Finite Group Databasebeta

← #13168328

Subgroups — #13168328

Order 1728.

1644 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89018C212yesmaximal, complemented
2C4×#6295C211yesmaximal, characteristic, complemented
3#11723494C211yesmaximal, characteristic, complemented
4C3×#63873C211yesmaximal, characteristic, complemented
5C3×#63871C212yesmaximal, complemented
6#11723000C214yesmaximal, complemented
7#11722968C212yesmaximal, complemented
8#11722912C212yesmaximal, complemented
9#10595341—32nomaximal
10#10594530—31nomaximal
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×C3×C4×C3⋊D3C2212yes—
13C2×C32×Dic6C2212yes—
14C2×#6295C2211yescharacteristic
15C2×#6292C2214yes—
16C2×#6289C2214yes—
17C2×#6289C2212yes—
18#89018—28no—
19C3×C4×C3⋊Dic3C2211yescharacteristic
20C32×C4⋊Dic3C2211yescharacteristic
21#88815C2211yescharacteristic
22#88814C2212yes—
23#88804C2212yes—
24C3×#3724C2212yes—
25C3×#3724C2212yes—
26C3×Dic3⋊Dic3C2214yes—
27C3×#3722C2214yes—
28C2×#3832—32no—
29C2×#3799—32no—
30C2×#3799—32no—
31C2×#3799—62no—
32#64102—31no—
33C3×#946—32no—
34C3×#944—32no—
35C3×#897—32no—
36#63912—32no—
37#63910—34no—
38C4×#626—32no—
39C4×#625—31no—
40#63900—32no—
41#63897—34no—
42#63894—32no—
43#63894—61no—
44#63883—34no—
45#63873D311yescharacteristic, complemented
46#63871D312yescomplemented
47#63854—34no—
48#63853—34no—
49C22×C3⋊(C3⋊D3)C2312yes—
50C22×C3×C3⋊D3C2312yes—
51C2×C33×C4C2311yescharacteristic
52C4⋊(C3⋊(C3⋊D3))D414yescomplemented
53C2×C3×C3⋊Dic3C2311yescharacteristic
54C2×C3×C3⋊Dic3C2312yes—
55C2×C3×C3⋊Dic3C2311yescharacteristic
56C3×C4×C3⋊D3—24no—
57C2×C32×Dic3C2314yes—
58C2×C32×Dic3C2312yes—
59C32×Dic6—24no—
60#6295—22no—
61#6292—28no—
62#6289—28no—
63#6289—24no—
64#5726—91no—
65#5631—92no—
66C2×C32×Q8—32no—
67#3832—68no—
68C2×C4⋊(C3⋊D3)—31no—
69C2×C4⋊(C3⋊D3)—32no—
70C2×C4⋊(C3⋊D3)—34no—
71C2×C4⋊(C3⋊D3)—62no—
72C2×C4⋊(C3⋊D3)—61no—
73C2×C4⋊(C3⋊D3)—121no—
74C2×C4×C3⋊D3—32no—
75C2×C4×C3⋊D3—34no—
76C2×C4×C3⋊D3C2×D312yes—
77C2×C3×C4×D3—32no—
78C2×C3×C4×D3—32no—
79C2×C3×C4×D3—62no—
80C2×C3×Dic6—22no—
81C2×C3×Dic6C2×D312yes—
82C2×C3×Dic6C2×D312yes—
83C2×#626—32no—
84C2×#626—34no—
85C2×#626—34no—
86C2×#626—64no—
87C2×#625—31no—
88C2×#624—34no—
89C2×#624—34no—
90C2×#624—34no—
91C2×#624—64no—
92C2×#624—62no—
93#3799—68no—
94#3799—68no—
95#3799—616no—
96C32×C4⋊C4—31no—
97#3752—32no—
98C4×C3⋊Dic3C2×D311yescharacteristic
99C3×(C2×D3)⋊C4—34no—
100C3×(C2×D3)⋊C4—34no—
101C3×C4⋊Dic3—21no—
102C3×C4⋊Dic3—34no—
103C3×C4⋊Dic3C2×D312yes—
104C3×Dic3⋊C4—34no—
105C3×Dic3⋊C4—34no—
106C3×Dic3⋊C4—34no—
107C3×C4×Dic3—32no—
108C3×C4×Dic3—61no—
109#3724—31no—
110#3724C2×D312yes—
111#3724C2×D312yes—
112Dic3⋊Dic3—32no—
113Dic3⋊Dic3C2×D314yes—
114#3722—32no—
115#3722—34no—
116#3722—62no—
117#3722C2×D314yes—
118(C2×D3)⋊Dic3—32no—
119(C2×D3)⋊Dic3—32no—
120C22×C33C2411yescharacteristic
121C2×C3⋊(C3⋊D3)—22no—
122C2×C3⋊(C3⋊D3)—22no—
123C2×C3⋊(C3⋊D3)C2×D414yes—
124C2×C3×C3⋊D3—22no—
125C2×C3×C3⋊D3—22no—
126C2×C3×C3⋊D3—22no—
127C33×C4C2×D412yes—
128C3×C3⋊Dic3—22no—
129C3×C3⋊Dic3—21no—
130C3×C3⋊Dic3#4112yescomplemented
131C32×Dic3—24no—
132C32×Dic3—22no—
133C2×#239—92no—
134C2×#239—92no—
135C2×#239—182no—
136C2×#235—92no—
137C3×C4⋊Q8—91no—
138C3×C22⋊Q8—92no—
139#946—32no—
140#944—32no—
141C4×#67—92no—
142#897—32no—
143#895—92no—
144#894—94no—
145#894—94no—
146#893—92no—
147#893—181no—
148D3×C4⋊C4—92no—
149D3×C4⋊C4—94no—
150#884—94no—
151#883—94no—
152C4×D12—91no—
153C4×D12—91no—
154C22×C3⋊D3—34no—
155C22×C3⋊D3—32no—
156C22×C3⋊D3—62no—
157C22×C3⋊D3—64no—
158C22×C3⋊D3—122no—
159C22×C3⋊D3C22×D312yes—
160C22×C3×D3—32no—
161C22×C3×D3—32no—
162C22×C3×D3—62no—
163C32×Q8—64no—
164C2×C32×C4—21no—
165C2×C32×C4—22no—
166C2×C32×C4—34no—
167C2×C32×C4—32no—
168C2×C32×C4—41no—
169C2×C32×C4C22×D312yes—
170C2×C32×C4C22×D311yescharacteristic
171C2×C3⋊Dic3C22×D311yescharacteristic
172C2×C3⋊Dic3C22×D312yes—
173C2×C3⋊Dic3C22×D311yescharacteristic
174C4⋊(C3⋊D3)—34no—
175C4⋊(C3⋊D3)—38no—
176C4⋊(C3⋊D3)—68no—
177C4⋊(C3⋊D3)—64no—
178C4⋊(C3⋊D3)—68no—
179C4⋊(C3⋊D3)—124no—
180C4×C3⋊D3—24no—
181C4×C3⋊D3—68no—
182C4×C3⋊D3—64no—
183C2×C3×Dic3—24no—
184C2×C3×Dic3—22no—
185C2×C3×Dic3—32no—
186C2×C3×Dic3—32no—
187C2×C3×Dic3—32no—
188C2×C3×Dic3—32no—
189C2×C3×Dic3—61no—
190C2×C3×Dic3—62no—
191C2×C3×Dic3—61no—
192C2×C3×Dic3C22×D314yes—
193C2×C3×Dic3C22×D314yes—
194C2×C3×Dic3C22×D314yes—
195C3×C4×D3—68no—
196C3×C4×D3—64no—
197C3×C4×D3—64no—
198C3×Dic6—28no—
199C3×Dic6—24no—
200C3×Dic6—24no—
201#626—68no—
202#626—68no—
203#626—616no—
204#626—64no—
205#625—62no—
206#624—68no—
207#624—68no—
208#624—68no—
209#624—616no—
210#624—124no—
211#547—271noSylow
212C2×C33C22×D411yescharacteristic
213C2×C33C2×#4111yescharacteristic
214C2×C332−1+411yescharacteristic
215C3⋊(C3⋊D3)—24no—
216C3×C3⋊D3—42no—
217C2×C3×Q8—32no—
218C2×C3×Q8—32no—
219C2×C3×Q8—62no—
220C22×C3×C4—92no—
221C2×#67—92no—
222C2×#67—94no—
223#239—188no—
224#239—188no—
225#239—1816no—
226#235—188no—
227C2×D12—91no—
228C2×D12—92no—
229C2×D12—94no—
230C2×D12—94no—
231C2×D12—91no—
232C2×D12—181no—
233C2×D12—182no—
234C2×D12—184no—
235C2×D12—361no—
236C2×C4×D3—32no—
237C2×C4×D3—32no—
238C2×C4×D3—62no—
239C2×C4×D3—94no—
240C2×C4×D3—94no—
241C2×C4×D3—94no—
242C2×C4×D3—92no—
243C2×C4×D3—94no—
244C2×C4×D3—184no—
245C2×C4×D3—182no—
246C2×Dic6D3212yes—
247C3×C4⋊C4—32no—
248C3×C4⋊C4—61no—
249C3×C4⋊C4—94no—
250C3×C4⋊C4—92no—
251C3×C4⋊C4—92no—
252C3×C22⋊C4—94no—
253C3×C42—91no—
254#217—92no—
255(C2×D3)⋊C4—34no—
256(C2×D3)⋊C4—34no—
257(C2×D3)⋊C4—94no—
258(C2×D3)⋊C4—92no—
259(C2×D3)⋊C4—92no—
260(C2×D3)⋊C4—92no—
261(C2×D3)⋊C4—182no—
262C4⋊Dic3—34no—
263C4⋊Dic3—91no—
264C4⋊Dic3D3211yescharacteristic
265Dic3⋊C4—34no—
266Dic3⋊C4—34no—
267Dic3⋊C4—34no—
268Dic3⋊C4—92no—
269C4×Dic3—32no—
270C4×Dic3—61no—
271C22×C32—21no—
272C22×C32—22no—
273C22×C32—41no—
274C22×C32C23×D311yescharacteristic
275C22×C32C23×D312yes—
276C2×C3⋊D3—22no—
277C2×C3⋊D3—22no—
278C2×C3⋊D3—22no—
279C2×C3⋊D3—34no—
280C2×C3⋊D3—38no—
281C2×C3⋊D3—68no—
282C2×C3⋊D3—64no—
283C2×C3⋊D3—62no—
284C2×C3⋊D3—62no—
285C2×C3⋊D3—64no—
286C2×C3⋊D3—64no—
287C2×C3⋊D3—124no—
288C2×C3⋊D3—122no—
289C2×C3⋊D3—122no—
290C2×C3⋊D3—124no—
291C2×C3⋊D3—124no—
292C2×C3⋊D3—242no—
293C2×C3⋊D3—242no—
294C2×C3×D3—64no—
295C2×C3×D3—62no—
296C2×C3×D3—62no—
297C2×C3×D3—62no—
298C2×C3×D3—62no—
299C2×C3×D3—62no—
300C2×C3×D3—62no—
301C2×C3×D3—122no—
302C2×C3×D3—122no—
303C32×C4—22no—
304C32×C4—24no—
305C32×C4—42no—
306C32×C4—64no—
307C32×C4—62no—
308C32×C4D3×D414yes—
309C32×C4D3×D412yes—
310C3⋊Dic3—22no—
311C3⋊Dic3—21no—
312C3⋊Dic3#23912yescomplemented
313C3×Dic3—28no—
314C3×Dic3—24no—
315C3×Dic3—24no—
316C3×Dic3—24no—
317C3×Dic3—34no—
318C3×Dic3—42no—
319C3×Dic3—62no—
320C3×Dic3—64no—
321C3×Dic3—62no—
322C3×Dic3—62no—
323C3×Dic3—62no—
324C3×Dic3—121no—
325C2×#41—272no—
326C4⋊Q8—91no—
327#123—274no—
328C22⋊Q8—92no—
329#121—272no—
330C4×D4—271no—
331C4×D4—271no—
332C2×C4⋊C4—272no—
333C33#54711yescharacteristic, complemented, Sylow
334C23×C3—92no—
335C22×D3—32no—
336C22×D3—32no—
337C22×D3—62no—
338C22×D3—92no—
339C22×D3—94no—
340C22×D3—182no—
341C22×D3—184no—
342C22×D3—362no—
343C3×Q8—68no—
344C3×Q8—64no—
345C3×Q8—64no—
346C2×C3×C4—21no—
347C2×C3×C4—22no—
348C2×C3×C4—34no—
349C2×C3×C4—34no—
350C2×C3×C4—34no—
351C2×C3×C4—41no—
352C2×C3×C4—64no—
353C2×C3×C4—62no—
354C2×C3×C4—91no—
355C2×C3×C4—92no—
356C2×C3×C4—91no—
357C2×C3×C4—184no—
358C2×C3×C4C2×D3211yescharacteristic
359C2×C3×C4C2×D3212yes—
360#67—188no—
361#67—184no—
362C2×Dic3—32no—
363C2×Dic3—32no—
364C2×Dic3—32no—
365C2×Dic3—32no—
366C2×Dic3—61no—
367C2×Dic3—62no—
368C2×Dic3—61no—
369C2×Dic3C2×D3214yes—
370C2×Dic3C2×D3212yes—
371D12—94no—
372D12—98no—
373D12—184no—
374D12—188no—
375D12—188no—
376D12—188no—
377D12—1816no—
378D12—182no—
379D12—364no—
380C4×D3—68no—
381C4×D3—64no—
382C4×D3—64no—
383C4×D3—184no—
384C4×D3—188no—
385C4×D3—188no—
386C4×D3—188no—
387C4×D3—188no—
388C4×D3—1816no—
389C4×D3—364no—
390Dic6—24no—
391C2×C32—21no—
392C2×C32—21no—
393C2×C32—21no—
394C2×C32—22no—
395C2×C32—22no—
396C2×C32—22no—
397C2×C32—41no—
398C2×C32—41no—
399C2×C32—41no—
400C2×C32C2×D3×D412yes—
401C2×C32C2×D3×D411yescharacteristic
402C2×C32C2×#23911yescharacteristic
403C2×C32#100711yescharacteristic
404C2×C32#100712yes—
405C2×C32D3×#4112yes—
406C3⋊D3—42no—
407C3⋊D3—64no—
408C3⋊D3—68no—
409C3⋊D3—128no—
410C3⋊D3—124no—
411C3⋊D3—244no—
412C3×D3—124no—
413C3×D3—122no—
414C3×D3—122no—
415#41—548no—
416C2×Q8—32no—
417C2×D4—274no—
418C2×D4—271no—
419C2×D4—271no—
420C22×C4—92no—
421C22×C4—272no—
422C22×C4—274no—
423C4⋊C4—31no—
424C4⋊C4—92no—
425C4⋊C4—94no—
426C4⋊C4—92no—
427C4⋊C4—271no—
428C22⋊C4—94no—
429C22⋊C4—272no—
430C22⋊C4—272no—
431C42—91no—
432C22×C3—21no—
433C22×C3—22no—
434C22×C3—41no—
435C22×C3—182no—
436C22×C3—182no—
437C22×C3—182no—
438C22×C3C22×D3211yescharacteristic
439C22×C3C22×D3212yes—
440C2×D3—64no—
441C2×D3—62no—
442C2×D3—62no—
443C2×D3—62no—
444C2×D3—62no—
445C2×D3—62no—
446C2×D3—62no—
447C2×D3—94no—
448C2×D3—98no—
449C2×D3—122no—
450C2×D3—122no—
451C2×D3—188no—
452C2×D3—184no—
453C2×D3—182no—
454C2×D3—184no—
455C2×D3—182no—
456C2×D3—184no—
457C2×D3—364no—
458C2×D3—362no—
459C2×D3—364no—
460C2×D3—362no—
461C2×D3—364no—
462C2×D3—722no—
463C2×D3—722no—
464C3×C4—24no—
465C3×C4—22no—
466C3×C4—42no—
467C3×C4—68no—
468C3×C4—64no—
469C3×C4—64no—
470C3×C4—64no—
471C3×C4—92no—
472C3×C4—122no—
473C3×C4—181no—
474C3×C4—182no—
475C3×C4#380212yes—
476C3×C4#381114yes—
477Dic3—22no—
478Dic3—24no—
479Dic3—34no—
480Dic3—62no—
481Dic3—64no—
482Dic3—62no—
483Dic3—62no—
484Dic3—62no—
485Dic3—121no—
486C32—22no—
487C32—21no—
488C32—41no—
489C32#563112yescomplemented
490C32#572611yescharacteristic, complemented
491C23—92no—
492C23—272no—
493Q8—64no—
494D4—274no—
495D4—548no—
496D4—542no—
497C2×C4—32no—
498C2×C4—34no—
499C2×C4—91no—
500C2×C4—91no—
501C2×C4—92no—
502C2×C4—184no—
503C2×C4—544no—
504C2×C4—548no—
505C2×C4D3311yescharacteristic
506C2×C3—21no—
507C2×C3—22no—
508C2×C3—21no—
509C2×C3—22no—
510C2×C3—22no—
511C2×C3—21no—
512C2×C3—41no—
513C2×C3—41no—
514C2×C3—41no—
515C2×C3—362no—
516C2×C3C2×#380211yescharacteristic
517C2×C3#6431211yescharacteristic
518C2×C3#6431612yes—
519C2×C3#6431611yescharacteristic
520C2×C3D3×#23912yes—
521C2×C3C2×#381112yes—
522D3—124no—
523D3—122no—
524D3—122no—
525D3—188no—
526D3—184no—
527D3—368no—
528D3—364no—
529D3—724no—
530C22—182no—
531C22—182no—
532C22—182no—
533C22—274no—
534C22—542no—
535C22—542no—
536C22C2×D3311yescharacteristic
537C4—62no—
538C4—64no—
539C4—92no—
540C4—181no—
541C4—182no—
542C4#8895412yes—
543C3—22no—
544C3—21no—
545C3—41no—
546C3#1059453011yescharacteristic, complemented
547C3#1059534112yescomplemented
548C2—362no—
549C2—544no—
550C2#1172437711yescharacteristic
551C2#1172439911yescharacteristic
552C2C2×#8895411yescharacteristic
subgroupG/H
center, FrattiniC22C2×D33
commutator, socle, cosocleC22×C33C24
FittingC2×C33×C4C23
hypercenterC2×C4D33
hypocenterC33#547
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×#89018C2semidirect
2. C4×#6295C2semidirect
3. #11723494C2semidirect
4. C3×#63873C2semidirect
5. C3×#63871C2semidirect
6. #11723000C2semidirect
7. #11722968C2semidirect
8. #11722912C2semidirect
45. #63873D3semidirect
46. #63871D3semidirect
52. C4⋊(C3⋊(C3⋊D3))D4semidirect
130. C3×C3⋊Dic3#41semidirect
312. C3⋊Dic3#239semidirect
333. C33#547semidirect
489. C32#5631semidirect
490. C32#5726semidirect
546. C3#10594530semidirect
547. C3#10595341semidirect

Maximal subgroups

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

subgroupindex
C2×#890182
C4×#62952
#117234942
C3×#638732
C3×#638712
#117230002
#117229682
#117229122
#105953413
#105945303