Finite Group Databasebeta

← #13175992

Subgroups — #13175992

Order 1728.

2156 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×#89019C211yesmaximal, characteristic
2C4⋊C4×C3⋊(C3⋊D3)C211yesmaximal, characteristic
3C3×#64096C211yesmaximal, characteristic, complemented
4C32×#887C211yesmaximal, characteristic, complemented
5#11723470C212yesmaximal
6C4×#6289C212yesmaximal
7C4×#6289C211yesmaximal, characteristic
8Dic3×#634C211yesmaximal, characteristic, complemented
9Dic6×C3⋊Dic3C211yesmaximal, characteristic, complemented
10#11723390C212yesmaximal, complemented
11#11723389C212yesmaximal, complemented
12C4×Q8×C3⋊D3—31nomaximal
13#10594363—34nomaximal
14C2×C4×C3⋊(C3⋊D3)C2212yes—
15C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
16C2×C3×#634C2211yescharacteristic
17C2×C32×Dic6C2211yescharacteristic
18C2×#6293C2212yes—
19C2×#6289C2211yescharacteristic
20C2×#6289C2212yes—
21C2×#6289C2211yescharacteristic
22#89019C418yescomplemented
23C33×C4⋊C4C2211yescharacteristic
24C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
25#88857C2212yes—
26C3×#3750C2212yes—
27C3×C4×C3⋊Dic3C2212yes—
28C3×C4×C3⋊Dic3C2211yescharacteristic
29C32×Dic3⋊C4C2212yes—
30C32×C4×Dic3C2212yes—
31C32×C4×Dic3C2211yescharacteristic
32#88807C2212yes—
33Dic3⋊(C3⋊Dic3)C2212yes—
34Dic3×C3⋊Dic3C2211yescharacteristic
35Dic3×C3⋊Dic3C2212yes—
36Dic3×C3⋊Dic3C2211yescharacteristic
37Dic3×C3⋊Dic3C2212yes—
38C2×Q8×C3⋊D3—31no—
39C2×#3797—34no—
40C32×C4×Q8—31no—
41Q8×C3⋊Dic3—31no—
42C4⋊C4×C3⋊D3—31no—
43C4⋊C4×C3⋊D3—32no—
44C4⋊C4×C3⋊D3—34no—
45C4⋊C4×C3⋊D3—64no—
46#64096—32no—
47#64096D311yescharacteristic, complemented
48C42×C3⋊D3—31no—
49C42×C3⋊D3—32no—
50C4×#634—31no—
51C4×#634—32no—
52C3×#887—34no—
53C3×#887D314yescomplemented
54#63883—38no—
55C4×#624—34no—
56C4×#624—38no—
57Dic3×Dic6—34no—
58Dic3×Dic6—34no—
59#63837—38no—
60#63837—38no—
61C22×C3⋊(C3⋊D3)C2311yescharacteristic
62C2×C33×C4C2312yes—
63C2×C33×C4C2311yescharacteristic
64C2×C3⋊(C3⋊Dic3)C2312yes—
65C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
66C4×C3⋊(C3⋊D3)—24no—
67C4×C3⋊(C3⋊D3)C2×C414yes—
68C2×C3×C3⋊Dic3C2311yescharacteristic
69C2×C3×C3⋊Dic3C2312yes—
70C2×C3×C3⋊Dic3C2311yescharacteristic
71C3×#634C2×C414yescomplemented
72C2×C32×Dic3C2311yescharacteristic
73C2×C32×Dic3C2312yes—
74C2×C32×Dic3C2311yescharacteristic
75C32×Dic6C2×C414yescomplemented
76#6293C2×C418yescomplemented
77#6289—22no—
78#6289C2×C418yes—
79#6289Q814yescomplemented
80C4×D3×Q8—94no—
81C4×D3×Q8—91no—
82C2×C32×Q8—31no—
83Q8×C3⋊D3—38no—
84C2×C4×C3⋊D3—31no—
85C2×C4×C3⋊D3—34no—
86C2×C4×C3⋊D3—31no—
87C2×C4×C3⋊D3—32no—
88C2×C4×C3⋊D3—32no—
89C2×C4×C3⋊D3—38no—
90C2×C4×C3⋊D3—31no—
91C2×C4×C3⋊D3—64no—
92C2×C4×C3⋊D3—68no—
93C2×#634—32no—
94C2×#634C2×D311yescharacteristic
95C2×C3×Dic6—34no—
96C2×C3×Dic6C2×D314yes—
97C2×#627—38no—
98C2×#624—38no—
99C2×#624—34no—
100C2×#624—34no—
101#3797—332no—
102C32×C4⋊C4—24no—
103C32×C4⋊C4—32no—
104C32×C4⋊C4C2×D314yes—
105C32×C4⋊C4C2×D311yescharacteristic
106C32×C42—32no—
107C32×C42—31no—
108C4⋊(C3⋊Dic3)—31no—
109C4⋊(C3⋊Dic3)—34no—
110C4⋊(C3⋊Dic3)—32no—
111C4⋊(C3⋊Dic3)—64no—
112#3750—32no—
113#3750—38no—
114#3750—32no—
115#3750—68no—
116#3750C2×D312yes—
117C4×C3⋊Dic3—31no—
118C4×C3⋊Dic3—32no—
119C4×C3⋊Dic3—31no—
120C4×C3⋊Dic3—32no—
121C4×C3⋊Dic3C2×D312yes—
122C4×C3⋊Dic3C2×D311yescharacteristic
123C3×Dic3⋊C4—38no—
124C3×Dic3⋊C4C2×D318yes—
125C3×C4×Dic3—38no—
126C3×C4×Dic3—34no—
127C3×C4×Dic3C2×D318yes—
128C3×C4×Dic3C2×D314yes—
129Dic3⋊Dic3—38no—
130Dic3⋊Dic3—38no—
131Dic32—34no—
132Dic32—34no—
133Dic32—38no—
134Dic32—38no—
135C22×C33C2411yescharacteristic
136C2×C3⋊(C3⋊D3)C22×C412yes—
137C2×C3⋊(C3⋊D3)C2×Q812yes—
138C2×C3⋊(C3⋊D3)#4112yes—
139C33×C4—22no—
140C33×C4C22×C412yes—
141C3⋊(C3⋊Dic3)—22no—
142C3⋊(C3⋊Dic3)C22×C412yes—
143C3×C3⋊Dic3—21no—
144C3×C3⋊Dic3C22×C414yes—
145C3×C3⋊Dic3C2×Q812yescomplemented
146C32×Dic3—21no—
147C32×Dic3C22×C414yes—
148C32×Dic3C2×Q812yescomplemented
149C2×D3×Q8—94no—
150C2×D3×Q8—91no—
151C3×C4×Q8—34no—
152C3×C4×Q8—91no—
153Q8×Dic3—94no—
154Q8×Dic3—91no—
155D3×C4⋊C4—98no—
156D3×C4⋊C4—94no—
157D3×C4⋊C4—91no—
158D3×C4⋊C4—92no—
159D3×C4⋊C4—184no—
160#887—34no—
161#887—98no—
162#887—92no—
163#887C3⋊D311yescharacteristic, complemented
164C42×D3—98no—
165C42×D3—94no—
166C42×D3—91no—
167C42×D3—92no—
168C4×Dic6—98no—
169C4×Dic6—94no—
170C4×Dic6—91no—
171C4×Dic6—92no—
172C22×C3⋊D3—31no—
173C22×C3⋊D3—34no—
174C22×C3⋊D3—64no—
175C32×Q8—34no—
176C2×C32×C4—24no—
177C2×C32×C4—28no—
178C2×C32×C4—31no—
179C2×C32×C4—32no—
180C2×C32×C4—31no—
181C2×C32×C4C22×D318yes—
182C2×C32×C4C22×D314yes—
183C2×C32×C4C22×D312yes—
184C2×C32×C4C22×D311yescharacteristic
185C2×C3⋊Dic3—31no—
186C2×C3⋊Dic3—32no—
187C2×C3⋊Dic3—34no—
188C2×C3⋊Dic3—38no—
189C2×C3⋊Dic3—64no—
190C2×C3⋊Dic3—68no—
191C2×C3⋊Dic3C22×D311yescharacteristic
192C2×C3⋊Dic3C22×D312yes—
193C2×C3⋊Dic3C22×D311yescharacteristic
194C4×C3⋊D3—34no—
195C4×C3⋊D3—38no—
196C4×C3⋊D3—34no—
197C4×C3⋊D3—316no—
198C4×C3⋊D3—616no—
199C4×C3⋊D3—62no—
200C4×C3⋊D3—64no—
201C4×C3⋊D3—616no—
202C4×C3⋊D3—1216no—
203#634—38no—
204#634C4×D314yescomplemented
205C2×C3×Dic3—34no—
206C2×C3×Dic3—38no—
207C2×C3×Dic3—34no—
208C2×C3×Dic3C22×D314yes—
209C2×C3×Dic3C22×D318yes—
210C2×C3×Dic3C22×D314yes—
211C3×Dic6—316no—
212C3×Dic6C4×D3116yescomplemented
213#627—332no—
214#624—332no—
215#624—316no—
216#624—68no—
217C2×C4×Q8—271noSylow
218C2×C33C23×C411yescharacteristic
219C2×C33C22×Q811yescharacteristic
220C2×C33C2×#4111yescharacteristic
221C3⋊(C3⋊D3)C4×Q814yescomplemented
222C2×C3×Q8—34no—
223C2×C3×Q8—91no—
224D3×Q8—932no—
225D3×Q8—98no—
226C2×C4×D3—94no—
227C2×C4×D3—98no—
228C2×C4×D3—94no—
229C2×C4×D3—91no—
230C2×C4×D3—91no—
231C2×C4×D3—92no—
232C2×C4×D3—94no—
233C2×C4×D3—98no—
234C2×C4×D3—92no—
235C2×C4×D3—91no—
236C2×C4×D3—184no—
237C2×C4×D3—188no—
238C2×Dic6—34no—
239C2×Dic6—98no—
240C2×Dic6—92no—
241C2×Dic6C2×C3⋊D311yescharacteristic
242C3×C4⋊C4—24no—
243C3×C4⋊C4—38no—
244C3×C4⋊C4—92no—
245C3×C4⋊C4D3214yes—
246C3×C4⋊C4C2×C3⋊D311yescharacteristic
247C3×C42—38no—
248C3×C42—34no—
249C3×C42—92no—
250C3×C42—91no—
251C4⋊Dic3—98no—
252C4⋊Dic3—94no—
253C4⋊Dic3—91no—
254C4⋊Dic3—92no—
255C4⋊Dic3—184no—
256Dic3⋊C4—38no—
257Dic3⋊C4—98no—
258Dic3⋊C4—98no—
259Dic3⋊C4—92no—
260Dic3⋊C4—92no—
261Dic3⋊C4—188no—
262Dic3⋊C4C2×C3⋊D312yes—
263C4×Dic3—38no—
264C4×Dic3—34no—
265C4×Dic3—98no—
266C4×Dic3—94no—
267C4×Dic3—94no—
268C4×Dic3—98no—
269C4×Dic3—91no—
270C4×Dic3—92no—
271C4×Dic3—91no—
272C4×Dic3—92no—
273C4×Dic3C2×C3⋊D312yes—
274C4×Dic3C2×C3⋊D311yescharacteristic
275C22×C32—24no—
276C22×C32C23×D314yes—
277C22×C32C23×D311yescharacteristic
278C2×C3⋊D3—32no—
279C2×C3⋊D3—32no—
280C2×C3⋊D3—32no—
281C2×C3⋊D3—38no—
282C2×C3⋊D3—38no—
283C2×C3⋊D3—38no—
284C2×C3⋊D3—68no—
285C2×C3⋊D3—68no—
286C2×C3⋊D3—68no—
287C32×C4—22no—
288C32×C4—28no—
289C32×C4—28no—
290C32×C4—34no—
291C32×C4—32no—
292C32×C4—48no—
293C32×C4—61no—
294C32×C4C2×C4×D318yes—
295C32×C4C2×C4×D312yes—
296C3⋊Dic3—21no—
297C3⋊Dic3—32no—
298C3⋊Dic3—38no—
299C3⋊Dic3—68no—
300C3⋊Dic3—62no—
301C3⋊Dic3—68no—
302C3⋊Dic3—128no—
303C3⋊Dic3C2×C4×D314yes—
304C3⋊Dic3D3×Q812yescomplemented
305C3×Dic3—24no—
306C3×Dic3—316no—
307C3×Dic3—38no—
308C3×Dic3—64no—
309C3×Dic3C2×C4×D3116yes—
310C3×Dic3D3×Q818yescomplemented
311C22×Q8—271no—
312C4×Q8—31no—
313C4×Q8—91no—
314C4×Q8—272no—
315C4×Q8—271no—
316C4×Q8—272no—
317C4×Q8—271no—
318C2×C4⋊C4—272no—
319C2×C4⋊C4—271no—
320C2×C42—272no—
321C2×C42—271no—
322C33C2×C4×Q811yescharacteristic, complemented, Sylow
323C22×D3—94no—
324C22×D3—91no—
325C22×D3—184no—
326C3×Q8—316no—
327C3×Q8—94no—
328C2×C3×C4—24no—
329C2×C3×C4—28no—
330C2×C3×C4—34no—
331C2×C3×C4—38no—
332C2×C3×C4—34no—
333C2×C3×C4—91no—
334C2×C3×C4—92no—
335C2×C3×C4—91no—
336C2×C3×C4C2×D3218yes—
337C2×C3×C4C2×D3214yes—
338C2×C3×C4C22×C3⋊D312yes—
339C2×C3×C4C22×C3⋊D311yescharacteristic
340C2×Dic3—34no—
341C2×Dic3—38no—
342C2×Dic3—34no—
343C2×Dic3—94no—
344C2×Dic3—91no—
345C2×Dic3—98no—
346C2×Dic3—92no—
347C2×Dic3—184no—
348C2×Dic3—188no—
349C2×Dic3C22×C3⋊D311yescharacteristic
350C2×Dic3C22×C3⋊D312yes—
351C2×Dic3C22×C3⋊D311yescharacteristic
352C4×D3—916no—
353C4×D3—932no—
354C4×D3—916no—
355C4×D3—94no—
356C4×D3—98no—
357C4×D3—94no—
358C4×D3—1816no—
359C4×D3—182no—
360C4×D3—188no—
361C4×D3—184no—
362C4×D3—1816no—
363C4×D3—3616no—
364Dic6—316no—
365Dic6—932no—
366Dic6—98no—
367Dic6C4×C3⋊D314yescomplemented
368C2×C32—24no—
369C2×C32—24no—
370C2×C32—24no—
371C2×C32C22×C4×D314yes—
372C2×C32C22×C4×D311yescharacteristic
373C2×C32C2×D3×Q814yes—
374C2×C32C2×D3×Q811yescharacteristic
375C2×C32D3×#4114yes—
376C2×C32D3×#4111yescharacteristic
377C3⋊D3—34no—
378C3⋊D3—316no—
379C3⋊D3—616no—
380C2×Q8—31no—
381C2×Q8—91no—
382C2×Q8—272no—
383C2×Q8—278no—
384C22×C4—271no—
385C22×C4—272no—
386C22×C4—271no—
387C22×C4—271no—
388C22×C4—272no—
389C4⋊C4—32no—
390C4⋊C4—92no—
391C4⋊C4—272no—
392C4⋊C4—272no—
393C4⋊C4—272no—
394C4⋊C4—271no—
395C4⋊C4D3×C3⋊D311yescharacteristic
396C42—32no—
397C42—31no—
398C42—92no—
399C42—91no—
400C42—272no—
401C42—271no—
402C42—272no—
403C42—271no—
404C22×C3—24no—
405C22×C3C22×D3214yes—
406C22×C3C23×C3⋊D311yescharacteristic
407C2×D3—98no—
408C2×D3—98no—
409C2×D3—98no—
410C2×D3—92no—
411C2×D3—92no—
412C2×D3—92no—
413C2×D3—188no—
414C2×D3—188no—
415C2×D3—188no—
416C3×C4—28no—
417C3×C4—22no—
418C3×C4—28no—
419C3×C4—316no—
420C3×C4—38no—
421C3×C4—48no—
422C3×C4—64no—
423C3×C4—94no—
424C3×C4—92no—
425C3×C4—181no—
426C3×C4C4×D3218yes—
427C3×C4C2×C4×C3⋊D312yes—
428Dic3—21no—
429Dic3—316no—
430Dic3—38no—
431Dic3—64no—
432Dic3—98no—
433Dic3—92no—
434Dic3—188no—
435Dic3—188no—
436Dic3—182no—
437Dic3—368no—
438Dic3C2×C4×C3⋊D314yes—
439Dic3Q8×C3⋊D312yescomplemented
440C32—24no—
441C32C4×D3×Q814yescomplemented
442C32C4×D3×Q811yescharacteristic, complemented
443C23—271no—
444Q8—34no—
445Q8—94no—
446Q8—278no—
447C2×C4—31no—
448C2×C4—32no—
449C2×C4—31no—
450C2×C4—91no—
451C2×C4—92no—
452C2×C4—91no—
453C2×C4—278no—
454C2×C4—274no—
455C2×C4—271no—
456C2×C4—274no—
457C2×C4—272no—
458C2×C4—544no—
459C2×C4—542no—
460C2×C4C2×D3×C3⋊D312yes—
461C2×C4C2×D3×C3⋊D311yescharacteristic
462C2×C3—24no—
463C2×C3—24no—
464C2×C3—24no—
465C2×C3C2×C4×D3214yes—
466C2×C3#6431214yes—
467C2×C3D32×Q814yes—
468C2×C3C22×C4×C3⋊D311yescharacteristic
469C2×C3C2×Q8×C3⋊D311yescharacteristic
470C2×C3#41×C3⋊D311yescharacteristic
471D3—916no—
472D3—94no—
473D3—1816no—
474C22—272no—
475C22—272no—
476C22—272no—
477C22C22×D3×C3⋊D311yescharacteristic
478C4—22no—
479C4—34no—
480C4—32no—
481C4—61no—
482C4—94no—
483C4—92no—
484C4—181no—
485C4—272no—
486C4—542no—
487C4C4×D3×C3⋊D312yes—
488C3—24no—
489C3#1059436314yescomplemented
490C3C4×Q8×C3⋊D311yescharacteristic, complemented
491C2—274no—
492C2C2×C4×D3×C3⋊D311yescharacteristic
493C2#1172458611yescharacteristic
494C2D3×Q8×C3⋊D311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutatorC2×C33C23×C4
FittingC33×C4⋊C4C22
socle, cosocleC22×C33C24
prime radicalC22×C3⋊(C3⋊D3)C23
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33C2×C4×Q8
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
3. C3×#64096C2semidirect
4. C32×#887C2semidirect
8. Dic3×#634C2semidirect
9. Dic6×C3⋊Dic3C2semidirect
10. #11723390C2semidirect
11. #11723389C2semidirect
22. #89019C4semidirect
47. #64096D3semidirect
53. C3×#887D3semidirect
71. C3×#634C2×C4semidirect
75. C32×Dic6C2×C4semidirect
76. #6293C2×C4semidirect
79. #6289Q8semidirect
145. C3×C3⋊Dic3C2×Q8semidirect
148. C32×Dic3C2×Q8semidirect
163. #887C3⋊D3semidirect
204. #634C4×D3semidirect
212. C3×Dic6C4×D3semidirect
221. C3⋊(C3⋊D3)C4×Q8semidirect
304. C3⋊Dic3D3×Q8semidirect
310. C3×Dic3D3×Q8semidirect
322. C33C2×C4×Q8semidirect
367. Dic6C4×C3⋊D3semidirect
439. Dic3Q8×C3⋊D3semidirect
441. C32C4×D3×Q8semidirect
442. C32C4×D3×Q8semidirect
489. C3#10594363semidirect
490. C3C4×Q8×C3⋊D3semidirect

Maximal subgroups

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

subgroupindex
C2×#890192
C4⋊C4×C3⋊(C3⋊D3)2
C3×#640962
C32×#8872
#117234702
C4×#62892
Dic3×#6342
Dic6×C3⋊Dic32
#117233902
#117233892
C4×Q8×C3⋊D33
#105943633