Finite Group Databasebeta

← C2×#10591600 #11867016

Subgroups — C2×#10591600 #11867016

Order 1152.

1145 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#64155C211yesmaximal, characteristic
2C22×C3×#216C211yesmaximal, characteristic, complemented
3C2×#63871C211yesmaximal, characteristic
4C22×#3712C211yesmaximal, characteristic
5C2×C3×#835C211yesmaximal, characteristic, complemented
6#10591600C218yesmaximal, direct product factor
7C2×#63573C211yesmaximal, characteristic, complemented
8C2×#63566C211yesmaximal, characteristic, complemented
9C2×#5189—31nomaximal
10C2×#4816—31nomaximal
11C22×C32×Q8C2211yescharacteristic
12C22×C4×C3⋊D3C2211yescharacteristic
13C22×C3×Dic6C2211yescharacteristic
14#64155—22no—
15#64155C2214yes—
16C2×#3752—21no—
17C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
18C2×#3750—21no—
19C2×C3×#216—24no—
20C2×C3×#216C2214yescomplemented
21C2×C3×C4⋊Dic3C2211yescharacteristic
22C22×C3×C3⋊C8C2211yescharacteristic
23C22×C3×C3⋊C8C2211yescharacteristic
24C2×Dic3⋊Dic3—21no—
25C2×#3722—21no—
26#63871—22no—
27#63871C2214yes—
28C2×#3712—24no—
29C2×#3712C2214yes—
30C3×#835C2214yescomplemented
31#63573C2214yescomplemented
32#63566C2214yescomplemented
33C22×C3×Q16—31no—
34C2×#946—31no—
35C2×#946—31no—
36C2×#946—31no—
37C2×#946—61no—
38C22×#216D311yescharacteristic, complemented
39C22×ΓL(1,𝔽25)—31no—
40C22×ΓL(1,𝔽25)—31no—
41C2×#897—31no—
42C2×C3×#103—31no—
43C2×#835—31no—
44C2×#835D311yescharacteristic, complemented
45#5189—38no—
46C2×#817—31no—
47C2×#816—31no—
48C2×#808—31no—
49#4816—38no—
50C23×C3⋊D3D411yescharacteristic
51C2×C32×Q8—21no—
52C2×C32×Q8—22no—
53C2×C32×Q8—22no—
54C2×C32×Q8C2312yes—
55C22×C32×C4—21no—
56C22×C32×C4C2311yescharacteristic
57C22×C3⋊Dic3—21no—
58C22×C3⋊Dic3D411yescharacteristic
59C2×C4×C3⋊D3—22no—
60C2×C4×C3⋊D3—22no—
61C2×C4×C3⋊D3D412yes—
62C2×C4×C3⋊D3C2312yes—
63C22×C3×Dic3—21no—
64C22×C3×Dic3—21no—
65C2×C3×Dic6—22no—
66C2×C3×Dic6—22no—
67C2×C3×Dic6—21no—
68C2×C3×Dic6C2312yes—
69#3752—24no—
70C4⋊(C3⋊Dic3)—21no—
71C4⋊(C3⋊Dic3)C2312yes—
72#3750—24no—
73C3×#216—28no—
74C3×C4⋊Dic3—21no—
75C3×C4⋊Dic3C2312yes—
76C2×C3×C3⋊C8—22no—
77C2×C3×C3⋊C8D414yes—
78C2×C3×C3⋊C8C2312yes—
79C2×C3×C3⋊C8C2312yes—
80Dic3⋊Dic3—24no—
81#3722—24no—
82#3712—28no—
83C2×#470—91noSylow
84C22×C3×Q8—21no—
85C22×C3×Q8—31no—
86C22×C3×Q8C2×D311yescharacteristic
87C22×C3×Q8C2×D311yescharacteristic
88C22×C4×D3—31no—
89C22×C4×D3—31no—
90C22×C4×D3—61no—
91C22×Dic6C2×D311yescharacteristic
92C2×C3×Q16—34no—
93C2×C3×Q16—64no—
94C22×C3×C8—31no—
95C22×C3×C8—31no—
96C2×C3×C4⋊C4—31no—
97#946—34no—
98#946—34no—
99#946—34no—
100#946—64no—
101#946—62no—
102#946—62no—
103#946—62no—
104#946—64no—
105C2×#216—24no—
106C2×#216C2×D314yescomplemented
107C2×(C2×D3)⋊C4—61no—
108C2×(C2×D3)⋊C4—61no—
109C2×(C2×D3)⋊C4—61no—
110C2×(C2×D3)⋊C4—61no—
111C2×(C2×D3)⋊C4—62no—
112C2×C4⋊Dic3—31no—
113C2×C4⋊Dic3—31no—
114C2×C4⋊Dic3—61no—
115C2×C4⋊Dic3—61no—
116C2×C4⋊Dic3C2×D311yescharacteristic
117C2×Dic3⋊C4—61no—
118C2×Dic3⋊C4—61no—
119C2×Dic3⋊C4—61no—
120C2×Dic3⋊C4—62no—
121C22×C3⋊C8C2×D311yescharacteristic
122C22×C3⋊C8C2×D311yescharacteristic
123C2×ΓL(1,𝔽25)—34no—
124C2×ΓL(1,𝔽25)—34no—
125C2×ΓL(1,𝔽25)—64no—
126C2×ΓL(1,𝔽25)—64no—
127#897—34no—
128#897—62no—
129C3×#103—34no—
130#835—34no—
131#835C2×D314yescomplemented
132#817—34no—
133#816—34no—
134#808—34no—
135C23×C32C2×D411yescharacteristic
136C22×C3⋊D3—22no—
137C22×C3⋊D3—22no—
138C22×C3⋊D3—22no—
139C22×C3⋊D3C2×D412yes—
140C32×Q8—22no—
141C32×Q8—22no—
142C32×Q8—42no—
143C2×C32×C4—22no—
144C2×C32×C4—42no—
145C2×C32×C4C2×D411yescharacteristic
146C2×C32×C4C2×D412yes—
147C2×C32×C4#4112yes—
148C2×C32×C4C2411yescharacteristic
149C2×C3⋊Dic3—22no—
150C2×C3⋊Dic3—22no—
151C2×C3⋊Dic3—42no—
152C2×C3⋊Dic3C2×D412yes—
153C4×C3⋊D3—24no—
154C4×C3⋊D3—22no—
155C4×C3⋊D3—22no—
156C2×C3×Dic3—22no—
157C2×C3×Dic3—22no—
158C2×C3×Dic3—42no—
159C2×C3×Dic3—42no—
160C3×Dic6—22no—
161C3×Dic6—22no—
162C3×Dic6—42no—
163C3×C3⋊C8—22no—
164C3×C3⋊C8C2×D414yes—
165C22×Q16—31no—
166C22×M16—91no—
167C2×C22⋊Q8—91no—
168C2×C22⋊Q8—91no—
169#470—98no—
170C2×#106—91no—
171C2×#103—31no—
172C2×#103—91no—
173C23×D3—31no—
174C23×D3—31no—
175C23×D3—61no—
176C2×C3×Q8—22no—
177C2×C3×Q8—21no—
178C2×C3×Q8—21no—
179C2×C3×Q8—22no—
180C2×C3×Q8—22no—
181C2×C3×Q8—22no—
182C2×C3×Q8—22no—
183C2×C3×Q8—22no—
184C2×C3×Q8—32no—
185C2×C3×Q8—42no—
186C2×C3×Q8—42no—
187C2×C3×Q8—61no—
188C2×C3×Q8—62no—
189C2×C3×Q8—62no—
190C2×C3×Q8C22×D312yes—
191C2×C3×Q8C22×D312yes—
192C22×C3×C4—22no—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4—61no—
197C22×C3×C4—61no—
198C22×C3×C4C22×D311yescharacteristic
199C22×C3×C4C22×D311yescharacteristic
200C22×Dic3—21no—
201C22×Dic3—21no—
202C22×Dic3—31no—
203C22×Dic3—31no—
204C22×Dic3—61no—
205C22×Dic3—62no—
206C22×Dic3—61no—
207C22×Dic3—61no—
208C2×C4×D3—32no—
209C2×C4×D3—32no—
210C2×C4×D3—32no—
211C2×C4×D3—32no—
212C2×C4×D3—62no—
213C2×C4×D3—62no—
214C2×C4×D3—62no—
215C2×C4×D3—62no—
216C2×C4×D3—62no—
217C2×C4×D3—62no—
218C2×C4×D3—122no—
219C2×C4×D3—122no—
220C2×Dic6—22no—
221C2×Dic6—22no—
222C2×Dic6—21no—
223C2×Dic6C22×D312yes—
224C3×Q16—68no—
225C2×C3×C8—34no—
226C2×C3×C8—32no—
227C2×C3×C8—32no—
228C2×C3×C8—62no—
229C3×C4⋊C4—32no—
230C3×C4⋊C4—61no—
231#216—28no—
232(C2×D3)⋊C4—64no—
233(C2×D3)⋊C4—64no—
234(C2×D3)⋊C4—64no—
235(C2×D3)⋊C4—64no—
236(C2×D3)⋊C4—68no—
237C4⋊Dic3—21no—
238C4⋊Dic3—32no—
239C4⋊Dic3—32no—
240C4⋊Dic3—62no—
241C4⋊Dic3—64no—
242C4⋊Dic3—62no—
243C4⋊Dic3—61no—
244C4⋊Dic3—61no—
245C4⋊Dic3C22×D312yes—
246Dic3⋊C4—64no—
247Dic3⋊C4—64no—
248Dic3⋊C4—64no—
249Dic3⋊C4—68no—
250C2×C3⋊C8—22no—
251C2×C3⋊C8#6714yes—
252C2×C3⋊C8C22×D312yes—
253C2×C3⋊C8C22×D312yes—
254ΓL(1,𝔽25)—68no—
255ΓL(1,𝔽25)—68no—
256C22×C32#12112yes—
257C22×C32#13712yes—
258C22×C32#13712yes—
259C22×C32C22×D411yescharacteristic
260C2×C3⋊D3—22no—
261C2×C3⋊D3—22no—
262C2×C3⋊D3—22no—
263C2×C3⋊D3—44no—
264C32×C4—42no—
265C32×C4#12112yes—
266C32×C4C22×D411yescharacteristic
267C32×C4C2×#4111yescharacteristic
268C3⋊Dic3—22no—
269C3⋊Dic3—42no—
270C3×Dic3—42no—
271C3×Dic3—42no—
272C22×Q8—31no—
273C22×Q8D3211yescharacteristic
274C23×C4—91no—
275C2×Q16—34no—
276C2×Q16—64no—
277C2×M16—94no—
278C2×M16—184no—
279C22×C8—31no—
280C22×C8—31no—
281C22⋊Q8—94no—
282C22⋊Q8—94no—
283C22⋊Q8—182no—
284C22⋊Q8—182no—
285C2×C4⋊C4—31no—
286C2×C4⋊C4—91no—
287C2×C4⋊C4—181no—
288C2×C4⋊C4—181no—
289C2×C22⋊C4—181no—
290C2×C22⋊C4—181no—
291#106—94no—
292#103—34no—
293#103—94no—
294C23×C3—21no—
295C23×C3D3×D411yescharacteristic
296C23×C3D3×D411yescharacteristic
297C22×D3—32no—
298C22×D3—32no—
299C22×D3—62no—
300C22×D3—64no—
301C22×D3—62no—
302C22×D3—62no—
303C22×D3—62no—
304C22×D3—62no—
305C22×D3—62no—
306C22×D3—62no—
307C22×D3—122no—
308C22×D3—122no—
309C3×Q8—22no—
310C3×Q8—22no—
311C3×Q8—22no—
312C3×Q8—22no—
313C3×Q8—42no—
314C3×Q8—42no—
315C3×Q8—44no—
316C3×Q8—42no—
317C3×Q8—42no—
318C3×Q8—62no—
319C3×Q8—62no—
320C3×Q8—122no—
321C2×C3×C4—24no—
322C2×C3×C4—22no—
323C2×C3×C4—22no—
324C2×C3×C4—21no—
325C2×C3×C4—22no—
326C2×C3×C4—22no—
327C2×C3×C4—21no—
328C2×C3×C4—44no—
329C2×C3×C4—42no—
330C2×C3×C4—42no—
331C2×C3×C4—62no—
332C2×C3×C4—62no—
333C2×C3×C4—122no—
334C2×C3×C4—122no—
335C2×C3×C4#23512yes—
336C2×C3×C4D3×D412yes—
337C2×C3×C4D3×D411yescharacteristic
338C2×C3×C4D3×D411yescharacteristic
339C2×C3×C4#23712yes—
340C2×C3×C4C2×#6712yes—
341C2×C3×C4C23×D311yescharacteristic
342C2×C3×C4C23×D311yescharacteristic
343C2×Dic3—22no—
344C2×Dic3—22no—
345C2×Dic3—32no—
346C2×Dic3—32no—
347C2×Dic3—42no—
348C2×Dic3—42no—
349C2×Dic3—62no—
350C2×Dic3—62no—
351C2×Dic3—62no—
352C2×Dic3—64no—
353C2×Dic3—62no—
354C2×Dic3—62no—
355C2×Dic3—122no—
356C2×Dic3—124no—
357C2×Dic3—122no—
358C2×Dic3—122no—
359C4×D3—64no—
360C4×D3—64no—
361C4×D3—62no—
362C4×D3—62no—
363C4×D3—62no—
364C4×D3—62no—
365C4×D3—122no—
366C4×D3—122no—
367C4×D3—124no—
368Dic6—22no—
369Dic6—22no—
370Dic6—42no—
371C3×C8—34no—
372C3×C8—62no—
373C3⋊C8—22no—
374C3⋊C8C2×#6714yes—
375C2×C32#47014yescomplemented
376C2×C32C2×#12111yescharacteristic
377C2×C32C2×#13711yescharacteristic
378C2×C32C2×#13711yescharacteristic
379C3⋊D3—42no—
380C24—91no—
381C2×Q8—21no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—32no—
385C2×Q8—61no—
386C2×Q8—62no—
387C2×Q8—62no—
388C2×Q8C2×D3212yes—
389C22×C4—21no—
390C22×C4—61no—
391C22×C4—61no—
392C22×C4—91no—
393C22×C4—92no—
394C22×C4—92no—
395C22×C4—182no—
396C22×C4—182no—
397C22×C4—181no—
398C22×C4C2×D3211yescharacteristic
399Q16—68no—
400M16—188no—
401C2×C8—32no—
402C2×C8—34no—
403C2×C8—32no—
404C2×C8—62no—
405C4⋊C4—32no—
406C4⋊C4—61no—
407C4⋊C4—92no—
408C4⋊C4—184no—
409C4⋊C4—184no—
410C4⋊C4—181no—
411C22⋊C4—184no—
412C22⋊C4—184no—
413C22×C3—21no—
414C22×C3—22no—
415C22×C3—22no—
416C22×C3—22no—
417C22×C3#88412yes—
418C22×C3#91512yes—
419C22×C3#91812yes—
420C22×C3#91812yes—
421C22×C3#91812yes—
422C22×C3#93812yes—
423C22×C3C2×D3×D411yescharacteristic
424C22×C3C2×D3×D411yescharacteristic
425C2×D3—64no—
426C2×D3—62no—
427C2×D3—62no—
428C2×D3—62no—
429C2×D3—62no—
430C2×D3—62no—
431C2×D3—62no—
432C2×D3—122no—
433C2×D3—122no—
434C2×D3—128no—
435C2×D3—124no—
436C2×D3—124no—
437C3×C4—21no—
438C3×C4—21no—
439C3×C4—22no—
440C3×C4—44no—
441C3×C4—42no—
442C3×C4—42no—
443C3×C4—122no—
444C3×C4—122no—
445C3×C4#88412yes—
446C3×C4#93812yes—
447C3×C4C2×#23511yescharacteristic
448C3×C4C2×D3×D411yescharacteristic
449C3×C4C2×#23711yescharacteristic
450C3×C4C22×#6711yescharacteristic
451Dic3—42no—
452Dic3—42no—
453Dic3—62no—
454Dic3—62no—
455Dic3—122no—
456Dic3—124no—
457Dic3—122no—
458Dic3—122no—
459C32C2×#47011yescharacteristic, complemented, Sylow
460C23—92no—
461C23—182no—
462C23—182no—
463C23—182no—
464C23#381111yescharacteristic
465Q8—22no—
466Q8—22no—
467Q8—42no—
468Q8—62no—
469Q8—62no—
470Q8—122no—
471C2×C4—22no—
472C2×C4—42no—
473C2×C4—62no—
474C2×C4—62no—
475C2×C4—92no—
476C2×C4—122no—
477C2×C4—122no—
478C2×C4—182no—
479C2×C4—182no—
480C2×C4—182no—
481C2×C4—184no—
482C2×C4—182no—
483C2×C4—362no—
484C2×C4#380412yes—
485C2×C4D3×#6712yes—
486C2×C4#381111yescharacteristic
487C2×C4C22×D3211yescharacteristic
488C8—34no—
489C8—62no—
490C2×C3—21no—
491C2×C3—21no—
492C2×C3—24no—
493C2×C3—21no—
494C2×C3#481614yescomplemented
495C2×C3#518914yescomplemented
496C2×C3C2×#88411yescharacteristic
497C2×C3C2×#91511yescharacteristic
498C2×C3C2×#91811yescharacteristic
499C2×C3C2×#91811yescharacteristic
500C2×C3C2×#91811yescharacteristic
501C2×C3C2×#93811yescharacteristic
502D3—124no—
503D3—122no—
504D3—122no—
505C22—182no—
506C22—182no—
507C22—182no—
508C22—364no—
509C22#6394512yes—
510C22#6395112yes—
511C22#6395812yes—
512C22C2×#381111yescharacteristic
513C4—42no—
514C4—122no—
515C4—122no—
516C4—182no—
517C4—362no—
518C4#6395812yes—
519C4C2×#380411yescharacteristic
520C4C2×D3×#6711yescharacteristic
521C3—21no—
522C3C2×#481611yescharacteristic, complemented
523C3C2×#518911yescharacteristic, complemented
524C2—362no—
525C2#1059160014yesdirect product factor
526C2C2×#6394511yescharacteristic
527C2C2×#6395111yescharacteristic
528C2C2×#6395811yescharacteristic
subgroupG/H
centerC23#3811
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC22×C32×Q8C22
socleC23×C32C2×D4
prime radicalC23×C3⋊D3D4
hypercenterC22×Q8D32
hypocenterC32C2×#470
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
2. C22×C3×#216C2semidirect
5. C2×C3×#835C2semidirect
6. #10591600C2direct (N × H)
7. C2×#63573C2semidirect
8. C2×#63566C2semidirect
20. C2×C3×#216C22semidirect
30. C3×#835C22semidirect
31. #63573C22semidirect
32. #63566C22semidirect
38. C22×#216D3semidirect
44. C2×#835D3semidirect
106. C2×#216C2×D3semidirect
131. #835C2×D3semidirect
375. C2×C32#470semidirect
459. C32C2×#470semidirect
494. C2×C3#4816semidirect
495. C2×C3#5189semidirect
522. C3C2×#4816semidirect
523. C3C2×#5189semidirect
525. C2#10591600direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#641552
C22×C3×#2162
C2×#638712
C22×#37122
C2×C3×#8352
#105916002
C2×#635732
C2×#635662
C2×#51893
C2×#48163