Finite Group Databasebeta

← #13175157

Subgroups — #13175157

Order 1728.

2068 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89017C211yesmaximal, characteristic
2#11723913C211yesmaximal, characteristic, complemented
3C3×C4×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
4C32×C4×Dic6C211yesmaximal, characteristic, complemented
5#11723512C212yesmaximal, complemented
6#11723503C211yesmaximal, characteristic, complemented
7#11723501C211yesmaximal, characteristic, complemented
8#11723471C212yesmaximal, complemented
9C3⋊D3×Dic3⋊C4C212yesmaximal, complemented
10#11723464C212yesmaximal, complemented
11#11723397C211yesmaximal, characteristic, complemented
12#10596045—31nomaximal
13#10593889—34nomaximal
14C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
15C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
16C2×C3×C4×C3⋊D3C2212yes—
17C2×C32×Dic6C2211yescharacteristic
18C2×#6292C2212yes—
19C2×#6289C2212yes—
20C2×Dic3×C3⋊D3C2212yes—
21C2×Dic3×C3⋊D3C2212yes—
22#89017C418yescomplemented
23C33×C42C2211yescharacteristic
24#88859C2211yescharacteristic
25#88859C2211yescharacteristic
26#88857C2211yescharacteristic
27#88857C2211yescharacteristic
28C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
29C3×#3752C2212yes—
30C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
31C32×C4⋊Dic3C2211yescharacteristic
32C32×Dic3⋊C4C2212yes—
33C32×C4×Dic3C2212yes—
34#88806C2212yes—
35Dic3⋊(C3⋊Dic3)C2212yes—
36#88803C2212yes—
37Dic3×C3⋊Dic3C2212yes—
38C2×#3832—31no—
39C2×#3796—34no—
40C32×C4×Q8—31no—
41Q8×C3⋊Dic3—31no—
42#64102—31no—
43#64102—32no—
44C4⋊C4×C3⋊D3—32no—
45C4⋊C4×C3⋊D3—31no—
46C4×C4⋊(C3⋊D3)—32no—
47C4×C4⋊(C3⋊D3)D311yescharacteristic, complemented
48#64081—31no—
49#64081—32no—
50#64081—34no—
51#64081—64no—
52C3×C4×D12—34no—
53C3×C4×Dic6D314yescomplemented
54#63904—38no—
55#63900—34no—
56#63898—34no—
57#63880—38no—
58#63877—38no—
59D3×Dic3⋊C4—38no—
60#63843—34no—
61C22×C3⋊(C3⋊D3)C2311yescharacteristic
62C22×C3×C3⋊D3C2312yes—
63C2×C33×C4C2311yescharacteristic
64C2×C33×C4C2311yescharacteristic
65C2×C33×C4C2311yescharacteristic
66C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
67C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
68C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
69C4×C3⋊(C3⋊D3)C2×C414yescomplemented
70C2×C3×C3⋊Dic3C2312yes—
71C3×C4⋊(C3⋊D3)C2×C414yescomplemented
72C3×C4×C3⋊D3—42no—
73C2×C32×Dic3C2312yes—
74C2×C32×Dic3C2312yes—
75C32×Dic6C2×C414yescomplemented
76#6292C2×C418yescomplemented
77#6289—42no—
78Dic3×C3⋊D3—42no—
79Dic3×C3⋊D3C2×C418yescomplemented
80#5574—94no—
81#5546—91no—
82C2×C32×Q8—31no—
83#3832—38no—
84C2×C4⋊(C3⋊D3)—32no—
85C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
86C2×C4×C3⋊D3—32no—
87C2×C4×C3⋊D3—31no—
88C2×C4×C3⋊D3—34no—
89C2×C4×C3⋊D3—32no—
90C2×C4×C3⋊D3—32no—
91C2×C4×C3⋊D3—64no—
92C2×C4×C3⋊D3C2×D312yes—
93C2×C3×D12—34no—
94C2×C3×C4×D3—38no—
95C2×C3×Dic6C2×D314yes—
96C2×#626—38no—
97C2×#624—38no—
98C2×D3×Dic3—38no—
99C2×D3×Dic3—38no—
100#3796—332no—
101C32×C4⋊C4—32no—
102C32×C4⋊C4—31no—
103C32×C42—24no—
104C32×C42—32no—
105C32×C42C2×D314yes—
106C32×C42C2×D311yescharacteristic
107#3752—32no—
108#3752—31no—
109#3752—31no—
110#3752—34no—
111#3752—34no—
112#3752—64no—
113#3752—64no—
114#3752C2×D312yes—
115C4⋊(C3⋊Dic3)—32no—
116C4⋊(C3⋊Dic3)C2×D311yescharacteristic
117#3750—31no—
118#3750—31no—
119#3750—34no—
120#3750—34no—
121#3750—32no—
122#3750—64no—
123#3750—64no—
124C4×C3⋊Dic3—31no—
125C4×C3⋊Dic3—34no—
126C4×C3⋊Dic3—32no—
127C4×C3⋊Dic3—64no—
128C3×(C2×D3)⋊C4—38no—
129C3×C4⋊Dic3—34no—
130C3×C4⋊Dic3C2×D314yes—
131C3×Dic3⋊C4C2×D318yes—
132C3×C4×Dic3C2×D318yes—
133#3724—38no—
134Dic3⋊Dic3—38no—
135(C2×D3)⋊Dic3—38no—
136Dic32—38no—
137C22×C33C2411yescharacteristic
138C2×C3⋊(C3⋊D3)—21no—
139C2×C3⋊(C3⋊D3)—21no—
140C2×C3⋊(C3⋊D3)C22×C412yes—
141C2×C3×C3⋊D3—22no—
142C2×C3×C3⋊D3—22no—
143C2×C3×C3⋊D3C22×C414yes—
144C33×C4—21no—
145C33×C4—21no—
146C33×C4C22×C412yes—
147C3⋊(C3⋊Dic3)—21no—
148C3⋊(C3⋊Dic3)—21no—
149C3⋊(C3⋊Dic3)C22×C412yes—
150C3×C3⋊Dic3—22no—
151C32×Dic3—22no—
152C32×Dic3C22×C414yes—
153C2×#239—94no—
154C2×#237—91no—
155C3×C4×Q8—34no—
156C3×C4×D4—91no—
157Q8×Dic3—94no—
158D4×Dic3—91no—
159C4×#67—92no—
160C2×Dic3⋊C4—92no—
161#893—98no—
162#893—94no—
163D3×C4⋊C4—94no—
164D3×C4⋊C4—98no—
165D3×C4⋊C4—91no—
166#887—91no—
167#67⋊C4—92no—
168#877—92no—
169C4×D12—34no—
170C4×D12—98no—
171#872—98no—
172#872—94no—
173#872—91no—
174#872—184no—
175C4×Dic6C3⋊D311yescharacteristic, complemented
176C22×C3⋊D3—31no—
177C22×C3⋊D3—34no—
178C22×C3⋊D3—64no—
179C22×C3⋊D3C22×D312yes—
180C22×C3×D3—38no—
181C32×Q8—34no—
182C2×C32×C4—24no—
183C2×C32×C4—24no—
184C2×C32×C4—24no—
185C2×C32×C4—32no—
186C2×C32×C4—32no—
187C2×C32×C4C22×D314yes—
188C2×C32×C4C22×D314yes—
189C2×C32×C4C22×D314yes—
190C2×C32×C4C22×D311yescharacteristic
191C2×C32×C4C22×D311yescharacteristic
192C2×C32×C4C22×D311yescharacteristic
193C2×C3⋊Dic3—31no—
194C2×C3⋊Dic3—31no—
195C2×C3⋊Dic3—31no—
196C2×C3⋊Dic3—34no—
197C2×C3⋊Dic3—34no—
198C2×C3⋊Dic3—34no—
199C2×C3⋊Dic3—64no—
200C2×C3⋊Dic3—64no—
201C2×C3⋊Dic3—64no—
202C2×C3⋊Dic3C22×D312yes—
203C4⋊(C3⋊D3)—38no—
204C4⋊(C3⋊D3)C4×D314yescomplemented
205C4×C3⋊D3—38no—
206C4×C3⋊D3—34no—
207C4×C3⋊D3—316no—
208C4×C3⋊D3—42no—
209C4×C3⋊D3—616no—
210C4×C3⋊D3—122no—
211C4×C3⋊D3—122no—
212C2×C3×Dic3—38no—
213C2×C3×Dic3C22×D318yes—
214C2×C3×Dic3C22×D318yes—
215C3×D12—316no—
216C3×C4×D3—128no—
217C3×Dic6C4×D3116yescomplemented
218#626—332no—
219#624—128no—
220D3×Dic3—332no—
221D3×Dic3—128no—
222#520—271noSylow
223C2×C33C23×C411yescharacteristic
224C2×C33H(5,𝔽2)11yescharacteristic
225C2×C332−1+411yescharacteristic
226C3⋊(C3⋊D3)—22no—
227C3×C3⋊D3—24no—
228C2×C3×Q8—34no—
229C2×C3×D4—91no—
230C22×C3×C4—92no—
231C2×#67—92no—
232C22×Dic3—92no—
233C22×Dic3—92no—
234#239—932no—
235#237—98no—
236C2×D12—34no—
237C2×D12—98no—
238C2×C4×D3—38no—
239C2×C4×D3—98no—
240C2×C4×D3—94no—
241C2×C4×D3—91no—
242C2×C4×D3—98no—
243C2×C4×D3—98no—
244C2×C4×D3—92no—
245C2×C4×D3—184no—
246C2×Dic6C2×C3⋊D311yescharacteristic
247C3×C4⋊C4—38no—
248C3×C4⋊C4—34no—
249C3×C4⋊C4—91no—
250C3×C22⋊C4—92no—
251C3×C42—24no—
252C3×C42—38no—
253C3×C42D3214yes—
254C3×C42C2×C3⋊D311yescharacteristic
255#217—92no—
256(C2×D3)⋊C4—38no—
257(C2×D3)⋊C4—98no—
258(C2×D3)⋊C4—94no—
259(C2×D3)⋊C4—94no—
260(C2×D3)⋊C4—91no—
261(C2×D3)⋊C4—91no—
262(C2×D3)⋊C4—184no—
263(C2×D3)⋊C4—184no—
264C4⋊Dic3—34no—
265C4⋊Dic3—98no—
266C4⋊Dic3C2×C3⋊D311yescharacteristic
267Dic3⋊C4—98no—
268Dic3⋊C4—94no—
269Dic3⋊C4—94no—
270Dic3⋊C4—91no—
271Dic3⋊C4—91no—
272Dic3⋊C4—92no—
273Dic3⋊C4—92no—
274Dic3⋊C4—184no—
275Dic3⋊C4—184no—
276Dic3⋊C4C2×C3⋊D312yes—
277C4×Dic3—98no—
278C4×Dic3—94no—
279C4×Dic3—91no—
280C4×Dic3—92no—
281C4×Dic3—184no—
282C4×Dic3C2×C3⋊D312yes—
283C22×C32—24no—
284C22×C32C23×D314yes—
285C22×C32C23×D311yescharacteristic
286C2×C3⋊D3—22no—
287C2×C3⋊D3—22no—
288C2×C3⋊D3—32no—
289C2×C3⋊D3—38no—
290C2×C3⋊D3—68no—
291C2×C3⋊D3—61no—
292C2×C3⋊D3—61no—
293C2×C3⋊D3—64no—
294C2×C3⋊D3—64no—
295C2×C3⋊D3—124no—
296C2×C3⋊D3—124no—
297C2×C3⋊D3C2×C4×D314yes—
298C2×C3×D3—316no—
299C2×C3×D3—68no—
300C2×C3×D3—68no—
301C32×C4—21no—
302C32×C4—21no—
303C32×C4—28no—
304C32×C4—24no—
305C32×C4—24no—
306C32×C4—34no—
307C32×C4—44no—
308C32×C4—44no—
309C32×C4—62no—
310C32×C4C2×C4×D318yes—
311C32×C4C2×C4×D312yes—
312C3⋊Dic3—22no—
313C3⋊Dic3—32no—
314C3⋊Dic3—38no—
315C3⋊Dic3—68no—
316C3⋊Dic3—61no—
317C3⋊Dic3—61no—
318C3⋊Dic3—64no—
319C3⋊Dic3—64no—
320C3⋊Dic3—124no—
321C3⋊Dic3—124no—
322C3×Dic3—28no—
323C3×Dic3—68no—
324C3×Dic3C2×C4×D3116yes—
325C2×#41—271no—
326C4×Q8—31no—
327C4×Q8—271no—
328C4×D4—91no—
329C4×D4—272no—
330C4×D4—272no—
331C4×D4—271no—
332#117—272no—
333#117—271no—
334C2×C4⋊C4—271no—
335C2×C4⋊C4—272no—
336C33#52011yescharacteristic, complemented, Sylow
337C23×C3—92no—
338C22×D3—38no—
339C22×D3—94no—
340C22×D3—91no—
341C22×D3—184no—
342C3×Q8—316no—
343C3×D4—94no—
344C2×C3×C4—24no—
345C2×C3×C4—24no—
346C2×C3×C4—24no—
347C2×C3×C4—38no—
348C2×C3×C4—38no—
349C2×C3×C4—92no—
350C2×C3×C4—362no—
351C2×C3×C4C2×D3214yes—
352C2×C3×C4C2×D3214yes—
353C2×C3×C4C2×D3214yes—
354C2×C3×C4C22×C3⋊D311yescharacteristic
355C2×C3×C4C22×C3⋊D311yescharacteristic
356C2×C3×C4C22×C3⋊D311yescharacteristic
357#67—98no—
358C2×Dic3—38no—
359C2×Dic3—94no—
360C2×Dic3—91no—
361C2×Dic3—98no—
362C2×Dic3—94no—
363C2×Dic3—94no—
364C2×Dic3—91no—
365C2×Dic3—91no—
366C2×Dic3—184no—
367C2×Dic3—184no—
368C2×Dic3—184no—
369C2×Dic3—362no—
370C2×Dic3C22×C3⋊D312yes—
371C2×Dic3C22×C3⋊D312yes—
372D12—316no—
373D12—932no—
374C4×D3—932no—
375C4×D3—916no—
376C4×D3—94no—
377C4×D3—128no—
378C4×D3—1816no—
379C4×D3—362no—
380C4×D3—368no—
381C4×D3—368no—
382Dic6C4×C3⋊D314yescomplemented
383C2×C32—24no—
384C2×C32—24no—
385C2×C32—24no—
386C2×C32C22×C4×D314yes—
387C2×C32C22×C4×D311yescharacteristic
388C2×C32#100411yescharacteristic
389C2×C32#100714yes—
390C2×C32#100914yes—
391C2×C32#101011yescharacteristic
392C3⋊D3—24no—
393C3⋊D3—62no—
394C3⋊D3—68no—
395C3⋊D3—128no—
396C3×D3—616no—
397#41—278no—
398C2×Q8—31no—
399C2×D4—91no—
400C2×D4—272no—
401C22×C4—92no—
402C22×C4—272no—
403C22×C4—272no—
404C22×C4—272no—
405C22×C4—271no—
406C4⋊C4—32no—
407C4⋊C4—31no—
408C4⋊C4—91no—
409C4⋊C4—272no—
410C4⋊C4—272no—
411C4⋊C4—271no—
412C4⋊C4—271no—
413C22⋊C4—92no—
414C22⋊C4—272no—
415C22⋊C4—271no—
416C22⋊C4—271no—
417C42—32no—
418C42—272no—
419C42—271no—
420C42D3×C3⋊D311yescharacteristic
421C22×C3—24no—
422C22×C3—94no—
423C22×C3—182no—
424C22×C3—182no—
425C22×C3C22×D3214yes—
426C22×C3C23×C3⋊D311yescharacteristic
427C2×D3—316no—
428C2×D3—68no—
429C2×D3—68no—
430C2×D3—98no—
431C2×D3—92no—
432C2×D3—188no—
433C2×D3—184no—
434C2×D3—184no—
435C2×D3—181no—
436C2×D3—181no—
437C2×D3—364no—
438C2×D3—364no—
439C3×C4—24no—
440C3×C4—24no—
441C3×C4—21no—
442C3×C4—21no—
443C3×C4—28no—
444C3×C4—316no—
445C3×C4—44no—
446C3×C4—44no—
447C3×C4—68no—
448C3×C4—182no—
449C3×C4C4×D3218yes—
450C3×C4C2×C4×C3⋊D312yes—
451Dic3—22no—
452Dic3—68no—
453Dic3—98no—
454Dic3—92no—
455Dic3—188no—
456Dic3—184no—
457Dic3—184no—
458Dic3—181no—
459Dic3—181no—
460Dic3—364no—
461Dic3—364no—
462Dic3C2×C4×C3⋊D314yes—
463C32—24no—
464C32#554611yescharacteristic, complemented
465C32#557414yescomplemented
466C23—92no—
467C23—271no—
468Q8—34no—
469D4—94no—
470D4—278no—
471C2×C4—32no—
472C2×C4—32no—
473C2×C4—92no—
474C2×C4—278no—
475C2×C4—271no—
476C2×C4—274no—
477C2×C4—271no—
478C2×C4—271no—
479C2×C4—362no—
480C2×C4—1082no—
481C2×C4—1082no—
482C2×C4C2×D3×C3⋊D311yescharacteristic
483C2×C4C2×D3×C3⋊D311yescharacteristic
484C2×C4C2×D3×C3⋊D311yescharacteristic
485C2×C3—24no—
486C2×C3—24no—
487C2×C3—24no—
488C2×C3—184no—
489C2×C3#6429714yes—
490C2×C3C2×C4×D3214yes—
491C2×C3#6430714yes—
492C2×C3C22×C4×C3⋊D311yescharacteristic
493C2×C3#6436411yescharacteristic
494C2×C3#6436611yescharacteristic
495D3—616no—
496D3—188no—
497D3—182no—
498D3—368no—
499C22—94no—
500C22—182no—
501C22—182no—
502C22—272no—
503C22—541no—
504C22—541no—
505C22C22×D3×C3⋊D311yescharacteristic
506C4—21no—
507C4—21no—
508C4—34no—
509C4—62no—
510C4—182no—
511C4—272no—
512C4—541no—
513C4—541no—
514C4C4×D3×C3⋊D312yes—
515C3—24no—
516C3#1059388914yescomplemented
517C3#1059604511yescharacteristic, complemented
518C2—184no—
519C2—542no—
520C2#1172456611yescharacteristic
521C2C2×C4×D3×C3⋊D311yescharacteristic
522C2#1172458011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutatorC2×C33C23×C4
FittingC33×C42C22
socle, cosocleC22×C33C24
prime radicalC2×#89017C2
hypercenterC42D3×C3⋊D3
hypocenterC33#520
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
2. #11723913C2semidirect
3. C3×C4×C4⋊(C3⋊D3)C2semidirect
4. C32×C4×Dic6C2semidirect
5. #11723512C2semidirect
6. #11723503C2semidirect
7. #11723501C2semidirect
8. #11723471C2semidirect
9. C3⋊D3×Dic3⋊C4C2semidirect
10. #11723464C2semidirect
11. #11723397C2semidirect
22. #89017C4semidirect
47. C4×C4⋊(C3⋊D3)D3semidirect
53. C3×C4×Dic6D3semidirect
69. C4×C3⋊(C3⋊D3)C2×C4semidirect
71. C3×C4⋊(C3⋊D3)C2×C4semidirect
75. C32×Dic6C2×C4semidirect
76. #6292C2×C4semidirect
79. Dic3×C3⋊D3C2×C4semidirect
175. C4×Dic6C3⋊D3semidirect
204. C4⋊(C3⋊D3)C4×D3semidirect
217. C3×Dic6C4×D3semidirect
336. C33#520semidirect
382. Dic6C4×C3⋊D3semidirect
464. C32#5546semidirect
465. C32#5574semidirect
516. C3#10593889semidirect
517. C3#10596045semidirect

Maximal subgroups

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

subgroupindex
C2×#890172
#117239132
C3×C4×C4⋊(C3⋊D3)2
C32×C4×Dic62
#117235122
#117235032
#117235012
#117234712
C3⋊D3×Dic3⋊C42
#117234642
#117233972
#105960453
#105938893