Finite Group Databasebeta

← #13176616

Subgroups — #13176616

Order 1728.

1796 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3⋊D3×#235C211yesmaximal, characteristic
2#11723942C211yesmaximal, characteristic, complemented
3C2×C3×#3743C211yesmaximal, characteristic, complemented
4C32×#901C211yesmaximal, characteristic, complemented
5#11723357C211yesmaximal, characteristic, complemented
6C3⋊D3×#208C211yesmaximal, characteristic, complemented
7#11723349C211yesmaximal, characteristic, complemented
8#11723334C212yesmaximal, complemented
9#11723332C212yesmaximal, complemented
10#11723312C212yesmaximal, complemented
11D3×#3743C212yesmaximal, complemented
12#10596221—31nomaximal
13#10594668—34nomaximal
14#89076C2211yescharacteristic
15C2×C3×C4×C3⋊D3C2211yescharacteristic
16C32×#235C2211yescharacteristic
17C3⋊D3×#67C412yes—
18#89028C412yes—
19C3⋊D3×D12C411yescharacteristic
20C4×D3×C3⋊D3C2212yes—
21#89020C2212yes—
22#89018C411yescharacteristic
23C3⋊D3×Dic6C411yescharacteristic
24#89014C411yescharacteristic
25C2×C33×C8C2211yescharacteristic, complemented
26#88855C2211yescharacteristic, complemented
27#88850C2212yescomplemented
28C8×C3⋊(C3⋊D3)C2212yescomplemented
29C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic, complemented
30C3×#3743C2211yescharacteristic
31C3×#3743C2211yescharacteristic
32C3×#3743C2211yescharacteristic, complemented
33C3×#3743C2211yescharacteristic, complemented
34C32×#208C2211yescharacteristic, complemented
35C32×ΓL(1,𝔽25)C2212yescomplemented
36C32×D3×C8C2212yescomplemented
37#88788C2212yescomplemented
38#88787C2212yes—
39#88786C2212yescomplemented
40#88785C2212yescomplemented
41C3⋊D3×C3⋊C8C2212yescomplemented
42D3×C3⋊(C3⋊C8)C2212yescomplemented
43#41×C3⋊D3—31no—
44D3×#235—34no—
45C32×#131—31no—
46#64157—31no—
47#64116—31no—
48#64116—32no—
49M16×C3⋊D3—31no—
50M16×C3⋊D3—32no—
51#64110—31no—
52#64110—32no—
53#64110—34no—
54#64110—64no—
55C2×#3743—32no—
56C2×#3743D311yescharacteristic, complemented
57C3×#901D314yescomplemented
58C2×C3×ΓL(1,𝔽25)—34no—
59#63817—34no—
60#63814—34no—
61D3×#208—34no—
62#63805—38no—
63#63804—38no—
64#63791—38no—
65D3×ΓL(1,𝔽25)—38no—
66C22×C3×C3⋊D3C2×C411yescharacteristic
67C2×D3×C3⋊D3C2×C412yes—
68C2×C33×C4C2311yescharacteristic
69#6312C2×C412yes—
70C4⋊(C3⋊(C3⋊D3))C2×C411yescharacteristic
71C4×C3⋊(C3⋊D3)C2312yes—
72#6308C2×C411yescharacteristic
73C2×C3×C3⋊Dic3C2×C411yescharacteristic
74C3×C4×C3⋊D3C2×C411yescharacteristic
75C3×C4×C3⋊D3C2×C411yescharacteristic
76C3×C4×C3⋊D3C2311yescharacteristic
77C3×C4×C3⋊D3C2311yescharacteristic
78C32×#67C2×C412yes—
79C32×D12C2×C411yescharacteristic
80C32×C4×D3C2312yes—
81C32×Dic6C2×C411yescharacteristic
82#6293C2×C412yes—
83#6292C2×C412yes—
84#6291C2×C412yes—
85#6290C2×C412yes—
86#6289C2×C412yes—
87Dic3×C3⋊D3C2×C412yes—
88D3×C3⋊Dic3C2×C412yes—
89C33×C8C2311yescharacteristic
90C33×C8C2311yescharacteristic
91C3⋊(C3⋊(C3⋊C8))C2312yes—
92C3×C3⋊(C3⋊C8)C2311yescharacteristic
93C3×C3⋊(C3⋊C8)C2311yescharacteristic
94C32×C3⋊C8C2312yes—
95#5750—94no—
96#5745—91no—
97C32×#41—31no—
98#3832—31no—
99Q8×C3⋊D3—31no—
100#3830—32no—
101#3830—31no—
102D4×C3⋊D3—32no—
103D4×C3⋊D3—31no—
104#3828—31no—
105#3828—34no—
106#3828—32no—
107#3828—64no—
108C2×C4×C3⋊D3—32no—
109C2×C4×C3⋊D3C2×D311yescharacteristic
110C3×#235C2×D314yes—
111C2×C3×C4×D3—34no—
112D3×#67—38no—
113#3804—38no—
114D3×D12—34no—
115C4×D32—38no—
116#3799—34no—
117#3798—38no—
118#3795—34no—
119D3×Dic6—34no—
120C32×M16—31no—
121C32×M16—32no—
122C2×C32×C8—24no—
123C2×C32×C8—32no—
124C2×C32×C8C2×D314yescomplemented
125C2×C32×C8C2×D311yescharacteristic, complemented
126#3748—31no—
127#3748—34no—
128#3748—32no—
129#3748—64no—
130C2×C3⋊(C3⋊C8)—32no—
131C2×C3⋊(C3⋊C8)C2×D311yescharacteristic, complemented
132#3743—32no—
133#3743—32no—
134#3743—38no—
135#3743—32no—
136#3743—68no—
137#3743C2×D311yescharacteristic
138#3743C2×D311yescharacteristic
139#3743C2×D311yescharacteristic, complemented
140#3743C2×D311yescharacteristic, complemented
141C8×C3⋊D3—32no—
142C8×C3⋊D3—32no—
143C8×C3⋊D3—38no—
144C8×C3⋊D3—32no—
145C8×C3⋊D3—68no—
146C3×#208C2×D314yescomplemented
147C2×C3×C3⋊C8—34no—
148C3×ΓL(1,𝔽25)—34no—
149C3×ΓL(1,𝔽25)—34no—
150C3×ΓL(1,𝔽25)—34no—
151C3×ΓL(1,𝔽25)—34no—
152C3×ΓL(1,𝔽25)C2×D318yescomplemented
153C3×D3×C8C2×D318yescomplemented
154#3712—38no—
155#3711—38no—
156#3711—38no—
157#3710—38no—
158D3×C3⋊C8—38no—
159D3×C3⋊C8—38no—
160C22×C33C22×C411yescharacteristic
161C2×C3⋊(C3⋊D3)C22×C412yes—
162C2×C3×C3⋊D3—41no—
163C2×C3×C3⋊D3C22×C411yescharacteristic
164C2×C3×C3⋊D3C22×C411yescharacteristic
165C2×C32×D3C22×C412yes—
166D3×C3⋊D3—42no—
167C33×C4C22×C411yescharacteristic
168C33×C4C2411yescharacteristic
169C3⋊(C3⋊Dic3)C22×C412yes—
170C3×C3⋊Dic3C22×C411yescharacteristic
171C3×C3⋊Dic3C22×C411yescharacteristic
172C32×Dic3C22×C412yes—
173D3×#41—94no—
174C2×#235—91no—
175C3×#131—34no—
176C2×C3×M16—91no—
177#948—94no—
178C2×#208—91no—
179#907—98no—
180#907—94no—
181#907—92no—
182#907—92no—
183D3×M16—98no—
184D3×M16—94no—
185D3×M16—92no—
186D3×M16—92no—
187#901—98no—
188#901—91no—
189#901—94no—
190#901—91no—
191#901—91no—
192#901—184no—
193#901C3⋊D311yescharacteristic, complemented
194C2×ΓL(1,𝔽25)—34no—
195C2×ΓL(1,𝔽25)—98no—
196C22×C3⋊D3—32no—
197C22×C3⋊D3C4×D311yescharacteristic
198C22×C3×D3—34no—
199C2×D32—38no—
200C32×Q8—31no—
201C32×D4—32no—
202C32×D4—31no—
203C2×C32×C4—24no—
204C2×C32×C4—32no—
205C2×C32×C4C22×D314yes—
206C2×C32×C4C22×D311yescharacteristic
207#638—32no—
208#638—32no—
209#638—32no—
210#638—38no—
211#638—68no—
212C2×C3⋊Dic3—32no—
213C2×C3⋊Dic3C4×D311yescharacteristic
214C4⋊(C3⋊D3)—32no—
215C4⋊(C3⋊D3)—31no—
216C4⋊(C3⋊D3)—34no—
217C4⋊(C3⋊D3)—64no—
218C4×C3⋊D3—32no—
219C4×C3⋊D3—32no—
220C4×C3⋊D3—32no—
221C4×C3⋊D3—38no—
222C4×C3⋊D3—68no—
223C4×C3⋊D3C4×D311yescharacteristic
224C4×C3⋊D3C4×D311yescharacteristic
225C4×C3⋊D3C22×D311yescharacteristic
226C4×C3⋊D3C22×D311yescharacteristic
227#634—31no—
228#634—32no—
229#634—34no—
230#634—64no—
231C3×#67C4×D318yes—
232C2×C3×Dic3—34no—
233C3×D12C4×D314yes—
234C3×C4×D3—34no—
235C3×C4×D3—34no—
236C3×C4×D3—34no—
237C3×C4×D3—34no—
238C3×C4×D3C22×D318yes—
239C3×Dic6C4×D314yes—
240#627—38no—
241#626—38no—
242#626—38no—
243#625—38no—
244#624—38no—
245D3×Dic3—38no—
246D3×Dic3—38no—
247C32×C8—24no—
248C32×C8—24no—
249C32×C8—32no—
250C32×C8C22×D314yes—
251C32×C8C22×D314yes—
252C32×C8C22×D311yescharacteristic
253C32×C8C22×D311yescharacteristic
254C3⋊(C3⋊C8)—32no—
255C3⋊(C3⋊C8)—38no—
256C3⋊(C3⋊C8)—68no—
257C3⋊(C3⋊C8)C22×D311yescharacteristic
258C3⋊(C3⋊C8)C22×D311yescharacteristic
259C3×C3⋊C8—34no—
260C3×C3⋊C8—34no—
261C3×C3⋊C8C22×D318yes—
262#568—271noSylow
263C2×C33—21no—
264C2×C33C23×C411yescharacteristic
265C3⋊(C3⋊D3)—22no—
266C3×C3⋊D3—21no—
267C3×C3⋊D3—21no—
268C32×D3—22no—
269C3×#41—34no—
270C22×C3×C4—91no—
271C2×#67—92no—
272#239—94no—
273D3×Q8—94no—
274#237—94no—
275#237—98no—
276D3×D4—94no—
277D3×D4—98no—
278#235—98no—
279#235—94no—
280#235—91no—
281#235—92no—
282#235—91no—
283#235—92no—
284#235—91no—
285#235—184no—
286#235C2×C3⋊D311yescharacteristic
287C2×D12—91no—
288C2×C4×D3—34no—
289C2×C4×D3—98no—
290C2×C4×D3—92no—
291C2×Dic6—91no—
292C3×M16—34no—
293C3×M16—38no—
294C3×M16—91no—
295C3×M16—91no—
296C3×M16—91no—
297C3×M16—91no—
298C2×C3×C8—24no—
299C2×C3×C8—38no—
300C2×C3×C8—91no—
301C2×C3×C8D3214yescomplemented
302C2×C3×C8C2×C3⋊D311yescharacteristic, complemented
303#208—98no—
304#208—94no—
305#208—91no—
306#208—92no—
307#208—184no—
308#208C2×C3⋊D311yescharacteristic, complemented
309C2×C3⋊C8—34no—
310C2×C3⋊C8—98no—
311C2×C3⋊C8—92no—
312ΓL(1,𝔽25)—34no—
313ΓL(1,𝔽25)—34no—
314ΓL(1,𝔽25)—34no—
315ΓL(1,𝔽25)—34no—
316ΓL(1,𝔽25)—98no—
317ΓL(1,𝔽25)—98no—
318ΓL(1,𝔽25)—92no—
319ΓL(1,𝔽25)—98no—
320ΓL(1,𝔽25)—92no—
321ΓL(1,𝔽25)—92no—
322ΓL(1,𝔽25)—188no—
323ΓL(1,𝔽25)C2×C3⋊D312yescomplemented
324D3×C8—98no—
325D3×C8—98no—
326D3×C8—92no—
327D3×C8—98no—
328D3×C8—92no—
329D3×C8—92no—
330D3×C8—188no—
331D3×C8C2×C3⋊D312yescomplemented
332C22×C32—24no—
333C22×C32—32no—
334C22×C32C2×C4×D314yes—
335C22×C32C2×C4×D311yescharacteristic
336C2×C3⋊D3—32no—
337C2×C3⋊D3—38no—
338C2×C3⋊D3—41no—
339C2×C3⋊D3—68no—
340C2×C3⋊D3—122no—
341C2×C3⋊D3C2×C4×D311yescharacteristic
342C2×C3⋊D3C2×C4×D311yescharacteristic
343C2×C3×D3—34no—
344C2×C3×D3—34no—
345C2×C3×D3—124no—
346C2×C3×D3C2×C4×D318yes—
347D32—128no—
348C32×C4—24no—
349C32×C4—24no—
350C32×C4—32no—
351C32×C4C2×C4×D314yes—
352C32×C4C2×C4×D311yescharacteristic
353C32×C4C23×D314yes—
354C32×C4C23×D311yescharacteristic
355C3⋊Dic3—32no—
356C3⋊Dic3—38no—
357C3⋊Dic3—68no—
358C3⋊Dic3C2×C4×D311yescharacteristic
359C3⋊Dic3C2×C4×D311yescharacteristic
360C3×Dic3—34no—
361C3×Dic3—34no—
362C3×Dic3C2×C4×D318yes—
363C2×#41—271no—
364#131—31no—
365#131—272no—
366#131—271no—
367#131—271no—
368#131—272no—
369#131—271no—
370C2×M16—91no—
371C2×M16—272no—
372C2×M16—272no—
373C2×M16—271no—
374C33#56811yescharacteristic, complemented, Sylow
375C23×C3—91no—
376C22×D3—34no—
377C22×D3—98no—
378C22×D3—92no—
379C3×Q8—34no—
380C3×D4—38no—
381C3×D4—34no—
382C2×C3×C4—24no—
383C2×C3×C4—38no—
384C2×C3×C4—91no—
385C2×C3×C4—91no—
386C2×C3×C4—91no—
387C2×C3×C4—91no—
388C2×C3×C4—91no—
389C2×C3×C4C2×D3214yes—
390C2×C3×C4C22×C3⋊D311yescharacteristic
391#67—98no—
392#67—98no—
393#67—98no—
394#67—92no—
395#67—92no—
396#67—92no—
397#67—188no—
398#67C4×C3⋊D312yes—
399C2×Dic3—34no—
400C2×Dic3—98no—
401C2×Dic3—92no—
402D12—98no—
403D12—92no—
404D12—94no—
405D12—91no—
406D12—184no—
407D12C4×C3⋊D311yescharacteristic
408C4×D3—34no—
409C4×D3—34no—
410C4×D3—34no—
411C4×D3—34no—
412C4×D3—98no—
413C4×D3—98no—
414C4×D3—92no—
415C4×D3—98no—
416C4×D3—92no—
417C4×D3—92no—
418C4×D3—188no—
419C4×D3C22×C3⋊D312yes—
420Dic6—98no—
421Dic6—94no—
422Dic6—91no—
423Dic6—92no—
424Dic6—184no—
425Dic6C4×C3⋊D311yescharacteristic
426C3×C8—24no—
427C3×C8—24no—
428C3×C8—38no—
429C3×C8—91no—
430C3×C8—91no—
431C3×C8C2×D3214yes—
432C3×C8C2×D3214yes—
433C3×C8C22×C3⋊D311yescharacteristic
434C3×C8C22×C3⋊D311yescharacteristic
435C3⋊C8—34no—
436C3⋊C8—34no—
437C3⋊C8—98no—
438C3⋊C8—92no—
439C3⋊C8—188no—
440C3⋊C8C22×C3⋊D312yes—
441C2×C32—21no—
442C2×C32—24no—
443C2×C32—24no—
444C2×C32—44no—
445C2×C32—62no—
446C2×C32C22×C4×D314yes—
447C2×C32C22×C4×D311yescharacteristic
448C3⋊D3—21no—
449C3⋊D3—21no—
450C3⋊D3—62no—
451C3⋊D3—68no—
452C3⋊D3—128no—
453C3×D3—28no—
454C3×D3—64no—
455C3×D3—64no—
456#41—31no—
457#41—272no—
458#41—272no—
459#41—271no—
460#41—271no—
461#41—271no—
462C2×Q8—271no—
463C2×D4—271no—
464C2×D4—272no—
465C22×C4—91no—
466C22×C4—272no—
467M16—31no—
468M16—32no—
469M16—91no—
470M16—91no—
471M16—91no—
472M16—91no—
473M16—272no—
474M16—272no—
475M16—272no—
476M16—271no—
477M16—272no—
478C2×C8—32no—
479C2×C8—91no—
480C2×C8—272no—
481C2×C8—272no—
482C2×C8—272no—
483C2×C8—272no—
484C2×C8D3×C3⋊D311yescharacteristic, complemented
485C22×C3—24no—
486C22×C3—38no—
487C22×C3—91no—
488C22×C3—91no—
489C22×C3—361no—
490C22×C3C4×D3214yes—
491C22×C3C2×C4×C3⋊D311yescharacteristic
492C2×D3—34no—
493C2×D3—34no—
494C2×D3—98no—
495C2×D3—92no—
496C2×D3—124no—
497C2×D3—188no—
498C2×D3—362no—
499C2×D3—368no—
500C2×D3C2×C4×C3⋊D312yes—
501C3×C4—24no—
502C3×C4—24no—
503C3×C4—38no—
504C3×C4—91no—
505C3×C4—91no—
506C3×C4C4×D3214yes—
507C3×C4C2×C4×C3⋊D311yescharacteristic
508C3×C4C22×D3214yes—
509C3×C4C23×C3⋊D311yescharacteristic
510Dic3—34no—
511Dic3—34no—
512Dic3—98no—
513Dic3—92no—
514Dic3—188no—
515Dic3C2×C4×C3⋊D312yes—
516C32—24no—
517C32#574511yescharacteristic, complemented
518C32#575014yescomplemented
519C23—91no—
520C23—272no—
521Q8—31no—
522Q8—272no—
523Q8—271no—
524D4—32no—
525D4—31no—
526D4—272no—
527D4—272no—
528D4—272no—
529D4—272no—
530D4—271no—
531C2×C4—32no—
532C2×C4—91no—
533C2×C4—91no—
534C2×C4—91no—
535C2×C4—91no—
536C2×C4—91no—
537C2×C4—272no—
538C2×C4—272no—
539C2×C4—272no—
540C2×C4—272no—
541C2×C4C2×D3×C3⋊D311yescharacteristic
542C8—32no—
543C8—91no—
544C8—91no—
545C8—272no—
546C8C2×D3×C3⋊D311yescharacteristic
547C8C2×D3×C3⋊D311yescharacteristic
548C2×C3—24no—
549C2×C3—21no—
550C2×C3—24no—
551C2×C3—44no—
552C2×C3—68no—
553C2×C3—181no—
554C2×C3—181no—
555C2×C3C2×C4×D3214yes—
556C2×C3C22×C4×C3⋊D311yescharacteristic
557D3—22no—
558D3—64no—
559D3—64no—
560D3—188no—
561D3—182no—
562D3—368no—
563C22—32no—
564C22—91no—
565C22—91no—
566C22—272no—
567C22—361no—
568C22—1082no—
569C22C4×D3×C3⋊D311yescharacteristic
570C4—32no—
571C4—91no—
572C4—91no—
573C4—272no—
574C4C4×D3×C3⋊D311yescharacteristic
575C4C22×D3×C3⋊D311yescharacteristic
576C3—24no—
577C3#1059466814yescomplemented
578C3#1059622111yescharacteristic, complemented
579C2—21no—
580C2—62no—
581C2—181no—
582C2—181no—
583C2—542no—
584C2C2×C4×D3×C3⋊D311yescharacteristic
subgroupG/H
center, FrattiniC4C22×D3×C3⋊D3
commutator, socleC2×C33C23×C4
FittingC2×C33×C8C22
prime radicalC3⋊D3×#235C2
cosocleC33×C4C24
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#568
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. #11723942C2semidirect
3. C2×C3×#3743C2semidirect
4. C32×#901C2semidirect
5. #11723357C2semidirect
6. C3⋊D3×#208C2semidirect
7. #11723349C2semidirect
8. #11723334C2semidirect
9. #11723332C2semidirect
10. #11723312C2semidirect
11. D3×#3743C2semidirect
25. C2×C33×C8C22semidirect
26. #88855C22semidirect
27. #88850C22semidirect
28. C8×C3⋊(C3⋊D3)C22semidirect
29. C2×C3×C3⋊(C3⋊C8)C22semidirect
32. C3×#3743C22semidirect
33. C3×#3743C22semidirect
34. C32×#208C22semidirect
35. C32×ΓL(1,𝔽25)C22semidirect
36. C32×D3×C8C22semidirect
37. #88788C22semidirect
39. #88786C22semidirect
40. #88785C22semidirect
41. C3⋊D3×C3⋊C8C22semidirect
42. D3×C3⋊(C3⋊C8)C22semidirect
56. C2×#3743D3semidirect
57. C3×#901D3semidirect
124. C2×C32×C8C2×D3semidirect
125. C2×C32×C8C2×D3semidirect
131. C2×C3⋊(C3⋊C8)C2×D3semidirect
139. #3743C2×D3semidirect
140. #3743C2×D3semidirect
146. C3×#208C2×D3semidirect
152. C3×ΓL(1,𝔽25)C2×D3semidirect
153. C3×D3×C8C2×D3semidirect
193. #901C3⋊D3semidirect
301. C2×C3×C8D32semidirect
302. C2×C3×C8C2×C3⋊D3semidirect
308. #208C2×C3⋊D3semidirect
323. ΓL(1,𝔽25)C2×C3⋊D3semidirect
331. D3×C8C2×C3⋊D3semidirect
374. C33#568semidirect
484. C2×C8D3×C3⋊D3semidirect
517. C32#5745semidirect
518. C32#5750semidirect
577. C3#10594668semidirect
578. C3#10596221semidirect

Maximal subgroups

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

subgroupindex
C3⋊D3×#2352
#117239422
C2×C3×#37432
C32×#9012
#117233572
C3⋊D3×#2082
#117233492
#117233342
#117233322
#117233122
D3×#37432
#105962213
#105946683