Finite Group Databasebeta

← #10596304⋊C3 #13153723

Subgroups — #10596304⋊C3 #13153723

Order 1728.

1193 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×H(3,𝔽3)C211yesmaximal, characteristic, complemented
2C2×#3828⋊C3C211yesmaximal, characteristic, complemented
3#64155⋊C3C212yesmaximal, complemented
4#64148⋊C3C214yesmaximal, complemented
5#64147⋊C3C212yesmaximal, complemented
6#64139⋊C3C212yesmaximal, complemented
7C2×C4⋊((C3⋊Dic3)⋊C3)C211yesmaximal, characteristic, complemented
8#64133⋊C3C212yesmaximal, complemented
9#10596304C311yesmaximal, characteristic, complemented
10C3×#5833—31nomaximal
11#41×H(3,𝔽3)—24no—
12C2×Q8×H(3,𝔽3)C2211yescharacteristic
13C2×D4×H(3,𝔽3)C2211yescharacteristic
14C2×D4×H(3,𝔽3)C2212yes—
15C22×C4×H(3,𝔽3)C2212yes—
16C22×C4×H(3,𝔽3)C2211yescharacteristic
17C2×#638⋊C3C2212yes—
18C2×#638⋊C3C2214yes—
19C22×(C3⋊Dic3)⋊C3C2212yes—
20#3828⋊C3—24no—
21C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
22C2×C4×(C3⋊D3)⋊C3C2212yes—
23C2×#634⋊C3C2211yescharacteristic
24#3757⋊C3C2214yes—
25#3752⋊C3C2214yes—
26C4⋊((C3⋊Dic3)⋊C3)C2212yes—
27C4⋊((C3⋊Dic3)⋊C3)C2212yes—
28#3750⋊C3C2214yes—
29C2×C32×#41—21no—
30C2×C32×#41D311yescharacteristic, complemented
31C2×C32×#41C2×C311yescharacteristic, complemented
32C2×#3828C2×C311yescharacteristic, complemented
33C2×C3×#235—31no—
34#64155C2×C312yescomplemented
35#64148C2×C314yescomplemented
36#64147C2×C312yescomplemented
37#64139C2×C312yescomplemented
38C2×C4⋊(C3⋊Dic3)C2×C311yescharacteristic, complemented
39#64133C2×C312yescomplemented
40C3×#946—32no—
41C3×#939—34no—
42C3×#938—32no—
43C3×#930—32no—
44C2×C3×C4⋊Dic3—31no—
45C3×#924—32no—
46C23×H(3,𝔽3)C2312yes—
47C23×H(3,𝔽3)C2311yescharacteristic
48C22×(C3⋊D3)⋊C3C2312yes—
49Q8×H(3,𝔽3)—22no—
50D4×H(3,𝔽3)—24no—
51D4×H(3,𝔽3)—22no—
52C2×C4×H(3,𝔽3)—24no—
53C2×C4×H(3,𝔽3)D414yescomplemented
54C2×C4×H(3,𝔽3)C2312yes—
55C2×C4×H(3,𝔽3)C2311yescharacteristic
56C2×C4×H(3,𝔽3)C2311yescharacteristic
57#638⋊C3—24no—
58#638⋊C3—44no—
59C2×(C3⋊Dic3)⋊C3—42no—
60C2×(C3⋊Dic3)⋊C3C2312yes—
61C2×(C3⋊Dic3)⋊C3C2314yes—
62C4⋊((C3⋊D3)⋊C3)—22no—
63C4×(C3⋊D3)⋊C3—24no—
64#634⋊C3—22no—
65C3×#537—91no—
66#5833—31no—
67#5833—91no—
68C32×#41—28no—
69C32×#41—24no—
70C32×#41—24no—
71C2×C32×Q8—21no—
72C2×C32×Q8C2×D311yescharacteristic
73C2×C32×Q8C22×C311yescharacteristic
74C2×C32×D4—21no—
75C2×C32×D4—22no—
76C2×C32×D4C2×D311yescharacteristic
77C2×C32×D4C2×D312yes—
78C2×C32×D4C22×C311yescharacteristic
79C2×C32×D4C22×C312yes—
80C22×C32×C4—22no—
81C22×C32×C4—21no—
82C22×C32×C4C2×D312yes—
83C22×C32×C4C2×D311yescharacteristic
84C22×C32×C4C22×C312yes—
85C22×C32×C4C22×C311yescharacteristic
86C2×#638C22×C312yes—
87C2×#638C22×C314yes—
88C22×C3⋊Dic3C22×C312yes—
89#3828—24no—
90C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
91C2×C4×C3⋊D3C22×C312yes—
92C2×#634C22×C311yescharacteristic
93C2×C3×#67—32no—
94C2×C3×#67—34no—
95C22×C3×Dic3—32no—
96C3×#235—64no—
97C2×C3×D12—31no—
98C2×C3×C4×D3—32no—
99C2×C3×Dic6—31no—
100#3757C22×C314yes—
101#3752C22×C314yes—
102C4⋊(C3⋊Dic3)C22×C312yes—
103C4⋊(C3⋊Dic3)C22×C312yes—
104#3750C22×C314yes—
105C3×#217—34no—
106C3×(C2×D3)⋊C4—34no—
107C3×C4⋊Dic3—32no—
108C3×C4⋊Dic3—32no—
109C3×Dic3⋊C4—34no—
110C22×H(3,𝔽3)—21no—
111C22×H(3,𝔽3)—21no—
112C22×H(3,𝔽3)—22no—
113C22×H(3,𝔽3)—22no—
114C22×H(3,𝔽3)—22no—
115C22×H(3,𝔽3)C2×D412yes—
116C22×H(3,𝔽3)C2411yescharacteristic
117C2×(C3⋊D3)⋊C3—22no—
118C2×(C3⋊D3)⋊C3—22no—
119C2×(C3⋊D3)⋊C3—22no—
120C4×H(3,𝔽3)—22no—
121C4×H(3,𝔽3)C2×D412yes—
122C4×H(3,𝔽3)C2×D412yes—
123(C3⋊Dic3)⋊C3—22no—
124(C3⋊Dic3)⋊C3—24no—
125C2×C3×#41—31no—
126C2×C3×#41—31no—
127C2×C3×#41—61no—
128C2×C3×#41—91no—
129C2×C3×#41C3×D311yescharacteristic
130C2×#235—31no—
131C2×#235—91no—
132C3×C22⋊Q8—92no—
133C3×C22⋊Q8—92no—
134C3×#121—92no—
135C3×#121—92no—
136C3×#121—94no—
137C2×C3×C4⋊C4—91no—
138#946—32no—
139#946—92no—
140#939—34no—
141#939—94no—
142#938—32no—
143#938—92no—
144#930—32no—
145#930—92no—
146C2×C4⋊Dic3—31no—
147C2×C4⋊Dic3—91no—
148#924—32no—
149#924—92no—
150C23×C32—22no—
151C23×C32—21no—
152C23×C32C22×D312yes—
153C23×C32C22×D311yescharacteristic
154C23×C32C23×C312yes—
155C23×C32C23×C311yescharacteristic
156C22×C3⋊D3C23×C312yes—
157C22×C3×D3—32no—
158C32×Q8—24no—
159C32×Q8—22no—
160C32×Q8—22no—
161C32×D4—28no—
162C32×D4—24no—
163C32×D4—24no—
164C32×D4—24no—
165C32×D4—22no—
166C32×D4—22no—
167C2×C32×C4—28no—
168C2×C32×C4—24no—
169C2×C32×C4—24no—
170C2×C32×C4—24no—
171C2×C32×C4—21no—
172C2×C32×C4—22no—
173C2×C32×C4—21no—
174C2×C32×C4#6714yescomplemented
175C2×C32×C4C3×D414yescomplemented
176C2×C32×C4C22×D312yes—
177C2×C32×C4C22×D311yescharacteristic
178C2×C32×C4C22×D311yescharacteristic
179C2×C32×C4C23×C312yes—
180C2×C32×C4C23×C311yescharacteristic
181C2×C32×C4C23×C311yescharacteristic
182#638—24no—
183#638—44no—
184C2×C3⋊Dic3—42no—
185C2×C3⋊Dic3C23×C312yes—
186C2×C3⋊Dic3C23×C314yes—
187C4⋊(C3⋊D3)—22no—
188C4×C3⋊D3—24no—
189#634—22no—
190C3×#67—64no—
191C3×#67—124no—
192C2×C3×Dic3—32no—
193C2×C3×Dic3—34no—
194C2×C3×Dic3—122no—
195C3×D12—62no—
196C3×C4×D3—64no—
197C3×Dic6—62no—
198#537—91noSylow
199C2×H(3,𝔽3)—22no—
200C2×H(3,𝔽3)—42no—
201C2×H(3,𝔽3)C22×D411yescharacteristic
202C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
203C2×H(3,𝔽3)2−1+411yescharacteristic
204(C3⋊D3)⋊C3—42no—
205C3×#41—24no—
206C3×#41—64no—
207C3×#41—64no—
208C3×#41—68no—
209C3×#41—184no—
210C2×C3×Q8—31no—
211C2×C3×Q8—31no—
212C2×C3×Q8—61no—
213C2×C3×Q8—91no—
214C2×C3×Q8C2×C3×D311yescharacteristic
215C2×C3×D4—31no—
216C2×C3×D4—32no—
217C2×C3×D4—31no—
218C2×C3×D4—32no—
219C2×C3×D4—62no—
220C2×C3×D4—61no—
221C2×C3×D4—91no—
222C2×C3×D4—92no—
223C2×C3×D4—94no—
224C2×C3×D4C2×C3×D311yescharacteristic
225C2×C3×D4C2×C3×D312yes—
226C22×C3×C4—31no—
227C22×C3×C4—31no—
228C22×C3×C4—32no—
229C22×C3×C4—32no—
230C22×C3×C4—62no—
231C22×C3×C4—61no—
232C22×C3×C4—92no—
233C22×C3×C4—92no—
234C22×C3×C4C2×C3×D312yes—
235C22×C3×C4C2×C3×D311yescharacteristic
236C2×#67—32no—
237C2×#67—34no—
238C2×#67—92no—
239C2×#67—94no—
240C22×Dic3—32no—
241C22×Dic3—92no—
242#235—64no—
243#235—184no—
244C2×D12—31no—
245C2×D12—91no—
246C2×C4×D3—32no—
247C2×C4×D3—92no—
248C2×Dic6—31no—
249C2×Dic6—91no—
250C3×C4⋊C4—94no—
251C3×C4⋊C4—92no—
252C3×C4⋊C4—92no—
253C3×C22⋊C4—94no—
254C3×C22⋊C4—94no—
255#217—34no—
256#217—94no—
257(C2×D3)⋊C4—34no—
258(C2×D3)⋊C4—94no—
259C4⋊Dic3—32no—
260C4⋊Dic3—32no—
261C4⋊Dic3—92no—
262C4⋊Dic3—92no—
263Dic3⋊C4—34no—
264Dic3⋊C4—94no—
265C22×C32—21no—
266C22×C32—21no—
267C22×C32—21no—
268C22×C32—21no—
269C22×C32—24no—
270C22×C32—22no—
271C22×C32—22no—
272C22×C32—22no—
273C22×C32—22no—
274C22×C32—22no—
275C22×C32—22no—
276C22×C32—21no—
277C22×C32—22no—
278C22×C32—41no—
279C22×C32—41no—
280C22×C32—42no—
281C22×C32—42no—
282C22×C32C2×#6712yes—
283C22×C32C2×C3×D412yes—
284C22×C32C23×D311yescharacteristic
285C22×C32C24×C311yescharacteristic
286C2×C3⋊D3—22no—
287C2×C3⋊D3—22no—
288C2×C3⋊D3—22no—
289C2×C3×D3—62no—
290C2×C3×D3—62no—
291C2×C3×D3—62no—
292C32×C4—24no—
293C32×C4—22no—
294C32×C4—22no—
295C32×C4—22no—
296C32×C4—22no—
297C32×C4C2×#6712yes—
298C32×C4C2×#6712yes—
299C32×C4C2×C3×D412yes—
300C32×C4C2×C3×D412yes—
301C3⋊Dic3—22no—
302C3⋊Dic3—24no—
303C3×Dic3—64no—
304C3×Dic3—62no—
305C2×#41—91no—
306C2×#41(C3⋊D3)⋊C311yescharacteristic, complemented
307C22⋊Q8—92no—
308C22⋊Q8—92no—
309#121—92no—
310#121—94no—
311#121—92no—
312C2×C4⋊C4—91no—
313H(3,𝔽3)#53711yescharacteristic, complemented, Sylow
314C23×C3—31no—
315C23×C3—31no—
316C23×C3—32no—
317C23×C3—32no—
318C23×C3—62no—
319C23×C3—61no—
320C23×C3—92no—
321C23×C3C22×C3×D312yes—
322C23×C3C22×C3×D311yescharacteristic
323C22×D3—32no—
324C22×D3—92no—
325C3×Q8—22no—
326C3×Q8—62no—
327C3×Q8—62no—
328C3×Q8—64no—
329C3×Q8—182no—
330C3×D4—24no—
331C3×D4—22no—
332C3×D4—62no—
333C3×D4—62no—
334C3×D4—64no—
335C3×D4—64no—
336C3×D4—68no—
337C3×D4—64no—
338C3×D4—182no—
339C3×D4—184no—
340C3×D4—364no—
341C2×C3×C4—24no—
342C2×C3×C4—34no—
343C2×C3×C4—34no—
344C2×C3×C4—31no—
345C2×C3×C4—31no—
346C2×C3×C4—31no—
347C2×C3×C4—31no—
348C2×C3×C4—32no—
349C2×C3×C4—32no—
350C2×C3×C4—64no—
351C2×C3×C4—64no—
352C2×C3×C4—68no—
353C2×C3×C4—64no—
354C2×C3×C4—61no—
355C2×C3×C4—62no—
356C2×C3×C4—61no—
357C2×C3×C4—94no—
358C2×C3×C4—92no—
359C2×C3×C4—184no—
360C2×C3×C4—362no—
361C2×C3×C4C3×#6714yes—
362C2×C3×C4C22×C3×D312yes—
363C2×C3×C4C22×C3×D311yescharacteristic
364C2×C3×C4C22×C3×D311yescharacteristic
365#67—64no—
366#67—124no—
367#67—184no—
368#67—364no—
369C2×Dic3—32no—
370C2×Dic3—34no—
371C2×Dic3—92no—
372C2×Dic3—94no—
373C2×Dic3—122no—
374C2×Dic3—362no—
375D12—62no—
376D12—182no—
377C4×D3—64no—
378C4×D3—184no—
379Dic6—62no—
380Dic6—182no—
381C2×C32—22no—
382C2×C32—22no—
383C2×C32—21no—
384C2×C32—21no—
385C2×C32—21no—
386C2×C32—42no—
387C2×C32—44no—
388C2×C32—42no—
389C2×C32—42no—
390C2×C32#100911yescharacteristic
391C2×C32#101011yescharacteristic
392C2×C32C22×#6711yescharacteristic
393C2×C32C22×C3×D411yescharacteristic
394C2×C32C3×H(5,𝔽2)11yescharacteristic
395C2×C32C3×2−1+411yescharacteristic
396C3⋊D3—42no—
397C3×D3—122no—
398#41—24no—
399#41—184no—
400C2×Q8—91no—
401C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
402C2×D4—92no—
403C2×D4—94no—
404C2×D4—91no—
405C2×D4C2×(C3⋊D3)⋊C312yes—
406C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
407C22×C4—92no—
408C22×C4—92no—
409C22×C4C2×(C3⋊D3)⋊C312yes—
410C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
411C4⋊C4—92no—
412C4⋊C4—92no—
413C4⋊C4—94no—
414C22⋊C4—94no—
415C22⋊C4—94no—
416C22×C3—21no—
417C22×C3—21no—
418C22×C3—22no—
419C22×C3—22no—
420C22×C3—22no—
421C22×C3—31no—
422C22×C3—31no—
423C22×C3—32no—
424C22×C3—32no—
425C22×C3—61no—
426C22×C3—61no—
427C22×C3—61no—
428C22×C3—61no—
429C22×C3—62no—
430C22×C3—62no—
431C22×C3—62no—
432C22×C3—62no—
433C22×C3—62no—
434C22×C3—62no—
435C22×C3—64no—
436C22×C3—61no—
437C22×C3—62no—
438C22×C3—121no—
439C22×C3—121no—
440C22×C3—122no—
441C22×C3—122no—
442C22×C3—182no—
443C22×C3—182no—
444C22×C3—182no—
445C22×C3C2×C3×#6712yes—
446C22×C3C23×C3×D311yescharacteristic
447C2×D3—62no—
448C2×D3—62no—
449C2×D3—62no—
450C2×D3—182no—
451C2×D3—182no—
452C2×D3—182no—
453C3×C4—22no—
454C3×C4—32no—
455C3×C4—32no—
456C3×C4—32no—
457C3×C4—32no—
458C3×C4—62no—
459C3×C4—62no—
460C3×C4—64no—
461C3×C4—62no—
462C3×C4—62no—
463C3×C4—182no—
464C3×C4—184no—
465C3×C4C2×C3×#6712yes—
466C3×C4C2×C3×#6712yes—
467Dic3—64no—
468Dic3—62no—
469Dic3—182no—
470Dic3—184no—
471C32—21no—
472C32#583311yescharacteristic, complemented
473C32C3×#53711yescharacteristic, complemented
474C23—92no—
475C23C22×(C3⋊D3)⋊C312yes—
476C23C22×(C3⋊D3)⋊C311yescharacteristic
477Q8—22no—
478Q8—182no—
479D4—24no—
480D4—22no—
481D4—184no—
482D4—182no—
483D4—364no—
484C2×C4—24no—
485C2×C4—94no—
486C2×C4—92no—
487C2×C4—184no—
488C2×C4—362no—
489C2×C4#638⋊C314yescomplemented
490C2×C4C22×(C3⋊D3)⋊C312yes—
491C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
492C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
493C2×C3—22no—
494C2×C3—31no—
495C2×C3—31no—
496C2×C3—31no—
497C2×C3—31no—
498C2×C3—31no—
499C2×C3—31no—
500C2×C3—42no—
501C2×C3—62no—
502C2×C3—62no—
503C2×C3—61no—
504C2×C3—61no—
505C2×C3—61no—
506C2×C3—122no—
507C2×C3—122no—
508C2×C3—122no—
509C2×C3—124no—
510C2×C3—362no—
511C2×C3C3×#100911yescharacteristic
512C2×C3C3×#101011yescharacteristic
513C2×C3C22×C3×#6711yescharacteristic
514D3—122no—
515D3—362no—
516C22—21no—
517C22—21no—
518C22—22no—
519C22—22no—
520C22—22no—
521C22—182no—
522C22—182no—
523C22—182no—
524C22C2×#638⋊C312yes—
525C22C23×(C3⋊D3)⋊C311yescharacteristic
526C4—22no—
527C4—184no—
528C4—182no—
529C4C2×#638⋊C312yes—
530C4C2×#638⋊C312yes—
531C3—31no—
532C3—31no—
533C3—61no—
534C3C3×#583311yescharacteristic
535C2—22no—
536C2—42no—
537C2—362no—
538C2#1172263811yescharacteristic
539C2#1172263911yescharacteristic
540C2C22×#638⋊C311yescharacteristic
subgroupG/H
centerC22C23×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingC2×#41×H(3,𝔽3)C2
hypercenterC2×#41(C3⋊D3)⋊C3
hypocenterC32C3×#537
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. C2×#41×H(3,𝔽3)C2semidirect
2. C2×#3828⋊C3C2semidirect
3. #64155⋊C3C2semidirect
4. #64148⋊C3C2semidirect
5. #64147⋊C3C2semidirect
6. #64139⋊C3C2semidirect
7. C2×C4⋊((C3⋊Dic3)⋊C3)C2semidirect
8. #64133⋊C3C2semidirect
9. #10596304C3semidirect
30. C2×C32×#41D3semidirect
31. C2×C32×#41C2×C3semidirect
32. C2×#3828C2×C3semidirect
34. #64155C2×C3semidirect
35. #64148C2×C3semidirect
36. #64147C2×C3semidirect
37. #64139C2×C3semidirect
38. C2×C4⋊(C3⋊Dic3)C2×C3semidirect
39. #64133C2×C3semidirect
53. C2×C4×H(3,𝔽3)D4semidirect
174. C2×C32×C4#67semidirect
175. C2×C32×C4C3×D4semidirect
306. C2×#41(C3⋊D3)⋊C3semidirect
313. H(3,𝔽3)#537semidirect
472. C32#5833semidirect
473. C32C3×#537semidirect
489. 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
C2×#3828⋊C32
#64155⋊C32
#64148⋊C32
#64147⋊C32
#64139⋊C32
C2×C4⋊((C3⋊Dic3)⋊C3)2
#64133⋊C32
#105963043
C3×#58333