Finite Group Databasebeta

← #13176516

Subgroups — #13176516

Order 1728.

1508 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723929C211yesmaximal, characteristic, complemented
2C3×C4⋊C4×C3⋊D3C211yesmaximal, characteristic
3C32×#890C211yesmaximal, characteristic, complemented
4#11723534C211yesmaximal, characteristic, complemented
5#11723450C211yesmaximal, characteristic, complemented
6#11723437C211yesmaximal, characteristic
7#11723431C212yesmaximal
8#11723418C212yesmaximal
9#11723412C212yesmaximal, complemented
10#11723404C211yesmaximal, characteristic
11#11723395C212yesmaximal, complemented
12#10596160—31nomaximal
13#10594344—34nomaximal
14C2×#6308C2212yes—
15C2×C3×C4×C3⋊D3C2212yes—
16C2×C3×C4×C3⋊D3C2211yescharacteristic
17C2×#6293C2212yes—
18C33×C4⋊C4C2211yescharacteristic
19C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
20#88857C2212yes—
21C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
22C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
23C3×#3750C2212yes—
24C32×C4⋊Dic3C2211yescharacteristic
25C32×C4⋊Dic3C2212yes—
26C32×Dic3⋊C4C2212yes—
27C32×C4×Dic3C2211yescharacteristic
28#88807C2211yescharacteristic
29#88807C2211yescharacteristic
30#88807C2212yes—
31Dic3⋊(C3⋊Dic3)C2212yes—
32Dic3⋊(C3⋊Dic3)C2212yes—
33#88803C2212yes—
34#88803C2211yescharacteristic
35#88803C2211yescharacteristic
36Dic3×C3⋊Dic3C2212yes—
37C32×#125—31no—
38#64106—32no—
39#64105—32no—
40#64101—31no—
41C4⋊C4×C3⋊D3D311yescharacteristic
42#64099—31no—
43#64098—32no—
44#64097—31no—
45#64097—32no—
46#64097—34no—
47#64097—64no—
48#64081—31no—
49#64078—31no—
50C3×D3×C4⋊C4—34no—
51C3×#890D314yescomplemented
52#63918—34no—
53#63872—34no—
54#63857—38no—
55#63855—34no—
56#63851—38no—
57#63846—38no—
58#63841—34no—
59#63840—38no—
60C22×C3×C3⋊D3C2311yescharacteristic
61C2×C33×C4C2312yes—
62C2×C33×C4C2311yescharacteristic
63C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
64C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
65C2×C3⋊(C3⋊Dic3)C2312yes—
66#6308—42no—
67C2×C3×C3⋊Dic3C2312yes—
68C2×C3×C3⋊Dic3C2311yescharacteristic
69C3×C4×C3⋊D3—42no—
70C3×C4×C3⋊D3Q814yescomplemented
71C2×C32×Dic3C2311yescharacteristic
72C2×C32×Dic3C2312yes—
73C2×C32×Dic3C2311yescharacteristic
74#6293—42no—
75#5689—94no—
76#5500—91no—
77C2×C4×C3⋊D3C2×D312yes—
78C2×C4×C3⋊D3C2×D311yescharacteristic
79C2×#634—32no—
80C2×#634—38no—
81C2×#634—32no—
82C2×#634—68no—
83C2×C3×C4×D3—34no—
84C2×C3×C4×D3—38no—
85C2×#627—38no—
86C32×C4⋊C4—24no—
87C32×C4⋊C4—32no—
88C32×C4⋊C4—32no—
89C32×C4⋊C4—31no—
90C32×C4⋊C4C2×D314yes—
91C32×C4⋊C4C2×D311yescharacteristic
92C32×C42—31no—
93#3752—31no—
94#3752—31no—
95#3752—32no—
96C4⋊(C3⋊Dic3)—31no—
97C4⋊(C3⋊Dic3)—34no—
98C4⋊(C3⋊Dic3)—32no—
99C4⋊(C3⋊Dic3)—32no—
100C4⋊(C3⋊Dic3)—64no—
101C4⋊(C3⋊Dic3)C2×D311yescharacteristic
102#3750—32no—
103#3750—38no—
104#3750—31no—
105#3750—31no—
106#3750—32no—
107#3750—68no—
108#3750C2×D312yes—
109C4×C3⋊Dic3—31no—
110C4×C3⋊Dic3—34no—
111C4×C3⋊Dic3—32no—
112C4×C3⋊Dic3—64no—
113C3×C4⋊Dic3—34no—
114C3×C4⋊Dic3C2×D314yes—
115C3×C4⋊Dic3C2×D318yes—
116C3×Dic3⋊C4—38no—
117C3×Dic3⋊C4C2×D318yes—
118C3×C4×Dic3C2×D314yes—
119Dic3⋊Dic3—34no—
120Dic3⋊Dic3—38no—
121Dic3⋊Dic3—34no—
122Dic3⋊Dic3—38no—
123Dic3⋊Dic3—38no—
124(C2×D3)⋊Dic3—38no—
125(C2×D3)⋊Dic3—34no—
126(C2×D3)⋊Dic3—34no—
127Dic32—38no—
128C22×C33C2411yescharacteristic
129C2×C3×C3⋊D3—21no—
130C2×C3×C3⋊D3—21no—
131C2×C3×C3⋊D3C2×Q812yes—
132C33×C4—22no—
133C33×C4C2×Q812yes—
134C3⋊(C3⋊Dic3)—21no—
135C3⋊(C3⋊Dic3)—22no—
136C3⋊(C3⋊Dic3)—21no—
137C3×C3⋊Dic3—22no—
138C3×C3⋊Dic3C2×Q812yes—
139C32×Dic3—21no—
140C32×Dic3—22no—
141C32×Dic3—21no—
142C3×#125—34no—
143C2×C3×C4⋊C4—91no—
144#926—91no—
145#926—91no—
146#924—92no—
147#924—92no—
148#897—98no—
149#896—98no—
150#892—94no—
151D3×C4⋊C4—34no—
152#890—94no—
153#890—92no—
154#890—91no—
155#890C3⋊D311yescharacteristic, complemented
156#889—98no—
157#888—98no—
158#888—94no—
159#888—91no—
160#888—92no—
161#888—91no—
162#888—184no—
163#872—94no—
164#869—94no—
165C22×C3⋊D3C22×D311yescharacteristic
166C22×C3×D3—34no—
167C2×C32×C4—24no—
168C2×C32×C4—28no—
169C2×C32×C4—32no—
170C2×C32×C4—31no—
171C2×C32×C4—31no—
172C2×C32×C4C22×D318yes—
173C2×C32×C4C22×D314yes—
174C2×C32×C4C22×D312yes—
175C2×C32×C4C22×D311yescharacteristic
176C2×C3⋊Dic3—31no—
177C2×C3⋊Dic3—31no—
178C2×C3⋊Dic3—32no—
179C2×C3⋊Dic3—38no—
180C2×C3⋊Dic3—34no—
181C2×C3⋊Dic3—34no—
182C2×C3⋊Dic3—64no—
183C2×C3⋊Dic3—64no—
184C2×C3⋊Dic3—68no—
185C2×C3⋊Dic3C22×D312yes—
186C2×C3⋊Dic3C22×D311yescharacteristic
187C4×C3⋊D3—42no—
188C4×C3⋊D3Dic614yescomplemented
189#634—122no—
190#634—122no—
191#634—128no—
192#634—1216no—
193C2×C3×Dic3—34no—
194C2×C3×Dic3—38no—
195C2×C3×Dic3C22×D318yes—
196C2×C3×Dic3C22×D314yes—
197C2×C3×Dic3C22×D314yes—
198C3×C4×D3—316no—
199C3×C4×D3—128no—
200#627—128no—
201#544—271noSylow
202C2×C33C22×Q811yescharacteristic
203C2×C33H(5,𝔽2)11yescharacteristic
204C2×C332−1+411yescharacteristic
205C3×C3⋊D3—22no—
206C22×C3×C4—91no—
207C22×C3×C4—92no—
208C2×C4×D3—34no—
209C2×C4×D3—38no—
210C2×Dic6—98no—
211C2×Dic6—98no—
212C2×Dic6—92no—
213C2×Dic6—92no—
214C2×Dic6—188no—
215C3×C4⋊C4—24no—
216C3×C4⋊C4—38no—
217C3×C4⋊C4—38no—
218C3×C4⋊C4—34no—
219C3×C4⋊C4—92no—
220C3×C4⋊C4—91no—
221C3×C4⋊C4D3214yes—
222C3×C4⋊C4C2×C3⋊D311yescharacteristic
223C3×C42—34no—
224#217—91no—
225#217—92no—
226#217—91no—
227(C2×D3)⋊C4—94no—
228(C2×D3)⋊C4—94no—
229(C2×D3)⋊C4—98no—
230C4⋊Dic3—34no—
231C4⋊Dic3—98no—
232C4⋊Dic3—98no—
233C4⋊Dic3—94no—
234C4⋊Dic3—91no—
235C4⋊Dic3—91no—
236C4⋊Dic3—92no—
237C4⋊Dic3—91no—
238C4⋊Dic3—184no—
239C4⋊Dic3C2×C3⋊D312yes—
240C4⋊Dic3C2×C3⋊D311yescharacteristic
241Dic3⋊C4—38no—
242Dic3⋊C4—94no—
243Dic3⋊C4—94no—
244Dic3⋊C4—98no—
245Dic3⋊C4—98no—
246Dic3⋊C4—92no—
247Dic3⋊C4—92no—
248Dic3⋊C4—92no—
249Dic3⋊C4—188no—
250Dic3⋊C4C2×C3⋊D312yes—
251C4×Dic3—98no—
252C4×Dic3—94no—
253C4×Dic3—91no—
254C4×Dic3—92no—
255C4×Dic3—184no—
256C4×Dic3C2×C3⋊D311yescharacteristic
257C22×C32—24no—
258C22×C32C23×D314yes—
259C22×C32C23×D311yescharacteristic
260C2×C3⋊D3—21no—
261C2×C3⋊D3—21no—
262C2×C3⋊D3C2×Dic612yes—
263C2×C3×D3—38no—
264C2×C3×D3—64no—
265C2×C3×D3—64no—
266C32×C4—22no—
267C32×C4—28no—
268C32×C4—28no—
269C32×C4—48no—
270C32×C4—61no—
271C32×C4—62no—
272C32×C4—61no—
273C32×C4C2×Dic612yes—
274C32×C4D3×Q818yes—
275C3⋊Dic3—22no—
276C3⋊Dic3—68no—
277C3⋊Dic3—68no—
278C3⋊Dic3—61no—
279C3⋊Dic3—62no—
280C3⋊Dic3—61no—
281C3⋊Dic3—64no—
282C3⋊Dic3—64no—
283C3⋊Dic3—68no—
284C3⋊Dic3—128no—
285C3⋊Dic3C2×Dic612yes—
286C3×Dic3—24no—
287C3×Dic3—28no—
288C3×Dic3—24no—
289C3×Dic3—38no—
290C3×Dic3—68no—
291C4⋊Q8—271no—
292C4⋊Q8—272no—
293C4⋊Q8—271no—
294#125—31no—
295#125—272no—
296#125—271no—
297C22⋊Q8—272no—
298C22⋊Q8—272no—
299#117—271no—
300#117—271no—
301C2×C4⋊C4—91no—
302C33#54411yescharacteristic, complemented, Sylow
303C23×C3—91no—
304C22×D3—34no—
305C2×C3×C4—24no—
306C2×C3×C4—28no—
307C2×C3×C4—38no—
308C2×C3×C4—34no—
309C2×C3×C4—34no—
310C2×C3×C4—91no—
311C2×C3×C4—94no—
312C2×C3×C4—92no—
313C2×C3×C4—362no—
314C2×C3×C4C2×D3218yes—
315C2×C3×C4C2×D3214yes—
316C2×C3×C4C22×C3⋊D312yes—
317C2×C3×C4C22×C3⋊D311yescharacteristic
318C2×Dic3—34no—
319C2×Dic3—38no—
320C2×Dic3—94no—
321C2×Dic3—94no—
322C2×Dic3—98no—
323C2×Dic3—91no—
324C2×Dic3—92no—
325C2×Dic3—91no—
326C2×Dic3—184no—
327C2×Dic3—184no—
328C2×Dic3—188no—
329C2×Dic3C22×C3⋊D312yes—
330C2×Dic3C22×C3⋊D311yescharacteristic
331C2×Dic3C22×C3⋊D311yescharacteristic
332C4×D3—316no—
333C4×D3—128no—
334Dic6—362no—
335Dic6—368no—
336Dic6—368no—
337Dic6—362no—
338Dic6—3616no—
339C2×C32—24no—
340C2×C32—24no—
341C2×C32—24no—
342C2×C32C22×Dic611yescharacteristic
343C2×C32#100411yescharacteristic
344C2×C32C2×D3×Q814yes—
345C2×C32#100711yescharacteristic
346C2×C32#100914yes—
347C2×C32#101014yes—
348C3⋊D3—22no—
349C3×D3—68no—
350C2×Q8—272no—
351C2×Q8—272no—
352C22×C4—91no—
353C22×C4—92no—
354C4⋊C4—31no—
355C4⋊C4—32no—
356C4⋊C4—32no—
357C4⋊C4—92no—
358C4⋊C4—91no—
359C4⋊C4—271no—
360C4⋊C4—271no—
361C4⋊C4—272no—
362C4⋊C4—272no—
363C4⋊C4—272no—
364C4⋊C4—271no—
365C4⋊C4—272no—
366C4⋊C4D3×C3⋊D311yescharacteristic
367C22⋊C4—271no—
368C22⋊C4—271no—
369C22⋊C4—272no—
370C42—31no—
371C42—272no—
372C42—271no—
373C22×C3—24no—
374C22×C3—92no—
375C22×C3—181no—
376C22×C3—181no—
377C22×C3C22×D3214yes—
378C22×C3C23×C3⋊D311yescharacteristic
379C2×D3—38no—
380C2×D3—64no—
381C2×D3—64no—
382C3×C4—28no—
383C3×C4—22no—
384C3×C4—28no—
385C3×C4—48no—
386C3×C4—64no—
387C3×C4—68no—
388C3×C4—64no—
389C3×C4—92no—
390C3×C4—182no—
391C3×C4D3×Dic618yes—
392C3×C4Q8×C3⋊D312yes—
393Dic3—21no—
394Dic3—22no—
395Dic3—21no—
396Dic3—38no—
397Dic3—68no—
398Dic3—188no—
399Dic3—188no—
400Dic3—184no—
401Dic3—188no—
402Dic3—184no—
403Dic3—181no—
404Dic3—182no—
405Dic3—181no—
406Dic3—368no—
407C32—24no—
408C32#550011yescharacteristic, complemented
409C32#568914yescomplemented
410C23—91no—
411Q8—1082no—
412Q8—1082no—
413C2×C4—31no—
414C2×C4—31no—
415C2×C4—32no—
416C2×C4—91no—
417C2×C4—94no—
418C2×C4—92no—
419C2×C4—271no—
420C2×C4—271no—
421C2×C4—272no—
422C2×C4—362no—
423C2×C4C2×D3×C3⋊D312yes—
424C2×C4C2×D3×C3⋊D311yescharacteristic
425C2×C3—24no—
426C2×C3—24no—
427C2×C3—24no—
428C2×C3—182no—
429C2×C3C2×D3×Dic614yes—
430C2×C3#6431414yes—
431C2×C3#6431514yes—
432C2×C3C2×Q8×C3⋊D311yescharacteristic
433C2×C3#6436611yescharacteristic
434C2×C3#6436711yescharacteristic
435D3—68no—
436C22—92no—
437C22—181no—
438C22—181no—
439C22C22×D3×C3⋊D311yescharacteristic
440C4—22no—
441C4—61no—
442C4—61no—
443C4—62no—
444C4—92no—
445C4—182no—
446C4—541no—
447C4—542no—
448C4—541no—
449C4C3⋊D3×Dic612yes—
450C3—24no—
451C3#1059434414yescomplemented
452C3#1059616011yescharacteristic, complemented
453C2—182no—
454C2C2×C3⋊D3×Dic611yescharacteristic
455C2#1172458911yescharacteristic
456C2#1172459011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C4⋊C4C22
prime radicalC22×C3×C3⋊D3C23
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#544
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #11723929C2semidirect
3. C32×#890C2semidirect
4. #11723534C2semidirect
5. #11723450C2semidirect
9. #11723412C2semidirect
11. #11723395C2semidirect
51. C3×#890D3semidirect
70. C3×C4×C3⋊D3Q8semidirect
155. #890C3⋊D3semidirect
188. C4×C3⋊D3Dic6semidirect
302. C33#544semidirect
408. C32#5500semidirect
409. C32#5689semidirect
451. C3#10594344semidirect
452. C3#10596160semidirect

Maximal subgroups

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

subgroupindex
#117239292
C3×C4⋊C4×C3⋊D32
C32×#8902
#117235342
#117234502
#117234372
#117234312
#117234182
#117234122
#117234042
#117233952
#105961603
#105943443