Finite Group Databasebeta

← C2×#10590990 #11867886

Subgroups — C2×#10590990 #11867886

Order 1152.

1085 conjugacy classes of subgroups.

The subgroup lattice has 633 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#64117C211yesmaximal, characteristic
2C2×#63833C211yesmaximal, characteristic, complemented
3C22×#3717C211yesmaximal, characteristic, complemented
4C2×#63817C211yesmaximal, characteristic, complemented
5C2×C3×#824C211yesmaximal, characteristic, complemented
6C2×C3×#822C211yesmaximal, characteristic, complemented
7C2×#63559C211yesmaximal, characteristic, complemented
8#10590990C218yesmaximal, direct product factor
9C2×#4898—31nomaximal
10C2×#4754—31nomaximal
11C2×#3828C2211yescharacteristic
12C22×C4⋊(C3⋊D3)C2211yescharacteristic
13C22×C3×Dic6C2211yescharacteristic
14C2×C32×M16C2211yescharacteristic
15#64117—22no—
16#64117C2214yes—
17C2×#3745—21no—
18C2×#3744—21no—
19C2×C3×#208C2211yescharacteristic
20C2×C3×#208C2211yescharacteristic
21C22×C3×C3⋊C8C2211yescharacteristic
22C2×#3719—21no—
23#63833—22no—
24#63833C2214yescomplemented
25C2×#3717—21no—
26C2×#3717—21no—
27C2×#3717—22no—
28C2×#3717—21no—
29C2×#3717C2214yescomplemented
30C2×#3712—21no—
31#63817—22no—
32#63817C2214yescomplemented
33C2×#3710—21no—
34C3×#824C2214yescomplemented
35C3×#822C2214yescomplemented
36#63559C2214yescomplemented
37C2×#942—31no—
38C22×#215—31no—
39C2×#909—31no—
40C2×#908—31no—
41C2×#908—31no—
42C2×#908—61no—
43C2×#907—31no—
44C22×#204—31no—
45C2×#901—31no—
46C2×C3×#108—31no—
47C2×C3×#101—31no—
48C2×#824D311yescharacteristic, complemented
49C2×#823—31no—
50C2×#823—31no—
51C2×#822D311yescharacteristic, complemented
52#4898—38no—
53#4754—38no—
54C23×C3⋊D3—21no—
55C22×C32×C4C2311yescharacteristic
56C2×#638—21no—
57#3828—21no—
58#3828—21no—
59#3828D412yes—
60#3828C2312yes—
61C2×C4⋊(C3⋊D3)—21no—
62C2×C4⋊(C3⋊D3)—21no—
63C2×C4⋊(C3⋊D3)—21no—
64C2×C4⋊(C3⋊D3)—21no—
65C2×C4⋊(C3⋊D3)—21no—
66C2×C4⋊(C3⋊D3)D411yescharacteristic
67C2×C4⋊(C3⋊D3)C2312yes—
68C2×C4×C3⋊D3—21no—
69C2×#634D411yescharacteristic
70C22×C3×Dic3—21no—
71C2×C3×Dic6—21no—
72C2×C3×Dic6—21no—
73C2×C3×Dic6—21no—
74C2×C3×Dic6—21no—
75C2×C3×Dic6—21no—
76C2×C3×Dic6C2312yes—
77C32×M16—21no—
78C32×M16C2312yes—
79C2×C32×C8—21no—
80#3745—24no—
81#3744—24no—
82C3×#208—21no—
83C3×#208—21no—
84C3×#208C2312yes—
85C3×#208C2312yes—
86C2×C3×C3⋊C8—21no—
87C2×C3×C3⋊C8—21no—
88C2×C3×C3⋊C8D411yescharacteristic
89C2×C3×C3⋊C8D412yes—
90C2×C3×C3⋊C8D411yescharacteristic
91C2×C3×C3⋊C8C2312yes—
92#3719—24no—
93#3717—24no—
94#3717—24no—
95#3717—24no—
96#3712—24no—
97#3710—24no—
98C2×#471—91noSylow
99C22×C3×Q8—31no—
100C2×#235—31no—
101C2×#235—31no—
102C2×#235—61no—
103C22×D12—31no—
104C22×D12—31no—
105C22×D12—61no—
106C22×Dic6C2×D311yescharacteristic
107C2×C3×M16—21no—
108C2×C3×M16—31no—
109C2×C3×M16—31no—
110C2×C3×M16C2×D311yescharacteristic
111C2×C3×M16C2×D311yescharacteristic
112C22×C3×C8—31no—
113C2×#216—61no—
114#942—34no—
115#942—62no—
116C2×#215—34no—
117C2×#215—61no—
118C2×#215—61no—
119C2×#215—61no—
120C2×#215—62no—
121C2×#208C2×D311yescharacteristic
122C2×#208C2×D311yescharacteristic
123C22×C3⋊C8C2×D311yescharacteristic
124#909—34no—
125#909—62no—
126#908—34no—
127#908—34no—
128#908—64no—
129#908—62no—
130#908—62no—
131#908—64no—
132#907—34no—
133#907—62no—
134C2×Dic12—61no—
135C2×D24—61no—
136C2×D24—61no—
137C2×D24—62no—
138C2×#204—34no—
139C2×#204—61no—
140C2×#204—61no—
141C2×#204—62no—
142C2×#204—61no—
143C2×#204—61no—
144C2×#204—62no—
145C2×#204—61no—
146#901—34no—
147#901—62no—
148C2×ΓL(1,𝔽25)—61no—
149C2×ΓL(1,𝔽25)—61no—
150C2×D3×C8—61no—
151C2×D3×C8—61no—
152C3×#108—34no—
153C3×#101—34no—
154#824C2×D314yescomplemented
155#823—34no—
156#823—34no—
157#822C2×D314yescomplemented
158C23×C32#4111yescharacteristic
159C22×C3⋊D3—21no—
160C22×C3⋊D3—22no—
161C22×C3⋊D3—41no—
162C22×C3⋊D3—41no—
163C22×C3⋊D3—82no—
164C2×C32×C4C2×D411yescharacteristic
165C2×C32×C4C2×D411yescharacteristic
166C2×C32×C4C2×D411yescharacteristic
167C2×C32×C4C2×D411yescharacteristic
168C2×C32×C4#4111yescharacteristic
169C2×C32×C4C2411yescharacteristic
170#638—22no—
171#638—22no—
172C2×C3⋊Dic3—21no—
173C4⋊(C3⋊D3)—22no—
174C4⋊(C3⋊D3)—21no—
175C4⋊(C3⋊D3)—22no—
176C4⋊(C3⋊D3)—41no—
177C4⋊(C3⋊D3)—41no—
178C4⋊(C3⋊D3)C2×D412yes—
179C4×C3⋊D3—22no—
180C4×C3⋊D3—22no—
181#634—21no—
182#634C2×D412yes—
183C2×C3×Dic3—22no—
184C2×C3×Dic3—41no—
185C2×C3×Dic3—41no—
186C3×Dic6—22no—
187C3×Dic6—22no—
188C3×Dic6—41no—
189C3×Dic6—41no—
190C32×C8—22no—
191C3×C3⋊C8—22no—
192C3×C3⋊C8—22no—
193C3×C3⋊C8C2×D412yes—
194C3×C3⋊C8C2×D412yes—
195C2×#137—91no—
196C2×AGL(1,ℤ/8)—91no—
197C22×SD16—91no—
198C2×#131—91no—
199#471—98no—
200C2×#108—31no—
201C2×#101—31no—
202C2×#100—91no—
203C23×D3—61no—
204C23×D3—61no—
205C23×D3—121no—
206C2×C3×Q8—32no—
207C2×C3×Q8—61no—
208C2×C3×Q8—61no—
209C2×C3×Q8—61no—
210C2×C3×Q8—61no—
211C2×C3×Q8—61no—
212C22×C3×C4—21no—
213C22×C3×C4—61no—
214C22×C3×C4C22×D311yescharacteristic
215C22×C3×C4C22×D311yescharacteristic
216C2×#67—61no—
217C2×#67—61no—
218C2×#67—121no—
219C22×Dic3—21no—
220#235—32no—
221#235—32no—
222#235—32no—
223#235—32no—
224#235—62no—
225#235—62no—
226#235—61no—
227#235—61no—
228#235—61no—
229#235—61no—
230#235—121no—
231#235—121no—
232C2×D12—31no—
233C2×D12—31no—
234C2×D12—32no—
235C2×D12—32no—
236C2×D12—61no—
237C2×D12—62no—
238C2×D12—62no—
239C2×D12—62no—
240C2×D12—61no—
241C2×D12—61no—
242C2×D12—61no—
243C2×D12—61no—
244C2×D12—61no—
245C2×D12—61no—
246C2×D12—61no—
247C2×D12—61no—
248C2×D12—61no—
249C2×D12—61no—
250C2×D12—121no—
251C2×D12—121no—
252C2×D12—121no—
253C2×C4×D3—61no—
254C2×C4×D3—61no—
255C2×C4×D3—121no—
256C2×Dic6—21no—
257C2×Dic6—21no—
258C2×Dic6—21no—
259C2×Dic6—21no—
260C2×Dic6—21no—
261C2×Dic6—31no—
262C2×Dic6—31no—
263C2×Dic6—61no—
264C2×Dic6C22×D312yes—
265C3×M16—22no—
266C3×M16—21no—
267C3×M16—21no—
268C3×M16—22no—
269C3×M16—32no—
270C3×M16—32no—
271C3×M16—61no—
272C3×M16—61no—
273C3×M16C22×D312yes—
274C3×M16C22×D312yes—
275C2×C3×C8—22no—
276C2×C3×C8—21no—
277C2×C3×C8—21no—
278C2×C3×C8—31no—
279C2×C3×C8—31no—
280C2×C3×C8—32no—
281C2×C3×C8—32no—
282C2×C3×C8—61no—
283C2×C3×C8—61no—
284#216—64no—
285#215—64no—
286#215—64no—
287#215—64no—
288#208—21no—
289#208—21no—
290#208C22×D312yes—
291#208C22×D312yes—
292C2×C3⋊C8—21no—
293C2×C3⋊C8—21no—
294C2×C3⋊C8D1211yescharacteristic
295C2×C3⋊C8D1211yescharacteristic
296C2×C3⋊C8D1212yes—
297C2×C3⋊C8C22×D312yes—
298Dic12—64no—
299D24—64no—
300D24—64no—
301D24—68no—
302#204—64no—
303#204—64no—
304#204—68no—
305#204—64no—
306#204—64no—
307#204—64no—
308ΓL(1,𝔽25)—64no—
309ΓL(1,𝔽25)—64no—
310D3×C8—64no—
311D3×C8—64no—
312C22×C32—22no—
313C22×C32#12111yescharacteristic
314C22×C32#12111yescharacteristic
315C22×C32C2×#4111yescharacteristic
316C2×C3⋊D3—22no—
317C2×C3⋊D3—42no—
318C2×C3⋊D3—42no—
319C2×C3⋊D3—82no—
320C2×C3⋊D3—82no—
321C2×C3⋊D3—82no—
322C32×C4#12111yescharacteristic
323C32×C4#12111yescharacteristic
324C32×C4C22×D411yescharacteristic
325C32×C4C22×D411yescharacteristic
326C3⋊Dic3—22no—
327C3×Dic3—42no—
328C2×#41—91no—
329C22×Q8—31no—
330C22×D4—91no—
331#137—94no—
332#137—182no—
333AGL(1,ℤ/8)—94no—
334AGL(1,ℤ/8)—182no—
335C2×Q16—181no—
336C2×SD16—94no—
337C2×SD16—181no—
338C2×SD16—181no—
339C2×SD16—181no—
340C2×SD16—182no—
341C2×SD16—181no—
342C2×D8—181no—
343#131—94no—
344#131—182no—
345C2×M16—31no—
346C2×M16—31no—
347C2×M16—181no—
348C2×M16D3211yescharacteristic
349C22×C8—31no—
350C22×C8—181no—
351#108—34no—
352#101—34no—
353#100—94no—
354C23×C3—21no—
355C23×C3#23511yescharacteristic
356C23×C3#23911yescharacteristic
357C22×D3—61no—
358C22×D3—61no—
359C22×D3—62no—
360C22×D3—62no—
361C22×D3—121no—
362C22×D3—122no—
363C22×D3—122no—
364C22×D3—122no—
365C22×D3—121no—
366C22×D3—121no—
367C22×D3—121no—
368C22×D3—121no—
369C22×D3—244no—
370C22×D3—242no—
371C22×D3—242no—
372C3×Q8—62no—
373C3×Q8—62no—
374C3×Q8—121no—
375C3×Q8—121no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4—62no—
383C2×C3×C4—121no—
384C2×C3×C4—121no—
385C2×C3×C4C2×D1211yescharacteristic
386C2×C3×C4C2×D1211yescharacteristic
387C2×C3×C4#23511yescharacteristic
388C2×C3×C4D3×D411yescharacteristic
389C2×C3×C4D3×D411yescharacteristic
390C2×C3×C4D3×D411yescharacteristic
391C2×C3×C4D3×D411yescharacteristic
392C2×C3×C4D3×D411yescharacteristic
393C2×C3×C4D3×D411yescharacteristic
394C2×C3×C4#23911yescharacteristic
395C2×C3×C4C23×D311yescharacteristic
396C2×C3×C4C23×D311yescharacteristic
397#67—62no—
398#67—62no—
399#67—62no—
400#67—62no—
401#67—122no—
402#67—122no—
403C2×Dic3—22no—
404C2×Dic3—41no—
405C2×Dic3—41no—
406C2×Dic3—61no—
407C2×Dic3—61no—
408C2×Dic3—121no—
409D12—32no—
410D12—32no—
411D12—62no—
412D12—64no—
413D12—61no—
414D12—61no—
415D12—62no—
416D12—62no—
417D12—62no—
418D12—62no—
419D12—122no—
420D12—121no—
421D12—122no—
422D12—122no—
423D12—121no—
424D12—121no—
425D12—121no—
426D12—121no—
427C4×D3—62no—
428C4×D3—62no—
429C4×D3—62no—
430C4×D3—62no—
431C4×D3—122no—
432C4×D3—122no—
433Dic6—22no—
434Dic6—22no—
435Dic6—32no—
436Dic6—32no—
437Dic6—41no—
438Dic6—41no—
439Dic6—62no—
440Dic6—61no—
441Dic6—61no—
442Dic6—121no—
443C3×C8—24no—
444C3×C8—22no—
445C3×C8—22no—
446C3×C8—32no—
447C3×C8—32no—
448C3×C8—62no—
449C3×C8—62no—
450C3⋊C8—22no—
451C3⋊C8—22no—
452C3⋊C8C2×D1212yes—
453C3⋊C8C2×D1212yes—
454C2×C32—21no—
455C2×C32—21no—
456C2×C32#47112yescomplemented
457C2×C32C2×#12111yescharacteristic
458C3⋊D3—42no—
459C3⋊D3—82no—
460C24—181no—
461#41—92no—
462#41—92no—
463#41—181no—
464#41—181no—
465C2×Q8—32no—
466C2×Q8—61no—
467C2×Q8—61no—
468C2×Q8—61no—
469C2×Q8—61no—
470C2×Q8—61no—
471C2×Q8—91no—
472C2×D4—92no—
473C2×D4—91no—
474C2×D4—181no—
475C2×D4—181no—
476C2×D4—181no—
477C2×D4—181no—
478C2×D4—181no—
479C2×D4—181no—
480C22×C4—61no—
481C22×C4—181no—
482C22×C4C2×D3211yescharacteristic
483Q16—184no—
484SD16—184no—
485SD16—184no—
486SD16—184no—
487SD16—184no—
488D8—184no—
489M16—21no—
490M16—32no—
491M16—32no—
492M16—61no—
493M16—61no—
494M16—184no—
495M16C2×D3212yes—
496C2×C8—21no—
497C2×C8—32no—
498C2×C8—31no—
499C2×C8—31no—
500C2×C8—32no—
501C2×C8—61no—
502C2×C8—61no—
503C2×C8—184no—
504C22×C3—22no—
505C22×C3—22no—
506C22×C3—21no—
507C22×C3—21no—
508C22×C3—21no—
509C22×C3—42no—
510C22×C3#88411yescharacteristic
511C22×C3#88411yescharacteristic
512C22×C3#89511yescharacteristic
513C22×C3#89511yescharacteristic
514C22×C3C2×#23511yescharacteristic
515C22×C3C2×#23911yescharacteristic
516C2×D3—62no—
517C2×D3—62no—
518C2×D3—122no—
519C2×D3—122no—
520C2×D3—122no—
521C2×D3—124no—
522C2×D3—122no—
523C2×D3—122no—
524C2×D3—242no—
525C2×D3—242no—
526C2×D3—242no—
527C2×D3—244no—
528C2×D3—244no—
529C2×D3—244no—
530C2×D3—242no—
531C2×D3—242no—
532C2×D3—242no—
533C2×D3—242no—
534C3×C4—21no—
535C3×C4—21no—
536C3×C4—21no—
537C3×C4—21no—
538C3×C4—122no—
539C3×C4#88411yescharacteristic
540C3×C4#88411yescharacteristic
541C3×C4#89511yescharacteristic
542C3×C4#89511yescharacteristic
543C3×C4C22×D1211yescharacteristic
544C3×C4C2×D3×D411yescharacteristic
545C3×C4C2×D3×D411yescharacteristic
546C3×C4C2×D3×D411yescharacteristic
547Dic3—42no—
548Dic3—62no—
549Dic3—62no—
550Dic3—122no—
551C32C2×#47111yescharacteristic, complemented, Sylow
552C23—181no—
553C23—182no—
554C23—361no—
555C23—361no—
556C23—722no—
557C23#379911yescharacteristic
558Q8—62no—
559Q8—62no—
560Q8—92no—
561Q8—121no—
562Q8—121no—
563Q8—181no—
564D4—92no—
565D4—182no—
566D4—182no—
567D4—182no—
568D4—182no—
569D4—181no—
570D4—361no—
571D4—361no—
572C2×C4—62no—
573C2×C4—121no—
574C2×C4—121no—
575C2×C4—181no—
576C2×C4—182no—
577C2×C4—182no—
578C2×C4#379911yescharacteristic
579C2×C4D3×D1211yescharacteristic
580C2×C4D3×D1211yescharacteristic
581C2×C4#381111yescharacteristic
582C2×C4#381111yescharacteristic
583C2×C4C22×D3211yescharacteristic
584C8—22no—
585C8—32no—
586C8—32no—
587C8—62no—
588C8—62no—
589C2×C3—21no—
590C2×C3—21no—
591C2×C3—21no—
592C2×C3—21no—
593C2×C3—22no—
594C2×C3—21no—
595C2×C3—41no—
596C2×C3—41no—
597C2×C3#475412yescomplemented
598C2×C3#489812yescomplemented
599C2×C3C2×#88411yescharacteristic
600C2×C3C2×#89511yescharacteristic
601D3—122no—
602D3—122no—
603D3—242no—
604D3—244no—
605D3—242no—
606D3—242no—
607C22—22no—
608C22—182no—
609C22—362no—
610C22—362no—
611C22—722no—
612C22—722no—
613C22—722no—
614C22#6391011yescharacteristic
615C22#6391011yescharacteristic
616C22C2×#379911yescharacteristic
617C4—122no—
618C4—182no—
619C4#6391011yescharacteristic
620C4#6391011yescharacteristic
621C4C2×D3×D1211yescharacteristic
622C4C2×#381111yescharacteristic
623C3—21no—
624C3C2×#475411yescharacteristic, complemented
625C3C2×#489811yescharacteristic, complemented
626C2—21no—
627C2—21no—
628C2—362no—
629C2—722no—
630C2#1059099012yesdirect product factor
631C2C2×#6391011yescharacteristic
subgroupG/H
centerC22#63910
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC2×C32×M16C22
socleC22×C32#121
prime radicalC2×#64117C2
hypercenterC2×M16D32
hypocenterC32C2×#471
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
2. C2×#63833C2semidirect
3. C22×#3717C2semidirect
4. C2×#63817C2semidirect
5. C2×C3×#824C2semidirect
6. C2×C3×#822C2semidirect
7. C2×#63559C2semidirect
8. #10590990C2direct (N × H)
24. #63833C22semidirect
29. C2×#3717C22semidirect
32. #63817C22semidirect
34. C3×#824C22semidirect
35. C3×#822C22semidirect
36. #63559C22semidirect
48. C2×#824D3semidirect
51. C2×#822D3semidirect
154. #824C2×D3semidirect
157. #822C2×D3semidirect
456. C2×C32#471semidirect
551. C32C2×#471semidirect
597. C2×C3#4754semidirect
598. C2×C3#4898semidirect
624. C3C2×#4754semidirect
625. C3C2×#4898semidirect
630. C2#10590990direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#641172
C2×#638332
C22×#37172
C2×#638172
C2×C3×#8242
C2×C3×#8222
C2×#635592
#105909902
C2×#48983
C2×#47543