Finite Group Databasebeta

← #13177700

Subgroups — #13177700

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×#64147C211yesmaximal, characteristic
3C32×#936C211yesmaximal, characteristic, complemented
4#11723642C212yesmaximal, complemented
5#11723609C212yesmaximal, complemented
6#11723597C212yesmaximal, complemented
7#11723591C212yesmaximal, complemented
8#11723534C211yesmaximal, characteristic, complemented
9#11723437C211yesmaximal, characteristic, complemented
10#11723435C211yesmaximal, characteristic, complemented
11#11723397C211yesmaximal, characteristic, complemented
12#10596153—31nomaximal
13#10595126—34nomaximal
14C2×C33×D4C2211yescharacteristic
15C22×C3⋊(C3⋊Dic3)C2212yes—
16C2×C3×#638C2212yes—
17C2×C3×C4×C3⋊D3C2211yescharacteristic
18C2×C32×Dic6C2211yescharacteristic
19C2×#6293C2212yes—
20#88864C2212yes—
21C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
22C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
23C3×#3757C2212yes—
24C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
25C32×#217C2212yes—
26C32×#217C2212yes—
27C32×C4×Dic3C2211yescharacteristic
28#88807C2211yescharacteristic
29#88807C2211yescharacteristic
30#88806C2212yes—
31Dic3⋊(C3⋊Dic3)C2212yes—
32Dic3⋊(C3⋊Dic3)C2212yes—
33#88803C2211yescharacteristic
34#88803C2212yes—
35#88803C2211yescharacteristic
36Dic3×C3⋊Dic3C2212yes—
37C32×#124—31no—
38#64155—31no—
39Q8×C3⋊Dic3—31no—
40#64147D311yescharacteristic
41D4×C3⋊Dic3—31no—
42D4×C3⋊Dic3—34no—
43D4×C3⋊Dic3—64no—
44#64095—32no—
45#64094—32no—
46#64088—32no—
47#64087—32no—
48#64081—31no—
49#64078—31no—
50C3×#938—34no—
51C3×#936D314yescomplemented
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×C3×C3⋊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—
68C3×#638—42no—
69C2×C3×C3⋊Dic3C2312yes—
70C2×C3×C3⋊Dic3C2311yescharacteristic
71C3×C4×C3⋊D3—22no—
72C2×C32×Dic3C2311yescharacteristic
73C2×C32×Dic3C2312yes—
74C2×C32×Dic3C2311yescharacteristic
75C32×Dic6—22no—
76#6293—42no—
77#5682—94no—
78#5603—91no—
79C2×C32×Q8—31no—
80C2×C32×D4—24no—
81C2×C32×D4C2×D314yes—
82C2×C32×D4C2×D311yescharacteristic
83C2×#638C2×D312yes—
84C22×C3⋊Dic3—32no—
85C22×C3⋊Dic3—38no—
86C22×C3⋊Dic3—68no—
87C2×C4×C3⋊D3C2×D311yescharacteristic
88C2×#634—32no—
89C2×C3×#67—38no—
90C2×C3×C4×D3—34no—
91C2×C3×Dic6C2×D314yes—
92C2×#627—38no—
93C32×C22⋊C4—32no—
94C32×C22⋊C4—32no—
95C32×C42—31no—
96#3757—32no—
97#3757—38no—
98#3757—68no—
99#3757C2×D312yes—
100#3752—31no—
101#3752—31no—
102#3752—32no—
103C4⋊(C3⋊Dic3)—31no—
104C4⋊(C3⋊Dic3)—34no—
105C4⋊(C3⋊Dic3)—32no—
106C4⋊(C3⋊Dic3)—32no—
107C4⋊(C3⋊Dic3)—64no—
108C4⋊(C3⋊Dic3)C2×D311yescharacteristic
109#3750—31no—
110#3750—31no—
111#3750—32no—
112C4×C3⋊Dic3—31no—
113C4×C3⋊Dic3—34no—
114C4×C3⋊Dic3—32no—
115C4×C3⋊Dic3—64no—
116C3×#217—38no—
117C3×#217C2×D318yes—
118C3×#217C2×D318yes—
119C3×C4⋊Dic3—34no—
120C3×C4×Dic3C2×D314yes—
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×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)—21no—
141C3×C3⋊Dic3—21no—
142C3×C3⋊Dic3—22no—
143C32×Dic3—22no—
144C32×Dic3—21no—
145C32×Dic3—21no—
146C3×#124—34no—
147C3×#121—91no—
148#946—94no—
149Q8×Dic3—94no—
150#938—34no—
151#936—92no—
152#936C3⋊D311yescharacteristic, complemented
153#935—92no—
154D4×Dic3—94no—
155D4×Dic3—91no—
156D4×Dic3—184no—
157#926—91no—
158#924—91no—
159#888—91no—
160#887—91no—
161#886—98no—
162#885—98no—
163#879—98no—
164#879—92no—
165#878—98no—
166#878—92no—
167#872—94no—
168#869—94no—
169C23×C32—28no—
170C23×C32C22×D318yes—
171C23×C32C22×D312yes—
172C22×C3⋊D3C22×D311yescharacteristic
173C22×C3×D3—34no—
174C32×Q8—62no—
175C32×D4—216no—
176C32×D4—28no—
177C32×D4—22no—
178C2×C32×C4—24no—
179C2×C32×C4—32no—
180C2×C32×C4—31no—
181C2×C32×C4—31no—
182C2×C32×C4C22×D314yes—
183C2×C32×C4C22×D311yescharacteristic
184#638—42no—
185C2×C3⋊Dic3—31no—
186C2×C3⋊Dic3—31no—
187C2×C3⋊Dic3—32no—
188C2×C3⋊Dic3—34no—
189C2×C3⋊Dic3—34no—
190C2×C3⋊Dic3—38no—
191C2×C3⋊Dic3—64no—
192C2×C3⋊Dic3—64no—
193C2×C3⋊Dic3—68no—
194C2×C3⋊Dic3—122no—
195C2×C3⋊Dic3—128no—
196C2×C3⋊Dic3—1216no—
197C2×C3⋊Dic3C22×D312yes—
198C2×C3⋊Dic3C22×D311yescharacteristic
199C4×C3⋊D3—22no—
200#634—122no—
201C3×#67—128no—
202C2×C3×Dic3—38no—
203C2×C3×Dic3—34no—
204C2×C3×Dic3C22×D318yes—
205C2×C3×Dic3C22×D314yes—
206C2×C3×Dic3C22×D314yes—
207C3×C4×D3—68no—
208C3×Dic6—28no—
209#627—128no—
210#542—271noSylow
211C2×C33—42no—
212C2×C33C2×#4111yescharacteristic
213C2×C33H(5,𝔽2)11yescharacteristic
214C2×C332−1+411yescharacteristic
215C3×C3⋊D3—41no—
216C2×C3×Q8—34no—
217C2×C3×D4—24no—
218C2×C3×D4—92no—
219C2×C3×D4D3214yes—
220C2×C3×D4C2×C3⋊D311yescharacteristic
221C22×C3×C4—91no—
222C2×#67—38no—
223C22×Dic3—98no—
224C22×Dic3—92no—
225C22×Dic3—188no—
226C2×C4×D3—34no—
227C2×Dic6—98no—
228C2×Dic6—92no—
229C2×Dic6C2×C3⋊D311yescharacteristic
230C3×C4⋊C4—91no—
231C3×C22⋊C4—38no—
232C3×C22⋊C4—38no—
233C3×C22⋊C4—92no—
234C3×C42—34no—
235#217—38no—
236#217—98no—
237#217—92no—
238#217—92no—
239#217—91no—
240#217—91no—
241#217—188no—
242#217C2×C3⋊D312yes—
243#217C2×C3⋊D312yes—
244(C2×D3)⋊C4—94no—
245(C2×D3)⋊C4—94no—
246(C2×D3)⋊C4—98no—
247C4⋊Dic3—34no—
248C4⋊Dic3—98no—
249C4⋊Dic3—98no—
250C4⋊Dic3—94no—
251C4⋊Dic3—91no—
252C4⋊Dic3—91no—
253C4⋊Dic3—91no—
254C4⋊Dic3—184no—
255Dic3⋊C4—94no—
256Dic3⋊C4—94no—
257Dic3⋊C4—98no—
258Dic3⋊C4—92no—
259Dic3⋊C4—92no—
260Dic3⋊C4—92no—
261C4×Dic3—98no—
262C4×Dic3—94no—
263C4×Dic3—91no—
264C4×Dic3—92no—
265C4×Dic3—184no—
266C4×Dic3C2×C3⋊D311yescharacteristic
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×C32C23×D314yes—
278C22×C32C23×D311yescharacteristic
279C2×C3⋊D3—21no—
280C2×C3⋊D3—21no—
281C2×C3⋊D3—21no—
282C2×C3×D3—64no—
283C2×C3×D3—64no—
284C2×C3×D3—64no—
285C32×C4—28no—
286C32×C4—62no—
287C32×C4—61no—
288C32×C4—61no—
289C32×C4#23718yes—
290C32×C4#23712yes—
291C3⋊Dic3—21no—
292C3⋊Dic3—22no—
293C3⋊Dic3—68no—
294C3⋊Dic3—62no—
295C3⋊Dic3—61no—
296C3⋊Dic3—64no—
297C3⋊Dic3—61no—
298C3⋊Dic3—68no—
299C3⋊Dic3—64no—
300C3⋊Dic3—124no—
301C3⋊Dic3—128no—
302C3×Dic3—28no—
303C3×Dic3—24no—
304C3×Dic3—24no—
305C3×Dic3—64no—
306C3×Dic3—68no—
307C4⋊Q8—271no—
308#126—272no—
309#124—31no—
310#124—272no—
311#123—272no—
312C22⋊Q8—272no—
313C22⋊Q8—271no—
314#121—91no—
315C4×Q8—271no—
316C4×D4—271no—
317#117—271no—
318C33#54211yescharacteristic, complemented, Sylow
319C23×C3—28no—
320C23×C3—91no—
321C23×C3C2×D3218yes—
322C23×C3C22×C3⋊D312yes—
323C22×D3—34no—
324C3×Q8—68no—
325C3×D4—216no—
326C3×D4—28no—
327C3×D4—22no—
328C3×D4—362no—
329C2×C3×C4—24no—
330C2×C3×C4—38no—
331C2×C3×C4—34no—
332C2×C3×C4—34no—
333C2×C3×C4—92no—
334C2×C3×C4—91no—
335C2×C3×C4—182no—
336C2×C3×C4C2×D3214yes—
337C2×C3×C4C22×C3⋊D311yescharacteristic
338#67—128no—
339C2×Dic3—38no—
340C2×Dic3—34no—
341C2×Dic3—94no—
342C2×Dic3—94no—
343C2×Dic3—98no—
344C2×Dic3—91no—
345C2×Dic3—91no—
346C2×Dic3—92no—
347C2×Dic3—184no—
348C2×Dic3—184no—
349C2×Dic3—188no—
350C2×Dic3—368no—
351C2×Dic3—362no—
352C2×Dic3—3616no—
353C2×Dic3C22×C3⋊D312yes—
354C2×Dic3C22×C3⋊D311yescharacteristic
355C2×Dic3C22×C3⋊D311yescharacteristic
356C4×D3—68no—
357Dic6—22no—
358Dic6—362no—
359Dic6—368no—
360C2×C32—24no—
361C2×C32—24no—
362C2×C32—24no—
363C2×C32—416no—
364C2×C32—48no—
365C2×C32—42no—
366C2×C32C2×#23714yes—
367C2×C32C2×#23711yescharacteristic
368C2×C32#100411yescharacteristic
369C2×C32#100914yes—
370C2×C32#101014yes—
371C2×C32#101011yescharacteristic
372C3⋊D3—41no—
373C3×D3—124no—
374C2×Q8—31no—
375C2×Q8—272no—
376C2×D4—92no—
377C2×D4D3×C3⋊D311yescharacteristic
378C22×C4—91no—
379C22×C4—272no—
380C4⋊C4—91no—
381C4⋊C4—271no—
382C4⋊C4—271no—
383C4⋊C4—272no—
384C4⋊C4—272no—
385C4⋊C4—272no—
386C4⋊C4—271no—
387C22⋊C4—32no—
388C22⋊C4—32no—
389C22⋊C4—92no—
390C22⋊C4—271no—
391C22⋊C4—271no—
392C22⋊C4—272no—
393C22⋊C4—272no—
394C42—31no—
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—181no—
408C22×C3—181no—
409C22×C3—181no—
410C22×C3C22×D3214yes—
411C22×C3C23×C3⋊D311yescharacteristic
412C2×D3—64no—
413C2×D3—64no—
414C2×D3—64no—
415C3×C4—28no—
416C3×C4—64no—
417C3×C4—64no—
418C3×C4—68no—
419C3×C4—181no—
420C3×C4—182no—
421C3×C4#380518yes—
422C3×C4#383012yes—
423Dic3—22no—
424Dic3—21no—
425Dic3—21no—
426Dic3—68no—
427Dic3—64no—
428Dic3—188no—
429Dic3—184no—
430Dic3—188no—
431Dic3—184no—
432Dic3—182no—
433Dic3—181no—
434Dic3—181no—
435Dic3—368no—
436Dic3—364no—
437C32—24no—
438C32#560311yescharacteristic, complemented
439C32#568214yescomplemented
440C23—91no—
441C23C2×D3×C3⋊D312yes—
442Q8—62no—
443Q8—1082no—
444D4—22no—
445D4—362no—
446C2×C4—31no—
447C2×C4—31no—
448C2×C4—32no—
449C2×C4—92no—
450C2×C4—91no—
451C2×C4—182no—
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—361no—
464C2×C3#6430914yes—
465C2×C3#6431414yes—
466C2×C3C2×#380514yes—
467C2×C3C2×#383011yescharacteristic
468C2×C3#6436611yescharacteristic
469C2×C3#6436711yescharacteristic
470D3—124no—
471C22—22no—
472C22—22no—
473C22—22no—
474C22—181no—
475C22—181no—
476C22—181no—
477C22C22×D3×C3⋊D311yescharacteristic
478C4—61no—
479C4—61no—
480C4—62no—
481C4—182no—
482C4—181no—
483C4—541no—
484C4—541no—
485C4—542no—
486C4#8902912yes—
487C3—24no—
488C3#1059512614yescomplemented
489C3#1059615311yescharacteristic, complemented
490C2—42no—
491C2—361no—
492C2#1172458211yescharacteristic
493C2#1172458911yescharacteristic
494C2C2×#8902911yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC2×C33×D4C22
prime radicalC3×#64147C2
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
3. C32×#936C2semidirect
4. #11723642C2semidirect
5. #11723609C2semidirect
6. #11723597C2semidirect
7. #11723591C2semidirect
8. #11723534C2semidirect
9. #11723437C2semidirect
10. #11723435C2semidirect
11. #11723397C2semidirect
51. C3×#936D3semidirect
152. #936C3⋊D3semidirect
318. C33#542semidirect
438. C32#5603semidirect
439. C32#5682semidirect
488. C3#10595126semidirect
489. C3#10596153semidirect

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×#641472
C32×#9362
#117236422
#117236092
#117235972
#117235912
#117235342
#117234372
#117234352
#117233972
#105961533
#105951263