Finite Group Databasebeta

← #13177701

Subgroups — #13177701

Order 1728.

1672 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723979C211yesmaximal, characteristic
2C3×#64145C211yesmaximal, characteristic, complemented
3C32×D4×Dic3C211yesmaximal, characteristic, complemented
4#11723640C212yesmaximal, complemented
5#11723612C212yesmaximal, complemented
6#11723597C212yesmaximal, complemented
7#11723592C212yesmaximal, complemented
8#11723535C211yesmaximal, characteristic, complemented
9#11723452C211yesmaximal, characteristic, complemented
10#11723449C211yesmaximal, characteristic, complemented
11Dic3×#634C211yesmaximal, characteristic, complemented
12#10596032—31nomaximal
13#10595121—34nomaximal
14C2×C33×D4C2211yescharacteristic
15C2×#6312C2212yes—
16C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
17C2×C3×#634C2211yescharacteristic
18C22×C32×Dic3C2212yes—
19C2×#6293C2212yes—
20#88864C2212yes—
21C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
22C3×#3757C2212yes—
23C3×#3757C2212yes—
24C3×C4×C3⋊Dic3C2211yescharacteristic
25C32×#217C2212yes—
26C32×C4⋊Dic3C2211yescharacteristic
27C32×C4×Dic3C2211yescharacteristic
28#88807C2212yes—
29#88807C2212yes—
30#88806C2211yescharacteristic
31#88806C2211yescharacteristic
32Dic3⋊(C3⋊Dic3)C2212yes—
33#88804C2211yescharacteristic
34#88804C2212yes—
35#88804C2211yescharacteristic
36Dic3×C3⋊Dic3C2212yes—
37C32×C4×D4—31no—
38#64147—31no—
39#64147—34no—
40#64147—64no—
41#64145D311yescharacteristic, complemented
42#64138—32no—
43#64133—32no—
44#64106—31no—
45#64097—31no—
46#64094—32no—
47#64088—32no—
48#64081—31no—
49C4×#634—31no—
50C3×#936—34no—
51C3×D4×Dic3D314yescomplemented
52#63966—38no—
53#63964—38no—
54#63956—38no—
55#63953—38no—
56#63918—34no—
57#63874—34no—
58#63871—34no—
59Dic3×Dic6—34no—
60C23×C33C2312yes—
61C22×C3⋊(C3⋊D3)C2311yescharacteristic
62C33×D4—22no—
63C2×C33×C4C2311yescharacteristic
64#6312—42no—
65C2×C3⋊(C3⋊Dic3)C2312yes—
66C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
67C4×C3⋊(C3⋊D3)—22no—
68C2×C3×C3⋊Dic3C2311yescharacteristic
69C2×C3×C3⋊Dic3C2311yescharacteristic
70C2×C3×C3⋊Dic3C2312yes—
71C3×#634—22no—
72C2×C32×Dic3—42no—
73C2×C32×Dic3C2311yescharacteristic
74C2×C32×Dic3C2312yes—
75C2×C32×Dic3C2311yescharacteristic
76#6293—42no—
77#5682—91no—
78#5561—94no—
79C2×C32×D4—24no—
80C2×C32×D4C2×D314yes—
81C2×C32×D4C2×D311yescharacteristic
82C22×C32×C4—32no—
83C2×#638—32no—
84C2×#638—38no—
85C2×#638—68no—
86C2×C4×C3⋊D3—31no—
87C2×C4×C3⋊D3—34no—
88C2×C4×C3⋊D3—64no—
89C2×#634—32no—
90C2×#634C2×D311yescharacteristic
91C22×C3×Dic3C2×D318yes—
92C2×C3×Dic6—34no—
93C2×#627—38no—
94C32×C4⋊C4—31no—
95C32×C22⋊C4—32no—
96C32×C42—31no—
97#3757—32no—
98#3757—38no—
99#3757—68no—
100#3757C2×D312yes—
101#3757C2×D312yes—
102#3752—31no—
103#3752—31no—
104#3752—32no—
105C4⋊(C3⋊Dic3)—31no—
106C4⋊(C3⋊Dic3)—34no—
107C4⋊(C3⋊Dic3)—32no—
108C4⋊(C3⋊Dic3)—64no—
109#3750—32no—
110#3750—31no—
111#3750—31no—
112#3750—32no—
113C4×C3⋊Dic3—32no—
114C4×C3⋊Dic3C2×D311yescharacteristic
115C3×#217—38no—
116C3×#217—38no—
117C3×#217C2×D318yes—
118C3×C4⋊Dic3C2×D314yes—
119C3×C4×Dic3—34no—
120C3×C4×Dic3C2×D314yes—
121#3724—34no—
122#3724—34no—
123Dic3⋊Dic3—38no—
124Dic3⋊Dic3—38no—
125Dic3⋊Dic3—38no—
126#3722—38no—
127#3722—34no—
128#3722—34no—
129Dic32—38no—
130C22×C33—22no—
131C22×C33—22no—
132C22×C33—22no—
133C22×C33C2411yescharacteristic
134C2×C3⋊(C3⋊D3)—21no—
135C2×C3⋊(C3⋊D3)—21no—
136C2×C3⋊(C3⋊D3)—21no—
137C33×C4#4112yes—
138C3⋊(C3⋊Dic3)—21no—
139C3⋊(C3⋊Dic3)—22no—
140C3×C3⋊Dic3—22no—
141C3×C3⋊Dic3—21no—
142C3×C3⋊Dic3—21no—
143C32×Dic3—22no—
144C32×Dic3—21no—
145C32×Dic3—21no—
146C3×#124—91no—
147C3×C4×D4—34no—
148#946—91no—
149Q8×Dic3—91no—
150#938—94no—
151#938—91no—
152#938—184no—
153#936—34no—
154D4×Dic3C3⋊D311yescharacteristic, complemented
155#929—98no—
156#924—98no—
157#897—94no—
158#888—94no—
159#886—92no—
160#885—98no—
161#885—92no—
162#879—98no—
163#879—92no—
164#878—92no—
165#872—94no—
166#872—91no—
167#869—91no—
168C4×Dic6—94no—
169C23×C32—28no—
170C23×C32C22×D318yes—
171C23×C32C22×D312yes—
172C22×C3⋊D3—31no—
173C22×C3⋊D3—34no—
174C22×C3⋊D3—64no—
175C32×D4—22no—
176C32×D4—28no—
177C32×D4—48no—
178C2×C32×C4—24no—
179C2×C32×C4—31no—
180C2×C32×C4—31no—
181C2×C32×C4—32no—
182C2×C32×C4—122no—
183C2×C32×C4C22×D314yes—
184C2×C32×C4C22×D311yescharacteristic
185#638—122no—
186#638—128no—
187#638—1216no—
188C2×C3⋊Dic3—32no—
189C2×C3⋊Dic3—31no—
190C2×C3⋊Dic3—38no—
191C2×C3⋊Dic3—34no—
192C2×C3⋊Dic3—68no—
193C2×C3⋊Dic3—64no—
194C2×C3⋊Dic3C22×D312yes—
195C2×C3⋊Dic3C22×D311yescharacteristic
196C2×C3⋊Dic3C22×D311yescharacteristic
197C4×C3⋊D3—62no—
198C4×C3⋊D3—68no—
199C4×C3⋊D3—128no—
200#634—22no—
201#634—122no—
202C2×C3×Dic3—38no—
203C2×C3×Dic3—34no—
204C2×C3×Dic3—34no—
205C2×C3×Dic3—48no—
206C2×C3×Dic3C22×D314yes—
207C2×C3×Dic3C22×D314yes—
208C2×C3×Dic3C22×D318yes—
209C3×Dic6—68no—
210#627—128no—
211#542—271noSylow
212C2×C33—42no—
213C2×C33C2×#4111yescharacteristic
214C2×C33H(5,𝔽2)11yescharacteristic
215C2×C332−1+411yescharacteristic
216C3⋊(C3⋊D3)—41no—
217C2×C3×Q8—91no—
218C2×C3×D4—24no—
219C2×C3×D4D3214yes—
220C2×C3×D4C2×C3⋊D311yescharacteristic
221C22×C3×C4—38no—
222C2×#67—98no—
223C2×#67—92no—
224C2×#67—188no—
225C22×Dic3C2×C3⋊D312yes—
226C2×C4×D3—94no—
227C2×C4×D3—91no—
228C2×C4×D3—184no—
229C2×Dic6—34no—
230C2×Dic6—98no—
231C2×Dic6—92no—
232C3×C4⋊C4—34no—
233C3×C22⋊C4—38no—
234C3×C22⋊C4—92no—
235C3×C22⋊C4—92no—
236C3×C42—34no—
237C3×C42—91no—
238#217—38no—
239#217—38no—
240#217—98no—
241#217—92no—
242#217—188no—
243#217C2×C3⋊D312yes—
244(C2×D3)⋊C4—94no—
245(C2×D3)⋊C4—94no—
246(C2×D3)⋊C4—98no—
247(C2×D3)⋊C4—91no—
248(C2×D3)⋊C4—91no—
249(C2×D3)⋊C4—92no—
250C4⋊Dic3—98no—
251C4⋊Dic3—94no—
252C4⋊Dic3—91no—
253C4⋊Dic3—92no—
254C4⋊Dic3—92no—
255C4⋊Dic3—184no—
256C4⋊Dic3C2×C3⋊D311yescharacteristic
257Dic3⋊C4—98no—
258Dic3⋊C4—94no—
259Dic3⋊C4—94no—
260Dic3⋊C4—98no—
261Dic3⋊C4—91no—
262Dic3⋊C4—91no—
263Dic3⋊C4—92no—
264C4×Dic3—34no—
265C4×Dic3—98no—
266C4×Dic3—92no—
267C4×Dic3C2×C3⋊D311yescharacteristic
268C22×C32—22no—
269C22×C32—22no—
270C22×C32—22no—
271C22×C32—24no—
272C22×C32—28no—
273C22×C32—28no—
274C22×C32—28no—
275C22×C32—48no—
276C22×C32—48no—
277C22×C32—48no—
278C22×C32C23×D314yes—
279C22×C32C23×D311yescharacteristic
280C2×C3⋊D3—68no—
281C2×C3⋊D3—61no—
282C2×C3⋊D3—64no—
283C2×C3⋊D3—61no—
284C2×C3⋊D3—61no—
285C2×C3⋊D3—64no—
286C2×C3⋊D3—64no—
287C2×C3⋊D3—124no—
288C2×C3⋊D3—124no—
289C32×C4—28no—
290C32×C4—62no—
291C32×C4—61no—
292C32×C4—61no—
293C32×C4#23518yes—
294C32×C4#23712yes—
295C3⋊Dic3—22no—
296C3⋊Dic3—21no—
297C3⋊Dic3—21no—
298C3⋊Dic3—61no—
299C3⋊Dic3—62no—
300C3⋊Dic3—64no—
301C3⋊Dic3—68no—
302C3⋊Dic3—124no—
303C3⋊Dic3—128no—
304C3×Dic3—28no—
305C3×Dic3—24no—
306C3×Dic3—24no—
307C3×Dic3—68no—
308C3×Dic3—64no—
309C3×Dic3—64no—
310C4⋊Q8—271no—
311#126—272no—
312#124—91no—
313#124—272no—
314#123—272no—
315C22⋊Q8—272no—
316C22⋊Q8—271no—
317#121—271no—
318C4×Q8—271no—
319C4×D4—31no—
320#117—271no—
321C33#54211yescharacteristic, complemented, Sylow
322C23×C3—28no—
323C23×C3C2×D3218yes—
324C23×C3C22×C3⋊D312yes—
325C22×D3—94no—
326C22×D3—91no—
327C22×D3—184no—
328C3×Q8—182no—
329C3×D4—28no—
330C3×D4—22no—
331C3×D4—48no—
332C2×C3×C4—24no—
333C2×C3×C4—34no—
334C2×C3×C4—34no—
335C2×C3×C4—38no—
336C2×C3×C4—92no—
337C2×C3×C4—91no—
338C2×C3×C4—91no—
339C2×C3×C4—128no—
340C2×C3×C4C2×D3214yes—
341C2×C3×C4C22×C3⋊D311yescharacteristic
342#67—368no—
343#67—362no—
344#67—3616no—
345C2×Dic3—34no—
346C2×Dic3—34no—
347C2×Dic3—38no—
348C2×Dic3—42no—
349C2×Dic3—98no—
350C2×Dic3—94no—
351C2×Dic3—92no—
352C2×Dic3—91no—
353C2×Dic3—188no—
354C2×Dic3—184no—
355C2×Dic3C22×C3⋊D311yescharacteristic
356C2×Dic3C22×C3⋊D311yescharacteristic
357C2×Dic3C22×C3⋊D312yes—
358C4×D3—188no—
359C4×D3—182no—
360C4×D3—368no—
361Dic6—68no—
362Dic6—362no—
363Dic6—368no—
364C2×C32—24no—
365C2×C32—24no—
366C2×C32—24no—
367C2×C32—42no—
368C2×C32—48no—
369C2×C32—88no—
370C2×C32C2×#23514yes—
371C2×C32C2×#23711yescharacteristic
372C2×C32#100414yes—
373C2×C32#100911yescharacteristic
374C2×C32#101014yes—
375C2×C32#101011yescharacteristic
376C3⋊D3—128no—
377C3⋊D3—121no—
378C3⋊D3—124no—
379C2×Q8—91no—
380C2×Q8—272no—
381C2×D4—272no—
382C2×D4D3×C3⋊D311yescharacteristic
383C22×C4—32no—
384C22×C4—271no—
385C4⋊C4—31no—
386C4⋊C4—271no—
387C4⋊C4—271no—
388C4⋊C4—272no—
389C4⋊C4—272no—
390C4⋊C4—272no—
391C4⋊C4—271no—
392C22⋊C4—32no—
393C22⋊C4—92no—
394C22⋊C4—92no—
395C22⋊C4—271no—
396C22⋊C4—271no—
397C22⋊C4—272no—
398C22⋊C4—272no—
399C42—31no—
400C42—91no—
401C42—272no—
402C22×C3—28no—
403C22×C3—28no—
404C22×C3—28no—
405C22×C3—22no—
406C22×C3—22no—
407C22×C3—22no—
408C22×C3—24no—
409C22×C3—48no—
410C22×C3—48no—
411C22×C3—48no—
412C22×C3C22×D3214yes—
413C22×C3C23×C3⋊D311yescharacteristic
414C2×D3—188no—
415C2×D3—184no—
416C2×D3—184no—
417C2×D3—184no—
418C2×D3—181no—
419C2×D3—181no—
420C2×D3—181no—
421C2×D3—364no—
422C2×D3—364no—
423C3×C4—28no—
424C3×C4—64no—
425C3×C4—68no—
426C3×C4—64no—
427C3×C4—182no—
428C3×C4—181no—
429C3×C4—181no—
430C3×C4#380418yes—
431C3×C4#382812yes—
432Dic3—22no—
433Dic3—21no—
434Dic3—21no—
435Dic3—64no—
436Dic3—64no—
437Dic3—68no—
438Dic3—188no—
439Dic3—184no—
440Dic3—181no—
441Dic3—182no—
442Dic3—364no—
443Dic3—368no—
444C32—24no—
445C32#556114yescomplemented
446C32#568211yescharacteristic, complemented
447C23—271no—
448C23C2×D3×C3⋊D312yes—
449Q8—182no—
450Q8—1082no—
451D4—22no—
452D4—1082no—
453C2×C4—31no—
454C2×C4—31no—
455C2×C4—32no—
456C2×C4—91no—
457C2×C4—91no—
458C2×C4—92no—
459C2×C4—122no—
460C2×C4—272no—
461C2×C4—271no—
462C2×C4—542no—
463C2×C4C2×D3×C3⋊D311yescharacteristic
464C2×C3—24no—
465C2×C3—24no—
466C2×C3—24no—
467C2×C3—48no—
468C2×C3—42no—
469C2×C3—88no—
470C2×C3#6430914yes—
471C2×C3#6431414yes—
472C2×C3C2×#380414yes—
473C2×C3C2×#382811yescharacteristic
474C2×C3#6436111yescharacteristic
475C2×C3#6436711yescharacteristic
476D3—368no—
477D3—364no—
478D3—361no—
479C22—22no—
480C22—22no—
481C22—22no—
482C22—541no—
483C22—541no—
484C22—541no—
485C22C22×D3×C3⋊D311yescharacteristic
486C4—61no—
487C4—61no—
488C4—62no—
489C4—181no—
490C4—181no—
491C4—182no—
492C4—542no—
493C4—541no—
494C4#8903112yes—
495C3—24no—
496C3#1059512114yescomplemented
497C3#1059603211yescharacteristic, complemented
498C2—42no—
499C2—1081no—
500C2#1172458211yescharacteristic
501C2#1172458811yescharacteristic
502C2C2×#8903111yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC2×C33×D4C22
prime radical#11723979C2
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#542
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C3×#64145C2semidirect
3. C32×D4×Dic3C2semidirect
4. #11723640C2semidirect
5. #11723612C2semidirect
6. #11723597C2semidirect
7. #11723592C2semidirect
8. #11723535C2semidirect
9. #11723452C2semidirect
10. #11723449C2semidirect
11. Dic3×#634C2semidirect
41. #64145D3semidirect
51. C3×D4×Dic3D3semidirect
154. D4×Dic3C3⋊D3semidirect
321. C33#542semidirect
445. C32#5561semidirect
446. C32#5682semidirect
496. C3#10595121semidirect
497. C3#10596032semidirect

Maximal subgroups

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

subgroupindex
#117239792
C3×#641452
C32×D4×Dic32
#117236402
#117236122
#117235972
#117235922
#117235352
#117234522
#117234492
Dic3×#6342
#105960323
#105951213