Finite Group Databasebeta

← C2×#64127⋊C3 #13153464

Subgroups — C2×#64127⋊C3 #13153464

Order 1728.

963 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88722C211yesmaximal, characteristic
2C2×Q8×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
3C2×Q16×H(3,𝔽3)C211yesmaximal, characteristic, complemented
4C2×#3755⋊C3C211yesmaximal, characteristic, complemented
5C2×#3756⋊C3C211yesmaximal, characteristic, complemented
6#64127⋊C3C218yesmaximal, direct product factor
7C2×#3744⋊C3C211yesmaximal, characteristic, complemented
8C2×#3743⋊C3C211yesmaximal, characteristic, complemented
9C2×#64127C311yesmaximal, characteristic, complemented
10C2×C3×#918—31nomaximal
11C2×Q8×H(3,𝔽3)C2211yescharacteristic
12C2×Q8×H(3,𝔽3)C2211yescharacteristic
13#88722—22no—
14#88722C2214yes—
15Q8×(C3⋊D3)⋊C3—22no—
16Q8×(C3⋊D3)⋊C3C2214yescomplemented
17C2×C4⋊((C3⋊D3)⋊C3)—21no—
18C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
19C2×C4×(C3⋊D3)⋊C3—21no—
20C2×C4×(C3⋊D3)⋊C3—21no—
21C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
22C2×#634⋊C3—21no—
23C2×#634⋊C3C2211yescharacteristic
24Q16×H(3,𝔽3)C2214yescomplemented
25C2×C8×H(3,𝔽3)C2211yescharacteristic
26#3755⋊C3C2214yescomplemented
27#3756⋊C3C2214yescomplemented
28C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
29#3744⋊C3C2214yescomplemented
30#3743⋊C3C2214yescomplemented
31C2×#3832C2×C311yescharacteristic
32C2×Q8×C3⋊D3C2×C311yescharacteristic, complemented
33C2×C3×#239—31no—
34C2×C3×D3×Q8—31no—
35C2×C32×Q16—21no—
36C2×C32×Q16D311yescharacteristic, complemented
37C2×C32×Q16C2×C311yescharacteristic, complemented
38C2×#3756C2×C311yescharacteristic, complemented
39C2×#3755C2×C311yescharacteristic, complemented
40#64127C2×C318yescomplemented
41C2×#3744C2×C311yescharacteristic, complemented
42C2×#3743C2×C311yescharacteristic, complemented
43C2×C3×#216—31no—
44C2×C3×#215—31no—
45C3×#918—38no—
46C2×C3×#204—31no—
47C2×C3×ΓL(1,𝔽25)—31no—
48C22×(C3⋊D3)⋊C3—21no—
49C22×(C3⋊D3)⋊C3D411yescharacteristic
50Q8×H(3,𝔽3)—21no—
51Q8×H(3,𝔽3)—21no—
52Q8×H(3,𝔽3)C2312yes—
53Q8×H(3,𝔽3)C2312yes—
54C2×C4×H(3,𝔽3)—21no—
55C2×C4×H(3,𝔽3)—21no—
56C2×C4×H(3,𝔽3)C2311yescharacteristic
57C2×(C3⋊Dic3)⋊C3—21no—
58C2×(C3⋊Dic3)⋊C3D411yescharacteristic
59C4⋊((C3⋊D3)⋊C3)—24no—
60C4⋊((C3⋊D3)⋊C3)—21no—
61C4⋊((C3⋊D3)⋊C3)C2312yes—
62C4×(C3⋊D3)⋊C3—24no—
63C4×(C3⋊D3)⋊C3—24no—
64C4×(C3⋊D3)⋊C3D412yes—
65C4×(C3⋊D3)⋊C3C2312yes—
66#634⋊C3—21no—
67#634⋊C3—24no—
68#634⋊C3C2312yes—
69C8×H(3,𝔽3)C2312yes—
70(C3⋊(C3⋊C8))⋊C3C2312yes—
71C2×C3×#137—91no—
72C2×#918—31no—
73C2×#918—91no—
74C2×C32×Q8—21no—
75C2×C32×Q8—21no—
76C2×C32×Q8C2×D311yescharacteristic
77C2×C32×Q8C2×D311yescharacteristic
78C2×C32×Q8C22×C311yescharacteristic
79C2×C32×Q8C22×C311yescharacteristic
80#3832—22no—
81#3832C22×C314yes—
82Q8×C3⋊D3—22no—
83Q8×C3⋊D3C22×C314yescomplemented
84C2×C4⋊(C3⋊D3)—21no—
85C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
86C2×C4×C3⋊D3—21no—
87C2×C4×C3⋊D3—21no—
88C2×C4×C3⋊D3C22×C311yescharacteristic
89C2×#634—21no—
90C2×#634C22×C311yescharacteristic
91C3×#239—34no—
92C3×#239—62no—
93C3×D3×Q8—34no—
94C3×D3×Q8—62no—
95C2×C3×D12—31no—
96C2×C3×D12—61no—
97C2×C3×C4×D3—31no—
98C2×C3×C4×D3—61no—
99C2×C3×C4×D3—61no—
100C2×C3×Dic6—31no—
101C2×C3×Dic6—61no—
102C32×Q16—24no—
103C32×Q16C2×D314yescomplemented
104C32×Q16C22×C314yescomplemented
105C2×C32×C8—21no—
106C2×C32×C8C2×D311yescharacteristic
107C2×C32×C8C22×C311yescharacteristic
108#3756C22×C314yescomplemented
109#3755C22×C314yescomplemented
110C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
111#3744C22×C314yescomplemented
112#3743C22×C314yescomplemented
113C3×#216—34no—
114C3×#215—34no—
115C2×C3×C3⋊C8—31no—
116C3×#204—34no—
117C3×ΓL(1,𝔽25)—34no—
118C22×H(3,𝔽3)C2×D411yescharacteristic
119C2×(C3⋊D3)⋊C3—22no—
120C2×(C3⋊D3)⋊C3—22no—
121C2×(C3⋊D3)⋊C3—42no—
122C2×(C3⋊D3)⋊C3C2×D412yes—
123C4×H(3,𝔽3)—22no—
124C4×H(3,𝔽3)—22no—
125C4×H(3,𝔽3)C2×D411yescharacteristic
126C4×H(3,𝔽3)C2411yescharacteristic
127(C3⋊Dic3)⋊C3—22no—
128(C3⋊Dic3)⋊C3C2×D412yes—
129C2×C3×#41—91no—
130C22×C3×Q8—91no—
131C2×#239—31no—
132C2×#239—91no—
133C2×D3×Q8—31no—
134C2×D3×Q8—91no—
135C3×#137—98no—
136C2×C3×Q16—31no—
137C2×C3×Q16—31no—
138C2×C3×Q16—61no—
139C2×C3×Q16—91no—
140C2×C3×Q16C3×D311yescharacteristic
141C2×C3×SD16—91no—
142C2×C3×SD16—91no—
143C2×C3×M16—91no—
144C2×#216—31no—
145C2×#216—91no—
146C2×#215—31no—
147C2×#215—91no—
148#918—38no—
149#918—98no—
150C2×#204—31no—
151C2×#204—91no—
152C2×ΓL(1,𝔽25)—31no—
153C2×ΓL(1,𝔽25)—91no—
154C22×C3⋊D3—21no—
155C22×C3⋊D3C3×D411yescharacteristic
156C22×C3×D3—31no—
157C22×C3×D3—61no—
158C32×Q8—21no—
159C32×Q8—21no—
160C32×Q8—21no—
161C32×Q8—21no—
162C32×Q8—22no—
163C32×Q8—22no—
164C32×Q8—41no—
165C32×Q8—41no—
166C32×Q8C22×D312yes—
167C32×Q8C22×D312yes—
168C32×Q8C23×C312yes—
169C32×Q8C23×C312yes—
170C2×C32×C4—21no—
171C2×C32×C4—21no—
172C2×C32×C4—21no—
173C2×C32×C4—21no—
174C2×C32×C4—21no—
175C2×C32×C4—41no—
176C2×C32×C4—41no—
177C2×C32×C4C22×D311yescharacteristic
178C2×C32×C4C23×C311yescharacteristic
179C2×C3⋊Dic3—21no—
180C2×C3⋊Dic3C3×D411yescharacteristic
181C4⋊(C3⋊D3)—24no—
182C4⋊(C3⋊D3)—21no—
183C4⋊(C3⋊D3)C23×C312yes—
184C4×C3⋊D3—24no—
185C4×C3⋊D3—24no—
186C4×C3⋊D3C3×D412yes—
187C4×C3⋊D3C23×C312yes—
188#634—21no—
189#634—24no—
190#634C23×C312yes—
191C2×C3×Dic3—31no—
192C2×C3×Dic3—61no—
193C3×D12—32no—
194C3×D12—64no—
195C3×D12—61no—
196C3×C4×D3—32no—
197C3×C4×D3—32no—
198C3×C4×D3—64no—
199C3×C4×D3—64no—
200C3×Dic6—32no—
201C3×Dic6—64no—
202C3×Dic6—61no—
203C32×C8—22no—
204C32×C8C22×D312yes—
205C32×C8C23×C312yes—
206C3⋊(C3⋊C8)C23×C312yes—
207C3×C3⋊C8—32no—
208C2×#137—91noSylow
209C2×H(3,𝔽3)#13712yescomplemented
210C2×H(3,𝔽3)C22×D411yescharacteristic
211(C3⋊D3)⋊C3—22no—
212(C3⋊D3)⋊C3—42no—
213C3×#41—94no—
214C3×#41—182no—
215C2×C3×Q8—31no—
216C2×C3×Q8—31no—
217C2×C3×Q8—31no—
218C2×C3×Q8—31no—
219C2×C3×Q8—61no—
220C2×C3×Q8—61no—
221C2×C3×Q8—91no—
222C2×C3×Q8—94no—
223C2×C3×Q8—181no—
224C2×C3×Q8—182no—
225C2×C3×Q8C2×C3×D311yescharacteristic
226C2×C3×Q8C2×C3×D311yescharacteristic
227C2×C3×D4—91no—
228C2×C3×D4—181no—
229C22×C3×C4—91no—
230C22×C3×C4—181no—
231C22×C3×C4—181no—
232#239—34no—
233#239—62no—
234#239—94no—
235#239—182no—
236D3×Q8—34no—
237D3×Q8—62no—
238D3×Q8—94no—
239D3×Q8—182no—
240C2×D12—31no—
241C2×D12—61no—
242C2×D12—91no—
243C2×D12—181no—
244C2×C4×D3—31no—
245C2×C4×D3—61no—
246C2×C4×D3—61no—
247C2×C4×D3—91no—
248C2×C4×D3—181no—
249C2×C4×D3—181no—
250C2×Dic6—31no—
251C2×Dic6—61no—
252C2×Dic6—91no—
253C2×Dic6—181no—
254C3×Q16—34no—
255C3×Q16—34no—
256C3×Q16—64no—
257C3×Q16—94no—
258C3×Q16C2×C3×D314yes—
259C3×SD16—94no—
260C3×SD16—94no—
261C3×M16—94no—
262C2×C3×C8—31no—
263C2×C3×C8—31no—
264C2×C3×C8—61no—
265C2×C3×C8—91no—
266C2×C3×C8C2×C3×D311yescharacteristic
267#216—34no—
268#216—94no—
269#215—34no—
270#215—94no—
271C2×C3⋊C8—31no—
272C2×C3⋊C8—91no—
273#204—34no—
274#204—94no—
275ΓL(1,𝔽25)—34no—
276ΓL(1,𝔽25)—94no—
277C22×C32—21no—
278C22×C32D3×D411yescharacteristic
279C22×C32C2×C3×D411yescharacteristic
280C2×C3⋊D3—22no—
281C2×C3⋊D3—22no—
282C2×C3⋊D3—42no—
283C2×C3⋊D3C2×C3×D412yes—
284C2×C3×D3—32no—
285C2×C3×D3—62no—
286C2×C3×D3—62no—
287C2×C3×D3—122no—
288C32×C4—22no—
289C32×C4—22no—
290C32×C4—22no—
291C32×C4—22no—
292C32×C4—21no—
293C32×C4—21no—
294C32×C4—42no—
295C32×C4—42no—
296C32×C4D3×D411yescharacteristic
297C32×C4C2×C3×D411yescharacteristic
298C32×C4C23×D311yescharacteristic
299C32×C4C24×C311yescharacteristic
300C3⋊Dic3—22no—
301C3⋊Dic3C2×C3×D412yes—
302C3×Dic3—32no—
303C3×Dic3—62no—
304C2×#41—91no—
305C22×Q8—91no—
306#137—98no—
307C2×Q16—91no—
308C2×Q16(C3⋊D3)⋊C311yescharacteristic, complemented
309C2×SD16—91no—
310C2×SD16—91no—
311C2×M16—91no—
312H(3,𝔽3)C2×#13711yescharacteristic, complemented, Sylow
313C23×C3—91no—
314C23×C3—181no—
315C22×D3—31no—
316C22×D3—61no—
317C22×D3—91no—
318C22×D3—181no—
319C3×Q8—21no—
320C3×Q8—21no—
321C3×Q8—32no—
322C3×Q8—32no—
323C3×Q8—32no—
324C3×Q8—32no—
325C3×Q8—61no—
326C3×Q8—61no—
327C3×Q8—61no—
328C3×Q8—61no—
329C3×Q8—62no—
330C3×Q8—62no—
331C3×Q8—92no—
332C3×Q8—121no—
333C3×Q8—121no—
334C3×Q8—181no—
335C3×Q8—184no—
336C3×Q8C22×C3×D312yes—
337C3×Q8C22×C3×D312yes—
338C3×D4—92no—
339C3×D4—181no—
340C3×D4—184no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—31no—
344C2×C3×C4—31no—
345C2×C3×C4—61no—
346C2×C3×C4—61no—
347C2×C3×C4—61no—
348C2×C3×C4—61no—
349C2×C3×C4—61no—
350C2×C3×C4—92no—
351C2×C3×C4—92no—
352C2×C3×C4—91no—
353C2×C3×C4—121no—
354C2×C3×C4—121no—
355C2×C3×C4—184no—
356C2×C3×C4—184no—
357C2×C3×C4—181no—
358C2×C3×C4C22×C3×D311yescharacteristic
359C2×Dic3—31no—
360C2×Dic3—61no—
361C2×Dic3—91no—
362C2×Dic3—181no—
363D12—32no—
364D12—64no—
365D12—61no—
366D12—92no—
367D12—181no—
368D12—184no—
369C4×D3—32no—
370C4×D3—32no—
371C4×D3—64no—
372C4×D3—64no—
373C4×D3—92no—
374C4×D3—92no—
375C4×D3—184no—
376C4×D3—184no—
377Dic6—32no—
378Dic6—64no—
379Dic6—61no—
380Dic6—92no—
381Dic6—184no—
382Dic6—181no—
383C3×C8—32no—
384C3×C8—32no—
385C3×C8—62no—
386C3×C8—92no—
387C3×C8C22×C3×D312yes—
388C3⋊C8—32no—
389C3⋊C8—92no—
390C2×C32—22no—
391C2×C32—21no—
392C2×C32#91812yescomplemented
393C2×C32C3×#13712yescomplemented
394C2×C32C2×D3×D411yescharacteristic
395C2×C32C22×C3×D411yescharacteristic
396C3⋊D3—22no—
397C3⋊D3—42no—
398C3×D3—62no—
399C3×D3—122no—
400#41—94no—
401#41—182no—
402C2×Q8—94no—
403C2×Q8—91no—
404C2×Q8—181no—
405C2×Q8—182no—
406C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
407C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
408C2×D4—91no—
409C2×D4—181no—
410C22×C4—91no—
411C22×C4—181no—
412C22×C4—181no—
413Q16—94no—
414Q16C2×(C3⋊D3)⋊C314yescomplemented
415SD16—94no—
416SD16—94no—
417M16—94no—
418C2×C8—91no—
419C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
420C22×C3—31no—
421C22×C3—31no—
422C22×C3—61no—
423C22×C3—92no—
424C22×C3—182no—
425C22×C3—182no—
426C22×C3—362no—
427C22×C3C3×D3×D411yescharacteristic
428C2×D3—32no—
429C2×D3—62no—
430C2×D3—62no—
431C2×D3—92no—
432C2×D3—122no—
433C2×D3—182no—
434C2×D3—182no—
435C2×D3—362no—
436C3×C4—22no—
437C3×C4—22no—
438C3×C4—31no—
439C3×C4—31no—
440C3×C4—31no—
441C3×C4—31no—
442C3×C4—62no—
443C3×C4—62no—
444C3×C4—62no—
445C3×C4—62no—
446C3×C4—61no—
447C3×C4—61no—
448C3×C4—92no—
449C3×C4—122no—
450C3×C4—122no—
451C3×C4—182no—
452C3×C4C3×D3×D411yescharacteristic
453C3×C4C23×C3×D311yescharacteristic
454Dic3—32no—
455Dic3—62no—
456Dic3—92no—
457Dic3—182no—
458C32—21no—
459C32C2×#91811yescharacteristic, complemented
460C32C2×C3×#13711yescharacteristic, complemented
461C23—91no—
462C23—181no—
463Q8—21no—
464Q8—21no—
465Q8—92no—
466Q8—184no—
467Q8—181no—
468Q8C22×(C3⋊D3)⋊C312yes—
469Q8C22×(C3⋊D3)⋊C312yes—
470D4—92no—
471D4—184no—
472D4—181no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—91no—
476C2×C4—92no—
477C2×C4—92no—
478C2×C4—184no—
479C2×C4—184no—
480C2×C4—181no—
481C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
482C8—92no—
483C8C22×(C3⋊D3)⋊C312yes—
484C2×C3—32no—
485C2×C3—32no—
486C2×C3—31no—
487C2×C3—31no—
488C2×C3—62no—
489C2×C3—61no—
490C2×C3—182no—
491C2×C3—362no—
492C2×C3C3×#91812yes—
493C2×C3C2×C3×D3×D411yescharacteristic
494D3—62no—
495D3—122no—
496D3—182no—
497D3—362no—
498C22—92no—
499C22—182no—
500C22—182no—
501C22—362no—
502C22D4×(C3⋊D3)⋊C311yescharacteristic
503C4—22no—
504C4—22no—
505C4—92no—
506C4—182no—
507C4D4×(C3⋊D3)⋊C311yescharacteristic
508C4C23×(C3⋊D3)⋊C311yescharacteristic
509C3—31no—
510C3—31no—
511C3—61no—
512C3C2×C3×#91811yescharacteristic
513C2—182no—
514C2—362no—
515C2#64127⋊C312yesdirect product factor
516C2C2×D4×(C3⋊D3)⋊C311yescharacteristic
subgroupG/H
centerC22D4×(C3⋊D3)⋊C3
commutator, cosocleC32×C4C24×C3
FrattiniC3×C4C23×C3×D3
FittingC2×Q16×H(3,𝔽3)C2
socleC22×C3C3×D3×D4
prime radicalC2×#88722C2
hypercenterC2×Q16(C3⋊D3)⋊C3
hypocenterC32C2×C3×#137
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q16

Complements

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

NH ≅ G/Ntype
2. C2×Q8×(C3⋊D3)⋊C3C2semidirect
3. C2×Q16×H(3,𝔽3)C2semidirect
4. C2×#3755⋊C3C2semidirect
5. C2×#3756⋊C3C2semidirect
6. #64127⋊C3C2direct (N × H)
7. C2×#3744⋊C3C2semidirect
8. C2×#3743⋊C3C2semidirect
9. C2×#64127C3semidirect
16. Q8×(C3⋊D3)⋊C3C22semidirect
24. Q16×H(3,𝔽3)C22semidirect
26. #3755⋊C3C22semidirect
27. #3756⋊C3C22semidirect
29. #3744⋊C3C22semidirect
30. #3743⋊C3C22semidirect
32. C2×Q8×C3⋊D3C2×C3semidirect
36. C2×C32×Q16D3semidirect
37. C2×C32×Q16C2×C3semidirect
38. C2×#3756C2×C3semidirect
39. C2×#3755C2×C3semidirect
40. #64127C2×C3semidirect
41. C2×#3744C2×C3semidirect
42. C2×#3743C2×C3semidirect
83. Q8×C3⋊D3C22×C3semidirect
103. C32×Q16C2×D3semidirect
104. C32×Q16C22×C3semidirect
108. #3756C22×C3semidirect
109. #3755C22×C3semidirect
111. #3744C22×C3semidirect
112. #3743C22×C3semidirect
209. C2×H(3,𝔽3)#137semidirect
308. C2×Q16(C3⋊D3)⋊C3semidirect
312. H(3,𝔽3)C2×#137semidirect
392. C2×C32#918semidirect
393. C2×C32C3×#137semidirect
414. Q16C2×(C3⋊D3)⋊C3semidirect
459. C32C2×#918semidirect
460. C32C2×C3×#137semidirect
515. C2#64127⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#887222
C2×Q8×(C3⋊D3)⋊C32
C2×Q16×H(3,𝔽3)2
C2×#3755⋊C32
C2×#3756⋊C32
#64127⋊C32
C2×#3744⋊C32
C2×#3743⋊C32
C2×#641273
C2×C3×#9183