Finite Group Databasebeta

← #13147881

Subgroups — #13147881

Order 1728.

920 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×C4⋊#272C211yesmaximal, characteristic, complemented
2#64084⋊C3C211yesmaximal, characteristic, complemented
3C4×#634⋊C3C211yesmaximal, characteristic, complemented
4#11721158C211yesmaximal, characteristic
5#11721155C212yesmaximal, complemented
6#11721132C211yesmaximal, characteristic, complemented
7#11721130C212yesmaximal, complemented
8#11721124C212yesmaximal, complemented
9#11721114C211yesmaximal, characteristic, complemented
10#11721104C212yesmaximal, complemented
11#11721082C211yesmaximal, characteristic, complemented
12#10593981—31nomaximal
13#10593978—31nomaximal
14C2×C4⋊#272C2211yescharacteristic
15C2×C4×#272C2212yes—
16C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
17C2×#634⋊C3C2211yescharacteristic
18C2×#634⋊C3C2211yescharacteristic
19C2×#6200C2212yes—
20C2×#6200C2212yes—
21C2×((C3⋊D3)⋊C3)⋊C4C2212yes—
22C42×H(3,𝔽3)C2211yescharacteristic
23#88542C2212yes—
24C4⋊#1084C2211yescharacteristic
25#3752⋊C3C2212yes—
26#3752⋊C3C2212yes—
27C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
28#3750⋊C3C2211yescharacteristic
29#3750⋊C3C2211yescharacteristic
30C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
31C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
32#88451C2212yes—
33#88451C2212yes—
34#88450C2212yes—
35#88448C2212yes—
36#88447C2212yes—
37#64084D311yescharacteristic, complemented
38C4×#634D311yescharacteristic, complemented
39C3×#875—31no—
40C3×C4×D12—31no—
41C3×C4×D12—31no—
42C3×C4×D12—61no—
43C3×C4×Dic6—31no—
44#63914—31no—
45#63913—32no—
46#63911—31no—
47#63907—32no—
48#63905—32no—
49#63904—32no—
50#63898—31no—
51#63894—31no—
52#63891—32no—
53#63879—32no—
54#63863—31no—
55#63860—31no—
56#63854—32no—
57#63850—32no—
58#63844—31no—
59#63843—31no—
60C22×#272C2312yes—
61C22×(C3⋊D3)⋊C3C2312yes—
62C2×C4×H(3,𝔽3)C2311yescharacteristic
63C2×C4×H(3,𝔽3)C2311yescharacteristic
64C2×C4×H(3,𝔽3)C2311yescharacteristic
65C2×#1084C2312yes—
66C4⋊#272—22no—
67C4×#272—42no—
68C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
69C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
70C2×(C3⋊Dic3)⋊C3C2312yes—
71C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
72C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
73C4⋊((C3⋊D3)⋊C3)—22no—
74#634⋊C3—22no—
75#634⋊C3—22no—
76#6200—24no—
77#6200—42no—
78((C3⋊D3)⋊C3)⋊C4—42no—
79#5681—91no—
80#5585—91no—
81#5559—91no—
82C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
83C2×#634C2×D311yescharacteristic
84C2×#634C2×D311yescharacteristic
85C2×C3×D12—31no—
86C2×C3×D12—31no—
87C2×C3×D12—31no—
88C2×C3×D12—61no—
89C2×C3×C4×D3—32no—
90C2×C3×C4×D3—32no—
91C2×C3×C4×D3—62no—
92C2×C3×Dic6—31no—
93C2×C3×Dic6—31no—
94C2×#626—32no—
95C2×#626—32no—
96C2×#625—32no—
97C2×#625—32no—
98C2×#624—32no—
99C2×D3×Dic3—32no—
100C32×C42—21no—
101C32×C42C2×D311yescharacteristic
102C32×C42C2×D311yescharacteristic
103#3752C2×D312yes—
104#3752C2×D312yes—
105C4⋊(C3⋊Dic3)C2×D311yescharacteristic
106#3750C2×D311yescharacteristic
107#3750C2×D311yescharacteristic
108C4×C3⋊Dic3C2×D311yescharacteristic
109C4×C3⋊Dic3C2×D311yescharacteristic
110C3×(C2×D3)⋊C4—32no—
111C3×(C2×D3)⋊C4—32no—
112C3×(C2×D3)⋊C4—32no—
113C3×(C2×D3)⋊C4—32no—
114C3×(C2×D3)⋊C4—62no—
115C3×C4⋊Dic3—31no—
116C3×C4⋊Dic3—31no—
117C3×C4⋊Dic3—31no—
118C3×C4⋊Dic3—61no—
119C3×Dic3⋊C4—31no—
120C3×Dic3⋊C4—31no—
121C3×C4×Dic3—31no—
122C3×C4×Dic3—31no—
123#3724—32no—
124Dic3⋊Dic3—32no—
125#3722—32no—
126(C2×D3)⋊Dic3—32no—
127(C2×D3)⋊Dic3—32no—
128(C2×D3)⋊Dic3—32no—
129(C2×D3)⋊Dic3—32no—
130(C2×D3)⋊Dic3—32no—
131Dic32—32no—
132Dic32—32no—
133C22×H(3,𝔽3)C2411yescharacteristic
134C2×#272—22no—
135C2×#272—22no—
136C2×#272—22no—
137C2×(C3⋊D3)⋊C3—22no—
138C2×(C3⋊D3)⋊C3—22no—
139C2×(C3⋊D3)⋊C3—22no—
140C4×H(3,𝔽3)—21no—
141C4×H(3,𝔽3)—21no—
142C4×H(3,𝔽3)#4112yes—
143#1084—22no—
144(C3⋊Dic3)⋊C3—21no—
145(C3⋊Dic3)⋊C3—22no—
146(C3⋊Dic3)⋊C3—21no—
147(C3⋊Dic3)⋊C3—21no—
148(C3⋊Dic3)⋊C3#4112yes—
149C3×#124—91no—
150C3×C4×Q8—91no—
151C3×C4×D4—91no—
152#947—92no—
153#947—91no—
154Q8×Dic3—91no—
155#940—91no—
156#940—91no—
157#936—91no—
158D4×Dic3—91no—
159#929—92no—
160C4×#67—92no—
161#894—92no—
162#894—92no—
163#893—91no—
164#893—92no—
165#887—91no—
166#886—92no—
167#885—92no—
168#885—92no—
169#883—92no—
170#883—92no—
171#67⋊C4—92no—
172#875—31no—
173#875—91no—
174#875—91no—
175#874—91no—
176C4×D12—31no—
177C4×D12—31no—
178C4×D12—61no—
179C4×D12—91no—
180C4×Dic6—31no—
181C4×Dic6—91no—
182C4×Dic6—91no—
183C22×C3⋊D3C22×D312yes—
184C22×C3×D3—32no—
185C22×C3×D3—32no—
186C22×C3×D3—32no—
187C22×C3×D3—62no—
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×D311yescharacteristic
199C2×C3⋊Dic3C22×D311yescharacteristic
200C2×C3⋊Dic3C22×D311yescharacteristic
201C2×C3⋊Dic3C22×D311yescharacteristic
202C4⋊(C3⋊D3)—22no—
203#634—22no—
204#634—22no—
205C2×C3×Dic3—31no—
206C2×C3×Dic3—32no—
207C2×C3×Dic3—31no—
208C2×C3×Dic3—31no—
209C2×C3×Dic3—31no—
210C2×C3×Dic3—32no—
211C2×C3×Dic3—32no—
212C2×C3×Dic3—62no—
213C3×D12—64no—
214C3×D12—62no—
215C3×D12—62no—
216C3×D12—62no—
217C3×C4×D3—122no—
218C3×C4×D3—122no—
219C3×C4×D3—124no—
220C3×Dic6—62no—
221C3×Dic6—62no—
222#626—64no—
223#626—122no—
224#625—64no—
225#625—122no—
226#624—122no—
227D3×Dic3—122no—
228#559—271noSylow
229C2×H(3,𝔽3)C2×#4111yescharacteristic
230C2×H(3,𝔽3)C2×#4111yescharacteristic
231C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
232#272—42no—
233(C3⋊D3)⋊C3—42no—
234C2×C3×Q8—91no—
235C2×C3×Q8—91no—
236C2×C3×D4—91no—
237C2×C3×D4—91no—
238C22×C3×C4—92no—
239C2×#67—92no—
240C2×#67—92no—
241C2×#67—92no—
242C2×#67—92no—
243C22×Dic3—92no—
244C2×D12—31no—
245C2×D12—31no—
246C2×D12—31no—
247C2×D12—61no—
248C2×D12—91no—
249C2×D12—92no—
250C2×D12—92no—
251C2×C4×D3—32no—
252C2×C4×D3—32no—
253C2×C4×D3—62no—
254C2×C4×D3—92no—
255C2×C4×D3—92no—
256C2×Dic6—31no—
257C2×Dic6—31no—
258C2×Dic6—91no—
259C2×Dic6—91no—
260C3×C4⋊C4—91no—
261C3×C4⋊C4—91no—
262C3×C4⋊C4—91no—
263C3×C4⋊C4—91no—
264C3×C22⋊C4—92no—
265C3×C22⋊C4—92no—
266C3×C22⋊C4—92no—
267C3×C42—31no—
268C3×C42—31no—
269C3×C42—61no—
270C3×C42—91no—
271C3×C42—91no—
272C3×C42D3211yescharacteristic
273#217—92no—
274#217—92no—
275#217—92no—
276(C2×D3)⋊C4—32no—
277(C2×D3)⋊C4—32no—
278(C2×D3)⋊C4—32no—
279(C2×D3)⋊C4—32no—
280(C2×D3)⋊C4—62no—
281(C2×D3)⋊C4—92no—
282(C2×D3)⋊C4—92no—
283(C2×D3)⋊C4—92no—
284(C2×D3)⋊C4—92no—
285(C2×D3)⋊C4—92no—
286(C2×D3)⋊C4—92no—
287(C2×D3)⋊C4—92no—
288(C2×D3)⋊C4—92no—
289C4⋊Dic3—31no—
290C4⋊Dic3—31no—
291C4⋊Dic3—31no—
292C4⋊Dic3—61no—
293C4⋊Dic3—91no—
294C4⋊Dic3—92no—
295Dic3⋊C4—31no—
296Dic3⋊C4—31no—
297Dic3⋊C4—91no—
298Dic3⋊C4—91no—
299Dic3⋊C4—92no—
300Dic3⋊C4—92no—
301C4×Dic3—31no—
302C4×Dic3—31no—
303C4×Dic3—91no—
304C4×Dic3—91no—
305C4×Dic3—92no—
306C4×Dic3—92no—
307C4×Dic3—92no—
308C22×C32—21no—
309C22×C32C23×D311yescharacteristic
310C22×C32C23×D311yescharacteristic
311C2×C3⋊D3—22no—
312C2×C3⋊D3—22no—
313C2×C3⋊D3—22no—
314C2×C3×D3—64no—
315C2×C3×D3—62no—
316C2×C3×D3—62no—
317C2×C3×D3—62no—
318C2×C3×D3—62no—
319C2×C3×D3—62no—
320C2×C3×D3—62no—
321C2×C3×D3—62no—
322C2×C3×D3—62no—
323C2×C3×D3—62no—
324C2×C3×D3—122no—
325C2×C3×D3—122no—
326C32×C4—22no—
327C32×C4—21no—
328C32×C4—21no—
329C32×C4—21no—
330C32×C4—21no—
331C32×C4—22no—
332C32×C4—41no—
333C32×C4#23712yes—
334C32×C4#23912yes—
335C3⋊Dic3—21no—
336C3⋊Dic3—22no—
337C3⋊Dic3—21no—
338C3⋊Dic3—21no—
339C3⋊Dic3#23512yes—
340C3×Dic3—32no—
341C3×Dic3—62no—
342C3×Dic3—61no—
343C3×Dic3—61no—
344C3×Dic3—61no—
345C3×Dic3—62no—
346C3×Dic3—62no—
347C3×Dic3—122no—
348#127—271no—
349#124—91no—
350#124—271no—
351#124—272no—
352#123—272no—
353#123—272no—
354C4×Q8—91no—
355C4×Q8—271no—
356C4×D4—91no—
357C4×D4—271no—
358C4×D4—272no—
359H(3,𝔽3)#55911yescharacteristic, complemented, Sylow
360C23×C3—92no—
361C23×C3—92no—
362C22×D3—32no—
363C22×D3—32no—
364C22×D3—32no—
365C22×D3—62no—
366C22×D3—92no—
367C3×Q8—182no—
368C3×Q8—182no—
369C3×D4—182no—
370C3×D4—182no—
371C2×C3×C4—31no—
372C2×C3×C4—31no—
373C2×C3×C4—31no—
374C2×C3×C4—31no—
375C2×C3×C4—31no—
376C2×C3×C4—31no—
377C2×C3×C4—61no—
378C2×C3×C4—61no—
379C2×C3×C4—61no—
380C2×C3×C4—91no—
381C2×C3×C4—92no—
382C2×C3×C4—91no—
383C2×C3×C4—91no—
384C2×C3×C4—91no—
385C2×C3×C4—92no—
386C2×C3×C4—362no—
387C2×C3×C4C2×D3211yescharacteristic
388C2×C3×C4C2×D3211yescharacteristic
389C2×C3×C4C2×D3211yescharacteristic
390#67—184no—
391#67—184no—
392#67—362no—
393#67—362no—
394C2×Dic3—31no—
395C2×Dic3—32no—
396C2×Dic3—32no—
397C2×Dic3—31no—
398C2×Dic3—31no—
399C2×Dic3—31no—
400C2×Dic3—32no—
401C2×Dic3—62no—
402C2×Dic3—91no—
403C2×Dic3—91no—
404C2×Dic3—91no—
405C2×Dic3—92no—
406C2×Dic3—91no—
407C2×Dic3—362no—
408D12—62no—
409D12—62no—
410D12—64no—
411D12—62no—
412D12—182no—
413D12—184no—
414D12—362no—
415C4×D3—122no—
416C4×D3—122no—
417C4×D3—124no—
418C4×D3—362no—
419C4×D3—362no—
420Dic6—62no—
421Dic6—62no—
422Dic6—182no—
423Dic6—182no—
424C2×C32—21no—
425C2×C32—21no—
426C2×C32—21no—
427C2×C32C2×#23511yescharacteristic
428C2×C32C2×#23711yescharacteristic
429C2×C32C2×#23911yescharacteristic
430C2×C32D3×#4111yescharacteristic
431C2×C32#100911yescharacteristic
432C2×C32#100911yescharacteristic
433C3⋊D3—42no—
434C3×D3—124no—
435C3×D3—122no—
436C3×D3—122no—
437C3×D3—122no—
438C2×Q8—91no—
439C2×Q8—91no—
440C2×D4—91no—
441C2×D4—91no—
442C2×D4—272no—
443C2×D4—272no—
444C22×C4—92no—
445C22×C4—272no—
446C4⋊C4—91no—
447C4⋊C4—91no—
448C4⋊C4—91no—
449C4⋊C4—91no—
450C4⋊C4—272no—
451C22⋊C4—92no—
452C22⋊C4—92no—
453C22⋊C4—92no—
454C22⋊C4—272no—
455C22⋊C4—272no—
456C22⋊C4—272no—
457C42—91no—
458C42—91no—
459C42—272no—
460C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
461C22×C3—31no—
462C22×C3—31no—
463C22×C3—61no—
464C22×C3—182no—
465C22×C3—182no—
466C22×C3—182no—
467C22×C3—182no—
468C22×C3—182no—
469C22×C3—182no—
470C22×C3C22×D3211yescharacteristic
471C2×D3—62no—
472C2×D3—62no—
473C2×D3—62no—
474C2×D3—62no—
475C2×D3—62no—
476C2×D3—62no—
477C2×D3—64no—
478C2×D3—62no—
479C2×D3—62no—
480C2×D3—62no—
481C2×D3—122no—
482C2×D3—122no—
483C2×D3—182no—
484C2×D3—182no—
485C2×D3—182no—
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—92no—
497C3×C4—121no—
498C3×C4—181no—
499C3×C4—181no—
500C3×C4—182no—
501C3×C4—181no—
502C3×C4—182no—
503C3×C4#379612yes—
504Dic3—32no—
505Dic3—62no—
506Dic3—62no—
507Dic3—62no—
508Dic3—61no—
509Dic3—61no—
510Dic3—61no—
511Dic3—92no—
512Dic3—122no—
513Dic3—182no—
514Dic3—181no—
515Dic3—181no—
516Dic3—181no—
517C32—21no—
518C32#558511yescharacteristic, complemented
519C32#568111yescharacteristic, complemented
520C23—92no—
521C23—92no—
522Q8—182no—
523Q8—182no—
524D4—182no—
525D4—182no—
526D4—544no—
527D4—1082no—
528C2×C4—91no—
529C2×C4—91no—
530C2×C4—91no—
531C2×C4—91no—
532C2×C4—92no—
533C2×C4—92no—
534C2×C4—362no—
535C2×C4—1082no—
536C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
537C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
538C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
539C2×C3—31no—
540C2×C3—31no—
541C2×C3—31no—
542C2×C3—31no—
543C2×C3—31no—
544C2×C3—31no—
545C2×C3—61no—
546C2×C3—61no—
547C2×C3—61no—
548C2×C3—362no—
549C2×C3—362no—
550C2×C3C2×#379611yescharacteristic
551C2×C3D3×#23511yescharacteristic
552C2×C3#6430811yescharacteristic
553D3—122no—
554D3—122no—
555D3—124no—
556D3—122no—
557D3—362no—
558C22—182no—
559C22—182no—
560C22—182no—
561C22—182no—
562C22—182no—
563C22—182no—
564C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
565C4—21no—
566C4—21no—
567C4—92no—
568C4—182no—
569C4—181no—
570C4—182no—
571C4—181no—
572C4—181no—
573C4#8864912yes—
574C3—31no—
575C3—31no—
576C3—61no—
577C3#1059393911yescharacteristic
578C2—362no—
579C2—362no—
580C2C2×#8864911yescharacteristic
581C2#1172249011yescharacteristic
582C2#1172249311yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC42×H(3,𝔽3)C22
prime radical#11721158C2
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#559
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
1. C4×C4⋊#272C2semidirect
2. #64084⋊C3C2semidirect
3. C4×#634⋊C3C2semidirect
5. #11721155C2semidirect
6. #11721132C2semidirect
7. #11721130C2semidirect
8. #11721124C2semidirect
9. #11721114C2semidirect
10. #11721104C2semidirect
11. #11721082C2semidirect
37. #64084D3semidirect
38. C4×#634D3semidirect
359. H(3,𝔽3)#559semidirect
518. C32#5585semidirect
519. C32#5681semidirect

Maximal subgroups

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

subgroupindex
C4×C4⋊#2722
#64084⋊C32
C4×#634⋊C32
#117211582
#117211552
#117211322
#117211302
#117211242
#117211142
#117211042
#117210822
#105939813
#105939783