Finite Group Databasebeta

← #13177693

Subgroups — #13177693

Order 1728.

1848 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723981C211yesmaximal, characteristic
2C3×#64144C211yesmaximal, characteristic, complemented
3C32×#935C211yesmaximal, characteristic, complemented
4#11723622C212yesmaximal, complemented
5#11723618C212yesmaximal, complemented
6#11723592C212yesmaximal, complemented
7#11723592C212yesmaximal, complemented
8#11723498C212yesmaximal, complemented
9#11723391C212yesmaximal, complemented
10#10596135—31nomaximal
11#10595212—34nomaximal
12C2×C33×D4C2211yescharacteristic
13C2×#6312C2212yes—
14C2×#6312C2212yes—
15C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
16C22×C3×C3⋊Dic3C2211yescharacteristic
17C22×C32×Dic3C2211yescharacteristic
18C2×#6293C2212yes—
19C2×#6289C2212yes—
20C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
21C3×#3757C2211yescharacteristic
22C3×#3757C2212yes—
23C3×#3750C2212yes—
24C32×#217C2211yescharacteristic
25C32×#217C2212yes—
26C32×Dic3⋊C4C2212yes—
27#88806C2212yes—
28#88804C2212yes—
29#88804C2212yes—
30#88804C2212yes—
31Dic3×C3⋊Dic3C2212yes—
32Dic3×C3⋊Dic3C2212yes—
33C32×#123—31no—
34#64149—31no—
35#64149—34no—
36#64149—64no—
37#64144D311yescharacteristic, complemented
38C4×#638—32no—
39#64103—32no—
40#64096—32no—
41#64094—32no—
42#64094—32no—
43#64090—32no—
44C3×#935—34no—
45C3×#935D314yescomplemented
46#63953—38no—
47#63953—38no—
48#63952—38no—
49#63952—38no—
50#63897—38no—
51#63838—38no—
52C23×C33C2311yescharacteristic
53C23×C33C2311yescharacteristic
54C22×C3⋊(C3⋊D3)C2312yes—
55C33×D4—41no—
56C2×C33×C4C2311yescharacteristic
57#6312—24no—
58#6312—24no—
59C2×C3⋊(C3⋊Dic3)C2312yes—
60C4⋊(C3⋊(C3⋊D3))—41no—
61C2×C3×C3⋊Dic3—41no—
62C2×C3×C3⋊Dic3C2312yes—
63C2×C3×C3⋊Dic3C2312yes—
64C2×C32×Dic3—41no—
65C2×C32×Dic3C2312yes—
66C2×C32×Dic3C2312yes—
67#6293—24no—
68#6289—42no—
69#5664—94no—
70#5664—91no—
71C2×C32×D4—24no—
72C2×C32×D4C2×D314yes—
73C2×C32×D4C2×D311yescharacteristic
74C22×C32×C4—31no—
75C2×#638—32no—
76C2×#638—32no—
77C2×#638—38no—
78C2×#638—38no—
79C2×#638—68no—
80C2×#638—68no—
81C22×C3⋊Dic3C2×D311yescharacteristic
82C2×C4⋊(C3⋊D3)—31no—
83C2×C4⋊(C3⋊D3)—34no—
84C2×C4⋊(C3⋊D3)—64no—
85C2×C4×C3⋊D3—32no—
86C2×#634—32no—
87C22×C3×Dic3—34no—
88C22×C3×Dic3C2×D314yes—
89C2×#627—38no—
90C2×#624—38no—
91C32×C4⋊C4—32no—
92C32×C22⋊C4—32no—
93C32×C22⋊C4—31no—
94#3757C2×D311yescharacteristic
95#3757C2×D312yes—
96#3752—32no—
97#3752—32no—
98#3752—32no—
99#3750—32no—
100#3750C2×D312yes—
101C4×C3⋊Dic3—31no—
102C4×C3⋊Dic3—34no—
103C4×C3⋊Dic3—32no—
104C4×C3⋊Dic3—32no—
105C4×C3⋊Dic3—64no—
106C3×#217—38no—
107C3×#217—34no—
108C3×#217C2×D318yes—
109C3×#217C2×D314yes—
110C3×Dic3⋊C4—38no—
111C3×Dic3⋊C4C2×D318yes—
112#3724—38no—
113#3722—38no—
114#3722—38no—
115#3722—38no—
116Dic32—38no—
117Dic32—38no—
118C22×C33—22no—
119C22×C33—22no—
120C22×C33—21no—
121C22×C33—21no—
122C22×C33C2411yescharacteristic
123C2×C3⋊(C3⋊D3)—22no—
124C2×C3⋊(C3⋊D3)—22no—
125C2×C3⋊(C3⋊D3)—22no—
126C33×C4—21no—
127C3⋊(C3⋊Dic3)#4114yes—
128C3×C3⋊Dic3—22no—
129C3×C3⋊Dic3—22no—
130C32×Dic3—22no—
131C32×Dic3—22no—
132C3×#123—34no—
133C3×#123—91no—
134#940—94no—
135#940—91no—
136#940—184no—
137#935—34no—
138#935C3⋊D311yescharacteristic, complemented
139C4×#67—98no—
140C4×#67—92no—
141#894—98no—
142#894—92no—
143#887—98no—
144#887—92no—
145#885—98no—
146#885—98no—
147#885—92no—
148#885—92no—
149#67⋊C4—98no—
150#67⋊C4—92no—
151C23×C32—24no—
152C23×C32—24no—
153C23×C32C22×D314yes—
154C23×C32C22×D314yes—
155C23×C32C22×D311yescharacteristic
156C23×C32C22×D311yescharacteristic
157C22×C3⋊D3—32no—
158C22×C3⋊D3—38no—
159C22×C3⋊D3—68no—
160C32×D4—41no—
161C32×D4—44no—
162C32×D4—48no—
163C2×C32×C4—24no—
164C2×C32×C4—32no—
165C2×C32×C4—32no—
166C2×C32×C4—121no—
167C2×C32×C4C22×D314yes—
168C2×C32×C4C22×D311yescharacteristic
169#638—64no—
170#638—64no—
171#638—616no—
172#638—616no—
173#638—1216no—
174#638—1216no—
175C2×C3⋊Dic3—32no—
176C2×C3⋊Dic3—38no—
177C2×C3⋊Dic3—41no—
178C2×C3⋊Dic3—68no—
179C2×C3⋊Dic3C22×D312yes—
180C2×C3⋊Dic3C22×D312yes—
181C4⋊(C3⋊D3)—121no—
182C4⋊(C3⋊D3)—124no—
183C4⋊(C3⋊D3)—128no—
184C4×C3⋊D3—122no—
185#634—64no—
186C2×C3×Dic3—38no—
187C2×C3×Dic3—38no—
188C2×C3×Dic3—44no—
189C2×C3×Dic3—124no—
190C2×C3×Dic3C22×D318yes—
191C2×C3×Dic3C22×D318yes—
192#627—616no—
193#624—128no—
194#559—271noSylow
195C2×C33—41no—
196C2×C33—41no—
197C2×C33C2×#4112yes—
198C2×C33H(5,𝔽2)11yescharacteristic
199C3⋊(C3⋊D3)—42no—
200C2×C3×D4—24no—
201C2×C3×D4D3214yes—
202C2×C3×D4C2×C3⋊D311yescharacteristic
203C22×C3×C4—34no—
204C22×C3×C4—91no—
205C2×#67—98no—
206C2×#67—98no—
207C2×#67—92no—
208C2×#67—92no—
209C2×#67—188no—
210C2×#67—188no—
211C22×Dic3—34no—
212C22×Dic3C2×C3⋊D311yescharacteristic
213C2×D12—94no—
214C2×D12—91no—
215C2×D12—184no—
216C2×C4×D3—98no—
217C2×C4×D3—92no—
218C2×Dic6—98no—
219C2×Dic6—92no—
220C3×C4⋊C4—38no—
221C3×C4⋊C4—92no—
222C3×C22⋊C4—38no—
223C3×C22⋊C4—34no—
224C3×C22⋊C4—92no—
225C3×C22⋊C4—91no—
226#217—34no—
227#217—38no—
228#217C2×C3⋊D311yescharacteristic
229#217C2×C3⋊D312yes—
230(C2×D3)⋊C4—98no—
231(C2×D3)⋊C4—98no—
232(C2×D3)⋊C4—98no—
233(C2×D3)⋊C4—92no—
234(C2×D3)⋊C4—92no—
235(C2×D3)⋊C4—92no—
236Dic3⋊C4—38no—
237Dic3⋊C4—98no—
238Dic3⋊C4—92no—
239Dic3⋊C4C2×C3⋊D312yes—
240C4×Dic3—98no—
241C4×Dic3—98no—
242C4×Dic3—94no—
243C4×Dic3—91no—
244C4×Dic3—92no—
245C4×Dic3—92no—
246C4×Dic3—184no—
247C22×C32—21no—
248C22×C32—21no—
249C22×C32—22no—
250C22×C32—22no—
251C22×C32—24no—
252C22×C32—24no—
253C22×C32—24no—
254C22×C32—28no—
255C22×C32—28no—
256C22×C32—48no—
257C22×C32—48no—
258C22×C32—44no—
259C22×C32—44no—
260C22×C32C23×D314yes—
261C22×C32C23×D311yescharacteristic
262C2×C3⋊D3—62no—
263C2×C3⋊D3—62no—
264C2×C3⋊D3—62no—
265C2×C3⋊D3—68no—
266C2×C3⋊D3—68no—
267C2×C3⋊D3—68no—
268C2×C3⋊D3—128no—
269C2×C3⋊D3—128no—
270C2×C3⋊D3—128no—
271C32×C4—28no—
272C32×C4—24no—
273C32×C4—21no—
274C32×C4—62no—
275C32×C4—62no—
276C3⋊Dic3—22no—
277C3⋊Dic3—22no—
278C3⋊Dic3—34no—
279C3⋊Dic3—316no—
280C3⋊Dic3—616no—
281C3×Dic3—28no—
282C3×Dic3—28no—
283C3×Dic3—68no—
284C3×Dic3—68no—
285#127—271no—
286#124—272no—
287#124—272no—
288#123—31no—
289#123—91no—
290#123—272no—
291C4×Q8—272no—
292C4×D4—272no—
293C4×D4—272no—
294C33#55911yescharacteristic, complemented, Sylow
295C23×C3—24no—
296C23×C3—24no—
297C23×C3C2×D3214yes—
298C23×C3C2×D3214yes—
299C23×C3C22×C3⋊D311yescharacteristic
300C23×C3C22×C3⋊D311yescharacteristic
301C22×D3—98no—
302C22×D3—92no—
303C22×D3—188no—
304C3×D4—44no—
305C3×D4—41no—
306C3×D4—48no—
307C2×C3×C4—24no—
308C2×C3×C4—38no—
309C2×C3×C4—38no—
310C2×C3×C4—92no—
311C2×C3×C4—92no—
312C2×C3×C4—124no—
313C2×C3×C4—361no—
314C2×C3×C4C2×D3214yes—
315C2×C3×C4C22×C3⋊D311yescharacteristic
316#67—184no—
317#67—184no—
318#67—1816no—
319#67—1816no—
320#67—3616no—
321#67—3616no—
322C2×Dic3—38no—
323C2×Dic3—38no—
324C2×Dic3—41no—
325C2×Dic3—98no—
326C2×Dic3—92no—
327C2×Dic3—124no—
328C2×Dic3—188no—
329C2×Dic3C22×C3⋊D312yes—
330C2×Dic3C22×C3⋊D312yes—
331D12—364no—
332D12—361no—
333D12—368no—
334C4×D3—362no—
335C4×D3—368no—
336Dic6—184no—
337Dic6—1816no—
338C2×C32—28no—
339C2×C32—24no—
340C2×C32—41no—
341C2×C32—41no—
342C2×C32—44no—
343C2×C32—44no—
344C2×C32—84no—
345C2×C32—84no—
346C2×C32#100414yes—
347C2×C32#100411yescharacteristic
348C2×C32D3×#4118yes—
349C2×C32D3×#4112yes—
350C3⋊D3—122no—
351C3⋊D3—128no—
352C3⋊D3—248no—
353C2×Q8—272no—
354C2×D4—272no—
355C2×D4—272no—
356C2×D4—271no—
357C2×D4D3×C3⋊D311yescharacteristic
358C22×C4—31no—
359C22×C4—91no—
360C22×C4—272no—
361C4⋊C4—32no—
362C4⋊C4—92no—
363C4⋊C4—272no—
364C22⋊C4—32no—
365C22⋊C4—31no—
366C22⋊C4—92no—
367C22⋊C4—91no—
368C22⋊C4—272no—
369C22⋊C4—272no—
370C22⋊C4—272no—
371C42—272no—
372C42—272no—
373C42—271no—
374C22×C3—28no—
375C22×C3—28no—
376C22×C3—24no—
377C22×C3—24no—
378C22×C3—21no—
379C22×C3—21no—
380C22×C3—22no—
381C22×C3—22no—
382C22×C3—24no—
383C22×C3—48no—
384C22×C3—48no—
385C22×C3—44no—
386C22×C3—44no—
387C22×C3C22×D3214yes—
388C22×C3C23×C3⋊D311yescharacteristic
389C2×D3—188no—
390C2×D3—188no—
391C2×D3—188no—
392C2×D3—182no—
393C2×D3—182no—
394C2×D3—182no—
395C2×D3—368no—
396C2×D3—368no—
397C2×D3—368no—
398C3×C4—28no—
399C3×C4—24no—
400C3×C4—21no—
401C3×C4—68no—
402C3×C4—68no—
403C3×C4—182no—
404C3×C4—182no—
405Dic3—22no—
406Dic3—22no—
407Dic3—68no—
408Dic3—68no—
409Dic3—916no—
410Dic3—94no—
411Dic3—1816no—
412C32—24no—
413C32#566414yescomplemented
414C32#566411yescharacteristic, complemented
415C23—272no—
416C23C2×D3×C3⋊D311yescharacteristic
417C23C2×D3×C3⋊D311yescharacteristic
418Q8—544no—
419D4—41no—
420D4—544no—
421D4—544no—
422D4—1081no—
423C2×C4—32no—
424C2×C4—32no—
425C2×C4—92no—
426C2×C4—92no—
427C2×C4—121no—
428C2×C4—272no—
429C2×C4—361no—
430C2×C4—1082no—
431C2×C4C2×D3×C3⋊D311yescharacteristic
432C2×C3—28no—
433C2×C3—24no—
434C2×C3—41no—
435C2×C3—41no—
436C2×C3—44no—
437C2×C3—44no—
438C2×C3—84no—
439C2×C3—84no—
440C2×C3#6431218yes—
441C2×C3#6433014yes—
442C2×C3#6436111yescharacteristic
443C2×C3#41×C3⋊D312yes—
444D3—362no—
445D3—368no—
446D3—728no—
447C22—22no—
448C22—22no—
449C22—21no—
450C22—21no—
451C22—542no—
452C22—542no—
453C22—542no—
454C22C22×D3×C3⋊D311yescharacteristic
455C4—21no—
456C4—62no—
457C4—62no—
458C4—182no—
459C4—182no—
460C4—274no—
461C3—24no—
462C3#1059521214yescomplemented
463C3#1059613511yescharacteristic, complemented
464C2—41no—
465C2—41no—
466C2—1082no—
467C2#1172458612yes—
468C2#1172461111yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC2×C33×D4C22
prime radical#11723981C2
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#559
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C3×#64144C2semidirect
3. C32×#935C2semidirect
4. #11723622C2semidirect
5. #11723618C2semidirect
6. #11723592C2semidirect
7. #11723592C2semidirect
8. #11723498C2semidirect
9. #11723391C2semidirect
37. #64144D3semidirect
45. C3×#935D3semidirect
138. #935C3⋊D3semidirect
294. C33#559semidirect
413. C32#5664semidirect
414. C32#5664semidirect
462. C3#10595212semidirect
463. C3#10596135semidirect

Maximal subgroups

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

subgroupindex
#117239812
C3×#641442
C32×#9352
#117236222
#117236182
#117235922
#117234982
#117233912
#105961353
#105952123