—
266#108—44no—
267#107—42no—
268#106—42no—
269#101—42no—
270#101—42no—
271#100—42no—
272#100—42no—
273#99—44no—
274#99—44no—
275C22⋊C8—24no—
276C22⋊C8—24no—
277#97—42no—
278C4×C8—42no—
279C4×C8—42no—
280C4×C8—42no—
281#95—24no—
282C24#11712yes—
283C24C2×#4111yescharacteristic
284#41—41no—
285#41—41no—
286#41—41no—
287#41—41no—
288#41—41no—
289#41—41no—
290#41—41no—
291#41—41no—
292#41—41no—
293#41—41no—
294#41—41no—
295#41—41no—
296#41—41no—
297#41—41no—
298#41—41no—
299#41—41no—
300C2×Q8—21no—
301C2×Q8—21no—
302C2×Q8—21no—
303C2×Q8—21no—
304C2×Q8—21no—
305C2×Q8—21no—
306C2×D4—22no—
307C2×D4—22no—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—22no—
311C2×D4—22no—
312C2×D4—21no—
313C2×D4—21no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—21no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—44no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4#11712yes—
334C22×C4C23×C411yescharacteristic
335C22×C4C2×#4111yescharacteristic
336C22×C4C2×#4111yescharacteristic
337C22×C4C2×#4111yescharacteristic
338C22×C4C2×#4111yescharacteristic
339C22×C4C2×#4111yescharacteristic
340C22×C4H(5,𝔽2)11yescharacteristic
341C22×C4H(5,𝔽2)11yescharacteristic
342C22×C4H(5,𝔽2)11yescharacteristic
343C22×C4H(5,𝔽2)11yescharacteristic
344C22×C4H(5,𝔽2)11yescharacteristic
345C22×C4H(5,𝔽2)11yescharacteristic
346C22×C42−1+411yescharacteristic
347C22×C42−1+411yescharacteristic
348M16—48no—
349M16—41no—
350M16—41no—
351M16—41no—
352M16—41no—
353M16—41no—
354M16—41no—
355M16—41no—
356M16—41no—
357M16—41no—
358M16—41no—
359M16—41no—
360M16—41no—
361M16—41no—
362M16—41no—
363M16—41no—
364M16—41no—
365M16—41no—
366M16—41no—
367M16—41no—
368M16—41no—
369M16—41no—
370M16—41no—
371M16—41no—
372M16—41no—
373C2×C8—21no—
374C2×C8—21no—
375C2×C8—22no—
376C2×C8—21no—
377C2×C8—21no—
378C2×C8—22no—
379C2×C8—21no—
380C2×C8—21no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—21no—
387C2×C8—21no—
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—48no—
396C2×C8—41no—
397C2×C8—41no—
398C2×C8—41no—
399C2×C8—41no—
400C2×C8—41no—
401C2×C8—41no—
402C2×C8—41no—
403C2×C8—41no—
404C2×C8—41no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8#11712yes—
409C2×C8#11712yes—
410C2×C8#11712yes—
411C2×C8#11712yes—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—41no—
419C4⋊C4—41no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C22⋊C4—48no—
423C22⋊C4—41no—
424C22⋊C4—41no—
425C22⋊C4—41no—
426C22⋊C4—41no—
427C22⋊C4—41no—
428C22⋊C4—41no—
429C22⋊C4—41no—
430C22⋊C4—41no—
431C22⋊C4—41no—
432C22⋊C4—41no—
433C42—22no—
434C42—22no—
435C42—22no—
436C42—22no—
437C42—41no—
438C42—41no—
439C42—41no—
440C42—41no—
441C42—41no—
442C42—41no—
443C42—41no—
444C42—41no—
445C42—41no—
446C42—41no—
447C23—22no—
448C23—22no—
449C23—22no—
450C23—42no—
451C23—44no—
452C23C2×#11711yescharacteristic
453C23#52911yescharacteristic
454C23#52911yescharacteristic
455C23#53811yescharacteristic
456C23#53811yescharacteristic
457C23#53911yescharacteristic
458C23#53911yescharacteristic
459Q8—41no—
460Q8—41no—
461Q8—41no—
462Q8—41no—
463D4—42no—
464D4—42no—
465D4—42no—
466D4—42no—
467D4—41no—
468D4—41no—
469D4—41no—
470D4—41no—
471C2×C4—24no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—42no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—81no—
490C2×C4—81no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4C2×#11711yescharacteristic
494C2×C4C2×#11711yescharacteristic
495C2×C4#51811yescharacteristic
496C2×C4#51811yescharacteristic
497C2×C4#52011yescharacteristic
498C2×C4#52011yescharacteristic
499C2×C4#52911yescharacteristic
500C2×C4#52911yescharacteristic
501C2×C4#53811yescharacteristic
502C2×C4#53811yescharacteristic
503C2×C4#53811yescharacteristic
504C2×C4#53811yescharacteristic
505C2×C4#53811yescharacteristic
506C2×C4#53811yescharacteristic
507C2×C4#53811yescharacteristic
508C2×C4#53811yescharacteristic
509C2×C4#53911yescharacteristic
510C2×C4#53911yescharacteristic
511C2×C4#53911yescharacteristic
512C2×C4#53911yescharacteristic
513C2×C4#54011yescharacteristic
514C2×C4#54011yescharacteristic
515C2×C4#54011yescharacteristic
516C2×C4#54111yescharacteristic
517C2×C4#54211yescharacteristic
518C2×C4#54211yescharacteristic
519C2×C4#54211yescharacteristic
520C2×C4#54211yescharacteristic
521C8—22no—
522C8—22no—
523C8—22no—
524C8—22no—
525C8—42no—
526C8—42no—
527C8—81no—
528C8—81no—
529C8—81no—
530C8—81no—
531C22—22no—
532C22—22no—
533C22—22no—
534C22—41no—
535C22—41no—
536C22—41no—
537C22—41no—
538C22—42no—
539C22—44no—
540C22—82no—
541C22—82no—
542C22#228311yescharacteristic
543C22#231811yescharacteristic
544C22#231811yescharacteristic
545C22#264811yescharacteristic
546C22#264911yescharacteristic
547C22#265211yescharacteristic
548C22#265311yescharacteristic
549C4—44no—
550C4—42no—
551C4—82no—
552C4—161no—
553C4—161no—
554C4—161no—
555C4—161no—
556C4#231211yescharacteristic
557C4#231411yescharacteristic
558C4#231811yescharacteristic
559C4#231911yescharacteristic
560C4#264811yescharacteristic
561C4#264911yescharacteristic
562C4#265211yescharacteristic
563C4#265311yescharacteristic
564C2—21no—
565C2—21no—
566C2—21no—
567C2—21no—
568C2—21no—
569C2—21no—
570C2—82no—
571C2#1435111yescharacteristic
572C2#2078712yes—
subgroupG/H
centerC2×C4C2×#117
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22#2283
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×#405 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #11300C2semidirect
3. #11299C2semidirect
4. #11260C2semidirect
5. #11259C2semidirect
10. #10859C2semidirect
11. #10818C2semidirect
12. #10800C2semidirect
13. #10799C2semidirect
27. #1753C4semidirect
28. C8×C22⋊C4C4semidirect

Maximal subgroups

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

subgroupindex
#203832
#113002
#112992
#112602
#112592
#111492
#111482
#110992
#110982
#108592
#108182
#108002
#107992
#107642