Finite Group Databasebeta

← #13147806

Subgroups — #13147806

Order 1728.

868 conjugacy classes of subgroups.

The subgroup lattice has 553 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#6229C211yesmaximal, characteristic, complemented
2C4×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic
3#64079⋊C3C211yesmaximal, characteristic, complemented
4#11721136C211yesmaximal, characteristic
5#11721136C211yesmaximal, characteristic
6#11721135C212yesmaximal, complemented
7#11721133C212yesmaximal, complemented
8#11721119C212yesmaximal, complemented
9#11721117C211yesmaximal, characteristic, complemented
10#11721101C212yesmaximal, complemented
11#11721095C211yesmaximal, characteristic, complemented
12#10593938—31nomaximal
13#10593929—31nomaximal
14C2×#6229C2211yescharacteristic
15C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
16C2×C4×(C3⋊D3)⋊C3C2212yes—
17C2×((C3⋊D3)⋊C3)⋊C4C2212yes—
18C2×((C3⋊D3)⋊C3)⋊C4C2212yes—
19C2×#6196C2212yes—
20C42×H(3,𝔽3)C2211yescharacteristic
21C4⋊#1084C2211yescharacteristic
22#88540C2211yescharacteristic
23#88540C2211yescharacteristic
24C4×#1084C2211yescharacteristic
25C4×#1084C2211yescharacteristic
26#3752⋊C3C2212yes—
27C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
28C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
29C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
30#3750⋊C3C2212yes—
31#3750⋊C3C2212yes—
32#88450C2212yes—
33#88450C2212yes—
34#88449C2212yes—
35#88448C2212yes—
36#88448C2212yes—
37C4×C4⋊(C3⋊D3)D311yescharacteristic
38#64079D311yescharacteristic, complemented
39C3×C4×D12—31no—
40C3×#870—31no—
41C3×C4×Dic6—31no—
42C3×C4×Dic6—31no—
43C3×C4×Dic6—61no—
44#63918—31no—
45#63918—31no—
46#63899—32no—
47#63895—32no—
48#63894—31no—
49#63894—31no—
50Dic3×D12—31no—
51Dic3×D12—31no—
52#63888—32no—
53#63887—32no—
54#63883—32no—
55D3×Dic3⋊C4—32no—
56#63869—31no—
57#63869—31no—
58#63865—32no—
59#63851—32no—
60C22×(C3⋊D3)⋊C3C2312yes—
61C2×C4×H(3,𝔽3)C2311yescharacteristic
62C2×C4×H(3,𝔽3)C2311yescharacteristic
63C2×C4×H(3,𝔽3)C2311yescharacteristic
64C2×#1084C2311yescharacteristic
65C2×#1084C2311yescharacteristic
66C2×#1084C2311yescharacteristic
67C2×#1084C2311yescharacteristic
68#6229—22no—
69C2×(C3⋊Dic3)⋊C3C2312yes—
70C2×(C3⋊Dic3)⋊C3C2312yes—
71C2×(C3⋊Dic3)⋊C3C2312yes—
72C4⋊((C3⋊D3)⋊C3)Q814yescomplemented
73C4×(C3⋊D3)⋊C3—42no—
74((C3⋊D3)⋊C3)⋊C4—24no—
75((C3⋊D3)⋊C3)⋊C4—42no—
76#6196—42no—
77#5690—91no—
78#5579—91no—
79#5539—91no—
80C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
81C2×C4×C3⋊D3C2×D312yes—
82C2×C3×D12—31no—
83C2×C3×C4×D3—32no—
84C2×C3×Dic6—31no—
85C2×C3×Dic6—31no—
86C2×C3×Dic6—61no—
87C2×#627—32no—
88C2×#627—32no—
89C2×#624—32no—
90C2×#624—32no—
91C2×D3×Dic3—32no—
92C2×D3×Dic3—32no—
93C32×C42—21no—
94C32×C42C2×D311yescharacteristic
95C32×C42C2×D311yescharacteristic
96#3752C2×D312yes—
97C4⋊(C3⋊Dic3)C2×D311yescharacteristic
98C4⋊(C3⋊Dic3)C2×D311yescharacteristic
99C4⋊(C3⋊Dic3)C2×D311yescharacteristic
100#3750C2×D312yes—
101#3750C2×D312yes—
102C3×(C2×D3)⋊C4—32no—
103C3×C4⋊Dic3—31no—
104C3×C4⋊Dic3—31no—
105C3×C4⋊Dic3—31no—
106C3×C4⋊Dic3—31no—
107C3×C4⋊Dic3—31no—
108C3×C4⋊Dic3—61no—
109C3×Dic3⋊C4—31no—
110C3×Dic3⋊C4—31no—
111C3×Dic3⋊C4—32no—
112C3×Dic3⋊C4—31no—
113C3×Dic3⋊C4—31no—
114C3×Dic3⋊C4—32no—
115C3×Dic3⋊C4—61no—
116C3×Dic3⋊C4—61no—
117C3×C4×Dic3—31no—
118C3×C4×Dic3—31no—
119C3×C4×Dic3—31no—
120C3×C4×Dic3—31no—
121C3×C4×Dic3—61no—
122C3×C4×Dic3—61no—
123#3724—32no—
124#3724—32no—
125Dic3⋊Dic3—32no—
126Dic3⋊Dic3—32no—
127Dic3⋊Dic3—32no—
128Dic3⋊Dic3—32no—
129#3722—32no—
130#3722—32no—
131(C2×D3)⋊Dic3—32no—
132(C2×D3)⋊Dic3—32no—
133C22×H(3,𝔽3)C2411yescharacteristic
134C2×(C3⋊D3)⋊C3—22no—
135C2×(C3⋊D3)⋊C3—22no—
136C2×(C3⋊D3)⋊C3C2×Q814yes—
137C4×H(3,𝔽3)—21no—
138C4×H(3,𝔽3)—21no—
139C4×H(3,𝔽3)C2×Q812yes—
140#1084—21no—
141#1084—21no—
142#1084—21no—
143#1084#4112yes—
144(C3⋊Dic3)⋊C3—22no—
145(C3⋊Dic3)⋊C3—22no—
146(C3⋊Dic3)⋊C3—22no—
147C3×#125—91no—
148C3×C4×Q8—91no—
149C3×C4×D4—91no—
150#946—92no—
151D4×Dic3—91no—
152D4×Dic3—91no—
153#924—92no—
154C2×Dic3⋊C4—92no—
155#897—92no—
156#896—92no—
157#896—92no—
158#896—92no—
159#896—92no—
160#893—91no—
161#893—91no—
162D3×C4⋊C4—92no—
163D3×C4⋊C4—92no—
164#890—91no—
165#890—91no—
166#888—91no—
167#888—91no—
168#878—92no—
169#878—92no—
170C4×D12—31no—
171C4×D12—91no—
172C4×D12—91no—
173C4×D12—91no—
174#870—31no—
175#870—91no—
176#870—91no—
177#869—91no—
178C4×Dic6—31no—
179C4×Dic6—31no—
180C4×Dic6—61no—
181C22×C3⋊D3C22×D312yes—
182C22×C3×D3—32no—
183C2×C32×C4—21no—
184C2×C32×C4—21no—
185C2×C32×C4—21no—
186C2×C32×C4C22×D311yescharacteristic
187C2×C32×C4C22×D311yescharacteristic
188C2×C32×C4C22×D311yescharacteristic
189C2×C32×C4C22×D311yescharacteristic
190C2×C32×C4C22×D311yescharacteristic
191C2×C32×C4C22×D311yescharacteristic
192C2×C3⋊Dic3C22×D312yes—
193C2×C3⋊Dic3C22×D312yes—
194C2×C3⋊Dic3C22×D312yes—
195C4⋊(C3⋊D3)Dic614yescomplemented
196C4×C3⋊D3—42no—
197C2×C3×Dic3—31no—
198C2×C3×Dic3—31no—
199C2×C3×Dic3—32no—
200C2×C3×Dic3—32no—
201C2×C3×Dic3—32no—
202C2×C3×Dic3—31no—
203C2×C3×Dic3—31no—
204C2×C3×Dic3—31no—
205C2×C3×Dic3—31no—
206C2×C3×Dic3—31no—
207C2×C3×Dic3—31no—
208C2×C3×Dic3—61no—
209C2×C3×Dic3—61no—
210C2×C3×Dic3—61no—
211C2×C3×Dic3—61no—
212C3×D12—34no—
213C3×C4×D3—122no—
214C3×Dic6—62no—
215C3×Dic6—62no—
216C3×Dic6—64no—
217#627—122no—
218#627—122no—
219#624—64no—
220#624—122no—
221D3×Dic3—64no—
222D3×Dic3—122no—
223#554—271noSylow
224C2×H(3,𝔽3)C22×Q811yescharacteristic
225C2×H(3,𝔽3)C2×#4111yescharacteristic
226C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
227(C3⋊D3)⋊C3—24no—
228C2×C3×Q8—91no—
229C2×C3×D4—91no—
230C22×C3×C4—92no—
231C22×Dic3—92no—
232C22×Dic3—92no—
233C2×D12—31no—
234C2×D12—91no—
235C2×C4×D3—32no—
236C2×C4×D3—92no—
237C2×C4×D3—92no—
238C2×C4×D3—92no—
239C2×C4×D3—92no—
240C2×C4×D3—92no—
241C2×Dic6—31no—
242C2×Dic6—31no—
243C2×Dic6—61no—
244C2×Dic6—92no—
245C2×Dic6—92no—
246C2×Dic6—92no—
247C3×C4⋊C4—91no—
248C3×C4⋊C4—91no—
249C3×C4⋊C4—91no—
250C3×C4⋊C4—92no—
251C3×C4⋊C4—92no—
252C3×C4⋊C4—91no—
253C3×C4⋊C4—91no—
254C3×C4⋊C4—91no—
255C3×C22⋊C4—92no—
256C3×C42—31no—
257C3×C42—31no—
258C3×C42—61no—
259C3×C42—91no—
260C3×C42—91no—
261C3×C42D3211yescharacteristic
262#217—92no—
263#217—92no—
264(C2×D3)⋊C4—32no—
265(C2×D3)⋊C4—92no—
266(C2×D3)⋊C4—92no—
267(C2×D3)⋊C4—92no—
268(C2×D3)⋊C4—92no—
269(C2×D3)⋊C4—92no—
270C4⋊Dic3—31no—
271C4⋊Dic3—31no—
272C4⋊Dic3—31no—
273C4⋊Dic3—31no—
274C4⋊Dic3—31no—
275C4⋊Dic3—61no—
276C4⋊Dic3—91no—
277C4⋊Dic3—91no—
278C4⋊Dic3—91no—
279C4⋊Dic3—92no—
280C4⋊Dic3—92no—
281C4⋊Dic3—92no—
282Dic3⋊C4—31no—
283Dic3⋊C4—31no—
284Dic3⋊C4—31no—
285Dic3⋊C4—31no—
286Dic3⋊C4—32no—
287Dic3⋊C4—32no—
288Dic3⋊C4—61no—
289Dic3⋊C4—61no—
290Dic3⋊C4—92no—
291Dic3⋊C4—92no—
292Dic3⋊C4—92no—
293Dic3⋊C4—92no—
294Dic3⋊C4—92no—
295Dic3⋊C4—92no—
296Dic3⋊C4—92no—
297Dic3⋊C4—92no—
298C4×Dic3—31no—
299C4×Dic3—31no—
300C4×Dic3—31no—
301C4×Dic3—31no—
302C4×Dic3—61no—
303C4×Dic3—61no—
304C22×C32—21no—
305C22×C32C23×D311yescharacteristic
306C22×C32C23×D311yescharacteristic
307C2×C3⋊D3—22no—
308C2×C3⋊D3—22no—
309C2×C3⋊D3C2×Dic614yes—
310C2×C3×D3—34no—
311C2×C3×D3—62no—
312C2×C3×D3—62no—
313C32×C4—22no—
314C32×C4—21no—
315C32×C4—21no—
316C32×C4—21no—
317C32×C4—21no—
318C32×C4—22no—
319C32×C4—41no—
320C32×C4C2×Dic612yes—
321C32×C4D3×Q812yes—
322C3⋊Dic3—22no—
323C3⋊Dic3—22no—
324C3⋊Dic3—22no—
325C3×Dic3—32no—
326C3×Dic3—32no—
327C3×Dic3—62no—
328C3×Dic3—62no—
329C3×Dic3—62no—
330C3×Dic3—61no—
331C3×Dic3—61no—
332C3×Dic3—62no—
333C3×Dic3—61no—
334C3×Dic3—61no—
335C3×Dic3—62no—
336C3×Dic3—61no—
337C3×Dic3—61no—
338C3×Dic3—121no—
339C3×Dic3—121no—
340C4⋊Q8—271no—
341#125—91no—
342#125—271no—
343C22⋊Q8—272no—
344C22⋊Q8—272no—
345C22⋊Q8—272no—
346C4×Q8—91no—
347C4×D4—91no—
348C4×D4—271no—
349C4×D4—271no—
350C2×C4⋊C4—272no—
351H(3,𝔽3)#55411yescharacteristic, complemented, Sylow
352C23×C3—92no—
353C22×D3—32no—
354C22×D3—92no—
355C3×Q8—182no—
356C3×D4—94no—
357C2×C3×C4—31no—
358C2×C3×C4—31no—
359C2×C3×C4—31no—
360C2×C3×C4—31no—
361C2×C3×C4—31no—
362C2×C3×C4—31no—
363C2×C3×C4—61no—
364C2×C3×C4—61no—
365C2×C3×C4—61no—
366C2×C3×C4—91no—
367C2×C3×C4—92no—
368C2×C3×C4—92no—
369C2×C3×C4—92no—
370C2×C3×C4—91no—
371C2×C3×C4—91no—
372C2×C3×C4—91no—
373C2×C3×C4—362no—
374C2×C3×C4C2×D3211yescharacteristic
375C2×C3×C4C2×D3211yescharacteristic
376C2×C3×C4C2×D3211yescharacteristic
377C2×Dic3—31no—
378C2×Dic3—31no—
379C2×Dic3—31no—
380C2×Dic3—31no—
381C2×Dic3—31no—
382C2×Dic3—31no—
383C2×Dic3—31no—
384C2×Dic3—31no—
385C2×Dic3—32no—
386C2×Dic3—32no—
387C2×Dic3—32no—
388C2×Dic3—61no—
389C2×Dic3—61no—
390C2×Dic3—61no—
391C2×Dic3—61no—
392C2×Dic3—92no—
393C2×Dic3—92no—
394C2×Dic3—92no—
395C2×Dic3—184no—
396C2×Dic3—362no—
397D12—34no—
398D12—94no—
399C4×D3—122no—
400C4×D3—184no—
401C4×D3—184no—
402C4×D3—362no—
403C4×D3—362no—
404C4×D3—362no—
405Dic6—62no—
406Dic6—62no—
407Dic6—64no—
408Dic6—362no—
409Dic6—362no—
410Dic6—362no—
411C2×C32—21no—
412C2×C32—21no—
413C2×C32—21no—
414C2×C32C22×Dic611yescharacteristic
415C2×C32#100411yescharacteristic
416C2×C32C2×D3×Q811yescharacteristic
417C2×C32D3×#4111yescharacteristic
418C2×C32D3×#4111yescharacteristic
419C2×C32#100911yescharacteristic
420C3⋊D3—24no—
421C3×D3—64no—
422C2×Q8—91no—
423C2×Q8—272no—
424C2×D4—91no—
425C22×C4—92no—
426C22×C4—272no—
427C22×C4—272no—
428C4⋊C4—91no—
429C4⋊C4—92no—
430C4⋊C4—91no—
431C4⋊C4—91no—
432C4⋊C4—91no—
433C4⋊C4—91no—
434C4⋊C4—91no—
435C4⋊C4—92no—
436C4⋊C4—272no—
437C4⋊C4—272no—
438C4⋊C4—272no—
439C22⋊C4—92no—
440C22⋊C4—272no—
441C22⋊C4—272no—
442C42—91no—
443C42—91no—
444C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
445C22×C3—31no—
446C22×C3—31no—
447C22×C3—61no—
448C22×C3—94no—
449C22×C3—182no—
450C22×C3—182no—
451C22×C3C22×D3211yescharacteristic
452C2×D3—34no—
453C2×D3—62no—
454C2×D3—62no—
455C2×D3—94no—
456C2×D3—182no—
457C2×D3—182no—
458C3×C4—21no—
459C3×C4—21no—
460C3×C4—32no—
461C3×C4—32no—
462C3×C4—61no—
463C3×C4—61no—
464C3×C4—61no—
465C3×C4—61no—
466C3×C4—62no—
467C3×C4—62no—
468C3×C4—92no—
469C3×C4—121no—
470C3×C4—182no—
471C3×C4—182no—
472C3×C4—182no—
473C3×C4—181no—
474C3×C4—181no—
475C3×C4—181no—
476C3×C4D3×Dic612yes—
477Dic3—32no—
478Dic3—32no—
479Dic3—62no—
480Dic3—61no—
481Dic3—61no—
482Dic3—61no—
483Dic3—61no—
484Dic3—61no—
485Dic3—61no—
486Dic3—62no—
487Dic3—62no—
488Dic3—62no—
489Dic3—62no—
490Dic3—121no—
491Dic3—121no—
492Dic3—182no—
493Dic3—182no—
494Dic3—182no—
495C32—21no—
496C32#553911yescharacteristic, complemented
497C32#569011yescharacteristic, complemented
498C23—92no—
499Q8—182no—
500Q8—1082no—
501D4—94no—
502C2×C4—91no—
503C2×C4—91no—
504C2×C4—91no—
505C2×C4—91no—
506C2×C4—92no—
507C2×C4—92no—
508C2×C4—92no—
509C2×C4—362no—
510C2×C4—544no—
511C2×C4—1082no—
512C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
513C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
514C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
515C2×C3—31no—
516C2×C3—31no—
517C2×C3—31no—
518C2×C3—31no—
519C2×C3—31no—
520C2×C3—31no—
521C2×C3—61no—
522C2×C3—61no—
523C2×C3—61no—
524C2×C3—184no—
525C2×C3C2×D3×Dic611yescharacteristic
526C2×C3#6430611yescharacteristic
527C2×C3#6430711yescharacteristic
528D3—64no—
529D3—184no—
530C22—94no—
531C22—182no—
532C22—182no—
533C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
534C4—21no—
535C4—21no—
536C4—92no—
537C4—182no—
538C4—181no—
539C4—182no—
540C4—181no—
541C4—182no—
542C4—181no—
543C4((C3⋊D3)⋊C3)⋊Q812yes—
544C3—31no—
545C3—31no—
546C3—61no—
547C3#1059394111yescharacteristic
548C2—184no—
549C2C2×((C3⋊D3)⋊C3)⋊Q811yescharacteristic
550C2#1172249111yescharacteristic
551C2#1172249211yescharacteristic
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⋊((C3⋊D3)⋊C3)C22
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#554
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×#6229C2semidirect
3. #64079⋊C3C2semidirect
6. #11721135C2semidirect
7. #11721133C2semidirect
8. #11721119C2semidirect
9. #11721117C2semidirect
10. #11721101C2semidirect
11. #11721095C2semidirect
38. #64079D3semidirect
72. C4⋊((C3⋊D3)⋊C3)Q8semidirect
195. C4⋊(C3⋊D3)Dic6semidirect
351. H(3,𝔽3)#554semidirect
496. C32#5539semidirect
497. C32#5690semidirect

Maximal subgroups

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

subgroupindex
C4×#62292
C4×C4⋊((C3⋊D3)⋊C3)2
#64079⋊C32
#117211362
#117211352
#117211332
#117211192
#117211172
#117211012
#117210952
#105939383
#105939293