Finite Group Databasebeta

← #10596299⋊C3 #13153718

Subgroups — #10596299⋊C3 #13153718

Order 1728.

1319 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×H(3,𝔽3)C211yesmaximal, characteristic, complemented
2C22×C4×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
3#64155⋊C3C212yesmaximal, complemented
4#64147⋊C3C212yesmaximal, complemented
5#64146⋊C3C212yesmaximal, complemented
6#64144⋊C3C212yesmaximal, complemented
7C4×#638⋊C3C214yesmaximal, complemented
8#64135⋊C3C211yesmaximal, characteristic, complemented
9#10596299C311yesmaximal, characteristic, complemented
10C3×#5828—31nomaximal
11C23×(C3⋊D3)⋊C3C2211yescharacteristic
12#41×H(3,𝔽3)—24no—
13C2×Q8×H(3,𝔽3)C2211yescharacteristic, complemented
14C2×D4×H(3,𝔽3)C2212yes—
15C2×D4×H(3,𝔽3)C2211yescharacteristic, complemented
16C22×C4×H(3,𝔽3)C2211yescharacteristic
17C22×C4×H(3,𝔽3)C2212yescomplemented
18C2×#638⋊C3C2214yes—
19C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
20C2×C4×(C3⋊D3)⋊C3—24no—
21C2×C4×(C3⋊D3)⋊C3C2212yes—
22C2×C4×(C3⋊D3)⋊C3C2212yes—
23#3757⋊C3C2212yes—
24#3757⋊C3C2214yescomplemented
25#3752⋊C3C2214yes—
26C4⋊((C3⋊Dic3)⋊C3)C2212yescomplemented
27#3750⋊C3C2214yescomplemented
28C4×(C3⋊Dic3)⋊C3C2212yescomplemented
29C2×C32×#41—21no—
30C2×C32×#41D311yescharacteristic, complemented
31C2×C32×#41C2×C311yescharacteristic, complemented
32C22×C4×C3⋊D3C2×C311yescharacteristic, complemented
33C22×C3×C4×D3—31no—
34#64155C2×C312yescomplemented
35#64147C2×C312yescomplemented
36#64146C2×C312yescomplemented
37#64144C2×C312yescomplemented
38C4×#638C2×C314yescomplemented
39#64135C2×C311yescharacteristic, complemented
40C3×#946—32no—
41C3×#938—32no—
42C3×#937—32no—
43C3×#935—32no—
44C3×C4×#67—34no—
45C3×#926—31no—
46C23×H(3,𝔽3)C2312yes—
47C23×H(3,𝔽3)C2311yescharacteristic
48C22×(C3⋊D3)⋊C3—22no—
49C22×(C3⋊D3)⋊C3—24no—
50C22×(C3⋊D3)⋊C3C2312yes—
51Q8×H(3,𝔽3)—22no—
52D4×H(3,𝔽3)—24no—
53D4×H(3,𝔽3)—22no—
54C2×C4×H(3,𝔽3)—24no—
55C2×C4×H(3,𝔽3)D414yescomplemented
56C2×C4×H(3,𝔽3)C2312yes—
57C2×C4×H(3,𝔽3)C2311yescharacteristic
58C2×C4×H(3,𝔽3)C2311yescharacteristic
59#638⋊C3—28no—
60C2×(C3⋊Dic3)⋊C3—22no—
61C2×(C3⋊Dic3)⋊C3C2312yes—
62C2×(C3⋊Dic3)⋊C3C2314yes—
63C4×(C3⋊D3)⋊C3—24no—
64C4×(C3⋊D3)⋊C3—24no—
65C3×#525—91no—
66#5828—31no—
67#5828—91no—
68C23×C3⋊D3C22×C311yescharacteristic
69C23×C3×D3—31no—
70C32×#41—28no—
71C32×#41—24no—
72C32×#41—24no—
73C2×C32×Q8—21no—
74C2×C32×Q8C2×D311yescharacteristic, complemented
75C2×C32×Q8C22×C311yescharacteristic, complemented
76C2×C32×D4—22no—
77C2×C32×D4—21no—
78C2×C32×D4C2×D312yes—
79C2×C32×D4C2×D311yescharacteristic, complemented
80C2×C32×D4C22×C312yes—
81C2×C32×D4C22×C311yescharacteristic, complemented
82C22×C32×C4—21no—
83C22×C32×C4—22no—
84C22×C32×C4C2×D311yescharacteristic
85C22×C32×C4C2×D312yescomplemented
86C22×C32×C4C22×C311yescharacteristic
87C22×C32×C4C22×C312yescomplemented
88C2×#638C22×C314yes—
89C22×C3⋊Dic3C22×C311yescharacteristic
90C2×C4×C3⋊D3—24no—
91C2×C4×C3⋊D3C22×C312yes—
92C2×C4×C3⋊D3C22×C312yes—
93C2×C3×#67—34no—
94C22×C3×Dic3—31no—
95C2×C3×C4×D3—32no—
96C2×C3×C4×D3—32no—
97C2×C3×C4×D3—64no—
98#3757C22×C312yes—
99#3757C22×C314yescomplemented
100#3752C22×C314yes—
101C4⋊(C3⋊Dic3)C22×C312yescomplemented
102#3750C22×C314yescomplemented
103C4×C3⋊Dic3C22×C312yescomplemented
104C3×#217—32no—
105C3×#217—34no—
106C3×(C2×D3)⋊C4—34no—
107C3×C4⋊Dic3—32no—
108C3×Dic3⋊C4—34no—
109C3×C4×Dic3—32no—
110C22×H(3,𝔽3)—22no—
111C22×H(3,𝔽3)—22no—
112C22×H(3,𝔽3)—24no—
113C22×H(3,𝔽3)C2×D412yes—
114C22×H(3,𝔽3)C2411yescharacteristic
115C2×(C3⋊D3)⋊C3—28no—
116C2×(C3⋊D3)⋊C3—22no—
117C2×(C3⋊D3)⋊C3—22no—
118C2×(C3⋊D3)⋊C3#4114yescomplemented
119C4×H(3,𝔽3)—22no—
120C4×H(3,𝔽3)C2×D412yes—
121C4×H(3,𝔽3)C2×D412yes—
122(C3⋊Dic3)⋊C3—22no—
123(C3⋊Dic3)⋊C3—24no—
124C2×C3×#41—31no—
125C2×C3×#41—31no—
126C2×C3×#41—61no—
127C2×C3×#41C3×D311yescharacteristic
128C23×C3×C4—91no—
129C22×C4×D3—31no—
130C22×C4×D3—91no—
131C3×#123—92no—
132C3×C22⋊Q8—92no—
133C3×#121—92no—
134C3×#120—92no—
135C3×C4×D4—94no—
136C3×#117—91no—
137#946—32no—
138#946—92no—
139#938—32no—
140#938—92no—
141#937—32no—
142#937—92no—
143#935—32no—
144#935—92no—
145C4×#67—34no—
146C4×#67—94no—
147#926—31no—
148#926—91no—
149C23×C32—22no—
150C23×C32—21no—
151C23×C32C22×D312yes—
152C23×C32C22×D311yescharacteristic
153C23×C32C23×C312yes—
154C23×C32C23×C311yescharacteristic
155C22×C3⋊D3—22no—
156C22×C3⋊D3—24no—
157C22×C3⋊D3C23×C312yes—
158C22×C3×D3—32no—
159C22×C3×D3—64no—
160C22×C3×D3—62no—
161C32×Q8—24no—
162C32×Q8—22no—
163C32×Q8—22no—
164C32×D4—28no—
165C32×D4—24no—
166C32×D4—24no—
167C32×D4—24no—
168C32×D4—22no—
169C32×D4—22no—
170C2×C32×C4—28no—
171C2×C32×C4—24no—
172C2×C32×C4—24no—
173C2×C32×C4—24no—
174C2×C32×C4—21no—
175C2×C32×C4—22no—
176C2×C32×C4—21no—
177C2×C32×C4#6714yescomplemented
178C2×C32×C4C3×D414yescomplemented
179C2×C32×C4C22×D312yes—
180C2×C32×C4C22×D311yescharacteristic
181C2×C32×C4C22×D311yescharacteristic
182C2×C32×C4C23×C312yes—
183C2×C32×C4C23×C311yescharacteristic
184C2×C32×C4C23×C311yescharacteristic
185#638—28no—
186C2×C3⋊Dic3—22no—
187C2×C3⋊Dic3C23×C314yes—
188C2×C3⋊Dic3C23×C312yes—
189C4×C3⋊D3—24no—
190C4×C3⋊D3—24no—
191C3×#67—68no—
192C2×C3×Dic3—34no—
193C2×C3×Dic3—32no—
194C2×C3×Dic3—62no—
195C3×C4×D3—64no—
196C3×C4×D3—64no—
197#525—91noSylow
198C2×H(3,𝔽3)—22no—
199C2×H(3,𝔽3)—42no—
200C2×H(3,𝔽3)C22×D411yescharacteristic
201C2×H(3,𝔽3)C2×#4112yes—
202(C3⋊D3)⋊C3—24no—
203C24×C3—91no—
204C23×D3—31no—
205C23×D3—91no—
206C3×#41—24no—
207C3×#41—64no—
208C3×#41—64no—
209C3×#41—68no—
210C2×C3×Q8—31no—
211C2×C3×Q8—31no—
212C2×C3×Q8—61no—
213C2×C3×Q8C2×C3×D311yescharacteristic
214C2×C3×D4—32no—
215C2×C3×D4—32no—
216C2×C3×D4—31no—
217C2×C3×D4—31no—
218C2×C3×D4—62no—
219C2×C3×D4—61no—
220C2×C3×D4—94no—
221C2×C3×D4C2×C3×D312yes—
222C2×C3×D4C2×C3×D311yescharacteristic
223C22×C3×C4—31no—
224C22×C3×C4—31no—
225C22×C3×C4—32no—
226C22×C3×C4—32no—
227C22×C3×C4—61no—
228C22×C3×C4—62no—
229C22×C3×C4—92no—
230C22×C3×C4—92no—
231C22×C3×C4—91no—
232C22×C3×C4—184no—
233C22×C3×C4C2×C3×D311yescharacteristic
234C22×C3×C4C2×C3×D312yes—
235C2×#67—34no—
236C2×#67—94no—
237C22×Dic3—31no—
238C22×Dic3—91no—
239C2×C4×D3—32no—
240C2×C4×D3—32no—
241C2×C4×D3—64no—
242C2×C4×D3—92no—
243C2×C4×D3—92no—
244C2×C4×D3—184no—
245C3×C4⋊C4—94no—
246C3×C4⋊C4—92no—
247C3×C22⋊C4—94no—
248C3×C22⋊C4—92no—
249C3×C22⋊C4—94no—
250C3×C42—92no—
251#217—32no—
252#217—34no—
253#217—92no—
254#217—94no—
255(C2×D3)⋊C4—34no—
256(C2×D3)⋊C4—94no—
257C4⋊Dic3—32no—
258C4⋊Dic3—92no—
259Dic3⋊C4—34no—
260Dic3⋊C4—94no—
261C4×Dic3—32no—
262C4×Dic3—92no—
263C22×C32—22no—
264C22×C32—22no—
265C22×C32—24no—
266C22×C32—24no—
267C22×C32—22no—
268C22×C32—22no—
269C22×C32—24no—
270C22×C32—21no—
271C22×C32—22no—
272C22×C32—42no—
273C22×C32—44no—
274C22×C32C2×#6712yes—
275C22×C32C2×C3×D412yes—
276C22×C32C23×D311yescharacteristic
277C22×C32C24×C311yescharacteristic
278C2×C3⋊D3—28no—
279C2×C3⋊D3—22no—
280C2×C3⋊D3—22no—
281C2×C3⋊D3C3×#4114yescomplemented
282C2×C3×D3—34no—
283C2×C3×D3—68no—
284C2×C3×D3—62no—
285C2×C3×D3—62no—
286C32×C4—24no—
287C32×C4—22no—
288C32×C4—22no—
289C32×C4—22no—
290C32×C4—22no—
291C32×C4C2×#6712yes—
292C32×C4C2×#6712yes—
293C32×C4C2×C3×D412yes—
294C32×C4C2×C3×D412yes—
295C3⋊Dic3—22no—
296C3⋊Dic3—24no—
297C3×Dic3—64no—
298C3×Dic3—62no—
299C2×#41(C3⋊D3)⋊C311yescharacteristic, complemented
300C23×C4—91no—
301#123—92no—
302C22⋊Q8—92no—
303#121—92no—
304#120—92no—
305C4×D4—94no—
306#117—91no—
307H(3,𝔽3)#52511yescharacteristic, complemented, Sylow
308C23×C3—31no—
309C23×C3—31no—
310C23×C3—32no—
311C23×C3—32no—
312C23×C3—62no—
313C23×C3—61no—
314C23×C3—92no—
315C23×C3—182no—
316C23×C3—184no—
317C23×C3C22×C3×D312yes—
318C23×C3C22×C3×D311yescharacteristic
319C22×D3—32no—
320C22×D3—64no—
321C22×D3—62no—
322C22×D3—92no—
323C22×D3—184no—
324C22×D3—182no—
325C3×Q8—22no—
326C3×Q8—62no—
327C3×Q8—62no—
328C3×Q8—64no—
329C3×D4—24no—
330C3×D4—22no—
331C3×D4—62no—
332C3×D4—62no—
333C3×D4—64no—
334C3×D4—64no—
335C3×D4—68no—
336C3×D4—64no—
337C3×D4—188no—
338C2×C3×C4—24no—
339C2×C3×C4—34no—
340C2×C3×C4—34no—
341C2×C3×C4—31no—
342C2×C3×C4—31no—
343C2×C3×C4—32no—
344C2×C3×C4—32no—
345C2×C3×C4—31no—
346C2×C3×C4—31no—
347C2×C3×C4—64no—
348C2×C3×C4—64no—
349C2×C3×C4—68no—
350C2×C3×C4—64no—
351C2×C3×C4—62no—
352C2×C3×C4—61no—
353C2×C3×C4—61no—
354C2×C3×C4—94no—
355C2×C3×C4—92no—
356C2×C3×C4—184no—
357C2×C3×C4—184no—
358C2×C3×C4—182no—
359C2×C3×C4C3×#6714yes—
360C2×C3×C4C22×C3×D312yes—
361C2×C3×C4C22×C3×D311yescharacteristic
362C2×C3×C4C22×C3×D311yescharacteristic
363#67—68no—
364#67—188no—
365C2×Dic3—34no—
366C2×Dic3—32no—
367C2×Dic3—62no—
368C2×Dic3—94no—
369C2×Dic3—92no—
370C2×Dic3—182no—
371C4×D3—64no—
372C4×D3—64no—
373C4×D3—184no—
374C4×D3—184no—
375C2×C32—22no—
376C2×C32—22no—
377C2×C32—22no—
378C2×C32—21no—
379C2×C32—42no—
380C2×C32—44no—
381C2×C32—42no—
382C2×C32—42no—
383C2×C32D3×#4112yes—
384C2×C32C22×#6711yescharacteristic
385C2×C32C22×C3×D411yescharacteristic
386C2×C32C2×C3×#4112yes—
387C3⋊D3—24no—
388C3×D3—64no—
389C24—91no—
390#41—24no—
391C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic, complemented
392C2×D4—94no—
393C2×D4C2×(C3⋊D3)⋊C312yes—
394C2×D4C2×(C3⋊D3)⋊C311yescharacteristic, complemented
395C22×C4—91no—
396C22×C4—92no—
397C22×C4—92no—
398C22×C4—184no—
399C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
400C22×C4C2×(C3⋊D3)⋊C312yescomplemented
401C4⋊C4—92no—
402C4⋊C4—94no—
403C22⋊C4—92no—
404C22⋊C4—94no—
405C22⋊C4—94no—
406C42—92no—
407C22×C3—22no—
408C22×C3—24no—
409C22×C3—22no—
410C22×C3—31no—
411C22×C3—31no—
412C22×C3—32no—
413C22×C3—32no—
414C22×C3—62no—
415C22×C3—62no—
416C22×C3—64no—
417C22×C3—62no—
418C22×C3—62no—
419C22×C3—64no—
420C22×C3—64no—
421C22×C3—61no—
422C22×C3—62no—
423C22×C3—94no—
424C22×C3—122no—
425C22×C3—124no—
426C22×C3—182no—
427C22×C3—182no—
428C22×C3—188no—
429C22×C3C2×C3×#6712yes—
430C22×C3C23×C3×D311yescharacteristic
431C2×D3—34no—
432C2×D3—68no—
433C2×D3—62no—
434C2×D3—62no—
435C2×D3—94no—
436C2×D3—182no—
437C2×D3—182no—
438C2×D3—188no—
439C3×C4—22no—
440C3×C4—32no—
441C3×C4—32no—
442C3×C4—32no—
443C3×C4—32no—
444C3×C4—62no—
445C3×C4—62no—
446C3×C4—64no—
447C3×C4—62no—
448C3×C4—62no—
449C3×C4—182no—
450C3×C4—184no—
451C3×C4C2×C3×#6712yes—
452C3×C4C2×C3×#6712yes—
453Dic3—64no—
454Dic3—62no—
455Dic3—182no—
456Dic3—184no—
457C32—21no—
458C32#582811yescharacteristic, complemented
459C32C3×#52511yescharacteristic, complemented
460C23—92no—
461C23—184no—
462C23—182no—
463C23C22×(C3⋊D3)⋊C312yes—
464C23C22×(C3⋊D3)⋊C311yescharacteristic
465Q8—22no—
466D4—24no—
467D4—22no—
468D4—188no—
469C2×C4—24no—
470C2×C4—94no—
471C2×C4—92no—
472C2×C4—182no—
473C2×C4—184no—
474C2×C4—184no—
475C2×C4#638⋊C314yescomplemented
476C2×C4C22×(C3⋊D3)⋊C312yes—
477C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
478C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
479C2×C3—22no—
480C2×C3—32no—
481C2×C3—32no—
482C2×C3—31no—
483C2×C3—31no—
484C2×C3—42no—
485C2×C3—62no—
486C2×C3—62no—
487C2×C3—62no—
488C2×C3—61no—
489C2×C3—122no—
490C2×C3—122no—
491C2×C3—122no—
492C2×C3—124no—
493C2×C3—184no—
494C2×C3C3×D3×#4112yes—
495C2×C3C22×C3×#6711yescharacteristic
496D3—64no—
497D3—184no—
498C22—22no—
499C22—24no—
500C22—22no—
501C22—94no—
502C22—188no—
503C22—182no—
504C22—182no—
505C22C2×#638⋊C312yes—
506C22C23×(C3⋊D3)⋊C311yescharacteristic
507C4—22no—
508C4—184no—
509C4—182no—
510C4C2×#638⋊C312yes—
511C4C2×#638⋊C312yes—
512C3—31no—
513C3—31no—
514C3—61no—
515C3C3×#582811yescharacteristic
516C2—22no—
517C2—42no—
518C2—184no—
519C2#41×(C3⋊D3)⋊C312yes—
520C2C22×#638⋊C311yescharacteristic
subgroupG/H
centerC2×C4C22×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingC2×#41×H(3,𝔽3)C2
hypercenterC2×#41(C3⋊D3)⋊C3
hypocenterC32C3×#525
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. C2×#41×H(3,𝔽3)C2semidirect
2. C22×C4×(C3⋊D3)⋊C3C2semidirect
3. #64155⋊C3C2semidirect
4. #64147⋊C3C2semidirect
5. #64146⋊C3C2semidirect
6. #64144⋊C3C2semidirect
7. C4×#638⋊C3C2semidirect
8. #64135⋊C3C2semidirect
9. #10596299C3semidirect
13. C2×Q8×H(3,𝔽3)C22semidirect
15. C2×D4×H(3,𝔽3)C22semidirect
17. C22×C4×H(3,𝔽3)C22semidirect
24. #3757⋊C3C22semidirect
26. C4⋊((C3⋊Dic3)⋊C3)C22semidirect
27. #3750⋊C3C22semidirect
28. C4×(C3⋊Dic3)⋊C3C22semidirect
30. C2×C32×#41D3semidirect
31. C2×C32×#41C2×C3semidirect
32. C22×C4×C3⋊D3C2×C3semidirect
34. #64155C2×C3semidirect
35. #64147C2×C3semidirect
36. #64146C2×C3semidirect
37. #64144C2×C3semidirect
38. C4×#638C2×C3semidirect
39. #64135C2×C3semidirect
55. C2×C4×H(3,𝔽3)D4semidirect
74. C2×C32×Q8C2×D3semidirect
75. C2×C32×Q8C22×C3semidirect
79. C2×C32×D4C2×D3semidirect
81. C2×C32×D4C22×C3semidirect
85. C22×C32×C4C2×D3semidirect
87. C22×C32×C4C22×C3semidirect
99. #3757C22×C3semidirect
101. C4⋊(C3⋊Dic3)C22×C3semidirect
102. #3750C22×C3semidirect
103. C4×C3⋊Dic3C22×C3semidirect
118. C2×(C3⋊D3)⋊C3#41semidirect
177. C2×C32×C4#67semidirect
178. C2×C32×C4C3×D4semidirect
281. C2×C3⋊D3C3×#41semidirect
299. C2×#41(C3⋊D3)⋊C3semidirect
307. H(3,𝔽3)#525semidirect
391. C2×Q8C2×(C3⋊D3)⋊C3semidirect
394. C2×D4C2×(C3⋊D3)⋊C3semidirect
400. C22×C4C2×(C3⋊D3)⋊C3semidirect
458. C32#5828semidirect
459. C32C3×#525semidirect
475. C2×C4#638⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×#41×H(3,𝔽3)2
C22×C4×(C3⋊D3)⋊C32
#64155⋊C32
#64147⋊C32
#64146⋊C32
#64144⋊C32
C4×#638⋊C32
#64135⋊C32
#105962993
C3×#58283