Finite Group Databasebeta

← #124×(C3⋊D3)⋊C3 #13153115

Subgroups — #124×(C3⋊D3)⋊C3 #13153115

Order 1728.

1231 conjugacy classes of subgroups.

The subgroup lattice has 525 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
2C2×D4×(C3⋊D3)⋊C3C211yesmaximal, characteristic
3H(3,𝔽3)×#124C211yesmaximal, characteristic, complemented
4#64156⋊C3C211yesmaximal, characteristic, complemented
5#64145⋊C3C211yesmaximal, characteristic, complemented
6#64094⋊C3C214yesmaximal, complemented
7C22⋊C4×(C3⋊D3)⋊C3C214yesmaximal, complemented
8#64084⋊C3C211yesmaximal, characteristic, complemented
9C42×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
10C3⋊D3×#124C311yesmaximal, characteristic, complemented
11C3×D3×#124—31nomaximal
12C23×(C3⋊D3)⋊C3C2212yes—
13C2×Q8×H(3,𝔽3)C2211yescharacteristic, complemented
14C2×D4×H(3,𝔽3)C2211yescharacteristic
15C2×#638⋊C3C2212yes—
16Q8×(C3⋊D3)⋊C3—24no—
17D4×(C3⋊D3)⋊C3—24no—
18C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
19C2×C4×(C3⋊D3)⋊C3C2212yes—
20C2×C4×(C3⋊D3)⋊C3C2212yes—
21C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
22C2×#634⋊C3C2211yescharacteristic, complemented
23C2×#634⋊C3C2212yescomplemented
24C22⋊C4×H(3,𝔽3)C2214yescomplemented
25C42×H(3,𝔽3)C2211yescharacteristic, complemented
26#3757⋊C3C2214yescomplemented
27#3752⋊C3C2214yescomplemented
28#3752⋊C3C2214yescomplemented
29C4×(C3⋊Dic3)⋊C3C2211yescharacteristic, complemented
30C4×(C3⋊Dic3)⋊C3C2212yescomplemented
31C2×Q8×C3⋊D3C2×C311yescharacteristic, complemented
32C2×D4×C3⋊D3C2×C311yescharacteristic
33C2×C3×D3×Q8—31no—
34C2×C3×D3×D4—31no—
35C32×#124—21no—
36C32×#124D311yescharacteristic, complemented
37C32×#124C2×C311yescharacteristic, complemented
38#64156C2×C311yescharacteristic, complemented
39#64145C2×C311yescharacteristic, complemented
40#64094C2×C314yescomplemented
41C22⋊C4×C3⋊D3C2×C314yescomplemented
42#64084C2×C311yescharacteristic, complemented
43C42×C3⋊D3C2×C311yescharacteristic, complemented
44C3×#947—31no—
45C3×#936—31no—
46C3×#885—34no—
47C3×D3×C22⋊C4—34no—
48C3×#875—31no—
49C3×C42×D3—31no—
50C23×H(3,𝔽3)C2312yes—
51C22×(C3⋊D3)⋊C3—24no—
52C22×(C3⋊D3)⋊C3—28no—
53C22×(C3⋊D3)⋊C3C2312yes—
54C22×(C3⋊D3)⋊C3C2311yescharacteristic
55Q8×H(3,𝔽3)—22no—
56D4×H(3,𝔽3)—22no—
57C2×C4×H(3,𝔽3)C2312yes—
58C2×C4×H(3,𝔽3)C2312yes—
59C2×C4×H(3,𝔽3)C2311yescharacteristic
60#638⋊C3—24no—
61C2×(C3⋊Dic3)⋊C3C2312yes—
62C2×(C3⋊Dic3)⋊C3C2312yes—
63C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
64C4⋊((C3⋊D3)⋊C3)—22no—
65C4×(C3⋊D3)⋊C3—24no—
66C4×(C3⋊D3)⋊C3—24no—
67C4×(C3⋊D3)⋊C3D414yes—
68#634⋊C3—22no—
69#634⋊C3—24no—
70C2×C3×#124—91no—
71D3×#124—31no—
72D3×#124—91no—
73C23×C3⋊D3C22×C312yes—
74C23×C3×D3—32no—
75C2×C32×Q8—21no—
76C2×C32×Q8C2×D311yescharacteristic, complemented
77C2×C32×Q8C22×C311yescharacteristic, complemented
78C2×C32×D4—21no—
79C2×C32×D4C2×D311yescharacteristic
80C2×C32×D4C22×C311yescharacteristic
81C2×#638C22×C312yes—
82Q8×C3⋊D3—24no—
83D4×C3⋊D3—24no—
84C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
85C2×C4×C3⋊D3C22×C312yes—
86C2×C4×C3⋊D3C22×C312yes—
87C2×C4×C3⋊D3C22×C311yescharacteristic
88C2×#634C22×C311yescharacteristic, complemented
89C2×#634C22×C312yescomplemented
90C2×C3×#67—32no—
91C3×D3×Q8—64no—
92C3×D3×D4—64no—
93C2×C3×D12—31no—
94C2×C3×C4×D3—31no—
95C2×C3×C4×D3—32no—
96C2×C3×C4×D3—32no—
97C2×C3×Dic6—32no—
98C2×C3×Dic6—31no—
99C32×C22⋊C4—24no—
100C32×C22⋊C4C2×D314yescomplemented
101C32×C22⋊C4C22×C314yescomplemented
102C32×C42—21no—
103C32×C42C2×D311yescharacteristic, complemented
104C32×C42C22×C311yescharacteristic, complemented
105#3757C22×C314yescomplemented
106#3752C22×C314yescomplemented
107#3752C22×C314yescomplemented
108C4×C3⋊Dic3C22×C312yescomplemented
109C4×C3⋊Dic3C22×C311yescharacteristic, complemented
110C3×#217—34no—
111C3×(C2×D3)⋊C4—34no—
112C3×(C2×D3)⋊C4—34no—
113C3×C4×Dic3—32no—
114C3×C4×Dic3—31no—
115C22×H(3,𝔽3)—22no—
116C22×H(3,𝔽3)—24no—
117C22×H(3,𝔽3)C2411yescharacteristic
118C2×(C3⋊D3)⋊C3—22no—
119C2×(C3⋊D3)⋊C3—24no—
120C2×(C3⋊D3)⋊C3—48no—
121C2×(C3⋊D3)⋊C3C2×D412yes—
122C2×(C3⋊D3)⋊C3#4114yes—
123C4×H(3,𝔽3)—22no—
124C4×H(3,𝔽3)—22no—
125C4×H(3,𝔽3)C2×D412yes—
126(C3⋊Dic3)⋊C3—22no—
127(C3⋊Dic3)⋊C3—22no—
128(C3⋊Dic3)⋊C3C2×D412yes—
129C22×C3×Q8—91no—
130C22×C3×D4—91no—
131C2×D3×Q8—31no—
132C2×D3×Q8—91no—
133C2×D3×D4—31no—
134C2×D3×D4—91no—
135C3×#124—31no—
136C3×#124—31no—
137C3×#124—61no—
138C3×#124—91no—
139C3×#124—94no—
140C3×#124—91no—
141C3×#124—91no—
142C3×#124C3×D311yescharacteristic
143C2×C3×C22⋊C4—94no—
144C2×C3×C42—91no—
145#947—31no—
146#947—91no—
147#936—31no—
148#936—91no—
149#885—34no—
150#885—94no—
151D3×C22⋊C4—34no—
152D3×C22⋊C4—94no—
153#875—31no—
154#875—91no—
155C42×D3—31no—
156C42×D3—91no—
157C23×C32—22no—
158C23×C32C22×D312yes—
159C23×C32C23×C312yes—
160C22×C3⋊D3—24no—
161C22×C3⋊D3—28no—
162C22×C3⋊D3C23×C312yes—
163C22×C3⋊D3C23×C311yescharacteristic
164C22×C3×D3—31no—
165C22×C3×D3—32no—
166C22×C3×D3—68no—
167C22×C3×D3—64no—
168C32×Q8—22no—
169C32×Q8—22no—
170C32×Q8—42no—
171C32×D4—22no—
172C32×D4—22no—
173C32×D4—42no—
174C2×C32×C4—21no—
175C2×C32×C4—22no—
176C2×C32×C4—22no—
177C2×C32×C4C22×D312yes—
178C2×C32×C4C22×D312yes—
179C2×C32×C4C22×D311yescharacteristic
180C2×C32×C4C23×C312yes—
181C2×C32×C4C23×C312yes—
182C2×C32×C4C23×C311yescharacteristic
183#638—24no—
184C2×C3⋊Dic3C23×C312yes—
185C2×C3⋊Dic3C23×C312yes—
186C2×C3⋊Dic3C23×C311yescharacteristic
187C4⋊(C3⋊D3)—22no—
188C4×C3⋊D3—24no—
189C4×C3⋊D3—24no—
190C4×C3⋊D3C3×D414yes—
191#634—22no—
192#634—24no—
193C3×#67—64no—
194C2×C3×Dic3—31no—
195C2×C3×Dic3—32no—
196C2×C3×Dic3—32no—
197C3×D12—62no—
198C3×C4×D3—34no—
199C3×C4×D3—64no—
200C3×C4×D3—64no—
201C3×Dic6—64no—
202C3×Dic6—62no—
203C2×#124—91noSylow
204C2×H(3,𝔽3)—42no—
205C2×H(3,𝔽3)C22×D411yescharacteristic
206C2×H(3,𝔽3)C2×#4112yes—
207(C3⋊D3)⋊C3—42no—
208(C3⋊D3)⋊C3#12414yesdirect product factor
209C24×C3—92no—
210C23×D3—32no—
211C23×D3—92no—
212C2×C3×Q8—31no—
213C2×C3×Q8—31no—
214C2×C3×Q8—61no—
215C2×C3×Q8—91no—
216C2×C3×Q8—92no—
217C2×C3×Q8—184no—
218C2×C3×Q8C2×C3×D311yescharacteristic
219C2×C3×D4—31no—
220C2×C3×D4—31no—
221C2×C3×D4—61no—
222C2×C3×D4—91no—
223C2×C3×D4—92no—
224C2×C3×D4—184no—
225C2×C3×D4C2×C3×D311yescharacteristic
226C22×C3×C4—91no—
227C22×C3×C4—92no—
228C22×C3×C4—92no—
229C2×#67—32no—
230C2×#67—92no—
231D3×Q8—64no—
232D3×Q8—184no—
233D3×D4—64no—
234D3×D4—184no—
235C2×D12—31no—
236C2×D12—91no—
237C2×C4×D3—31no—
238C2×C4×D3—32no—
239C2×C4×D3—32no—
240C2×C4×D3—91no—
241C2×C4×D3—92no—
242C2×C4×D3—92no—
243C2×Dic6—31no—
244C2×Dic6—32no—
245C2×Dic6—91no—
246C2×Dic6—92no—
247C3×C22⋊C4—34no—
248C3×C22⋊C4—34no—
249C3×C22⋊C4—64no—
250C3×C22⋊C4—94no—
251C3×C22⋊C4—94no—
252C3×C22⋊C4—94no—
253C3×C22⋊C4C2×C3×D314yes—
254C3×C42—31no—
255C3×C42—31no—
256C3×C42—61no—
257C3×C42—91no—
258C3×C42—92no—
259C3×C42C2×C3×D311yescharacteristic
260#217—34no—
261#217—94no—
262(C2×D3)⋊C4—34no—
263(C2×D3)⋊C4—34no—
264(C2×D3)⋊C4—94no—
265(C2×D3)⋊C4—94no—
266C4×Dic3—32no—
267C4×Dic3—31no—
268C4×Dic3—92no—
269C4×Dic3—91no—
270C22×C32—22no—
271C22×C32—24no—
272C22×C32—22no—
273C22×C32—24no—
274C22×C32—21no—
275C22×C32—42no—
276C22×C32—44no—
277C22×C32C23×D311yescharacteristic
278C22×C32C24×C311yescharacteristic
279C2×C3⋊D3—22no—
280C2×C3⋊D3—24no—
281C2×C3⋊D3—48no—
282C2×C3⋊D3C2×C3×D412yes—
283C2×C3⋊D3C3×#4114yes—
284C2×C3×D3—32no—
285C2×C3×D3—34no—
286C2×C3×D3—62no—
287C2×C3×D3—64no—
288C2×C3×D3—128no—
289C32×C4—22no—
290C32×C4—22no—
291C32×C4—22no—
292C32×C4—22no—
293C32×C4—22no—
294C32×C4—42no—
295C32×C4—42no—
296C32×C4D3×D412yes—
297C32×C4C2×C3×D412yes—
298C3⋊Dic3—22no—
299C3⋊Dic3—22no—
300C3⋊Dic3C2×C3×D412yes—
301C3×Dic3—32no—
302C3×Dic3—62no—
303C3×Dic3—62no—
304C22×Q8—91no—
305C22×D4—91no—
306#124—91no—
307#124—91no—
308#124—94no—
309#124—91no—
310#124(C3⋊D3)⋊C311yescharacteristic, direct product factor
311C2×C22⋊C4—94no—
312C2×C42—91no—
313H(3,𝔽3)C2×#12411yescharacteristic, complemented, Sylow
314C23×C3—32no—
315C23×C3—32no—
316C23×C3—62no—
317C23×C3—91no—
318C23×C3—92no—
319C23×C3—184no—
320C23×C3—188no—
321C23×C3C22×C3×D312yes—
322C22×D3—31no—
323C22×D3—32no—
324C22×D3—64no—
325C22×D3—68no—
326C22×D3—91no—
327C22×D3—92no—
328C22×D3—184no—
329C22×D3—188no—
330C3×Q8—22no—
331C3×Q8—62no—
332C3×Q8—62no—
333C3×Q8—122no—
334C3×Q8—182no—
335C3×Q8—184no—
336C3×D4—22no—
337C3×D4—62no—
338C3×D4—62no—
339C3×D4—122no—
340C3×D4—182no—
341C3×D4—184no—
342C2×C3×C4—31no—
343C2×C3×C4—31no—
344C2×C3×C4—32no—
345C2×C3×C4—32no—
346C2×C3×C4—32no—
347C2×C3×C4—32no—
348C2×C3×C4—61no—
349C2×C3×C4—62no—
350C2×C3×C4—62no—
351C2×C3×C4—94no—
352C2×C3×C4—91no—
353C2×C3×C4—92no—
354C2×C3×C4—92no—
355C2×C3×C4—184no—
356C2×C3×C4—184no—
357C2×C3×C4C22×C3×D312yes—
358C2×C3×C4C22×C3×D312yes—
359C2×C3×C4C22×C3×D311yescharacteristic
360#67—64no—
361#67—184no—
362C2×Dic3—31no—
363C2×Dic3—32no—
364C2×Dic3—32no—
365C2×Dic3—91no—
366C2×Dic3—92no—
367C2×Dic3—92no—
368D12—62no—
369D12—182no—
370C4×D3—34no—
371C4×D3—64no—
372C4×D3—64no—
373C4×D3—94no—
374C4×D3—184no—
375C4×D3—184no—
376Dic6—64no—
377Dic6—62no—
378Dic6—184no—
379Dic6—182no—
380C2×C32—21no—
381C2×C32—22no—
382C2×C32—42no—
383C2×C32—42no—
384C2×C32—82no—
385C2×C32C2×D3×D411yescharacteristic
386C2×C32D3×#4112yes—
387C2×C32C22×C3×D411yescharacteristic
388C2×C32C2×C3×#4112yes—
389C3⋊D3—42no—
390C3⋊D3C3×#12414yescomplemented
391C3×D3—34no—
392C3×D3—122no—
393C24—92no—
394C2×Q8—92no—
395C2×Q8—91no—
396C2×Q8—184no—
397C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic, complemented
398C2×D4—92no—
399C2×D4—91no—
400C2×D4—184no—
401C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
402C22×C4—91no—
403C22×C4—92no—
404C22×C4—92no—
405C22⋊C4—94no—
406C22⋊C4—94no—
407C22⋊C4—94no—
408C22⋊C4C2×(C3⋊D3)⋊C314yescomplemented
409C42—92no—
410C42—91no—
411C42C2×(C3⋊D3)⋊C311yescharacteristic, complemented
412C22×C3—22no—
413C22×C3—24no—
414C22×C3—31no—
415C22×C3—31no—
416C22×C3—62no—
417C22×C3—62no—
418C22×C3—64no—
419C22×C3—64no—
420C22×C3—61no—
421C22×C3—92no—
422C22×C3—94no—
423C22×C3—122no—
424C22×C3—124no—
425C22×C3—182no—
426C22×C3—184no—
427C22×C3—368no—
428C22×C3C23×C3×D311yescharacteristic
429C2×D3—32no—
430C2×D3—34no—
431C2×D3—62no—
432C2×D3—64no—
433C2×D3—92no—
434C2×D3—94no—
435C2×D3—128no—
436C2×D3—182no—
437C2×D3—184no—
438C2×D3—368no—
439C3×C4—22no—
440C3×C4—22no—
441C3×C4—32no—
442C3×C4—32no—
443C3×C4—62no—
444C3×C4—62no—
445C3×C4—62no—
446C3×C4—62no—
447C3×C4—62no—
448C3×C4—92no—
449C3×C4—122no—
450C3×C4—122no—
451C3×C4—182no—
452C3×C4—182no—
453C3×C4C3×D3×D412yes—
454Dic3—32no—
455Dic3—62no—
456Dic3—62no—
457Dic3—92no—
458Dic3—182no—
459Dic3—182no—
460C32—21no—
461C32D3×#12411yescharacteristic, complemented
462C32C2×C3×#12411yescharacteristic, complemented
463C23—91no—
464C23—92no—
465C23—184no—
466C23—188no—
467C23C22×(C3⋊D3)⋊C312yes—
468Q8—22no—
469Q8—184no—
470Q8—182no—
471D4—22no—
472D4—184no—
473D4—182no—
474C2×C4—91no—
475C2×C4—94no—
476C2×C4—92no—
477C2×C4—92no—
478C2×C4—184no—
479C2×C4—184no—
480C2×C4C22×(C3⋊D3)⋊C312yes—
481C2×C4C22×(C3⋊D3)⋊C312yes—
482C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
483C2×C3—31no—
484C2×C3—31no—
485C2×C3—32no—
486C2×C3—32no—
487C2×C3—42no—
488C2×C3—61no—
489C2×C3—62no—
490C2×C3—94no—
491C2×C3—122no—
492C2×C3—122no—
493C2×C3—242no—
494C2×C3—362no—
495C2×C3C2×C3×D3×D411yescharacteristic
496C2×C3C3×D3×#4112yes—
497D3—34no—
498D3—94no—
499D3—122no—
500D3—362no—
501C22—22no—
502C22—24no—
503C22—92no—
504C22—94no—
505C22—182no—
506C22—184no—
507C22—368no—
508C22C23×(C3⋊D3)⋊C311yescharacteristic
509C4—22no—
510C4—22no—
511C4—92no—
512C4—182no—
513C4—182no—
514C4D4×(C3⋊D3)⋊C312yes—
515C3—31no—
516C3—31no—
517C3—61no—
518C3C3×D3×#12411yescharacteristic
519C2—42no—
520C2—94no—
521C2—362no—
522C2C2×D4×(C3⋊D3)⋊C311yescharacteristic
523C2#41×(C3⋊D3)⋊C312yes—
subgroupG/H
centerC22C23×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingH(3,𝔽3)×#124C2
prime radicalC2×D4×(C3⋊D3)⋊C3C2
hypercenter#124(C3⋊D3)⋊C3
hypocenterC32C2×C3×#124
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ #124

Complements

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

NH ≅ G/Ntype
1. C2×Q8×(C3⋊D3)⋊C3C2semidirect
3. H(3,𝔽3)×#124C2semidirect
4. #64156⋊C3C2semidirect
5. #64145⋊C3C2semidirect
6. #64094⋊C3C2semidirect
7. C22⋊C4×(C3⋊D3)⋊C3C2semidirect
8. #64084⋊C3C2semidirect
9. C42×(C3⋊D3)⋊C3C2semidirect
10. C3⋊D3×#124C3semidirect
13. C2×Q8×H(3,𝔽3)C22semidirect
22. C2×#634⋊C3C22semidirect
23. C2×#634⋊C3C22semidirect
24. C22⋊C4×H(3,𝔽3)C22semidirect
25. C42×H(3,𝔽3)C22semidirect
26. #3757⋊C3C22semidirect
27. #3752⋊C3C22semidirect
28. #3752⋊C3C22semidirect
29. C4×(C3⋊Dic3)⋊C3C22semidirect
30. C4×(C3⋊Dic3)⋊C3C22semidirect
31. C2×Q8×C3⋊D3C2×C3semidirect
36. C32×#124D3semidirect
37. C32×#124C2×C3semidirect
38. #64156C2×C3semidirect
39. #64145C2×C3semidirect
40. #64094C2×C3semidirect
41. C22⋊C4×C3⋊D3C2×C3semidirect
42. #64084C2×C3semidirect
43. C42×C3⋊D3C2×C3semidirect
76. C2×C32×Q8C2×D3semidirect
77. C2×C32×Q8C22×C3semidirect
88. C2×#634C22×C3semidirect
89. C2×#634C22×C3semidirect
100. C32×C22⋊C4C2×D3semidirect
101. C32×C22⋊C4C22×C3semidirect
103. C32×C42C2×D3semidirect
104. C32×C42C22×C3semidirect
105. #3757C22×C3semidirect
106. #3752C22×C3semidirect
107. #3752C22×C3semidirect
108. C4×C3⋊Dic3C22×C3semidirect
109. C4×C3⋊Dic3C22×C3semidirect
208. (C3⋊D3)⋊C3#124direct (N × H)
310. #124(C3⋊D3)⋊C3direct (N × H)
313. H(3,𝔽3)C2×#124semidirect
390. C3⋊D3C3×#124semidirect
397. C2×Q8C2×(C3⋊D3)⋊C3semidirect
408. C22⋊C4C2×(C3⋊D3)⋊C3semidirect
411. C42C2×(C3⋊D3)⋊C3semidirect
461. C32D3×#124semidirect
462. C32C2×C3×#124semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×(C3⋊D3)⋊C32
C2×D4×(C3⋊D3)⋊C32
H(3,𝔽3)×#1242
#64156⋊C32
#64145⋊C32
#64094⋊C32
C22⋊C4×(C3⋊D3)⋊C32
#64084⋊C32
C42×(C3⋊D3)⋊C32
C3⋊D3×#1243
C3×D3×#1243