Finite Group Databasebeta

← #13147877

Subgroups — #13147877

Order 1728.

913 conjugacy classes of subgroups.

The subgroup lattice has 577 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×C4⋊#272C211yesmaximal, characteristic, complemented
2C42×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
3C4×#634⋊C3C211yesmaximal, characteristic, complemented
4#11721156C211yesmaximal, characteristic
5C4×#6200C212yesmaximal, complemented
6#11721114C211yesmaximal, characteristic, complemented
7#11721106C211yesmaximal, characteristic, complemented
8#11721100C212yesmaximal, complemented
9#11721097C212yesmaximal, complemented
10#11721093C211yesmaximal, characteristic, complemented
11#11721092C212yesmaximal, complemented
12#10593974—31nomaximal
13#10593875—31nomaximal
14C2×C4⋊#272C2211yescharacteristic
15C2×C4×#272C2212yes—
16C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
17C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
18C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
19C2×#634⋊C3C2211yescharacteristic
20C2×#6200C2212yes—
21C42×H(3,𝔽3)C2211yescharacteristic
22#88542C2212yes—
23C4⋊#1084C2211yescharacteristic
24C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
25#3750⋊C3C2212yes—
26C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
27C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
28C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
29C4×(C3⋊Dic3)⋊C3C2212yes—
30#88451C2212yes—
31#88451C2212yes—
32#88451C2212yes—
33#88450C2212yes—
34#88449C2212yes—
35#88449C2212yes—
36#88448C2212yes—
37C42×C3⋊D3D311yescharacteristic, complemented
38C4×#634D311yescharacteristic, complemented
39C3×C4×D12—31no—
40C3×C4×D12—31no—
41C3×C4×D12—61no—
42C3×C42×D3—31no—
43C3×C4×Dic6—31no—
44#63916—31no—
45#63908—31no—
46#63905—32no—
47C4×#626—32no—
48C4×#625—32no—
49#63872—31no—
50#63871—31no—
51#63869—31no—
52#63863—31no—
53#63862—32no—
54#63860—31no—
55#63859—32no—
56#63857—31no—
57#63855—32no—
58#63855—32no—
59#63853—32no—
60C22×#272C2312yes—
61C22×(C3⋊D3)⋊C3C2311yescharacteristic
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—24no—
68C2×(C3⋊Dic3)⋊C3C2312yes—
69C2×(C3⋊Dic3)⋊C3C2312yes—
70C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
71C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
72C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
73C4×(C3⋊D3)⋊C3—22no—
74C4×(C3⋊D3)⋊C3—22no—
75C4×(C3⋊D3)⋊C3—22no—
76#634⋊C3—22no—
77#6200—24no—
78#5580—91no—
79#5557—91no—
80#5479—91no—
81C2×C4×C3⋊D3C2×D311yescharacteristic
82C2×C4×C3⋊D3C2×D311yescharacteristic
83C2×C4×C3⋊D3C2×D311yescharacteristic
84C2×#634C2×D311yescharacteristic
85C2×C3×D12—31no—
86C2×C3×D12—31no—
87C2×C3×D12—61no—
88C2×C3×C4×D3—32no—
89C2×C3×C4×D3—32no—
90C2×C3×C4×D3—31no—
91C2×C3×C4×D3—31no—
92C2×C3×C4×D3—31no—
93C2×C3×C4×D3—62no—
94C2×C3×Dic6—31no—
95C2×#626—32no—
96C2×#625—32no—
97C32×C42—21no—
98C32×C42C2×D311yescharacteristic
99C32×C42C2×D311yescharacteristic
100C4⋊(C3⋊Dic3)C2×D311yescharacteristic
101#3750C2×D312yes—
102C4×C3⋊Dic3C2×D312yes—
103C4×C3⋊Dic3C2×D311yescharacteristic
104C4×C3⋊Dic3C2×D311yescharacteristic
105C4×C3⋊Dic3C2×D311yescharacteristic
106C3×(C2×D3)⋊C4—32no—
107C3×(C2×D3)⋊C4—32no—
108C3×(C2×D3)⋊C4—62no—
109C3×C4⋊Dic3—31no—
110C3×C4⋊Dic3—31no—
111C3×C4⋊Dic3—31no—
112C3×C4⋊Dic3—61no—
113C3×Dic3⋊C4—32no—
114C3×C4×Dic3—31no—
115C3×C4×Dic3—32no—
116C3×C4×Dic3—31no—
117C3×C4×Dic3—31no—
118#3724—32no—
119Dic3⋊Dic3—32no—
120Dic3⋊Dic3—32no—
121Dic3⋊Dic3—32no—
122Dic3⋊Dic3—32no—
123Dic3⋊Dic3—32no—
124#3722—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—
130(C2×D3)⋊Dic3—32no—
131(C2×D3)⋊Dic3—32no—
132C22×H(3,𝔽3)C2411yescharacteristic
133C2×#272—22no—
134C2×#272—22no—
135C2×#272—22no—
136C2×(C3⋊D3)⋊C3—21no—
137C2×(C3⋊D3)⋊C3—21no—
138C2×(C3⋊D3)⋊C3#4112yes—
139C4×H(3,𝔽3)—21no—
140C4×H(3,𝔽3)#4112yes—
141C4×H(3,𝔽3)#4112yes—
142#1084—22no—
143(C3⋊Dic3)⋊C3—22no—
144(C3⋊Dic3)⋊C3—22no—
145(C3⋊Dic3)⋊C3—21no—
146(C3⋊Dic3)⋊C3—21no—
147(C3⋊Dic3)⋊C3—21no—
148C3×C4×Q8—91no—
149C3×C4×D4—91no—
150C2×C3×C42—91no—
151#947—91no—
152#946—91no—
153#938—91no—
154#936—91no—
155#930—91no—
156#929—92no—
157C4×#67—92no—
158C4×#67—92no—
159#926—92no—
160#924—91no—
161#898—92no—
162#897—91no—
163#895—91no—
164#894—92no—
165#890—91no—
166#890—91no—
167#883—92no—
168#879—92no—
169#876—92no—
170#875—91no—
171C4×D12—31no—
172C4×D12—31no—
173C4×D12—61no—
174C4×D12—92no—
175#872—92no—
176#872—92no—
177C42×D3—31no—
178C42×D3—91no—
179#870—91no—
180C4×Dic6—31no—
181C4×Dic6—91no—
182C22×C3⋊D3C22×D311yescharacteristic
183C22×C3×D3—31no—
184C22×C3×D3—32no—
185C22×C3×D3—32no—
186C22×C3×D3—62no—
187C2×C32×C4—21no—
188C2×C32×C4—21no—
189C2×C32×C4—21no—
190C2×C32×C4C22×D311yescharacteristic
191C2×C32×C4C22×D311yescharacteristic
192C2×C32×C4C22×D311yescharacteristic
193C2×C32×C4C22×D311yescharacteristic
194C2×C32×C4C22×D311yescharacteristic
195C2×C32×C4C22×D311yescharacteristic
196C2×C3⋊Dic3C22×D312yes—
197C2×C3⋊Dic3C22×D311yescharacteristic
198C2×C3⋊Dic3C22×D311yescharacteristic
199C2×C3⋊Dic3C22×D311yescharacteristic
200C2×C3⋊Dic3C22×D312yes—
201C4×C3⋊D3—22no—
202C4×C3⋊D3—22no—
203C4×C3⋊D3—22no—
204#634—22no—
205C2×C3×Dic3—32no—
206C2×C3×Dic3—31no—
207C2×C3×Dic3—31no—
208C2×C3×Dic3—31no—
209C2×C3×Dic3—32no—
210C2×C3×Dic3—32no—
211C2×C3×Dic3—32no—
212C2×C3×Dic3—62no—
213C3×D12—64no—
214C3×D12—62no—
215C3×D12—62no—
216C3×C4×D3—62no—
217C3×C4×D3—62no—
218C3×C4×D3—64no—
219C3×C4×D3—62no—
220C3×C4×D3—64no—
221C3×C4×D3—124no—
222C3×Dic6—62no—
223#626—64no—
224#625—64no—
225#529—271noSylow
226C2×H(3,𝔽3)C2×#4111yescharacteristic
227C2×H(3,𝔽3)C2×#4111yescharacteristic
228C2×H(3,𝔽3)C2×#4111yescharacteristic
229#272—42no—
230(C3⋊D3)⋊C3—22no—
231C2×C3×Q8—91no—
232C2×C3×D4—91no—
233C22×C3×C4—91no—
234C22×C3×C4—91no—
235C22×C3×C4—91no—
236C22×C3×C4—92no—
237C2×#67—92no—
238C2×#67—92no—
239C2×D12—31no—
240C2×D12—31no—
241C2×D12—61no—
242C2×D12—92no—
243C2×C4×D3—32no—
244C2×C4×D3—32no—
245C2×C4×D3—31no—
246C2×C4×D3—31no—
247C2×C4×D3—31no—
248C2×C4×D3—62no—
249C2×C4×D3—91no—
250C2×C4×D3—91no—
251C2×C4×D3—91no—
252C2×Dic6—31no—
253C2×Dic6—91no—
254C3×C4⋊C4—91no—
255C3×C4⋊C4—92no—
256C3×C4⋊C4—91no—
257C3×C22⋊C4—92no—
258C3×C42—31no—
259C3×C42—31no—
260C3×C42—61no—
261C3×C42—91no—
262C3×C42—91no—
263C3×C42—92no—
264C3×C42—91no—
265C3×C42D3211yescharacteristic
266#217—92no—
267#217—92no—
268#217—92no—
269#217—92no—
270(C2×D3)⋊C4—32no—
271(C2×D3)⋊C4—32no—
272(C2×D3)⋊C4—62no—
273(C2×D3)⋊C4—92no—
274(C2×D3)⋊C4—92no—
275(C2×D3)⋊C4—92no—
276(C2×D3)⋊C4—92no—
277(C2×D3)⋊C4—92no—
278(C2×D3)⋊C4—92no—
279(C2×D3)⋊C4—92no—
280(C2×D3)⋊C4—92no—
281C4⋊Dic3—31no—
282C4⋊Dic3—31no—
283C4⋊Dic3—31no—
284C4⋊Dic3—61no—
285C4⋊Dic3—91no—
286C4⋊Dic3—92no—
287C4⋊Dic3—92no—
288C4⋊Dic3—92no—
289Dic3⋊C4—32no—
290Dic3⋊C4—92no—
291Dic3⋊C4—92no—
292Dic3⋊C4—92no—
293Dic3⋊C4—92no—
294Dic3⋊C4—92no—
295Dic3⋊C4—92no—
296Dic3⋊C4—92no—
297C4×Dic3—32no—
298C4×Dic3—31no—
299C4×Dic3—31no—
300C4×Dic3—31no—
301C4×Dic3—92no—
302C4×Dic3—91no—
303C4×Dic3—91no—
304C4×Dic3—91no—
305C22×C32—21no—
306C22×C32C23×D311yescharacteristic
307C22×C32C23×D311yescharacteristic
308C2×C3⋊D3—21no—
309C2×C3⋊D3—21no—
310C2×C3⋊D3#23512yes—
311C2×C3×D3—32no—
312C2×C3×D3—61no—
313C2×C3×D3—61no—
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—122no—
322C2×C3×D3—122no—
323C32×C4—21no—
324C32×C4—21no—
325C32×C4—22no—
326C32×C4—22no—
327C32×C4—41no—
328C32×C4#23512yes—
329C32×C4#23512yes—
330C32×C4#23512yes—
331C32×C4#23912yes—
332C3⋊Dic3—22no—
333C3⋊Dic3—21no—
334C3⋊Dic3—21no—
335C3⋊Dic3—21no—
336C3⋊Dic3—22no—
337C3×Dic3—62no—
338C3×Dic3—62no—
339C3×Dic3—61no—
340C3×Dic3—62no—
341C3×Dic3—61no—
342C3×Dic3—62no—
343C3×Dic3—61no—
344C3×Dic3—122no—
345#126—272no—
346#125—271no—
347#124—271no—
348#123—272no—
349C22⋊Q8—271no—
350#121—271no—
351C4×Q8—91no—
352C4×D4—91no—
353C4×D4—272no—
354#117—272no—
355C2×C42—91no—
356H(3,𝔽3)#52911yescharacteristic, complemented, Sylow
357C23×C3—91no—
358C23×C3—92no—
359C22×D3—31no—
360C22×D3—32no—
361C22×D3—32no—
362C22×D3—62no—
363C22×D3—91no—
364C3×Q8—182no—
365C3×D4—182no—
366C2×C3×C4—31no—
367C2×C3×C4—31no—
368C2×C3×C4—31no—
369C2×C3×C4—31no—
370C2×C3×C4—31no—
371C2×C3×C4—31no—
372C2×C3×C4—61no—
373C2×C3×C4—61no—
374C2×C3×C4—61no—
375C2×C3×C4—91no—
376C2×C3×C4—91no—
377C2×C3×C4—91no—
378C2×C3×C4—92no—
379C2×C3×C4—92no—
380C2×C3×C4—92no—
381C2×C3×C4—182no—
382C2×C3×C4—182no—
383C2×C3×C4—182no—
384C2×C3×C4—184no—
385C2×C3×C4C2×D3211yescharacteristic
386C2×C3×C4C2×D3211yescharacteristic
387C2×C3×C4C2×D3211yescharacteristic
388#67—184no—
389#67—184no—
390C2×Dic3—32no—
391C2×Dic3—32no—
392C2×Dic3—31no—
393C2×Dic3—31no—
394C2×Dic3—31no—
395C2×Dic3—32no—
396C2×Dic3—32no—
397C2×Dic3—62no—
398C2×Dic3—91no—
399C2×Dic3—91no—
400C2×Dic3—91no—
401C2×Dic3—92no—
402C2×Dic3—92no—
403D12—62no—
404D12—62no—
405D12—64no—
406D12—184no—
407C4×D3—62no—
408C4×D3—64no—
409C4×D3—64no—
410C4×D3—62no—
411C4×D3—62no—
412C4×D3—124no—
413C4×D3—182no—
414C4×D3—182no—
415C4×D3—182no—
416Dic6—62no—
417Dic6—182no—
418C2×C32—21no—
419C2×C32—21no—
420C2×C32—21no—
421C2×C32C2×#23511yescharacteristic
422C2×C32C2×#23511yescharacteristic
423C2×C32C2×#23511yescharacteristic
424C2×C32C2×#23511yescharacteristic
425C2×C32C2×#23911yescharacteristic
426C2×C32D3×#4111yescharacteristic
427C3⋊D3—22no—
428C3×D3—62no—
429C3×D3—124no—
430C3×D3—122no—
431C3×D3—122no—
432C2×Q8—91no—
433C2×D4—91no—
434C2×D4—272no—
435C22×C4—91no—
436C22×C4—91no—
437C22×C4—91no—
438C22×C4—92no—
439C4⋊C4—91no—
440C4⋊C4—91no—
441C4⋊C4—92no—
442C4⋊C4—272no—
443C4⋊C4—272no—
444C4⋊C4—272no—
445C22⋊C4—92no—
446C22⋊C4—272no—
447C22⋊C4—272no—
448C22⋊C4—272no—
449C22⋊C4—272no—
450C42—91no—
451C42—91no—
452C42—91no—
453C42—92no—
454C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
455C22×C3—31no—
456C22×C3—31no—
457C22×C3—61no—
458C22×C3—92no—
459C22×C3—181no—
460C22×C3—181no—
461C22×C3—182no—
462C22×C3—182no—
463C22×C3—182no—
464C22×C3C22×D3211yescharacteristic
465C2×D3—32no—
466C2×D3—61no—
467C2×D3—61no—
468C2×D3—62no—
469C2×D3—62no—
470C2×D3—62no—
471C2×D3—62no—
472C2×D3—62no—
473C2×D3—62no—
474C2×D3—64no—
475C2×D3—92no—
476C2×D3—122no—
477C2×D3—122no—
478C2×D3—181no—
479C2×D3—181no—
480C3×C4—21no—
481C3×C4—32no—
482C3×C4—32no—
483C3×C4—32no—
484C3×C4—32no—
485C3×C4—61no—
486C3×C4—61no—
487C3×C4—62no—
488C3×C4—62no—
489C3×C4—121no—
490C3×C4—182no—
491C3×C4—181no—
492C3×C4—181no—
493C3×C4—181no—
494C3×C4—182no—
495C3×C4—182no—
496C3×C4#379812yes—
497C3×C4#379912yes—
498Dic3—62no—
499Dic3—62no—
500Dic3—62no—
501Dic3—61no—
502Dic3—61no—
503Dic3—61no—
504Dic3—62no—
505Dic3—122no—
506Dic3—182no—
507Dic3—181no—
508Dic3—181no—
509Dic3—181no—
510Dic3—182no—
511C32—21no—
512C32#547911yescharacteristic, complemented
513C32#558011yescharacteristic, complemented
514C23—91no—
515C23—92no—
516Q8—182no—
517D4—182no—
518D4—544no—
519C2×C4—91no—
520C2×C4—91no—
521C2×C4—91no—
522C2×C4—92no—
523C2×C4—92no—
524C2×C4—92no—
525C2×C4—182no—
526C2×C4—182no—
527C2×C4—184no—
528C2×C4—182no—
529C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
530C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
531C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
532C2×C3—31no—
533C2×C3—31no—
534C2×C3—31no—
535C2×C3—31no—
536C2×C3—31no—
537C2×C3—31no—
538C2×C3—61no—
539C2×C3—61no—
540C2×C3—61no—
541C2×C3—182no—
542C2×C3—362no—
543C2×C3C2×#379811yescharacteristic
544C2×C3C2×#379911yescharacteristic
545C2×C3D3×#23511yescharacteristic
546D3—62no—
547D3—122no—
548D3—122no—
549D3—124no—
550D3—182no—
551C22—92no—
552C22—181no—
553C22—181no—
554C22—182no—
555C22—182no—
556C22—182no—
557C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
558C4—21no—
559C4—181no—
560C4—182no—
561C4—182no—
562C4—181no—
563C4—181no—
564C4—182no—
565C4#8865012yes—
566C4#8865212yes—
567C3—31no—
568C3—31no—
569C3—61no—
570C3#1059387011yescharacteristic
571C2—182no—
572C2—362no—
573C2C2×#8865011yescharacteristic
574C2C2×#8865211yescharacteristic
575C2#1172249011yescharacteristic
subgroupG/H
centerC2×C4C2×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC42×H(3,𝔽3)C22
prime radical#11721156C2
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#529
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C4×C4⋊#272C2semidirect
2. C42×(C3⋊D3)⋊C3C2semidirect
3. C4×#634⋊C3C2semidirect
5. C4×#6200C2semidirect
6. #11721114C2semidirect
7. #11721106C2semidirect
8. #11721100C2semidirect
9. #11721097C2semidirect
10. #11721093C2semidirect
11. #11721092C2semidirect
37. C42×C3⋊D3D3semidirect
38. C4×#634D3semidirect
356. H(3,𝔽3)#529semidirect
512. C32#5479semidirect
513. C32#5580semidirect

Maximal subgroups

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

subgroupindex
C4×C4⋊#2722
C42×(C3⋊D3)⋊C32
C4×#634⋊C32
#117211562
C4×#62002
#117211142
#117211062
#117211002
#117210972
#117210932
#117210922
#105939743
#105938753