Finite Group Databasebeta

← #13140552

Subgroups — #13140552

Order 1728.

540 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 77 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#64149⋊C3C211yesmaximal, characteristic, complemented
2C2×#3753⋊C3C211yesmaximal, characteristic, complemented
3#11721132C211yesmaximal, characteristic, complemented
4C2×#88432C211yesmaximal, characteristic, complemented
5#11720578C211yesmaximal, characteristic, complemented
6#11720311C211yesmaximal, characteristic, complemented
7#11720308C211yesmaximal, characteristic, complemented
8#10591433—31nomaximal
9#10591429—31nomaximal
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C2×#638⋊C3—21no—
12C2×#638⋊C3—21no—
13C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
14C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
15C2×((C3⋊D3)⋊C3)⋊C4—21no—
16C4⋊#1084C2211yescharacteristic
17C2×#6180C2211yescharacteristic
18#3753⋊C3—22no—
19C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21#88450—21no—
22#88432—22no—
23#64149D311yescharacteristic, complemented
24C2×#3753D311yescharacteristic, complemented
25C3×#940—31no—
26C2×C3×#213—31no—
27#63898—31no—
28#63894—31no—
29C2×#3714—31no—
30C2×#3714—31no—
31C3×#832—31no—
32C3×#832—31no—
33C3×#832—61no—
34#63572—31no—
35Dic3⋊(C3⋊C8)—31no—
36#63564—31no—
37#63562—31no—
38C23×H(3,𝔽3)—21no—
39C22×(C3⋊D3)⋊C3—21no—
40C22×(C3⋊D3)⋊C3—21no—
41D4×H(3,𝔽3)—21no—
42D4×H(3,𝔽3)—21no—
43C2×C4×H(3,𝔽3)C2311yescharacteristic
44C2×#1084—21no—
45#638⋊C3—42no—
46#638⋊C3—42no—
47C2×(C3⋊Dic3)⋊C3D411yescharacteristic
48C2×(C3⋊Dic3)⋊C3D411yescharacteristic
49C4⋊((C3⋊D3)⋊C3)—21no—
50C4⋊((C3⋊D3)⋊C3)—21no—
51C4⋊((C3⋊D3)⋊C3)—21no—
52C4⋊((C3⋊D3)⋊C3)D412yescomplemented
53((C3⋊D3)⋊C3)⋊C4—42no—
54#6180—21no—
55(C3⋊(C3⋊C8))⋊C3—21no—
56#5150—91no—
57#5072—91no—
58#4786—91no—
59C2×C32×D4—21no—
60C2×C32×D4C2×D311yescharacteristic
61C2×C32×D4C2×D311yescharacteristic
62C2×#638—21no—
63C2×#638—21no—
64C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
65C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
66C2×C3×#67—61no—
67C2×C3×#67—61no—
68C2×C3×D12—31no—
69C2×C3×D12—31no—
70C2×#624—61no—
71C2×D3×Dic3—61no—
72#3753—22no—
73C4×C3⋊Dic3C2×D311yescharacteristic
74C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
75C3×#213—62no—
76C3×C4⋊Dic3—31no—
77C3×C4⋊Dic3—31no—
78C3×C4⋊Dic3—61no—
79C3×C4×Dic3—31no—
80C2×C3×C3⋊C8—31no—
81C2×C3×C3⋊C8—31no—
82C2×C3×C3⋊C8—31no—
83C2×C3×C3⋊C8—61no—
84#3722—61no—
85(C2×D3)⋊Dic3—61no—
86#3714—62no—
87#3714—62no—
88C22×H(3,𝔽3)—41no—
89C22×H(3,𝔽3)—41no—
90C22×H(3,𝔽3)—41no—
91C22×H(3,𝔽3)C2×D411yescharacteristic
92C2×(C3⋊D3)⋊C3—22no—
93C2×(C3⋊D3)⋊C3—41no—
94C2×(C3⋊D3)⋊C3—41no—
95C2×(C3⋊D3)⋊C3—41no—
96C2×(C3⋊D3)⋊C3—41no—
97C2×(C3⋊D3)⋊C3—41no—
98C4×H(3,𝔽3)C2×D411yescharacteristic
99C4×H(3,𝔽3)#4111yescharacteristic
100#1084—41no—
101(C3⋊Dic3)⋊C3—21no—
102(C3⋊Dic3)⋊C3D812yescomplemented
103C2×C3×D8—91no—
104C3×#127—91no—
105#940—31no—
106#940—91no—
107D4×Dic3—91no—
108C2×#213—31no—
109C2×#213—91no—
110C2×#213—91no—
111C2×#213—91no—
112C2×D24—91no—
113#893—91no—
114C4×D12—91no—
115C3×#102—91no—
116#832—31no—
117#832—31no—
118#832—61no—
119#832—91no—
120#821—91no—
121Dic3⋊C8—91no—
122Dic3⋊C8—91no—
123#809—91no—
124C4⋊(C3⋊C8)—91no—
125C23×C32—21no—
126C23×C32—21no—
127C23×C32—41no—
128C22×C3⋊D3—21no—
129C22×C3⋊D3—21no—
130C22×C3×D3—61no—
131C22×C3×D3—61no—
132C32×D4—22no—
133C32×D4—21no—
134C32×D4—21no—
135C32×D4—21no—
136C32×D4—21no—
137C32×D4—41no—
138C2×C32×C4—21no—
139C2×C32×C4C22×D311yescharacteristic
140C2×C32×C4C22×D311yescharacteristic
141#638—42no—
142#638—42no—
143C2×C3⋊Dic3#6711yescharacteristic
144C2×C3⋊Dic3#6711yescharacteristic
145C4⋊(C3⋊D3)—21no—
146C4⋊(C3⋊D3)—21no—
147C4⋊(C3⋊D3)—21no—
148C4⋊(C3⋊D3)#6712yescomplemented
149C3×#67—122no—
150C3×#67—122no—
151C2×C3×Dic3—31no—
152C2×C3×Dic3—31no—
153C2×C3×Dic3—61no—
154C2×C3×Dic3—61no—
155C2×C3×Dic3—121no—
156C3×D12—32no—
157C3×D12—61no—
158C3×D12—61no—
159C3×D12—61no—
160#624—122no—
161D3×Dic3—122no—
162C3⋊(C3⋊C8)—21no—
163C3×C3⋊C8—61no—
164C3×C3⋊C8—61no—
165C3×C3⋊C8—61no—
166C3×C3⋊C8—121no—
167#459—271noSylow
168C2×H(3,𝔽3)—81no—
169C2×H(3,𝔽3)#12111yescharacteristic
170C2×H(3,𝔽3)C2×D811yescharacteristic
171C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
172(C3⋊D3)⋊C3—42no—
173(C3⋊D3)⋊C3—81no—
174C2×C3×D4—31no—
175C2×C3×D4—31no—
176C2×C3×D4—61no—
177C2×C3×D4—91no—
178C2×C3×D4—91no—
179C2×C3×D4—181no—
180C2×C3×D4—181no—
181C2×C3×D4D3211yescharacteristic
182C2×#67—61no—
183C2×#67—61no—
184C2×#67—181no—
185C2×#67—181no—
186C22×Dic3—181no—
187C2×D12—31no—
188C2×D12—31no—
189C2×D12—91no—
190C2×D12—91no—
191C2×C4×D3—181no—
192C2×C4×D3—181no—
193C3×D8—182no—
194C2×C3×C8—91no—
195C2×C3×C8—91no—
196C3×C4⋊C4—91no—
197C3×C42—91no—
198#217—181no—
199#213—62no—
200#213—182no—
201#213—182no—
202#213—182no—
203(C2×D3)⋊C4—181no—
204(C2×D3)⋊C4—181no—
205C4⋊Dic3—31no—
206C4⋊Dic3—31no—
207C4⋊Dic3—61no—
208C4×Dic3—31no—
209C4×Dic3—91no—
210C2×C3⋊C8—31no—
211C2×C3⋊C8—31no—
212C2×C3⋊C8—31no—
213C2×C3⋊C8—61no—
214C2×C3⋊C8—91no—
215D24—182no—
216C22×C32—21no—
217C22×C32—42no—
218C22×C32—41no—
219C22×C32—41no—
220C22×C32—41no—
221C22×C32—41no—
222C22×C32—41no—
223C22×C32—41no—
224C22×C32—81no—
225C22×C32—81no—
226C22×C32D3×D411yescharacteristic
227C22×C32C2×#6711yescharacteristic
228C2×C3⋊D3—22no—
229C2×C3⋊D3—41no—
230C2×C3⋊D3—41no—
231C2×C3⋊D3—41no—
232C2×C3⋊D3—41no—
233C2×C3⋊D3—41no—
234C2×C3×D3—62no—
235C2×C3×D3—121no—
236C2×C3×D3—121no—
237C2×C3×D3—121no—
238C2×C3×D3—121no—
239C2×C3×D3—121no—
240C32×C4—21no—
241C32×C4—21no—
242C32×C4D3×D411yescharacteristic
243C32×C4#23711yescharacteristic
244C32×C4#23711yescharacteristic
245C32×C4C2×#6711yescharacteristic
246C3⋊Dic3—21no—
247C3⋊Dic3#21312yescomplemented
248C3×Dic3—32no—
249C3×Dic3—61no—
250C3×Dic3—121no—
251C3×Dic3—121no—
252C3×Dic3—241no—
253C2×D8—91no—
254C2×D8—271no—
255#127—91no—
256C4×D4—271no—
257C4⋊C8—271no—
258#102—91no—
259#102—271no—
260H(3,𝔽3)#45911yescharacteristic, complemented, Sylow
261C23×C3—21no—
262C23×C3—61no—
263C23×C3—61no—
264C23×C3—121no—
265C23×C3—181no—
266C23×C3—181no—
267C22×D3—61no—
268C22×D3—61no—
269C22×D3—181no—
270C22×D3—181no—
271C3×D4—21no—
272C3×D4—21no—
273C3×D4—61no—
274C3×D4—61no—
275C3×D4—61no—
276C3×D4—61no—
277C3×D4—62no—
278C3×D4—92no—
279C3×D4—121no—
280C3×D4—181no—
281C3×D4—181no—
282C3×D4—181no—
283C3×D4—362no—
284C3×D4—362no—
285C2×C3×C4—31no—
286C2×C3×C4—31no—
287C2×C3×C4—61no—
288C2×C3×C4—91no—
289C2×C3×C4—91no—
290C2×C3×C4—181no—
291C2×C3×C4C2×D3211yescharacteristic
292#67—122no—
293#67—122no—
294#67—362no—
295#67—362no—
296C2×Dic3—31no—
297C2×Dic3—31no—
298C2×Dic3—61no—
299C2×Dic3—61no—
300C2×Dic3—91no—
301C2×Dic3—91no—
302C2×Dic3—121no—
303C2×Dic3—362no—
304D12—32no—
305D12—61no—
306D12—61no—
307D12—61no—
308D12—92no—
309D12—181no—
310D12—181no—
311D12—181no—
312C4×D3—362no—
313C4×D3—362no—
314C3×C8—181no—
315C3×C8—181no—
316C3⋊C8—61no—
317C3⋊C8—61no—
318C3⋊C8—61no—
319C3⋊C8—121no—
320C3⋊C8—181no—
321C2×C32—21no—
322C2×C32—21no—
323C2×C32—21no—
324C2×C32—82no—
325C2×C32—81no—
326C2×C32—81no—
327C2×C32D3×D811yescharacteristic
328C2×C32#91111yescharacteristic
329C2×C32C2×#21311yescharacteristic
330C2×C32#93211yescharacteristic
331C2×C32#93811yescharacteristic
332C2×C32#93911yescharacteristic
333C3⋊D3—42no—
334C3⋊D3—81no—
335C3×D3—122no—
336C3×D3—241no—
337C2×D4—91no—
338C2×D4—91no—
339C2×D4—181no—
340C2×D4—181no—
341C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
342C22×C4—541no—
343D8—182no—
344D8—542no—
345C2×C8—91no—
346C2×C8—91no—
347C4⋊C4—91no—
348C22⋊C4—541no—
349C42—91no—
350C22×C3—31no—
351C22×C3—31no—
352C22×C3—41no—
353C22×C3—41no—
354C22×C3—41no—
355C22×C3—61no—
356C22×C3—121no—
357C22×C3—121no—
358C22×C3—121no—
359C22×C3—121no—
360C22×C3—121no—
361C22×C3—121no—
362C22×C3—122no—
363C22×C3—182no—
364C22×C3—241no—
365C22×C3—241no—
366C22×C3—361no—
367C22×C3—361no—
368C22×C3—361no—
369C22×C3—361no—
370C22×C3—361no—
371C22×C3D3×#6711yescharacteristic
372C2×D3—62no—
373C2×D3—121no—
374C2×D3—121no—
375C2×D3—121no—
376C2×D3—121no—
377C2×D3—121no—
378C2×D3—182no—
379C2×D3—361no—
380C2×D3—361no—
381C2×D3—361no—
382C2×D3—361no—
383C2×D3—361no—
384C3×C4—31no—
385C3×C4—31no—
386C3×C4—31no—
387C3×C4—31no—
388C3×C4—61no—
389C3×C4—61no—
390C3×C4—92no—
391C3×C4—181no—
392C3×C4—361no—
393C3×C4#380511yescharacteristic
394C3×C4D3×#6711yescharacteristic
395Dic3—32no—
396Dic3—61no—
397Dic3—92no—
398Dic3—121no—
399Dic3—121no—
400Dic3—181no—
401Dic3—241no—
402C32—21no—
403C32#507211yescharacteristic, complemented
404C32#515011yescharacteristic, complemented
405C23—21no—
406C23—181no—
407C23—181no—
408D4—21no—
409D4—21no—
410D4—92no—
411D4—181no—
412D4—181no—
413D4—181no—
414D4—362no—
415D4—362no—
416C2×C4—91no—
417C2×C4—91no—
418C2×C4—181no—
419C2×C4—1082no—
420C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
421C8—181no—
422C8—181no—
423C2×C3—31no—
424C2×C3—31no—
425C2×C3—31no—
426C2×C3—31no—
427C2×C3—31no—
428C2×C3—31no—
429C2×C3—61no—
430C2×C3—61no—
431C2×C3—61no—
432C2×C3—81no—
433C2×C3—241no—
434C2×C3—241no—
435C2×C3—242no—
436C2×C3—362no—
437C2×C3—721no—
438C2×C3D3×#21311yescharacteristic
439C2×C3#6393211yescharacteristic
440C2×C3#6397011yescharacteristic
441D3—122no—
442D3—241no—
443D3—362no—
444D3—721no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22—182no—
449C22—361no—
450C22—361no—
451C22—361no—
452C22—361no—
453C22—361no—
454C22#8867611yescharacteristic
455C4—92no—
456C4—181no—
457C4—361no—
458C4#8867211yescharacteristic
459C4#8867611yescharacteristic
460C3—31no—
461C3—31no—
462C3—61no—
463C3#1059142811yescharacteristic
464C2—81no—
465C2—362no—
466C2—721no—
467C2#1172119411yescharacteristic
468C2#1172120011yescharacteristic
469C2#1172131311yescharacteristic
subgroupG/H
centerC22#88676
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×D4×H(3,𝔽3)C22
socleC22×C3D3×#67
hypercenterC2×D4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#459
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. #64149⋊C3C2semidirect
2. C2×#3753⋊C3C2semidirect
3. #11721132C2semidirect
4. C2×#88432C2semidirect
5. #11720578C2semidirect
6. #11720311C2semidirect
7. #11720308C2semidirect
23. #64149D3semidirect
24. C2×#3753D3semidirect
52. C4⋊((C3⋊D3)⋊C3)D4semidirect
102. (C3⋊Dic3)⋊C3D8semidirect
148. C4⋊(C3⋊D3)#67semidirect
247. C3⋊Dic3#213semidirect
260. H(3,𝔽3)#459semidirect
403. C32#5072semidirect
404. C32#5150semidirect

Maximal subgroups

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

subgroupindex
#64149⋊C32
C2×#3753⋊C32
#117211322
C2×#884322
#117205782
#117203112
#117203082
#105914333
#105914293