Finite Group Databasebeta

← #13177702

Subgroups — #13177702

Order 1728.

1672 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
1D4×C3⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
2C3×#64145C211yesmaximal, characteristic, complemented
3C32×#938C211yesmaximal, characteristic
4#11723642C212yesmaximal, complemented
5#11723603C212yesmaximal, complemented
6#11723600C212yesmaximal, complemented
7#11723596C212yesmaximal, complemented
8#11723535C211yesmaximal, characteristic, complemented
9#11723425C211yesmaximal, characteristic, complemented
10#11723423C211yesmaximal, characteristic, complemented
11#11723410C211yesmaximal, characteristic, complemented
12#10596074—31nomaximal
13#10595126—34nomaximal
14C2×C33×D4C2211yescharacteristic
15C22×C3⋊(C3⋊Dic3)C2212yes—
16C2×C3×#634C2211yescharacteristic
17C2×C32×#67C2212yes—
18C2×C32×C4×D3C2211yescharacteristic
19C2×#6293C2212yes—
20#88864C2212yes—
21C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
22C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
23C3×#3757C2212yes—
24C3×#3757C2212yes—
25C3×C4×C3⋊Dic3C2211yescharacteristic
26C32×#217C2212yes—
27C32×C4⋊Dic3C2211yescharacteristic
28#88807C2212yes—
29#88807C2212yes—
30#88806C2212yes—
31Dic3⋊(C3⋊Dic3)C2211yescharacteristic
32Dic3⋊(C3⋊Dic3)C2211yescharacteristic
33(C2×D3)⋊(C3⋊Dic3)C2211yescharacteristic
34(C2×D3)⋊(C3⋊Dic3)C2212yes—
35(C2×D3)⋊(C3⋊Dic3)C2211yescharacteristic
36Dic3×C3⋊Dic3C2212yes—
37C32×#121—31no—
38#64145—32no—
39#64145D311yescharacteristic, complemented
40#64144—32no—
41D4×C3⋊Dic3—31no—
42D4×C3⋊Dic3—34no—
43D4×C3⋊Dic3—64no—
44#64135—31no—
45#64133—31no—
46#64097—31no—
47#64096—31no—
48#64088—32no—
49#64087—32no—
50C3×#938D314yes—
51C3×#936—34no—
52#63982—38no—
53#63962—38no—
54#63959—38no—
55#63956—38no—
56#63918—34no—
57#63856—34no—
58#63855—34no—
59#63843—34no—
60C23×C33C2312yes—
61C22×C32×D3C2311yescharacteristic
62C33×D4—22no—
63C2×C33×C4C2311yescharacteristic
64C2×C3⋊(C3⋊Dic3)—42no—
65C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
66C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
67C2×C3⋊(C3⋊Dic3)C2312yes—
68C2×C3×C3⋊Dic3C2311yescharacteristic
69C2×C3×C3⋊Dic3C2311yescharacteristic
70C2×C3×C3⋊Dic3C2312yes—
71C3×#634—22no—
72C32×#67—42no—
73C2×C32×Dic3C2311yescharacteristic
74C2×C32×Dic3C2312yes—
75C32×C4×D3—22no—
76#6293—42no—
77#5682—91no—
78#5603—94no—
79C2×C32×D4—24no—
80C2×C32×D4—32no—
81C2×C32×D4C2×D314yes—
82C2×C32×D4C2×D311yescharacteristic
83C22×C32×C4—31no—
84C22×C3⋊Dic3—32no—
85C22×C3⋊Dic3—38no—
86C22×C3⋊Dic3—68no—
87C2×#634—32no—
88C2×#634C2×D311yescharacteristic
89C2×C3×#67C2×D318yes—
90C2×C3×C4×D3C2×D314yes—
91C2×C3×Dic6—34no—
92C2×#627—38no—
93C32×C4⋊C4—31no—
94C32×C22⋊C4—32no—
95#3757—32no—
96#3757—38no—
97#3757—32no—
98#3757—31no—
99#3757—31no—
100#3757—68no—
101#3757C2×D312yes—
102#3757C2×D312yes—
103C4⋊(C3⋊Dic3)—31no—
104C4⋊(C3⋊Dic3)—34no—
105C4⋊(C3⋊Dic3)—31no—
106C4⋊(C3⋊Dic3)—31no—
107C4⋊(C3⋊Dic3)—64no—
108#3750—32no—
109#3750—32no—
110#3750—32no—
111C4×C3⋊Dic3—31no—
112C4×C3⋊Dic3—34no—
113C4×C3⋊Dic3—32no—
114C4×C3⋊Dic3—64no—
115C4×C3⋊Dic3C2×D311yescharacteristic
116C3×#217—38no—
117C3×#217—38no—
118C3×#217C2×D318yes—
119C3×C4⋊Dic3C2×D314yes—
120C3×C4×Dic3—34no—
121#3724—38no—
122Dic3⋊Dic3—38no—
123Dic3⋊Dic3—34no—
124Dic3⋊Dic3—34no—
125Dic3⋊Dic3—38no—
126(C2×D3)⋊Dic3—34no—
127(C2×D3)⋊Dic3—34no—
128(C2×D3)⋊Dic3—38no—
129Dic32—38no—
130C22×C33—22no—
131C22×C33—22no—
132C22×C33—22no—
133C22×C33C2411yescharacteristic
134C2×C32×D3—21no—
135C2×C32×D3—21no—
136C2×C32×D3—21no—
137C33×C4#4112yes—
138C3⋊(C3⋊Dic3)—21no—
139C3⋊(C3⋊Dic3)—22no—
140C3⋊(C3⋊Dic3)—21no—
141C3×C3⋊Dic3—22no—
142C3×C3⋊Dic3—21no—
143C3×C3⋊Dic3—21no—
144C32×Dic3—21no—
145C32×Dic3—22no—
146C3×#124—91no—
147C3×#121—34no—
148#946—91no—
149Q8×Dic3—91no—
150#938C3⋊D311yescharacteristic
151#936—34no—
152#936—98no—
153#935—98no—
154D4×Dic3—94no—
155D4×Dic3—91no—
156D4×Dic3—184no—
157#926—94no—
158#924—94no—
159#888—94no—
160#887—94no—
161#886—92no—
162#885—92no—
163#879—98no—
164#879—92no—
165#878—98no—
166#878—92no—
167#872—91no—
168#869—91no—
169C23×C32—28no—
170C23×C32—31no—
171C23×C32C22×D318yes—
172C23×C32C22×D312yes—
173C22×C3×D3C22×D314yes—
174C32×D4—216no—
175C32×D4—28no—
176C32×D4—22no—
177C32×D4—122no—
178C2×C32×C4—24no—
179C2×C32×C4—32no—
180C2×C32×C4—31no—
181C2×C32×C4—62no—
182C2×C32×C4C22×D314yes—
183C2×C32×C4C22×D311yescharacteristic
184C2×C3⋊Dic3—31no—
185C2×C3⋊Dic3—31no—
186C2×C3⋊Dic3—32no—
187C2×C3⋊Dic3—34no—
188C2×C3⋊Dic3—34no—
189C2×C3⋊Dic3—38no—
190C2×C3⋊Dic3—64no—
191C2×C3⋊Dic3—64no—
192C2×C3⋊Dic3—68no—
193C2×C3⋊Dic3—122no—
194C2×C3⋊Dic3—128no—
195C2×C3⋊Dic3—1216no—
196C2×C3⋊Dic3C22×D312yes—
197C2×C3⋊Dic3C22×D311yescharacteristic
198C2×C3⋊Dic3C22×D311yescharacteristic
199#634—22no—
200#634—122no—
201C3×#67—48no—
202C2×C3×Dic3—38no—
203C2×C3×Dic3—34no—
204C2×C3×Dic3—34no—
205C2×C3×Dic3C22×D318yes—
206C2×C3×Dic3C22×D314yes—
207C3×C4×D3—28no—
208C3×Dic6—68no—
209#627—128no—
210#542—271noSylow
211C2×C33—42no—
212C2×C33C2×#4111yescharacteristic
213C2×C33H(5,𝔽2)11yescharacteristic
214C2×C332−1+411yescharacteristic
215C32×D3—41no—
216C2×C3×Q8—91no—
217C2×C3×D4—24no—
218C2×C3×D4—38no—
219C2×C3×D4D3214yes—
220C2×C3×D4C2×C3⋊D311yescharacteristic
221C22×C3×C4—34no—
222C2×#67C2×C3⋊D312yes—
223C22×Dic3—98no—
224C22×Dic3—92no—
225C22×Dic3—188no—
226C2×C4×D3C2×C3⋊D311yescharacteristic
227C2×Dic6—34no—
228C2×Dic6—98no—
229C2×Dic6—92no—
230C3×C4⋊C4—34no—
231C3×C22⋊C4—38no—
232C3×C22⋊C4—92no—
233C3×C22⋊C4—92no—
234C3×C42—91no—
235#217—38no—
236#217—38no—
237#217—98no—
238#217—94no—
239#217—94no—
240#217—98no—
241#217—92no—
242#217—188no—
243#217C2×C3⋊D312yes—
244(C2×D3)⋊C4—91no—
245(C2×D3)⋊C4—91no—
246(C2×D3)⋊C4—92no—
247C4⋊Dic3—94no—
248C4⋊Dic3—94no—
249C4⋊Dic3—94no—
250C4⋊Dic3—91no—
251C4⋊Dic3—92no—
252C4⋊Dic3—92no—
253C4⋊Dic3—184no—
254C4⋊Dic3C2×C3⋊D311yescharacteristic
255Dic3⋊C4—98no—
256Dic3⋊C4—98no—
257Dic3⋊C4—98no—
258Dic3⋊C4—91no—
259Dic3⋊C4—91no—
260Dic3⋊C4—92no—
261C4×Dic3—34no—
262C4×Dic3—98no—
263C4×Dic3—94no—
264C4×Dic3—91no—
265C4×Dic3—92no—
266C4×Dic3—184no—
267C22×C32—216no—
268C22×C32—28no—
269C22×C32—22no—
270C22×C32—22no—
271C22×C32—22no—
272C22×C32—24no—
273C22×C32—28no—
274C22×C32—28no—
275C22×C32—48no—
276C22×C32—48no—
277C22×C32—61no—
278C22×C32—61no—
279C22×C32—61no—
280C22×C32C23×D314yes—
281C22×C32C23×D311yescharacteristic
282C2×C3×D3—24no—
283C2×C3×D3—24no—
284C2×C3×D3—24no—
285C32×C4—28no—
286C32×C4—61no—
287C32×C4—62no—
288C32×C4#23718yes—
289C32×C4#23712yes—
290C3⋊Dic3—22no—
291C3⋊Dic3—21no—
292C3⋊Dic3—21no—
293C3⋊Dic3—68no—
294C3⋊Dic3—64no—
295C3⋊Dic3—62no—
296C3⋊Dic3—61no—
297C3⋊Dic3—61no—
298C3⋊Dic3—68no—
299C3⋊Dic3—64no—
300C3⋊Dic3—124no—
301C3⋊Dic3—128no—
302C3×Dic3—24no—
303C3×Dic3—28no—
304C3×Dic3—68no—
305C3×Dic3—64no—
306C3×Dic3—64no—
307C4⋊Q8—271no—
308#126—272no—
309#124—91no—
310#124—272no—
311#123—272no—
312C22⋊Q8—272no—
313C22⋊Q8—271no—
314#121—31no—
315C4×Q8—271no—
316C4×D4—271no—
317#117—271no—
318C33#54211yescharacteristic, complemented, Sylow
319C23×C3—28no—
320C23×C3—34no—
321C23×C3C2×D3218yes—
322C23×C3C22×C3⋊D312yes—
323C22×D3C22×C3⋊D311yescharacteristic
324C3×Q8—182no—
325C3×D4—216no—
326C3×D4—28no—
327C3×D4—22no—
328C3×D4—128no—
329C2×C3×C4—24no—
330C2×C3×C4—38no—
331C2×C3×C4—34no—
332C2×C3×C4—68no—
333C2×C3×C4—92no—
334C2×C3×C4—91no—
335C2×C3×C4—91no—
336C2×C3×C4C2×D3214yes—
337C2×C3×C4C22×C3⋊D311yescharacteristic
338#67—42no—
339C2×Dic3—34no—
340C2×Dic3—34no—
341C2×Dic3—38no—
342C2×Dic3—94no—
343C2×Dic3—94no—
344C2×Dic3—98no—
345C2×Dic3—91no—
346C2×Dic3—91no—
347C2×Dic3—92no—
348C2×Dic3—184no—
349C2×Dic3—184no—
350C2×Dic3—188no—
351C2×Dic3—368no—
352C2×Dic3—362no—
353C2×Dic3—3616no—
354C2×Dic3C22×C3⋊D312yes—
355C2×Dic3C22×C3⋊D311yescharacteristic
356C4×D3—22no—
357Dic6—68no—
358Dic6—362no—
359Dic6—368no—
360C2×C32—24no—
361C2×C32—24no—
362C2×C32—24no—
363C2×C32—416no—
364C2×C32—42no—
365C2×C32—48no—
366C2×C32—121no—
367C2×C32C2×#23714yes—
368C2×C32C2×#23711yescharacteristic
369C2×C32#100414yes—
370C2×C32#100911yescharacteristic
371C2×C32#101014yes—
372C2×C32#101011yescharacteristic
373C3×D3—44no—
374C2×Q8—91no—
375C2×Q8—272no—
376C2×D4—32no—
377C2×D4D3×C3⋊D311yescharacteristic
378C22×C4—31no—
379C22×C4—272no—
380C4⋊C4—31no—
381C4⋊C4—271no—
382C4⋊C4—271no—
383C4⋊C4—272no—
384C4⋊C4—272no—
385C4⋊C4—272no—
386C4⋊C4—271no—
387C22⋊C4—32no—
388C22⋊C4—92no—
389C22⋊C4—92no—
390C22⋊C4—271no—
391C22⋊C4—271no—
392C22⋊C4—272no—
393C22⋊C4—272no—
394C42—91no—
395C42—272no—
396C42—271no—
397C22×C3—216no—
398C22×C3—28no—
399C22×C3—28no—
400C22×C3—28no—
401C22×C3—22no—
402C22×C3—22no—
403C22×C3—22no—
404C22×C3—24no—
405C22×C3—48no—
406C22×C3—48no—
407C22×C3—64no—
408C22×C3—64no—
409C22×C3—64no—
410C22×C3C22×D3214yes—
411C22×C3C23×C3⋊D311yescharacteristic
412C2×D3—21no—
413C2×D3—21no—
414C2×D3—21no—
415C3×C4—28no—
416C3×C4—68no—
417C3×C4—64no—
418C3×C4—182no—
419C3×C4—181no—
420C3×C4—181no—
421C3×C4#380518yes—
422C3×C4#383012yes—
423Dic3—21no—
424Dic3—22no—
425Dic3—64no—
426Dic3—64no—
427Dic3—68no—
428Dic3—188no—
429Dic3—184no—
430Dic3—188no—
431Dic3—184no—
432Dic3—182no—
433Dic3—181no—
434Dic3—181no—
435Dic3—368no—
436Dic3—364no—
437C32—24no—
438C32#560314yescomplemented
439C32#568211yescharacteristic, complemented
440C23—31no—
441C23C2×D3×C3⋊D312yes—
442Q8—182no—
443Q8—1082no—
444D4—22no—
445D4—122no—
446C2×C4—32no—
447C2×C4—31no—
448C2×C4—62no—
449C2×C4—91no—
450C2×C4—91no—
451C2×C4—92no—
452C2×C4—271no—
453C2×C4—271no—
454C2×C4—272no—
455C2×C4—1082no—
456C2×C4C2×D3×C3⋊D311yescharacteristic
457C2×C3—24no—
458C2×C3—24no—
459C2×C3—24no—
460C2×C3—416no—
461C2×C3—48no—
462C2×C3—42no—
463C2×C3—124no—
464C2×C3#6430914yes—
465C2×C3#6431414yes—
466C2×C3C2×#380514yes—
467C2×C3C2×#383011yescharacteristic
468C2×C3#6436111yescharacteristic
469C2×C3#6436711yescharacteristic
470D3—41no—
471C22—22no—
472C22—22no—
473C22—22no—
474C22—61no—
475C22—61no—
476C22—61no—
477C22C22×D3×C3⋊D311yescharacteristic
478C4—62no—
479C4—61no—
480C4—181no—
481C4—181no—
482C4—182no—
483C4—541no—
484C4—541no—
485C4—542no—
486C4#8902912yes—
487C3—24no—
488C3#1059512614yescomplemented
489C3#1059607411yescharacteristic, complemented
490C2—42no—
491C2—121no—
492C2#1172458211yescharacteristic
493C2#1172458811yescharacteristic
494C2C2×#8902911yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC2×C33×D4C22
prime radicalC32×#938C2
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
1. D4×C3⋊(C3⋊Dic3)C2semidirect
2. C3×#64145C2semidirect
4. #11723642C2semidirect
5. #11723603C2semidirect
6. #11723600C2semidirect
7. #11723596C2semidirect
8. #11723535C2semidirect
9. #11723425C2semidirect
10. #11723423C2semidirect
11. #11723410C2semidirect
39. #64145D3semidirect
318. C33#542semidirect
438. C32#5603semidirect
439. C32#5682semidirect
488. C3#10595126semidirect
489. C3#10596074semidirect

Maximal subgroups

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

subgroupindex
D4×C3⋊(C3⋊Dic3)2
C3×#641452
C32×#9382
#117236422
#117236032
#117236002
#117235962
#117235352
#117234252
#117234232
#117234102
#105960743
#105951263