Finite Group Databasebeta

← #13175354

Subgroups — #13175354

Order 1728.

1576 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#6308C211yesmaximal, characteristic, complemented
2C3×C4×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
3C32×#872C211yesmaximal, characteristic, complemented
4#11723534C211yesmaximal, characteristic, complemented
5#11723530C212yesmaximal, complemented
6#11723518C212yesmaximal, complemented
7#11723515C211yesmaximal, characteristic
8#11723439C211yesmaximal, characteristic, complemented
9#11723433C212yesmaximal, complemented
10#11723426C211yesmaximal, characteristic, complemented
11#11723418C212yesmaximal, complemented
12#10596005—31nomaximal
13#10593983—34nomaximal
14C2×#6308C2211yescharacteristic
15C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
16C2×C3×C4×C3⋊D3C2212yes—
17C2×C32×C4×D3C2211yescharacteristic
18C2×#6293C2212yes—
19C2×#6290C2212yes—
20C33×C42C2211yescharacteristic
21C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
22#88857C2212yes—
23C4×C3⋊(C3⋊Dic3)C2212yes—
24C3×#3752C2212yes—
25C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
26C32×(C2×D3)⋊C4C2211yescharacteristic
27C32×(C2×D3)⋊C4C2211yescharacteristic
28C32×Dic3⋊C4C2211yescharacteristic
29C32×Dic3⋊C4C2211yescharacteristic
30C32×C4×Dic3C2211yescharacteristic
31#88807C2212yes—
32Dic3⋊(C3⋊Dic3)C2212yes—
33#88803C2212yes—
34#88803C2212yes—
35#88803C2212yes—
36(C2×D3)⋊(C3⋊Dic3)C2212yes—
37C32×#117—31no—
38#64139—31no—
39#64133—31no—
40#64107—32no—
41#64105—32no—
42#64094—32no—
43#64092—32no—
44#64084—31no—
45C4×C4⋊(C3⋊D3)D311yescharacteristic, complemented
46#64078—31no—
47C4×#634—31no—
48C4×#634—34no—
49C4×#634—64no—
50C3×C4×D12—34no—
51C3×#872D314yescomplemented
52#63918—34no—
53#63915—38no—
54#63908—34no—
55#63905—38no—
56#63862—38no—
57#63860—34no—
58#63857—34no—
59#63851—38no—
60C22×C3×C3⋊D3C2312yes—
61C22×C32×D3C2311yescharacteristic
62C2×C33×C4C2311yescharacteristic
63C2×C33×C4C2311yescharacteristic
64C2×C33×C4C2311yescharacteristic
65C2×C3⋊(C3⋊Dic3)C2312yes—
66C2×C3⋊(C3⋊Dic3)C2312yes—
67#6308—22no—
68C2×C3×C3⋊Dic3C2312yes—
69C3×C4⋊(C3⋊D3)—22no—
70C3×C4×C3⋊D3—42no—
71C2×C32×Dic3C2311yescharacteristic
72C2×C32×Dic3C2311yescharacteristic
73C2×C32×Dic3C2311yescharacteristic
74C32×C4×D3—22no—
75#6293—42no—
76#6290—42no—
77#5561—91no—
78#5534—94no—
79C22×C32×C4—31no—
80C2×#638—32no—
81C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
82C2×C4×C3⋊D3C2×D312yes—
83C2×#634—31no—
84C2×#634—34no—
85C2×#634—32no—
86C2×#634—64no—
87C2×C3×D12—34no—
88C2×C3×C4×D3—38no—
89C2×C3×C4×D3C2×D314yes—
90C2×#627—38no—
91C2×#625—38no—
92C32×C4⋊C4—31no—
93C32×C4⋊C4—31no—
94C32×C22⋊C4—31no—
95C32×C22⋊C4—31no—
96C32×C42—24no—
97C32×C42—31no—
98C32×C42C2×D314yes—
99C32×C42C2×D311yescharacteristic
100#3757—32no—
101#3752—32no—
102#3752—32no—
103#3752—32no—
104#3752C2×D312yes—
105C4⋊(C3⋊Dic3)—31no—
106C4⋊(C3⋊Dic3)—34no—
107C4⋊(C3⋊Dic3)—32no—
108C4⋊(C3⋊Dic3)—64no—
109C4⋊(C3⋊Dic3)C2×D311yescharacteristic
110#3750—32no—
111#3750—38no—
112#3750—32no—
113#3750—68no—
114C4×C3⋊Dic3—32no—
115C4×C3⋊Dic3—38no—
116C4×C3⋊Dic3—68no—
117C3×(C2×D3)⋊C4—38no—
118C3×(C2×D3)⋊C4C2×D314yes—
119C3×(C2×D3)⋊C4C2×D314yes—
120C3×C4⋊Dic3—34no—
121C3×Dic3⋊C4C2×D314yes—
122C3×Dic3⋊C4C2×D314yes—
123C3×C4×Dic3C2×D314yes—
124Dic3⋊Dic3—38no—
125Dic3⋊Dic3—38no—
126(C2×D3)⋊Dic3—38no—
127(C2×D3)⋊Dic3—38no—
128(C2×D3)⋊Dic3—38no—
129(C2×D3)⋊Dic3—38no—
130C22×C33C2411yescharacteristic
131C2×C3×C3⋊D3—22no—
132C2×C3×C3⋊D3—22no—
133C2×C3×C3⋊D3—22no—
134C2×C32×D3—21no—
135C2×C32×D3—21no—
136C2×C32×D3—21no—
137C33×C4—21no—
138C33×C4—21no—
139C33×C4#4112yes—
140C3⋊(C3⋊Dic3)—22no—
141C3⋊(C3⋊Dic3)—22no—
142C3×C3⋊Dic3—22no—
143C32×Dic3—21no—
144C32×Dic3—21no—
145C32×Dic3—21no—
146C3×C4×D4—91no—
147C3×#117—34no—
148#938—91no—
149#936—91no—
150#930—94no—
151#929—92no—
152#924—94no—
153#924—92no—
154#898—98no—
155#897—91no—
156#896—98no—
157#888—91no—
158#885—98no—
159#885—92no—
160#883—98no—
161#879—92no—
162#875—94no—
163C4×D12—34no—
164#872C3⋊D311yescharacteristic, complemented
165#869—94no—
166C4×Dic6—94no—
167C4×Dic6—91no—
168C4×Dic6—184no—
169C23×C32—31no—
170C22×C3⋊D3C22×D312yes—
171C22×C3×D3—38no—
172C22×C3×D3C22×D314yes—
173C2×C32×C4—24no—
174C2×C32×C4—24no—
175C2×C32×C4—24no—
176C2×C32×C4—31no—
177C2×C32×C4—31no—
178C2×C32×C4—31no—
179C2×C32×C4—62no—
180C2×C32×C4C22×D314yes—
181C2×C32×C4C22×D314yes—
182C2×C32×C4C22×D314yes—
183C2×C32×C4C22×D311yescharacteristic
184C2×C32×C4C22×D311yescharacteristic
185C2×C32×C4C22×D311yescharacteristic
186#638—122no—
187C2×C3⋊Dic3—32no—
188C2×C3⋊Dic3—32no—
189C2×C3⋊Dic3—38no—
190C2×C3⋊Dic3—38no—
191C2×C3⋊Dic3—68no—
192C2×C3⋊Dic3—68no—
193C2×C3⋊Dic3C22×D312yes—
194C4⋊(C3⋊D3)—22no—
195C4×C3⋊D3—42no—
196#634—616no—
197#634—68no—
198#634—62no—
199#634—122no—
200C2×C3×Dic3—38no—
201C2×C3×Dic3C22×D314yes—
202C2×C3×Dic3C22×D314yes—
203C2×C3×Dic3C22×D314yes—
204C3×D12—68no—
205C3×C4×D3—28no—
206C3×C4×D3—128no—
207#627—128no—
208#625—128no—
209#542—271noSylow
210C2×C33C2×#4111yescharacteristic
211C2×C33H(5,𝔽2)11yescharacteristic
212C2×C332−1+411yescharacteristic
213C3×C3⋊D3—42no—
214C32×D3—41no—
215C2×C3×D4—91no—
216C22×C3×C4—34no—
217C22×C3×C4—92no—
218C2×#67—98no—
219C2×#67—92no—
220C2×D12—34no—
221C2×C4×D3—38no—
222C2×C4×D3C2×C3⋊D311yescharacteristic
223C2×Dic6—94no—
224C2×Dic6—91no—
225C2×Dic6—98no—
226C2×Dic6—92no—
227C2×Dic6—184no—
228C3×C4⋊C4—34no—
229C3×C4⋊C4—34no—
230C3×C4⋊C4—91no—
231C3×C22⋊C4—34no—
232C3×C22⋊C4—34no—
233C3×C22⋊C4—92no—
234C3×C42—24no—
235C3×C42—34no—
236C3×C42D3214yes—
237C3×C42C2×C3⋊D311yescharacteristic
238#217—98no—
239#217—92no—
240#217—92no—
241#217—92no—
242(C2×D3)⋊C4—38no—
243(C2×D3)⋊C4—98no—
244(C2×D3)⋊C4—98no—
245(C2×D3)⋊C4—98no—
246(C2×D3)⋊C4—92no—
247(C2×D3)⋊C4C2×C3⋊D311yescharacteristic
248(C2×D3)⋊C4C2×C3⋊D311yescharacteristic
249C4⋊Dic3—34no—
250C4⋊Dic3—98no—
251C4⋊Dic3—94no—
252C4⋊Dic3—91no—
253C4⋊Dic3—92no—
254C4⋊Dic3—184no—
255Dic3⋊C4—98no—
256Dic3⋊C4—98no—
257Dic3⋊C4—92no—
258Dic3⋊C4—92no—
259Dic3⋊C4—188no—
260Dic3⋊C4C2×C3⋊D311yescharacteristic
261Dic3⋊C4C2×C3⋊D311yescharacteristic
262C4×Dic3—98no—
263C4×Dic3—92no—
264C4×Dic3—188no—
265C4×Dic3C2×C3⋊D311yescharacteristic
266C22×C32—24no—
267C22×C32—61no—
268C22×C32—61no—
269C22×C32—61no—
270C22×C32C23×D314yes—
271C22×C32C23×D311yescharacteristic
272C2×C3⋊D3—22no—
273C2×C3⋊D3—22no—
274C2×C3⋊D3—22no—
275C2×C3×D3—24no—
276C2×C3×D3—24no—
277C2×C3×D3—24no—
278C2×C3×D3—68no—
279C2×C3×D3—68no—
280C2×C3×D3—68no—
281C32×C4—28no—
282C32×C4—21no—
283C32×C4—24no—
284C32×C4—21no—
285C32×C4—28no—
286C32×C4—24no—
287C32×C4—44no—
288C32×C4—61no—
289C32×C4—61no—
290C32×C4—61no—
291C32×C4#23518yes—
292C32×C4#23512yes—
293C3⋊Dic3—22no—
294C3⋊Dic3—616no—
295C3⋊Dic3—68no—
296C3⋊Dic3—62no—
297C3⋊Dic3—62no—
298C3⋊Dic3—68no—
299C3⋊Dic3—128no—
300C3×Dic3—24no—
301C3×Dic3—24no—
302C3×Dic3—24no—
303C3×Dic3—68no—
304C4⋊Q8—271no—
305#126—272no—
306#124—271no—
307#124—272no—
308#123—272no—
309C22⋊Q8—272no—
310C22⋊Q8—271no—
311#121—271no—
312C4×Q8—271no—
313C4×D4—91no—
314#117—31no—
315C33#54211yescharacteristic, complemented, Sylow
316C23×C3—34no—
317C23×C3—92no—
318C22×D3—38no—
319C22×D3C22×C3⋊D311yescharacteristic
320C3×D4—182no—
321C2×C3×C4—24no—
322C2×C3×C4—24no—
323C2×C3×C4—24no—
324C2×C3×C4—34no—
325C2×C3×C4—34no—
326C2×C3×C4—34no—
327C2×C3×C4—68no—
328C2×C3×C4—92no—
329C2×C3×C4—362no—
330C2×C3×C4C2×D3214yes—
331C2×C3×C4C2×D3214yes—
332C2×C3×C4C2×D3214yes—
333C2×C3×C4C22×C3⋊D311yescharacteristic
334C2×C3×C4C22×C3⋊D311yescharacteristic
335C2×C3×C4C22×C3⋊D311yescharacteristic
336#67—362no—
337#67—368no—
338C2×Dic3—38no—
339C2×Dic3—98no—
340C2×Dic3—98no—
341C2×Dic3—92no—
342C2×Dic3—92no—
343C2×Dic3—188no—
344C2×Dic3—188no—
345C2×Dic3C22×C3⋊D311yescharacteristic
346C2×Dic3C22×C3⋊D311yescharacteristic
347C2×Dic3C22×C3⋊D311yescharacteristic
348D12—68no—
349C4×D3—22no—
350C4×D3—128no—
351Dic6—1816no—
352Dic6—188no—
353Dic6—182no—
354Dic6—362no—
355Dic6—368no—
356C2×C32—24no—
357C2×C32—24no—
358C2×C32—24no—
359C2×C32—121no—
360C2×C32C2×#23514yes—
361C2×C32C2×#23511yescharacteristic
362C2×C32#100411yescharacteristic
363C2×C32#100914yes—
364C2×C32#101014yes—
365C2×C32#101011yescharacteristic
366C3⋊D3—42no—
367C3×D3—44no—
368C3×D3—128no—
369C2×Q8—272no—
370C2×Q8—271no—
371C2×D4—91no—
372C2×D4—272no—
373C22×C4—31no—
374C22×C4—92no—
375C4⋊C4—31no—
376C4⋊C4—31no—
377C4⋊C4—91no—
378C4⋊C4—272no—
379C4⋊C4—272no—
380C4⋊C4—271no—
381C4⋊C4—272no—
382C22⋊C4—31no—
383C22⋊C4—31no—
384C22⋊C4—92no—
385C22⋊C4—272no—
386C22⋊C4—272no—
387C22⋊C4—272no—
388C22⋊C4—272no—
389C42—31no—
390C42—272no—
391C42D3×C3⋊D311yescharacteristic
392C22×C3—24no—
393C22×C3—64no—
394C22×C3—64no—
395C22×C3—64no—
396C22×C3—182no—
397C22×C3—182no—
398C22×C3—182no—
399C22×C3C22×D3214yes—
400C22×C3C23×C3⋊D311yescharacteristic
401C2×D3—21no—
402C2×D3—21no—
403C2×D3—21no—
404C2×D3—68no—
405C2×D3—68no—
406C2×D3—68no—
407C3×C4—28no—
408C3×C4—24no—
409C3×C4—24no—
410C3×C4—21no—
411C3×C4—21no—
412C3×C4—28no—
413C3×C4—44no—
414C3×C4—64no—
415C3×C4—64no—
416C3×C4—64no—
417C3×C4—182no—
418C3×C4#379818yes—
419C3×C4#382812yes—
420Dic3—21no—
421Dic3—21no—
422Dic3—21no—
423Dic3—68no—
424Dic3—1816no—
425Dic3—188no—
426Dic3—188no—
427Dic3—182no—
428Dic3—182no—
429Dic3—368no—
430C32—24no—
431C32#553414yescomplemented
432C32#556111yescharacteristic, complemented
433C23—31no—
434C23—92no—
435Q8—542no—
436Q8—1082no—
437D4—182no—
438D4—1082no—
439C2×C4—31no—
440C2×C4—31no—
441C2×C4—31no—
442C2×C4—62no—
443C2×C4—92no—
444C2×C4—272no—
445C2×C4—272no—
446C2×C4—362no—
447C2×C4C2×D3×C3⋊D311yescharacteristic
448C2×C4C2×D3×C3⋊D311yescharacteristic
449C2×C4C2×D3×C3⋊D311yescharacteristic
450C2×C3—24no—
451C2×C3—24no—
452C2×C3—24no—
453C2×C3—124no—
454C2×C3—362no—
455C2×C3#6429814yes—
456C2×C3C2×#379814yes—
457C2×C3#6430714yes—
458C2×C3C2×#382811yescharacteristic
459C2×C3#6436611yescharacteristic
460C2×C3#6436711yescharacteristic
461D3—41no—
462D3—128no—
463C22—61no—
464C22—61no—
465C22—61no—
466C22—182no—
467C22—182no—
468C22—182no—
469C22C22×D3×C3⋊D311yescharacteristic
470C4—21no—
471C4—21no—
472C4—61no—
473C4—61no—
474C4—61no—
475C4—182no—
476C4—542no—
477C4—542no—
478C4#8902012yes—
479C3—24no—
480C3#1059398314yescomplemented
481C3#1059600511yescharacteristic, complemented
482C2—121no—
483C2—362no—
484C2#1172456311yescharacteristic
485C2C2×#8902011yescharacteristic
486C2#1172458011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C42C22
prime radical#11723515C2
hypercenterC42D3×C3⋊D3
hypocenterC33#542
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C4×#6308C2semidirect
2. C3×C4×C4⋊(C3⋊D3)C2semidirect
3. C32×#872C2semidirect
4. #11723534C2semidirect
5. #11723530C2semidirect
6. #11723518C2semidirect
8. #11723439C2semidirect
9. #11723433C2semidirect
10. #11723426C2semidirect
11. #11723418C2semidirect
45. C4×C4⋊(C3⋊D3)D3semidirect
51. C3×#872D3semidirect
164. #872C3⋊D3semidirect
315. C33#542semidirect
431. C32#5534semidirect
432. C32#5561semidirect
480. C3#10593983semidirect
481. C3#10596005semidirect

Maximal subgroups

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

subgroupindex
C4×#63082
C3×C4×C4⋊(C3⋊D3)2
C32×#8722
#117235342
#117235302
#117235182
#117235152
#117234392
#117234332
#117234262
#117234182
#105960053
#105939833