Finite Group Databasebeta

← C2×#11721101 #13150654

Subgroups — C2×#11721101 #13150654

Order 1728.

1219 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
1C22×C4×(C3⋊D3)⋊C3C211yesmaximal, characteristic
2C22×#6196C211yesmaximal, characteristic, complemented
3C2×#88540C211yesmaximal, characteristic, complemented
4C2×#3750⋊C3C211yesmaximal, characteristic, complemented
5C2×#88450C211yesmaximal, characteristic
6C2×#88450C211yesmaximal, characteristic
7C2×#88448C211yesmaximal, characteristic, complemented
8#11721101C218yesmaximal, direct product factor
9C2×#63865—31nomaximal
10C2×#63851—31nomaximal
11C23×(C3⋊D3)⋊C3C2211yescharacteristic
12C22×C4×H(3,𝔽3)C2211yescharacteristic
13C22×#1084C2211yescharacteristic
14C22×#1084C2211yescharacteristic
15C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
16C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
17C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
18C2×C4×(C3⋊D3)⋊C3—24no—
19C2×C4×(C3⋊D3)⋊C3C2214yes—
20C2×#6196—24no—
21C2×#6196C2214yescomplemented
22#88540C2214yescomplemented
23#3750⋊C3C2214yescomplemented
24#88450C2214yes—
25#88450C2214yes—
26#88448C2214yescomplemented
27C22×C4×C3⋊D3D311yescharacteristic
28C22×C3×C4×D3—31no—
29C22×#627—31no—
30C22×#627—31no—
31C2×#3750D311yescharacteristic, complemented
32C2×C3×Dic3⋊C4—31no—
33C2×C3×Dic3⋊C4—31no—
34C2×C3×Dic3⋊C4—31no—
35C2×C3×Dic3⋊C4—61no—
36C2×#3724—31no—
37C2×Dic3⋊Dic3—31no—
38C2×#3722—31no—
39C2×#3722—31no—
40C2×(C2×D3)⋊Dic3—31no—
41C2×(C2×D3)⋊Dic3—31no—
42#63865—38no—
43#63851—38no—
44C23×H(3,𝔽3)C2311yescharacteristic
45C22×(C3⋊D3)⋊C3—22no—
46C22×(C3⋊D3)⋊C3—22no—
47C22×(C3⋊D3)⋊C3Q814yescomplemented
48C22×(C3⋊D3)⋊C3C2312yes—
49C2×C4×H(3,𝔽3)—22no—
50C2×C4×H(3,𝔽3)C2312yes—
51C2×#1084—22no—
52C2×#1084—22no—
53C2×#1084C2312yes—
54C2×#1084C2312yes—
55C2×(C3⋊Dic3)⋊C3—22no—
56C2×(C3⋊Dic3)⋊C3—22no—
57C2×(C3⋊Dic3)⋊C3D414yes—
58C2×(C3⋊Dic3)⋊C3C2312yes—
59C2×(C3⋊Dic3)⋊C3C2312yes—
60C2×(C3⋊Dic3)⋊C3C2312yes—
61C4×(C3⋊D3)⋊C3—28no—
62#6196—28no—
63C2×#924—91no—
64C2×#896—91no—
65C2×#896—91no—
66C23×C3⋊D3C2×D311yescharacteristic
67C23×C3×D3—31no—
68C22×C32×C4—21no—
69C22×C32×C4C2×D311yescharacteristic
70C22×C32×C4C2×D311yescharacteristic
71C22×C3⋊Dic3C2×D311yescharacteristic
72C22×C3⋊Dic3C2×D311yescharacteristic
73C22×C3⋊Dic3C2×D311yescharacteristic
74C2×C4×C3⋊D3—24no—
75C2×C4×C3⋊D3C2×D314yes—
76C22×C3×Dic3—31no—
77C22×C3×Dic3—31no—
78C22×C3×Dic3—31no—
79C22×C3×Dic3—31no—
80C22×C3×Dic3—31no—
81C22×C3×Dic3—31no—
82C22×C3×Dic3—31no—
83C22×C3×Dic3—61no—
84C22×C3×Dic3—61no—
85C2×C3×C4×D3—34no—
86C2×C3×C4×D3—64no—
87C2×#627—34no—
88C2×#627—34no—
89C2×#627—64no—
90C2×#627—64no—
91#3750C2×D314yescomplemented
92C3×Dic3⋊C4—34no—
93C3×Dic3⋊C4—34no—
94C3×Dic3⋊C4—34no—
95C3×Dic3⋊C4—64no—
96#3724—34no—
97Dic3⋊Dic3—34no—
98#3722—34no—
99#3722—34no—
100(C2×D3)⋊Dic3—34no—
101(C2×D3)⋊Dic3—34no—
102C22×H(3,𝔽3)C2×D412yes—
103C22×H(3,𝔽3)C2×Q812yes—
104C22×H(3,𝔽3)#4112yes—
105C22×H(3,𝔽3)C2411yescharacteristic
106C2×(C3⋊D3)⋊C3—28no—
107C2×(C3⋊D3)⋊C3—22no—
108C2×(C3⋊D3)⋊C3—22no—
109C2×(C3⋊D3)⋊C3C2×Q814yescomplemented
110C4×H(3,𝔽3)—22no—
111#1084—22no—
112#1084—22no—
113(C3⋊Dic3)⋊C3—22no—
114(C3⋊Dic3)⋊C3—22no—
115(C3⋊Dic3)⋊C3C2×D414yes—
116C23×C3×C4—91no—
117C22×C4×D3—31no—
118C22×C4×D3—91no—
119C22×Dic6—91no—
120C22×Dic6—91no—
121C22×Dic6—91no—
122C2×C3×C4⋊C4—91no—
123C2×C3×C4⋊C4—91no—
124C2×#217—91no—
125C2×#217—91no—
126C2×(C2×D3)⋊C4—91no—
127C2×(C2×D3)⋊C4—91no—
128C2×(C2×D3)⋊C4—91no—
129C2×(C2×D3)⋊C4—91no—
130C2×C4⋊Dic3—91no—
131#924—98no—
132C2×Dic3⋊C4—31no—
133C2×Dic3⋊C4—31no—
134C2×Dic3⋊C4—31no—
135C2×Dic3⋊C4—61no—
136C2×Dic3⋊C4—91no—
137C2×Dic3⋊C4—91no—
138C2×Dic3⋊C4—91no—
139#896—98no—
140#896—98no—
141C23×C32—21no—
142C23×C32C22×D311yescharacteristic
143C23×C32C22×D311yescharacteristic
144C22×C3⋊D3—22no—
145C22×C3⋊D3—22no—
146C22×C3⋊D3Dic614yescomplemented
147C22×C3⋊D3C22×D312yes—
148C22×C3×D3—34no—
149C22×C3×D3—32no—
150C22×C3×D3—62no—
151C22×C3×D3—62no—
152C2×C32×C4—22no—
153C2×C32×C4—22no—
154C2×C32×C4—22no—
155C2×C32×C4—42no—
156C2×C32×C4C22×D312yes—
157C2×C32×C4C22×D312yes—
158C2×C3⋊Dic3—22no—
159C2×C3⋊Dic3—22no—
160C2×C3⋊Dic3#6714yes—
161C2×C3⋊Dic3C22×D312yes—
162C2×C3⋊Dic3C22×D312yes—
163C2×C3⋊Dic3C22×D312yes—
164C4×C3⋊D3—28no—
165C2×C3×Dic3—34no—
166C2×C3×Dic3—32no—
167C2×C3×Dic3—32no—
168C2×C3×Dic3—32no—
169C2×C3×Dic3—32no—
170C2×C3×Dic3—32no—
171C2×C3×Dic3—32no—
172C2×C3×Dic3—32no—
173C2×C3×Dic3—62no—
174C2×C3×Dic3—62no—
175C2×C3×Dic3—64no—
176C2×C3×Dic3—62no—
177C2×C3×Dic3—62no—
178C2×C3×Dic3—62no—
179C2×C3×Dic3—62no—
180C2×C3×Dic3—62no—
181C2×C3×Dic3—62no—
182C2×C3×Dic3—122no—
183C3×C4×D3—68no—
184#627—68no—
185#627—68no—
186C2×C22⋊Q8—271noSylow
187C2×H(3,𝔽3)C22⋊Q814yescomplemented
188C2×H(3,𝔽3)C22×D411yescharacteristic
189C2×H(3,𝔽3)C22×Q811yescharacteristic
190C2×H(3,𝔽3)C2×#4111yescharacteristic
191(C3⋊D3)⋊C3—24no—
192C24×C3—91no—
193C23×D3—31no—
194C23×D3—91no—
195C22×C3×C4—31no—
196C22×C3×C4—31no—
197C22×C3×C4—61no—
198C22×C3×C4—94no—
199C22×C3×C4—91no—
200C22×C3×C4—91no—
201C22×C3×C4—91no—
202C22×C3×C4—91no—
203C22×C3×C4—91no—
204C22×C3×C4—184no—
205C22×C3×C4D3211yescharacteristic
206C22×Dic3—31no—
207C22×Dic3—31no—
208C22×Dic3—31no—
209C22×Dic3—31no—
210C22×Dic3—31no—
211C22×Dic3—31no—
212C22×Dic3—31no—
213C22×Dic3—61no—
214C22×Dic3—61no—
215C22×Dic3—91no—
216C22×Dic3—91no—
217C22×Dic3—91no—
218C2×C4×D3—34no—
219C2×C4×D3—64no—
220C2×C4×D3—94no—
221C2×C4×D3—184no—
222C2×Dic6—94no—
223C2×Dic6—94no—
224C2×Dic6—94no—
225C2×Dic6—184no—
226C2×Dic6—184no—
227C2×Dic6—184no—
228C3×C4⋊C4—94no—
229C3×C4⋊C4—94no—
230#217—94no—
231#217—94no—
232(C2×D3)⋊C4—94no—
233(C2×D3)⋊C4—94no—
234(C2×D3)⋊C4—94no—
235(C2×D3)⋊C4—94no—
236C4⋊Dic3—94no—
237Dic3⋊C4—34no—
238Dic3⋊C4—34no—
239Dic3⋊C4—34no—
240Dic3⋊C4—64no—
241Dic3⋊C4—94no—
242Dic3⋊C4—94no—
243Dic3⋊C4—94no—
244C22×C32—21no—
245C22×C32—22no—
246C22×C32—22no—
247C22×C32—22no—
248C22×C32C2×Dic612yes—
249C22×C32#23512yes—
250C22×C32#23512yes—
251C22×C32D3×D412yes—
252C22×C32D3×Q812yes—
253C22×C32C2×#6712yes—
254C22×C32C23×D311yescharacteristic
255C22×C32C23×D311yescharacteristic
256C2×C3⋊D3—28no—
257C2×C3⋊D3—22no—
258C2×C3⋊D3—22no—
259C2×C3⋊D3C2×Dic614yescomplemented
260C2×C3×D3—34no—
261C2×C3×D3—68no—
262C2×C3×D3—62no—
263C2×C3×D3—62no—
264C32×C4—22no—
265C32×C4—22no—
266C32×C4—42no—
267C3⋊Dic3—22no—
268C3⋊Dic3—22no—
269C3⋊Dic3C2×#6714yes—
270C3×Dic3—34no—
271C3×Dic3—64no—
272C3×Dic3—62no—
273C3×Dic3—62no—
274C3×Dic3—62no—
275C3×Dic3—62no—
276C3×Dic3—62no—
277C3×Dic3—62no—
278C3×Dic3—122no—
279C22×Q8—271no—
280C23×C4—91no—
281C22⋊Q8—278no—
282C2×C4⋊C4—91no—
283C2×C4⋊C4—91no—
284C2×C4⋊C4—271no—
285C2×C22⋊C4—271no—
286C2×C22⋊C4—271no—
287H(3,𝔽3)C2×C22⋊Q811yescharacteristic, complemented, Sylow
288C23×C3—31no—
289C23×C3—31no—
290C23×C3—61no—
291C23×C3—92no—
292C23×C3—94no—
293C23×C3—182no—
294C23×C3—182no—
295C23×C3C2×D3211yescharacteristic
296C22×D3—32no—
297C22×D3—34no—
298C22×D3—62no—
299C22×D3—62no—
300C22×D3—94no—
301C22×D3—92no—
302C22×D3—182no—
303C22×D3—182no—
304C2×C3×C4—22no—
305C2×C3×C4—32no—
306C2×C3×C4—32no—
307C2×C3×C4—62no—
308C2×C3×C4—62no—
309C2×C3×C4—62no—
310C2×C3×C4—92no—
311C2×C3×C4—94no—
312C2×C3×C4—92no—
313C2×C3×C4—92no—
314C2×C3×C4—92no—
315C2×C3×C4—92no—
316C2×C3×C4—122no—
317C2×C3×C4—188no—
318C2×C3×C4—182no—
319C2×C3×C4—182no—
320C2×C3×C4—182no—
321C2×C3×C4—182no—
322C2×C3×C4C2×D3212yes—
323C2×Dic3—32no—
324C2×Dic3—34no—
325C2×Dic3—32no—
326C2×Dic3—32no—
327C2×Dic3—32no—
328C2×Dic3—32no—
329C2×Dic3—32no—
330C2×Dic3—32no—
331C2×Dic3—62no—
332C2×Dic3—62no—
333C2×Dic3—62no—
334C2×Dic3—62no—
335C2×Dic3—62no—
336C2×Dic3—62no—
337C2×Dic3—64no—
338C2×Dic3—62no—
339C2×Dic3—62no—
340C2×Dic3—94no—
341C2×Dic3—92no—
342C2×Dic3—92no—
343C2×Dic3—92no—
344C2×Dic3—122no—
345C2×Dic3—182no—
346C2×Dic3—182no—
347C4×D3—68no—
348C4×D3—188no—
349Dic6—188no—
350Dic6—188no—
351Dic6—188no—
352C2×C32—21no—
353C2×C32—21no—
354C2×C32—21no—
355C2×C32—24no—
356C2×C32#89614yescomplemented
357C2×C32#92414yescomplemented
358C2×C32C22×Dic611yescharacteristic
359C2×C32C2×#23511yescharacteristic
360C2×C32C2×#23511yescharacteristic
361C2×C32C2×D3×D411yescharacteristic
362C2×C32C2×D3×Q811yescharacteristic
363C2×C32C22×#6711yescharacteristic
364C3⋊D3—24no—
365C3×D3—64no—
366C24—91no—
367C2×Q8—274no—
368C2×Q8—544no—
369C22×C4—91no—
370C22×C4—91no—
371C22×C4—91no—
372C22×C4—91no—
373C22×C4—91no—
374C22×C4—94no—
375C22×C4—184no—
376C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
377C4⋊C4—94no—
378C4⋊C4—94no—
379C4⋊C4—274no—
380C22⋊C4—274no—
381C22⋊C4—274no—
382C22×C3—31no—
383C22×C3—31no—
384C22×C3—32no—
385C22×C3—32no—
386C22×C3—32no—
387C22×C3—32no—
388C22×C3—32no—
389C22×C3—32no—
390C22×C3—61no—
391C22×C3—62no—
392C22×C3—62no—
393C22×C3—62no—
394C22×C3—94no—
395C22×C3—182no—
396C22×C3—188no—
397C22×C3—182no—
398C22×C3D3×Dic612yes—
399C22×C3#379812yes—
400C22×C3D3×#6712yes—
401C22×C3C22×D3211yescharacteristic
402C2×D3—34no—
403C2×D3—62no—
404C2×D3—62no—
405C2×D3—68no—
406C2×D3—94no—
407C2×D3—182no—
408C2×D3—188no—
409C2×D3—182no—
410C3×C4—22no—
411C3×C4—62no—
412C3×C4—62no—
413C3×C4—94no—
414C3×C4—122no—
415C3×C4—182no—
416C3×C4—182no—
417C3×C4—182no—
418C3×C4—182no—
419Dic3—34no—
420Dic3—62no—
421Dic3—62no—
422Dic3—62no—
423Dic3—62no—
424Dic3—64no—
425Dic3—62no—
426Dic3—62no—
427Dic3—94no—
428Dic3—122no—
429Dic3—182no—
430Dic3—182no—
431C32—21no—
432C32C2×#89611yescharacteristic, complemented
433C32C2×#92411yescharacteristic, complemented
434C23—92no—
435C23—94no—
436C23—182no—
437C23—182no—
438C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
439Q8—548no—
440C2×C4—22no—
441C2×C4—92no—
442C2×C4—94no—
443C2×C4—92no—
444C2×C4—92no—
445C2×C4—92no—
446C2×C4—92no—
447C2×C4—188no—
448C2×C4—182no—
449C2×C4—182no—
450C2×C4—182no—
451C2×C4—182no—
452C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
453C2×C3—31no—
454C2×C3—31no—
455C2×C3—34no—
456C2×C3—34no—
457C2×C3—31no—
458C2×C3—31no—
459C2×C3—31no—
460C2×C3—31no—
461C2×C3—61no—
462C2×C3—64no—
463C2×C3—61no—
464C2×C3—61no—
465C2×C3—184no—
466C2×C3#6385114yes—
467C2×C3C2×D3×Dic611yescharacteristic
468C2×C3C2×#379811yescharacteristic
469C2×C3C2×D3×#6711yescharacteristic
470D3—64no—
471D3—184no—
472C22—94no—
473C22—182no—
474C22—188no—
475C22—182no—
476C22((C3⋊D3)⋊C3)⋊Q812yes—
477C22#8865012yes—
478C22#8867612yes—
479C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
480C4—22no—
481C4—94no—
482C4—182no—
483C4—182no—
484C4—182no—
485C4—182no—
486C3—31no—
487C3—31no—
488C3—61no—
489C3C2×#6385111yescharacteristic
490C2—184no—
491C2#1172110114yesdirect product factor
492C2C2×((C3⋊D3)⋊C3)⋊Q811yescharacteristic
493C2C2×#8865011yescharacteristic
494C2C2×#8867611yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
FrattiniC22×C3C22×D32
FittingC22×C4×H(3,𝔽3)C22
socleC23×C3C2×D32
prime radicalC23×(C3⋊D3)⋊C3C22
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×C22⋊Q8
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C22×#6196C2semidirect
3. C2×#88540C2semidirect
4. C2×#3750⋊C3C2semidirect
7. C2×#88448C2semidirect
8. #11721101C2direct (N × H)
21. C2×#6196C22semidirect
22. #88540C22semidirect
23. #3750⋊C3C22semidirect
26. #88448C22semidirect
31. C2×#3750D3semidirect
47. C22×(C3⋊D3)⋊C3Q8semidirect
91. #3750C2×D3semidirect
109. C2×(C3⋊D3)⋊C3C2×Q8semidirect
146. C22×C3⋊D3Dic6semidirect
187. C2×H(3,𝔽3)C22⋊Q8semidirect
259. C2×C3⋊D3C2×Dic6semidirect
287. H(3,𝔽3)C2×C22⋊Q8semidirect
356. C2×C32#896semidirect
357. C2×C32#924semidirect
432. C32C2×#896semidirect
433. C32C2×#924semidirect
491. C2#11721101direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×(C3⋊D3)⋊C32
C22×#61962
C2×#885402
C2×#3750⋊C32
C2×#884502
C2×#884482
#117211012
C2×#638653
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