Finite Group Databasebeta

← #13176479

Subgroups — #13176479

Order 1728.

2056 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89019C212yesmaximal, complemented
2#11723934C211yesmaximal, characteristic
3C3×#64097C211yesmaximal, characteristic, complemented
4C32×#888C211yesmaximal, characteristic, complemented
5#11723536C211yesmaximal, characteristic, complemented
6C4×#6293C211yesmaximal, characteristic, complemented
7#11723494C211yesmaximal, characteristic
8#11723489C212yesmaximal, complemented
9#11723484C211yesmaximal, characteristic
10#11723477C212yesmaximal, complemented
11#11723442C212yesmaximal, complemented
12#10596198—31nomaximal
13#10594379—34nomaximal
14C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
15C2×C4×C3⋊(C3⋊D3)C2212yes—
16C2×C3×#634C2212yes—
17C2×C32×Dic6C2212yes—
18C2×#6293C2212yes—
19C2×#6293C2211yescharacteristic
20C2×#6289C2212yes—
21C2×#6289C2212yes—
22#89019—28no—
23C33×C4⋊C4C2211yescharacteristic
24#88859C2212yes—
25C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
26C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
27C3×#3750C2212yes—
28C3×C4×C3⋊Dic3C2211yescharacteristic
29C32×C4⋊Dic3C2211yescharacteristic
30C32×Dic3⋊C4C2212yes—
31C32×C4×Dic3C2211yescharacteristic
32#88807C2211yescharacteristic
33#88806C2211yescharacteristic
34#88806C2212yes—
35Dic3⋊(C3⋊Dic3)C2211yescharacteristic
36#88804C2212yes—
37#88804C2212yes—
38C2×Q8×C3⋊D3—32no—
39C2×#3797—38no—
40C32×C4⋊Q8—31no—
41#64155—32no—
42#64105—32no—
43#64105—32no—
44#64102—31no—
45#64102—31no—
46#64102—34no—
47#64102—64no—
48#64097—31no—
49#64097D311yescharacteristic, complemented
50C4×C4⋊(C3⋊D3)—31no—
51C4×#634—31no—
52C3×#888—34no—
53C3×#888D314yescomplemented
54#63919—34no—
55C4×#627—34no—
56#63894—34no—
57#63894—34no—
58#63888—38no—
59#63888—38no—
60#63865—38no—
61C22×C3⋊(C3⋊D3)C2312yes—
62C2×C33×C4C2312yes—
63C2×C33×C4C2311yescharacteristic
64C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
65C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
66C4⋊(C3⋊(C3⋊D3))Q814yescomplemented
67C4×C3⋊(C3⋊D3)—24no—
68C2×C3×C3⋊Dic3C2311yescharacteristic
69C2×C3×C3⋊Dic3C2312yes—
70C2×C3×C3⋊Dic3C2311yescharacteristic
71C3×#634—24no—
72C2×C32×Dic3C2312yes—
73C2×C32×Dic3C2311yescharacteristic
74C2×C32×Dic3C2311yescharacteristic
75C32×Dic6—24no—
76#6293—24no—
77#6293D414yescomplemented
78#6289—24no—
79#6289—24no—
80#5727—94no—
81#5727—91no—
82C2×C32×Q8—32no—
83Q8×C3⋊D3—68no—
84C2×C4⋊(C3⋊D3)—31no—
85C2×C4⋊(C3⋊D3)—34no—
86C2×C4⋊(C3⋊D3)—64no—
87C2×C4×C3⋊D3—32no—
88C2×C4×C3⋊D3—38no—
89C2×C4×C3⋊D3—32no—
90C2×C4×C3⋊D3—32no—
91C2×C4×C3⋊D3—68no—
92C2×#634—31no—
93C2×#634—32no—
94C2×#634C2×D312yes—
95C2×C3×Dic6—38no—
96C2×C3×Dic6C2×D318yes—
97C2×#627—34no—
98C2×#627—38no—
99C2×#624—38no—
100C2×#624—38no—
101#3797—632no—
102C32×C4⋊C4—24no—
103C32×C4⋊C4—31no—
104C32×C4⋊C4—32no—
105C32×C4⋊C4C2×D314yes—
106C32×C4⋊C4C2×D311yescharacteristic
107C32×C42—31no—
108#3752—32no—
109#3752—38no—
110#3752—32no—
111#3752—32no—
112#3752—68no—
113C4⋊(C3⋊Dic3)—31no—
114C4⋊(C3⋊Dic3)C2×D311yescharacteristic
115#3750—31no—
116#3750—31no—
117#3750—32no—
118#3750C2×D312yes—
119C4×C3⋊Dic3—31no—
120C4×C3⋊Dic3—34no—
121C4×C3⋊Dic3—64no—
122C4×C3⋊Dic3C2×D311yescharacteristic
123C3×C4⋊Dic3—34no—
124C3×C4⋊Dic3C2×D314yes—
125C3×Dic3⋊C4—38no—
126C3×Dic3⋊C4C2×D318yes—
127C3×C4×Dic3—34no—
128C3×C4×Dic3C2×D314yes—
129#3724—34no—
130#3724—38no—
131Dic3⋊Dic3—34no—
132Dic3⋊Dic3—34no—
133#3722—38no—
134#3722—38no—
135C22×C33C2411yescharacteristic
136C2×C3⋊(C3⋊D3)—22no—
137C2×C3⋊(C3⋊D3)—22no—
138C2×C3⋊(C3⋊D3)C2×Q814yescomplemented
139C33×C4—22no—
140C33×C4C2×Q812yescomplemented
141C3⋊(C3⋊Dic3)—21no—
142C3⋊(C3⋊Dic3)C2×D412yes—
143C3×C3⋊Dic3—22no—
144C3×C3⋊Dic3—21no—
145C3×C3⋊Dic3C2×D412yes—
146C32×Dic3—22no—
147C32×Dic3—21no—
148C32×Dic3C2×D412yes—
149C2×D3×Q8—98no—
150C2×D3×Q8—92no—
151C3×C4⋊Q8—34no—
152C3×C4⋊Q8—91no—
153#946—98no—
154#946—92no—
155#896—98no—
156#896—98no—
157#896—92no—
158#896—92no—
159#893—94no—
160#893—94no—
161#893—91no—
162#893—91no—
163#893—184no—
164#888—34no—
165#888—94no—
166#888—91no—
167#888C3⋊D311yescharacteristic, complemented
168C4×D12—94no—
169C4×D12—91no—
170C4×Dic6—94no—
171C4×Dic6—91no—
172C22×C3⋊D3—32no—
173C22×C3⋊D3—38no—
174C22×C3⋊D3—68no—
175C32×Q8—64no—
176C2×C32×C4—24no—
177C2×C32×C4—28no—
178C2×C32×C4—31no—
179C2×C32×C4—31no—
180C2×C32×C4—32no—
181C2×C32×C4C22×D318yes—
182C2×C32×C4C22×D314yes—
183C2×C32×C4C22×D312yes—
184C2×C32×C4C22×D311yescharacteristic
185C2×C3⋊Dic3—31no—
186C2×C3⋊Dic3—31no—
187C2×C3⋊Dic3—34no—
188C2×C3⋊Dic3—34no—
189C2×C3⋊Dic3—64no—
190C2×C3⋊Dic3—64no—
191C2×C3⋊Dic3C22×D311yescharacteristic
192C2×C3⋊Dic3C22×D312yes—
193C2×C3⋊Dic3C22×D311yescharacteristic
194C4⋊(C3⋊D3)—34no—
195C4⋊(C3⋊D3)—316no—
196C4⋊(C3⋊D3)—616no—
197C4×C3⋊D3—64no—
198C4×C3⋊D3—64no—
199C4×C3⋊D3—64no—
200C4×C3⋊D3—616no—
201C4×C3⋊D3—1216no—
202#634—24no—
203#634—34no—
204#634—64no—
205C2×C3×Dic3—34no—
206C2×C3×Dic3—34no—
207C2×C3×Dic3—38no—
208C2×C3×Dic3C22×D314yes—
209C2×C3×Dic3C22×D318yes—
210C2×C3×Dic3C22×D314yes—
211C3×Dic6—216no—
212C3×Dic6—616no—
213#627—316no—
214#627—616no—
215#624—616no—
216#624—616no—
217D4×Q8—271noSylow
218C2×C33C22×D411yescharacteristic
219C2×C33C22×Q811yescharacteristic
220C2×C332−1+411yescharacteristic
221C3⋊(C3⋊D3)—24no—
222C2×C3×Q8—38no—
223C2×C3×Q8—92no—
224D3×Q8—188no—
225D3×Q8—1832no—
226C2×D12—94no—
227C2×D12—91no—
228C2×D12—184no—
229C2×C4×D3—98no—
230C2×C4×D3—98no—
231C2×C4×D3—98no—
232C2×C4×D3—92no—
233C2×C4×D3—92no—
234C2×C4×D3—92no—
235C2×C4×D3—188no—
236C2×Dic6—38no—
237C2×Dic6—94no—
238C2×Dic6—91no—
239C2×Dic6—98no—
240C2×Dic6—92no—
241C2×Dic6C2×C3⋊D312yes—
242C3×C4⋊C4—24no—
243C3×C4⋊C4—34no—
244C3×C4⋊C4—38no—
245C3×C4⋊C4—91no—
246C3×C4⋊C4—92no—
247C3×C4⋊C4D3214yes—
248C3×C4⋊C4C2×C3⋊D311yescharacteristic
249C3×C42—34no—
250C3×C42—91no—
251(C2×D3)⋊C4—98no—
252(C2×D3)⋊C4—98no—
253(C2×D3)⋊C4—98no—
254(C2×D3)⋊C4—92no—
255(C2×D3)⋊C4—92no—
256(C2×D3)⋊C4—92no—
257(C2×D3)⋊C4—188no—
258C4⋊Dic3—34no—
259C4⋊Dic3—94no—
260C4⋊Dic3—91no—
261C4⋊Dic3C2×C3⋊D311yescharacteristic
262Dic3⋊C4—38no—
263Dic3⋊C4—94no—
264Dic3⋊C4—94no—
265Dic3⋊C4—98no—
266Dic3⋊C4—91no—
267Dic3⋊C4—91no—
268Dic3⋊C4—92no—
269Dic3⋊C4C2×C3⋊D312yes—
270C4×Dic3—34no—
271C4×Dic3—94no—
272C4×Dic3—91no—
273C4×Dic3—184no—
274C4×Dic3C2×C3⋊D311yescharacteristic
275C22×C32—24no—
276C22×C32C23×D314yes—
277C22×C32C23×D311yescharacteristic
278C2×C3⋊D3—34no—
279C2×C3⋊D3—316no—
280C2×C3⋊D3—616no—
281C2×C3⋊D3—62no—
282C2×C3⋊D3—62no—
283C2×C3⋊D3—68no—
284C2×C3⋊D3—68no—
285C2×C3⋊D3—128no—
286C2×C3⋊D3—128no—
287C32×C4—22no—
288C32×C4—28no—
289C32×C4—28no—
290C32×C4—32no—
291C32×C4—48no—
292C32×C4—62no—
293C32×C4—61no—
294C32×C4D3×Q818yescomplemented
295C32×C4D3×Q812yescomplemented
296C3⋊Dic3—22no—
297C3⋊Dic3—21no—
298C3⋊Dic3—32no—
299C3⋊Dic3—38no—
300C3⋊Dic3—68no—
301C3⋊Dic3—61no—
302C3⋊Dic3—64no—
303C3⋊Dic3—124no—
304C3⋊Dic3D3×D412yes—
305C3×Dic3—28no—
306C3×Dic3—24no—
307C3×Dic3—38no—
308C3×Dic3—68no—
309C3×Dic3—64no—
310C3×Dic3D3×D418yes—
311C22×Q8—272no—
312C4⋊Q8—31no—
313C4⋊Q8—91no—
314C4⋊Q8—271no—
315C22⋊Q8—272no—
316C22⋊Q8—272no—
317C22⋊Q8—272no—
318C4×Q8—271no—
319C4×D4—271no—
320C4×D4—271no—
321C4×D4—271no—
322C33D4×Q811yescharacteristic, complemented, Sylow
323C22×D3—98no—
324C22×D3—92no—
325C22×D3—188no—
326C3×Q8—616no—
327C3×Q8—184no—
328C2×C3×C4—24no—
329C2×C3×C4—28no—
330C2×C3×C4—34no—
331C2×C3×C4—34no—
332C2×C3×C4—38no—
333C2×C3×C4—91no—
334C2×C3×C4—91no—
335C2×C3×C4—92no—
336C2×C3×C4C2×D3218yes—
337C2×C3×C4C2×D3214yes—
338C2×C3×C4C22×C3⋊D312yes—
339C2×C3×C4C22×C3⋊D311yescharacteristic
340C2×Dic3—34no—
341C2×Dic3—34no—
342C2×Dic3—38no—
343C2×Dic3—94no—
344C2×Dic3—91no—
345C2×Dic3—94no—
346C2×Dic3—91no—
347C2×Dic3—184no—
348C2×Dic3—184no—
349C2×Dic3C22×C3⋊D311yescharacteristic
350C2×Dic3C22×C3⋊D312yes—
351C2×Dic3C22×C3⋊D311yescharacteristic
352D12—916no—
353D12—94no—
354D12—1816no—
355C4×D3—184no—
356C4×D3—1816no—
357C4×D3—1816no—
358C4×D3—184no—
359C4×D3—1816no—
360C4×D3—184no—
361C4×D3—3616no—
362Dic6—24no—
363Dic6—616no—
364Dic6—916no—
365Dic6—94no—
366Dic6—184no—
367Dic6—1816no—
368C2×C32—24no—
369C2×C32—24no—
370C2×C32—24no—
371C2×C32C2×D3×D414yes—
372C2×C32C2×D3×D411yescharacteristic
373C2×C32C2×D3×Q814yes—
374C2×C32C2×D3×Q811yescharacteristic
375C2×C32#100714yes—
376C2×C32#100711yescharacteristic
377C3⋊D3—64no—
378C3⋊D3—616no—
379C3⋊D3—1216no—
380C2×Q8—32no—
381C2×Q8—92no—
382C2×Q8—271no—
383C2×Q8—272no—
384C2×Q8—548no—
385C2×D4—271no—
386C22×C4—272no—
387C22×C4—272no—
388C22×C4—272no—
389C4⋊C4—31no—
390C4⋊C4—32no—
391C4⋊C4—91no—
392C4⋊C4—92no—
393C4⋊C4—271no—
394C4⋊C4—271no—
395C4⋊C4—271no—
396C4⋊C4—272no—
397C4⋊C4D3×C3⋊D311yescharacteristic
398C22⋊C4—272no—
399C22⋊C4—272no—
400C22⋊C4—272no—
401C42—31no—
402C42—91no—
403C42—271no—
404C22×C3—24no—
405C22×C3C22×D3214yes—
406C22×C3C23×C3⋊D311yescharacteristic
407C2×D3—916no—
408C2×D3—94no—
409C2×D3—1816no—
410C2×D3—188no—
411C2×D3—188no—
412C2×D3—182no—
413C2×D3—182no—
414C2×D3—368no—
415C2×D3—368no—
416C3×C4—28no—
417C3×C4—22no—
418C3×C4—28no—
419C3×C4—38no—
420C3×C4—48no—
421C3×C4—68no—
422C3×C4—64no—
423C3×C4—92no—
424C3×C4—182no—
425C3×C4—181no—
426C3×C4#379718yescomplemented
427C3×C4Q8×C3⋊D312yescomplemented
428Dic3—22no—
429Dic3—21no—
430Dic3—38no—
431Dic3—68no—
432Dic3—64no—
433Dic3—98no—
434Dic3—92no—
435Dic3—188no—
436Dic3—184no—
437Dic3—181no—
438Dic3—364no—
439Dic3D4×C3⋊D312yes—
440C32—24no—
441C32#572714yescomplemented
442C32#572711yescharacteristic, complemented
443C23—272no—
444Q8—64no—
445Q8—184no—
446Q8—274no—
447Q8—544no—
448D4—274no—
449C2×C4—31no—
450C2×C4—31no—
451C2×C4—32no—
452C2×C4—91no—
453C2×C4—91no—
454C2×C4—92no—
455C2×C4—271no—
456C2×C4—271no—
457C2×C4—544no—
458C2×C4—544no—
459C2×C4—544no—
460C2×C4C2×D3×C3⋊D312yes—
461C2×C4C2×D3×C3⋊D311yescharacteristic
462C2×C3—24no—
463C2×C3—24no—
464C2×C3—24no—
465C2×C3C2×#379714yes—
466C2×C3D32×D414yes—
467C2×C3#6431614yes—
468C2×C3C2×D4×C3⋊D311yescharacteristic
469C2×C3C2×Q8×C3⋊D311yescharacteristic
470C2×C3#6436411yescharacteristic
471D3—1816no—
472D3—184no—
473D3—3616no—
474C22—274no—
475C22—542no—
476C22—542no—
477C22C22×D3×C3⋊D311yescharacteristic
478C4—22no—
479C4—32no—
480C4—62no—
481C4—61no—
482C4—92no—
483C4—182no—
484C4—181no—
485C4—272no—
486C4—541no—
487C4#8901912yescomplemented
488C3—24no—
489C3#1059437914yescomplemented
490C3#1059619811yescharacteristic, complemented
491C2—544no—
492C2C2×#8901911yescharacteristic
493C2D3×D4×C3⋊D311yescharacteristic
494C2#1172459111yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C4⋊C4C22
prime radicalC2×C4⋊(C3⋊(C3⋊D3))C22
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33D4×Q8
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#89019C2semidirect
3. C3×#64097C2semidirect
4. C32×#888C2semidirect
5. #11723536C2semidirect
6. C4×#6293C2semidirect
8. #11723489C2semidirect
10. #11723477C2semidirect
11. #11723442C2semidirect
49. #64097D3semidirect
53. C3×#888D3semidirect
66. C4⋊(C3⋊(C3⋊D3))Q8semidirect
77. #6293D4semidirect
138. C2×C3⋊(C3⋊D3)C2×Q8semidirect
140. C33×C4C2×Q8semidirect
167. #888C3⋊D3semidirect
294. C32×C4D3×Q8semidirect
295. C32×C4D3×Q8semidirect
322. C33D4×Q8semidirect
426. C3×C4#3797semidirect
427. C3×C4Q8×C3⋊D3semidirect
441. C32#5727semidirect
442. C32#5727semidirect
487. C4#89019semidirect
489. C3#10594379semidirect
490. C3#10596198semidirect

Maximal subgroups

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

subgroupindex
C2×#890192
#117239342
C3×#640972
C32×#8882
#117235362
C4×#62932
#117234942
#117234892
#117234842
#117234772
#117234422
#105961983
#105943793