Finite Group Databasebeta

← #13141072

Subgroups — #13141072

Order 1728.

703 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 703 conjugacy classes of subgroups — too many to draw. Only its 81 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#41×#272C211yesmaximal, characteristic
2#11721097C211yesmaximal, characteristic, complemented
3#11721065C211yesmaximal, characteristic, complemented
4#63664⋊C3C212yesmaximal, complemented
5#11720328C212yesmaximal, complemented
6#10591670—32nomaximal
7#41×H(3,𝔽3)C2211yescharacteristic
8#88748C411yescharacteristic, complemented
9Q8×#272C411yescharacteristic, complemented
10#88744—21no—
11#88744C411yescharacteristic, complemented
12D4×#272—21no—
13D4×#272C411yescharacteristic, complemented
14#88740—21no—
15#88740C2211yescharacteristic
16C2×C4×#272—21no—
17C2×C4×#272C2211yescharacteristic
18C4×(C3⋊Dic3)⋊C3C2212yescomplemented
19#3748⋊C3C2212yescomplemented
20#88451—21no—
21#88449—21no—
22#88435—21no—
23#88434—21no—
24C3×D3×#41—32no—
25C3×D3×#41—61no—
26#63855—32no—
27D3×#208—32no—
28#63664D312yescomplemented
29C3×#837—32no—
30#63569—32no—
31#63568—32no—
32C22×#272—21no—
33C22×#272D411yescharacteristic
34Q8×H(3,𝔽3)C2×C411yescharacteristic
35D4×H(3,𝔽3)—21no—
36D4×H(3,𝔽3)C2×C411yescharacteristic
37C2×C4×H(3,𝔽3)—21no—
38C2×C4×H(3,𝔽3)C2311yescharacteristic
39#6235—21no—
40#6235—21no—
41#6235—21no—
42C2×#1084—21no—
43C2×#1084D411yescharacteristic
44C4⋊#272—21no—
45C4⋊#272C2×C411yescharacteristic
46C4×#272—21no—
47C4×#272—21no—
48C4×#272—21no—
49C4×#272C2×C411yescharacteristic
50C4×#272C2×C411yescharacteristic
51C4×#272D411yescharacteristic
52C4×#272D411yescharacteristic
53#6229—21no—
54#6229C2×C411yescharacteristic
55C2×(C3⋊Dic3)⋊C3—22no—
56(C3⋊(C3⋊C8))⋊C3—22no—
57#5236—91no—
58#4821—92no—
59C32×#41—21no—
60C32×#41C2×D312yes—
61C3×#239—32no—
62C3×#239—61no—
63C3×D3×Q8—32no—
64C3×D3×Q8—61no—
65C3×#237—32no—
66C3×#237—62no—
67C3×#237—61no—
68C3×#237—62no—
69C3×D3×D4—32no—
70C3×D3×D4—62no—
71C3×D3×D4—61no—
72C3×D3×D4—62no—
73C3×#235—32no—
74C3×#235—62no—
75C3×#235—61no—
76C3×#235—62no—
77C2×C3×C4×D3—32no—
78C2×C3×C4×D3—62no—
79C2×C3×C4×D3—61no—
80C2×C3×C4×D3—62no—
81C4×C3⋊Dic3C2×D312yescomplemented
82#3748C2×D312yescomplemented
83C3×C4×Dic3—32no—
84C3×#208—32no—
85Dic3⋊Dic3—62no—
86(C2×D3)⋊Dic3—62no—
87#3711—62no—
88D3×C3⋊C8—62no—
89C22×H(3,𝔽3)—21no—
90C22×H(3,𝔽3)C2×D411yescharacteristic
91C2×#272—21no—
92C2×#272—41no—
93C2×#272—42no—
94C2×#272C22⋊C411yescharacteristic
95C2×#272C22⋊C411yescharacteristic
96C4×H(3,𝔽3)—21no—
97C4×H(3,𝔽3)C22×C411yescharacteristic
98C4×H(3,𝔽3)C2×D411yescharacteristic
99#1084—21no—
100#1084C22⋊C411yescharacteristic
101#1084C22⋊C411yescharacteristic
102(C3⋊Dic3)⋊C3—42no—
103C2×C3×#41—91no—
104D3×#41—32no—
105D3×#41—61no—
106#926—91no—
107C2×#208—91no—
108D3×M16—92no—
109#872—92no—
110C3×#104—92no—
111#837—32no—
112#837—92no—
113#837—92no—
114#825—92no—
115#805—92no—
116C22×C3×D3—32no—
117C22×C3×D3—61no—
118C22×C3×D3—62no—
119C22×C3×D3—62no—
120C32×Q8—21no—
121C32×Q8C4×D312yes—
122C32×D4—22no—
123C32×D4—22no—
124C32×D4—21no—
125C32×D4C4×D312yes—
126C2×C32×C4—22no—
127C2×C32×C4—22no—
128C2×C32×C4—21no—
129C2×C32×C4C22×D312yes—
130C2×C3⋊Dic3—22no—
131C3×#67—62no—
132C3×#67—62no—
133C3×#67—62no—
134C3×#67—62no—
135C3×#67—62no—
136C3×#67—62no—
137C2×C3×Dic3—32no—
138C2×C3×Dic3—61no—
139C2×C3×Dic3—62no—
140C2×C3×Dic3—62no—
141C2×C3×Dic3—62no—
142C3×D12—32no—
143C3×D12—61no—
144C3×D12—62no—
145C3×D12—62no—
146C3×C4×D3—32no—
147C3×C4×D3—32no—
148C3×C4×D3—32no—
149C3×C4×D3—32no—
150C3×C4×D3—61no—
151C3×C4×D3—61no—
152C3×C4×D3—61no—
153C3×C4×D3—61no—
154C3×C4×D3—62no—
155C3×C4×D3—62no—
156C3×C4×D3—62no—
157C3×C4×D3—62no—
158C3×C4×D3—62no—
159C3×C4×D3—62no—
160C3×Dic6—32no—
161C3×Dic6—61no—
162C3×Dic6—62no—
163C3×Dic6—62no—
164C3⋊(C3⋊C8)—22no—
165C3×C3⋊C8—62no—
166#421—271noSylow
167C2×H(3,𝔽3)—21no—
168C2×H(3,𝔽3)—41no—
169C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
170#272—21no—
171#272—21no—
172#272—41no—
173C3×#41—32no—
174C3×#41—61no—
175C3×#41—91no—
176C3×#41—91no—
177C3×#41—91no—
178C3×#41—181no—
179C3×#41—181no—
180C3×#41D3211yescharacteristic
181C2×C3×Q8—91no—
182C2×C3×D4—91no—
183C2×C3×D4—181no—
184C22×C3×C4—91no—
185C22×C3×C4—181no—
186#239—32no—
187#239—61no—
188D3×Q8—32no—
189D3×Q8—61no—
190#237—32no—
191#237—61no—
192#237—62no—
193#237—62no—
194D3×D4—32no—
195D3×D4—61no—
196D3×D4—62no—
197D3×D4—62no—
198#235—32no—
199#235—61no—
200#235—62no—
201#235—62no—
202C2×C4×D3—32no—
203C2×C4×D3—61no—
204C2×C4×D3—62no—
205C2×C4×D3—62no—
206C3×M16—92no—
207C3×C42—92no—
208#217—181no—
209(C2×D3)⋊C4—182no—
210C4⋊Dic3—181no—
211Dic3⋊C4—182no—
212C4×Dic3—32no—
213C4×Dic3—92no—
214#208—32no—
215#208—92no—
216#208—181no—
217C2×C3⋊C8—181no—
218ΓL(1,𝔽25)—182no—
219D3×C8—182no—
220C22×C32—22no—
221C22×C32—22no—
222C22×C32—21no—
223C22×C32D3×D412yes—
224C2×C3×D3—32no—
225C2×C3×D3—32no—
226C2×C3×D3—61no—
227C2×C3×D3—61no—
228C2×C3×D3—62no—
229C2×C3×D3—62no—
230C2×C3×D3—122no—
231C2×C3×D3—122no—
232C2×C3×D3—124no—
233C2×C3×D3—124no—
234C32×C4—22no—
235C32×C4—22no—
236C32×C4—21no—
237C32×C4—21no—
238C32×C4C2×C4×D312yes—
239C32×C4D3×D412yes—
240C3⋊Dic3—42no—
241C3×Dic3—32no—
242C3×Dic3—32no—
243C3×Dic3—61no—
244C3×Dic3—61no—
245C3×Dic3—62no—
246C3×Dic3—62no—
247C3×Dic3—122no—
248C2×#41—91no—
249C2×M16—271no—
250#117—271no—
251#104—92no—
252#104—272no—
253H(3,𝔽3)#42111yescharacteristic, complemented, Sylow
254C23×C3—91no—
255C23×C3—181no—
256C22×D3—32no—
257C22×D3—61no—
258C22×D3—62no—
259C22×D3—62no—
260C3×Q8—32no—
261C3×Q8—61no—
262C3×Q8—91no—
263C3×Q8—181no—
264C3×Q8#62411yescharacteristic
265C3×D4—21no—
266C3×D4—32no—
267C3×D4—62no—
268C3×D4—62no—
269C3×D4—61no—
270C3×D4—91no—
271C3×D4—181no—
272C3×D4—181no—
273C3×D4—181no—
274C3×D4—181no—
275C3×D4#62411yescharacteristic
276C2×C3×C4—21no—
277C2×C3×C4—32no—
278C2×C3×C4—62no—
279C2×C3×C4—62no—
280C2×C3×C4—61no—
281C2×C3×C4—91no—
282C2×C3×C4—91no—
283C2×C3×C4—91no—
284C2×C3×C4—91no—
285C2×C3×C4—91no—
286C2×C3×C4—181no—
287C2×C3×C4—181no—
288C2×C3×C4—181no—
289C2×C3×C4—181no—
290C2×C3×C4—182no—
291C2×C3×C4C2×D3211yescharacteristic
292#67—62no—
293#67—62no—
294#67—62no—
295#67—62no—
296#67—62no—
297#67—62no—
298C2×Dic3—32no—
299C2×Dic3—61no—
300C2×Dic3—62no—
301C2×Dic3—62no—
302C2×Dic3—62no—
303C2×Dic3—182no—
304D12—32no—
305D12—61no—
306D12—62no—
307D12—62no—
308C4×D3—32no—
309C4×D3—32no—
310C4×D3—32no—
311C4×D3—32no—
312C4×D3—61no—
313C4×D3—61no—
314C4×D3—61no—
315C4×D3—61no—
316C4×D3—62no—
317C4×D3—62no—
318C4×D3—62no—
319C4×D3—62no—
320C4×D3—62no—
321C4×D3—62no—
322Dic6—32no—
323Dic6—61no—
324Dic6—62no—
325Dic6—62no—
326C3×C8—182no—
327C3⋊C8—62no—
328C3⋊C8—182no—
329C2×C32—22no—
330C2×C32—21no—
331C2×C32—41no—
332C2×C32—42no—
333C2×C32—42no—
334C2×C32D3×C22⋊C412yes—
335C3×D3—62no—
336C3×D3—62no—
337C3×D3—62no—
338C3×D3—121no—
339C3×D3—122no—
340C3×D3—122no—
341#41—91no—
342#41—91no—
343#41—91no—
344#41—181no—
345#41—181no—
346#41((C3⋊D3)⋊C3)⋊C211yescharacteristic
347C2×Q8—91no—
348C2×D4—91no—
349C2×D4—181no—
350C22×C4—91no—
351C22×C4—181no—
352M16—92no—
353M16—541no—
354C2×C8—541no—
355C4⋊C4—541no—
356C22⋊C4—541no—
357C42—92no—
358C22×C3—21no—
359C22×C3—32no—
360C22×C3—62no—
361C22×C3—62no—
362C22×C3—61no—
363C22×C3—91no—
364C22×C3—91no—
365C22×C3—181no—
366C22×C3—361no—
367C22×C3—362no—
368C22×C3#381111yescharacteristic
369C2×D3—32no—
370C2×D3—32no—
371C2×D3—61no—
372C2×D3—61no—
373C2×D3—62no—
374C2×D3—62no—
375C2×D3—122no—
376C2×D3—122no—
377C2×D3—124no—
378C2×D3—124no—
379C3×C4—21no—
380C3×C4—32no—
381C3×C4—32no—
382C3×C4—62no—
383C3×C4—62no—
384C3×C4—61no—
385C3×C4—61no—
386C3×C4—91no—
387C3×C4—91no—
388C3×C4—181no—
389C3×C4—362no—
390C3×C4C2×#62411yescharacteristic
391C3×C4#381111yescharacteristic
392Dic3—32no—
393Dic3—32no—
394Dic3—61no—
395Dic3—61no—
396Dic3—62no—
397Dic3—62no—
398Dic3—122no—
399Dic3—362no—
400C32—21no—
401C32#482112yescomplemented
402C23—91no—
403C23—181no—
404Q8—91no—
405Q8—181no—
406Q8#619911yescharacteristic
407D4—21no—
408D4—91no—
409D4—181no—
410D4—181no—
411D4—181no—
412D4—181no—
413D4#619911yescharacteristic
414C2×C4—21no—
415C2×C4—91no—
416C2×C4—91no—
417C2×C4—91no—
418C2×C4—91no—
419C2×C4—91no—
420C2×C4—181no—
421C2×C4—181no—
422C2×C4—181no—
423C2×C4—181no—
424C2×C4—182no—
425C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
426C8—182no—
427C2×C3—21no—
428C2×C3—32no—
429C2×C3—41no—
430C2×C3—62no—
431C2×C3—61no—
432C2×C3—121no—
433C2×C3—122no—
434C2×C3—122no—
435C2×C3—181no—
436C2×C3—181no—
437C2×C3—361no—
438C2×C3#6397411yescharacteristic
439D3—62no—
440D3—62no—
441D3—62no—
442D3—121no—
443D3—122no—
444D3—122no—
445C22—21no—
446C22—91no—
447C22—91no—
448C22—181no—
449C22—361no—
450C22—362no—
451C22#8867711yescharacteristic
452C4—21no—
453C4—91no—
454C4—91no—
455C4—181no—
456C4—362no—
457C4C2×#619911yescharacteristic
458C4#8867711yescharacteristic
459C3—32no—
460C3—61no—
461C3#1059167211yescharacteristic
462C2—21no—
463C2—41no—
464C2—181no—
465C2—181no—
466C2—361no—
467C2#1172132011yescharacteristic
subgroupG/H
centerC4#88677
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
Fitting#41×H(3,𝔽3)C22
socleC2×C3#63974
prime radical#41×#272C2
cosocleC2×C4×H(3,𝔽3)C23
hypercenter#41((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#421
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C4 ◁ C2×C4 ◁ #41

Complements

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

NH ≅ G/Ntype
2. #11721097C2semidirect
3. #11721065C2semidirect
4. #63664⋊C3C2semidirect
5. #11720328C2semidirect
8. #88748C4semidirect
9. Q8×#272C4semidirect
11. #88744C4semidirect
13. D4×#272C4semidirect
18. C4×(C3⋊Dic3)⋊C3C22semidirect
19. #3748⋊C3C22semidirect
28. #63664D3semidirect
81. C4×C3⋊Dic3C2×D3semidirect
82. #3748C2×D3semidirect
253. H(3,𝔽3)#421semidirect
401. C32#4821semidirect

Maximal subgroups

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

subgroupindex
#41×#2722
#117210972
#117210652
#63664⋊C32
#117203282
#105916703