Finite Group Databasebeta

← #13147866

Subgroups — #13147866

Order 1728.

843 conjugacy classes of subgroups.

The subgroup lattice has 583 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×C4⋊#272C211yesmaximal, characteristic
2#64079⋊C3C211yesmaximal, characteristic, complemented
3C4×#634⋊C3C211yesmaximal, characteristic, complemented
4#11721152C211yesmaximal, characteristic, complemented
5#11721151C212yesmaximal
6#11721094C211yesmaximal, characteristic, complemented
7#11721094C211yesmaximal, characteristic, complemented
8#11721092C212yesmaximal, complemented
9#11721089C211yesmaximal, characteristic, complemented
10#11721085C212yesmaximal, complemented
11#11721076C212yesmaximal, complemented
12#10593920—31nomaximal
13#10593917—31nomaximal
14C2×C4⋊#272C2211yescharacteristic
15C2×C4×#272C2211yescharacteristic
16C2×C4×#272C2211yescharacteristic
17C2×#634⋊C3C2211yescharacteristic
18C2×#6199C2212yes—
19C42×H(3,𝔽3)C2211yescharacteristic
20#88542C2211yescharacteristic
21#88542C2211yescharacteristic
22C4⋊#1084C2211yescharacteristic
23C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
24C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
25C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
26#3750⋊C3C2212yes—
27#3750⋊C3C2212yes—
28#3750⋊C3C2212yes—
29C4×(C3⋊Dic3)⋊C3C2212yes—
30#88451C2212yes—
31#88451C2212yes—
32#88451C2212yes—
33#88449C2212yes—
34#88448C2212yes—
35#88448C2212yes—
36#88447C2212yes—
37#64079D311yescharacteristic, complemented
38C4×#634D311yescharacteristic, complemented
39C3×C4×D12—31no—
40C3×C4×D12—31no—
41C3×C4×D12—61no—
42C3×#870—31no—
43C3×C4×Dic6—31no—
44D3×C4⋊Dic3—31no—
45D3×C4⋊Dic3—31no—
46#63884—32no—
47#63884—32no—
48#63878—32no—
49#63877—32no—
50#63862—32no—
51#63859—32no—
52#63857—31no—
53#63857—31no—
54#63856—31no—
55#63856—31no—
56#63847—32no—
57#63846—32no—
58#63841—31no—
59#63841—31no—
60C22×#272C2311yescharacteristic
61C22×#272C2311yescharacteristic
62C2×C4×H(3,𝔽3)C2311yescharacteristic
63C2×C4×H(3,𝔽3)C2311yescharacteristic
64C2×C4×H(3,𝔽3)C2311yescharacteristic
65C2×#1084C2311yescharacteristic
66C2×#1084C2311yescharacteristic
67C4⋊#272—22no—
68C4×#272—22no—
69C4×#272—22no—
70C2×(C3⋊Dic3)⋊C3C2312yes—
71C2×(C3⋊Dic3)⋊C3C2312yes—
72C2×(C3⋊Dic3)⋊C3C2312yes—
73C2×(C3⋊Dic3)⋊C3C2312yes—
74#634⋊C3—22no—
75#6199—24no—
76#5694—91no—
77#5581—91no—
78#5541—91no—
79C2×#634C2×D311yescharacteristic
80C2×C3×D12—31no—
81C2×C3×D12—31no—
82C2×C3×D12—61no—
83C2×C3×C4×D3—31no—
84C2×C3×C4×D3—31no—
85C2×C3×C4×D3—31no—
86C2×C3×C4×D3—31no—
87C2×C3×C4×D3—61no—
88C2×C3×C4×D3—61no—
89C2×C3×Dic6—31no—
90C2×D3×Dic3—32no—
91C2×D3×Dic3—32no—
92C32×C42—21no—
93C32×C42C2×D311yescharacteristic
94C32×C42C2×D311yescharacteristic
95C4⋊(C3⋊Dic3)C2×D311yescharacteristic
96C4⋊(C3⋊Dic3)C2×D311yescharacteristic
97C4⋊(C3⋊Dic3)C2×D311yescharacteristic
98#3750C2×D312yes—
99#3750C2×D312yes—
100#3750C2×D312yes—
101C4×C3⋊Dic3C2×D312yes—
102C3×(C2×D3)⋊C4—31no—
103C3×(C2×D3)⋊C4—31no—
104C3×(C2×D3)⋊C4—31no—
105C3×(C2×D3)⋊C4—31no—
106C3×(C2×D3)⋊C4—61no—
107C3×(C2×D3)⋊C4—61no—
108C3×C4⋊Dic3—31no—
109C3×C4⋊Dic3—31no—
110C3×C4⋊Dic3—31no—
111C3×C4⋊Dic3—31no—
112C3×C4⋊Dic3—31no—
113C3×C4⋊Dic3—61no—
114C3×Dic3⋊C4—32no—
115C3×Dic3⋊C4—32no—
116C3×Dic3⋊C4—32no—
117C3×C4×Dic3—32no—
118#3724—32no—
119#3724—32no—
120Dic3⋊Dic3—32no—
121Dic3⋊Dic3—32no—
122Dic3⋊Dic3—32no—
123Dic3⋊Dic3—32no—
124(C2×D3)⋊Dic3—32no—
125(C2×D3)⋊Dic3—32no—
126(C2×D3)⋊Dic3—32no—
127(C2×D3)⋊Dic3—32no—
128(C2×D3)⋊Dic3—32no—
129(C2×D3)⋊Dic3—32no—
130Dic32—32no—
131Dic32—32no—
132C22×H(3,𝔽3)C2411yescharacteristic
133C2×#272—21no—
134C2×#272—21no—
135C2×#272—21no—
136C2×#272—21no—
137C2×#272—21no—
138C2×#272#4112yes—
139C4×H(3,𝔽3)—21no—
140C4×H(3,𝔽3)—21no—
141C4×H(3,𝔽3)#4112yes—
142#1084—21no—
143#1084—21no—
144(C3⋊Dic3)⋊C3—22no—
145(C3⋊Dic3)⋊C3—22no—
146(C3⋊Dic3)⋊C3—22no—
147(C3⋊Dic3)⋊C3—22no—
148C3×#125—91no—
149C3×C4×Q8—91no—
150C3×C4×D4—91no—
151#946—91no—
152#946—91no—
153#935—92no—
154#926—91no—
155C2×C4⋊Dic3—91no—
156#924—91no—
157#924—91no—
158#898—92no—
159#897—91no—
160#897—91no—
161#894—92no—
162#894—92no—
163#892—92no—
164#892—91no—
165#892—91no—
166D3×C4⋊C4—91no—
167D3×C4⋊C4—91no—
168#890—92no—
169#889—92no—
170#889—92no—
171#879—92no—
172#877—92no—
173#876—92no—
174C4×D12—31no—
175C4×D12—31no—
176C4×D12—61no—
177#872—92no—
178#870—31no—
179#870—91no—
180C4×Dic6—31no—
181C4×Dic6—91no—
182C22×C3×D3—31no—
183C22×C3×D3—31no—
184C22×C3×D3—31no—
185C22×C3×D3—31no—
186C22×C3×D3—61no—
187C22×C3×D3—61no—
188C2×C32×C4—21no—
189C2×C32×C4—21no—
190C2×C32×C4—21no—
191C2×C32×C4C22×D311yescharacteristic
192C2×C32×C4C22×D311yescharacteristic
193C2×C32×C4C22×D311yescharacteristic
194C2×C32×C4C22×D311yescharacteristic
195C2×C32×C4C22×D311yescharacteristic
196C2×C32×C4C22×D311yescharacteristic
197C2×C3⋊Dic3C22×D312yes—
198C2×C3⋊Dic3C22×D312yes—
199C2×C3⋊Dic3C22×D312yes—
200C2×C3⋊Dic3C22×D312yes—
201#634—22no—
202C2×C3×Dic3—32no—
203C2×C3×Dic3—32no—
204C2×C3×Dic3—32no—
205C2×C3×Dic3—32no—
206C2×C3×Dic3—31no—
207C2×C3×Dic3—31no—
208C2×C3×Dic3—31no—
209C2×C3×Dic3—31no—
210C2×C3×Dic3—61no—
211C2×C3×Dic3—61no—
212C3×D12—62no—
213C3×D12—62no—
214C3×D12—64no—
215C3×C4×D3—62no—
216C3×C4×D3—62no—
217C3×C4×D3—62no—
218C3×C4×D3—62no—
219C3×C4×D3—122no—
220C3×C4×D3—122no—
221C3×Dic6—62no—
222D3×Dic3—64no—
223D3×Dic3—64no—
224#552—271noSylow
225C2×H(3,𝔽3)C2×#4111yescharacteristic
226C2×H(3,𝔽3)C2×#4111yescharacteristic
227C2×H(3,𝔽3)2−1+411yescharacteristic
228#272—22no—
229#272—41no—
230C2×C3×Q8—91no—
231C2×C3×D4—91no—
232C22×C3×C4—91no—
233C22×C3×C4—91no—
234C22×Dic3—92no—
235C2×D12—31no—
236C2×D12—31no—
237C2×D12—61no—
238C2×C4×D3—31no—
239C2×C4×D3—31no—
240C2×C4×D3—31no—
241C2×C4×D3—31no—
242C2×C4×D3—61no—
243C2×C4×D3—61no—
244C2×C4×D3—92no—
245C2×C4×D3—92no—
246C2×Dic6—31no—
247C2×Dic6—91no—
248C3×C4⋊C4—92no—
249C3×C4⋊C4—92no—
250C3×C4⋊C4—91no—
251C3×C4⋊C4—91no—
252C3×C4⋊C4—91no—
253C3×C4⋊C4—92no—
254C3×C4⋊C4—91no—
255C3×C22⋊C4—91no—
256C3×C22⋊C4—91no—
257C3×C42—31no—
258C3×C42—31no—
259C3×C42—61no—
260C3×C42—92no—
261C3×C42D3211yescharacteristic
262#217—92no—
263#217—92no—
264#217—92no—
265(C2×D3)⋊C4—31no—
266(C2×D3)⋊C4—31no—
267(C2×D3)⋊C4—31no—
268(C2×D3)⋊C4—31no—
269(C2×D3)⋊C4—61no—
270(C2×D3)⋊C4—61no—
271(C2×D3)⋊C4—92no—
272(C2×D3)⋊C4—92no—
273(C2×D3)⋊C4—92no—
274(C2×D3)⋊C4—92no—
275(C2×D3)⋊C4—92no—
276(C2×D3)⋊C4—92no—
277C4⋊Dic3—31no—
278C4⋊Dic3—31no—
279C4⋊Dic3—31no—
280C4⋊Dic3—31no—
281C4⋊Dic3—31no—
282C4⋊Dic3—61no—
283C4⋊Dic3—91no—
284C4⋊Dic3—91no—
285C4⋊Dic3—91no—
286C4⋊Dic3—92no—
287C4⋊Dic3—92no—
288C4⋊Dic3—92no—
289Dic3⋊C4—32no—
290Dic3⋊C4—32no—
291Dic3⋊C4—32no—
292Dic3⋊C4—92no—
293Dic3⋊C4—92no—
294Dic3⋊C4—92no—
295Dic3⋊C4—92no—
296Dic3⋊C4—92no—
297Dic3⋊C4—92no—
298Dic3⋊C4—92no—
299Dic3⋊C4—92no—
300Dic3⋊C4—92no—
301C4×Dic3—32no—
302C4×Dic3—92no—
303C4×Dic3—92no—
304C4×Dic3—92no—
305C4×Dic3—92no—
306C22×C32—21no—
307C22×C32C23×D311yescharacteristic
308C22×C32C23×D311yescharacteristic
309C2×C3×D3—32no—
310C2×C3×D3—32no—
311C2×C3×D3—62no—
312C2×C3×D3—61no—
313C2×C3×D3—61no—
314C2×C3×D3—61no—
315C2×C3×D3—61no—
316C2×C3×D3—62no—
317C2×C3×D3—61no—
318C2×C3×D3—61no—
319C2×C3×D3—61no—
320C2×C3×D3—61no—
321C2×C3×D3—61no—
322C2×C3×D3—61no—
323C2×C3×D3—121no—
324C2×C3×D3—121no—
325C2×C3×D3—121no—
326C2×C3×D3—121no—
327C32×C4—22no—
328C32×C4—21no—
329C32×C4—21no—
330C32×C4—21no—
331C32×C4—21no—
332C32×C4—22no—
333C32×C4—41no—
334C32×C4#23512yes—
335C32×C4#23912yes—
336C3⋊Dic3—22no—
337C3⋊Dic3—22no—
338C3⋊Dic3—22no—
339C3⋊Dic3—22no—
340C3×Dic3—61no—
341C3×Dic3—61no—
342C3×Dic3—62no—
343C3×Dic3—61no—
344C3×Dic3—61no—
345C3×Dic3—62no—
346C3×Dic3—62no—
347C3×Dic3—62no—
348C3×Dic3—121no—
349C3×Dic3—121no—
350#126—272no—
351#125—91no—
352#125—272no—
353#123—272no—
354C22⋊Q8—271no—
355C22⋊Q8—271no—
356C4×Q8—91no—
357C4×D4—91no—
358#117—272no—
359#117—271no—
360C2×C4⋊C4—271no—
361H(3,𝔽3)#55211yescharacteristic, complemented, Sylow
362C23×C3—91no—
363C23×C3—91no—
364C22×D3—31no—
365C22×D3—31no—
366C22×D3—31no—
367C22×D3—31no—
368C22×D3—61no—
369C22×D3—61no—
370C3×Q8—182no—
371C3×D4—182no—
372C2×C3×C4—31no—
373C2×C3×C4—31no—
374C2×C3×C4—31no—
375C2×C3×C4—31no—
376C2×C3×C4—31no—
377C2×C3×C4—31no—
378C2×C3×C4—61no—
379C2×C3×C4—61no—
380C2×C3×C4—61no—
381C2×C3×C4—92no—
382C2×C3×C4—92no—
383C2×C3×C4—92no—
384C2×C3×C4—92no—
385C2×C3×C4—91no—
386C2×C3×C4—91no—
387C2×C3×C4—182no—
388C2×C3×C4—182no—
389C2×C3×C4C2×D3211yescharacteristic
390C2×C3×C4C2×D3211yescharacteristic
391C2×C3×C4C2×D3211yescharacteristic
392C2×Dic3—31no—
393C2×Dic3—31no—
394C2×Dic3—31no—
395C2×Dic3—31no—
396C2×Dic3—32no—
397C2×Dic3—32no—
398C2×Dic3—32no—
399C2×Dic3—32no—
400C2×Dic3—61no—
401C2×Dic3—61no—
402C2×Dic3—92no—
403C2×Dic3—92no—
404C2×Dic3—92no—
405C2×Dic3—92no—
406C2×Dic3—184no—
407D12—62no—
408D12—62no—
409D12—64no—
410C4×D3—62no—
411C4×D3—62no—
412C4×D3—62no—
413C4×D3—62no—
414C4×D3—122no—
415C4×D3—122no—
416C4×D3—184no—
417C4×D3—184no—
418Dic6—62no—
419Dic6—182no—
420C2×C32—21no—
421C2×C32—21no—
422C2×C32—21no—
423C2×C32C2×#23511yescharacteristic
424C2×C32C2×#23911yescharacteristic
425C2×C32#100711yescharacteristic
426C2×C32D3×#4111yescharacteristic
427C2×C32D3×#4111yescharacteristic
428C2×C32#101011yescharacteristic
429C3×D3—62no—
430C3×D3—62no—
431C3×D3—122no—
432C3×D3—122no—
433C3×D3—121no—
434C3×D3—121no—
435C2×Q8—91no—
436C2×D4—91no—
437C22×C4—91no—
438C22×C4—91no—
439C22×C4—272no—
440C4⋊C4—91no—
441C4⋊C4—91no—
442C4⋊C4—91no—
443C4⋊C4—92no—
444C4⋊C4—91no—
445C4⋊C4—92no—
446C4⋊C4—92no—
447C4⋊C4—272no—
448C4⋊C4—272no—
449C4⋊C4—272no—
450C22⋊C4—91no—
451C22⋊C4—91no—
452C22⋊C4—272no—
453C22⋊C4—272no—
454C22⋊C4—272no—
455C42—92no—
456C42—272no—
457C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
458C22×C3—31no—
459C22×C3—31no—
460C22×C3—61no—
461C22×C3—92no—
462C22×C3—181no—
463C22×C3—181no—
464C22×C3—181no—
465C22×C3—181no—
466C22×C3—181no—
467C22×C3C22×D3211yescharacteristic
468C2×D3—32no—
469C2×D3—32no—
470C2×D3—62no—
471C2×D3—61no—
472C2×D3—61no—
473C2×D3—61no—
474C2×D3—61no—
475C2×D3—61no—
476C2×D3—61no—
477C2×D3—61no—
478C2×D3—61no—
479C2×D3—61no—
480C2×D3—61no—
481C2×D3—62no—
482C2×D3—121no—
483C2×D3—121no—
484C2×D3—121no—
485C2×D3—121no—
486C3×C4—21no—
487C3×C4—21no—
488C3×C4—32no—
489C3×C4—32no—
490C3×C4—61no—
491C3×C4—61no—
492C3×C4—61no—
493C3×C4—61no—
494C3×C4—62no—
495C3×C4—62no—
496C3×C4—121no—
497C3×C4—182no—
498C3×C4—182no—
499C3×C4—182no—
500C3×C4—182no—
501C3×C4—181no—
502C3×C4—181no—
503C3×C4#379912yes—
504Dic3—61no—
505Dic3—61no—
506Dic3—61no—
507Dic3—61no—
508Dic3—62no—
509Dic3—62no—
510Dic3—62no—
511Dic3—62no—
512Dic3—121no—
513Dic3—121no—
514Dic3—182no—
515Dic3—182no—
516Dic3—182no—
517Dic3—182no—
518C32—21no—
519C32#558111yescharacteristic, complemented
520C32#569411yescharacteristic, complemented
521C23—91no—
522C23—91no—
523Q8—182no—
524D4—182no—
525C2×C4—92no—
526C2×C4—92no—
527C2×C4—91no—
528C2×C4—92no—
529C2×C4—91no—
530C2×C4—92no—
531C2×C4—182no—
532C2×C4—182no—
533C2×C4—544no—
534C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
535C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
536C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
537C2×C3—31no—
538C2×C3—31no—
539C2×C3—31no—
540C2×C3—31no—
541C2×C3—31no—
542C2×C3—31no—
543C2×C3—61no—
544C2×C3—61no—
545C2×C3—61no—
546C2×C3—182no—
547C2×C3—361no—
548C2×C3#6429711yescharacteristic
549C2×C3C2×#379911yescharacteristic
550C2×C3#6430611yescharacteristic
551D3—62no—
552D3—62no—
553D3—122no—
554D3—121no—
555D3—121no—
556D3—122no—
557C22—92no—
558C22—181no—
559C22—181no—
560C22—181no—
561C22—181no—
562C22—181no—
563C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
564C4—21no—
565C4—21no—
566C4—182no—
567C4—181no—
568C4—181no—
569C4—182no—
570C4—182no—
571C4—182no—
572C4#8865212yes—
573C3—31no—
574C3—31no—
575C3—61no—
576C3#1059394711yescharacteristic
577C2—182no—
578C2—361no—
579C2#1172248211yescharacteristic
580C2C2×#8865211yescharacteristic
581C2#1172249111yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC42×H(3,𝔽3)C22
prime radicalC2×C4⋊#272C22
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#552
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
2. #64079⋊C3C2semidirect
3. C4×#634⋊C3C2semidirect
4. #11721152C2semidirect
6. #11721094C2semidirect
7. #11721094C2semidirect
8. #11721092C2semidirect
9. #11721089C2semidirect
10. #11721085C2semidirect
11. #11721076C2semidirect
37. #64079D3semidirect
38. C4×#634D3semidirect
361. H(3,𝔽3)#552semidirect
519. C32#5581semidirect
520. C32#5694semidirect

Maximal subgroups

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

subgroupindex
C4×C4⋊#2722
#64079⋊C32
C4×#634⋊C32
#117211522
#117211512
#117210942
#117210922
#117210892
#117210852
#117210762
#105939203
#105939173