Finite Group Databasebeta

← #13138947

Subgroups — #13138947

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
1C2×#3745⋊C3C211yesmaximal, characteristic, complemented
2#11721158C211yesmaximal, characteristic, complemented
3#11721132C211yesmaximal, characteristic, complemented
4C2×#88436C211yesmaximal, characteristic, complemented
5#11720554C211yesmaximal, characteristic, complemented
6#63641⋊C3C211yesmaximal, characteristic, complemented
7#11720308C211yesmaximal, characteristic, complemented
8#10590765—31nomaximal
9#10590749—31nomaximal
10C2×C4⋊#272C2211yescharacteristic
11C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
12C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
13C2×#6200—21no—
14C2×#6200—21no—
15C2×((C3⋊D3)⋊C3)⋊C4—21no—
16C2×C8×H(3,𝔽3)C2211yescharacteristic
17C4⋊#1084C2211yescharacteristic
18C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20#3745⋊C3—22no—
21#88450—21no—
22#88436—22no—
23C2×#3745D311yescharacteristic, complemented
24C2×C3×D24—31no—
25#63914—31no—
26#63911—31no—
27#63898—31no—
28#63894—31no—
29C2×#3714—31no—
30C2×#3713—31no—
31#63641D311yescharacteristic, complemented
32C3×#821—31no—
33C3×#821—31no—
34C3×#821—61no—
35C3×Dic3⋊C8—31no—
36#63564—31no—
37#63562—31no—
38C22×#272—21no—
39C22×(C3⋊D3)⋊C3—21no—
40C22×(C3⋊D3)⋊C3—21no—
41C2×C4×H(3,𝔽3)C2311yescharacteristic
42C2×#1084—21no—
43C4⋊#272—21no—
44C4⋊#272—21no—
45C2×(C3⋊Dic3)⋊C3D411yescharacteristic
46C2×(C3⋊Dic3)⋊C3D411yescharacteristic
47C4⋊((C3⋊D3)⋊C3)—21no—
48C4⋊((C3⋊D3)⋊C3)—21no—
49C4⋊((C3⋊D3)⋊C3)—21no—
50C4⋊((C3⋊D3)⋊C3)D412yescomplemented
51#6200—42no—
52#6200—42no—
53((C3⋊D3)⋊C3)⋊C4—42no—
54C8×H(3,𝔽3)—21no—
55(C3⋊(C3⋊C8))⋊C3—21no—
56#5150—91no—
57#4843—91no—
58#4786—91no—
59C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
60C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
61C2×C3×D12—31no—
62C2×C3×D12—31no—
63C2×C3×D12—31no—
64C2×C3×D12—31no—
65C2×C3×D12—61no—
66C2×#626—61no—
67C2×#626—61no—
68C2×#625—61no—
69C2×#625—61no—
70C2×#624—61no—
71C2×D3×Dic3—61no—
72C2×C32×C8—21no—
73C2×C32×C8C2×D311yescharacteristic
74C2×C32×C8C2×D311yescharacteristic
75C4×C3⋊Dic3C2×D311yescharacteristic
76C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
77#3745—22no—
78C3×C4⋊Dic3—31no—
79C3×C4⋊Dic3—31no—
80C3×C4⋊Dic3—61no—
81C3×C4×Dic3—31no—
82C2×C3×C3⋊C8—31no—
83C3×D24—62no—
84#3722—61no—
85(C2×D3)⋊Dic3—61no—
86#3714—62no—
87#3713—62no—
88C22×H(3,𝔽3)C2×D411yescharacteristic
89C2×#272—41no—
90C2×#272—41no—
91C2×#272—41no—
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—
104#940—91no—
105#940—91no—
106D4×Dic3—91no—
107C2×#213—91no—
108C2×#213—91no—
109C2×D24—31no—
110C2×D24—91no—
111C2×D24—91no—
112#893—91no—
113#874—91no—
114C4×D12—91no—
115C3×C4⋊C8—91no—
116C3×#102—91no—
117#832—91no—
118#821—31no—
119#821—31no—
120#821—61no—
121#821—91no—
122Dic3⋊C8—31no—
123Dic3⋊C8—91no—
124#809—91no—
125C22×C3⋊D3—21no—
126C22×C3⋊D3—21no—
127C22×C3×D3—61no—
128C22×C3×D3—61no—
129C22×C3×D3—61no—
130C22×C3×D3—61no—
131C22×C3×D3—121no—
132C2×C32×C4—21no—
133C2×C32×C4C22×D311yescharacteristic
134C2×C32×C4C22×D311yescharacteristic
135C2×C3⋊Dic3D1211yescharacteristic
136C2×C3⋊Dic3D1211yescharacteristic
137C4⋊(C3⋊D3)—21no—
138C4⋊(C3⋊D3)—21no—
139C4⋊(C3⋊D3)—21no—
140C4⋊(C3⋊D3)#6712yescomplemented
141C2×C3×Dic3—31no—
142C2×C3×Dic3—31no—
143C2×C3×Dic3—61no—
144C2×C3×Dic3—61no—
145C2×C3×Dic3—121no—
146C3×D12—32no—
147C3×D12—62no—
148C3×D12—61no—
149C3×D12—61no—
150C3×D12—61no—
151C3×D12—61no—
152C3×D12—61no—
153C3×D12—61no—
154C3×D12—61no—
155C3×D12—121no—
156#626—122no—
157#626—122no—
158#625—122no—
159#625—122no—
160#624—122no—
161D3×Dic3—122no—
162C32×C8—21no—
163C32×C8—21no—
164C32×C8—41no—
165C3⋊(C3⋊C8)—21no—
166C3×C3⋊C8—61no—
167#459—271noSylow
168C2×H(3,𝔽3)#12111yescharacteristic
169C2×H(3,𝔽3)C2×D811yescharacteristic
170C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
171#272—81no—
172(C3⋊D3)⋊C3—42no—
173(C3⋊D3)⋊C3—81no—
174C2×C3×D4—91no—
175C2×C3×D4—91no—
176C2×C3×D4—91no—
177C2×#67—181no—
178C2×#67—181no—
179C2×#67—181no—
180C2×#67—181no—
181C22×Dic3—181no—
182C2×D12—31no—
183C2×D12—31no—
184C2×D12—31no—
185C2×D12—31no—
186C2×D12—61no—
187C2×D12—91no—
188C2×D12—91no—
189C2×D12—181no—
190C2×D12—181no—
191C2×C4×D3—181no—
192C2×C4×D3—181no—
193C3×D8—182no—
194C2×C3×C8—31no—
195C2×C3×C8—31no—
196C2×C3×C8—61no—
197C2×C3×C8—91no—
198C2×C3×C8D3211yescharacteristic
199C3×C4⋊C4—91no—
200C3×C42—91no—
201#217—181no—
202#213—182no—
203#213—182no—
204(C2×D3)⋊C4—181no—
205(C2×D3)⋊C4—181no—
206C4⋊Dic3—31no—
207C4⋊Dic3—31no—
208C4⋊Dic3—61no—
209C4×Dic3—31no—
210C4×Dic3—91no—
211C2×C3⋊C8—31no—
212C2×C3⋊C8—91no—
213D24—62no—
214D24—182no—
215D24—182no—
216C22×C32—21no—
217C22×C32C2×D1211yescharacteristic
218C22×C32D3×D411yescharacteristic
219C2×C3⋊D3—22no—
220C2×C3⋊D3—41no—
221C2×C3⋊D3—41no—
222C2×C3⋊D3—41no—
223C2×C3⋊D3—41no—
224C2×C3⋊D3—41no—
225C2×C3×D3—62no—
226C2×C3×D3—121no—
227C2×C3×D3—121no—
228C2×C3×D3—122no—
229C2×C3×D3—121no—
230C2×C3×D3—121no—
231C2×C3×D3—121no—
232C2×C3×D3—121no—
233C2×C3×D3—121no—
234C2×C3×D3—121no—
235C2×C3×D3—121no—
236C2×C3×D3—121no—
237C2×C3×D3—121no—
238C2×C3×D3—241no—
239C2×C3×D3—241no—
240C32×C4—21no—
241C32×C4—21no—
242C32×C4D3×D411yescharacteristic
243C32×C4#23711yescharacteristic
244C32×C4#23911yescharacteristic
245C32×C4C2×#6711yescharacteristic
246C3⋊Dic3—21no—
247C3⋊Dic3D2412yescomplemented
248C3×Dic3—32no—
249C3×Dic3—61no—
250C3×Dic3—121no—
251C3×Dic3—121no—
252C3×Dic3—241no—
253C2×D8—91no—
254C2×D8—271no—
255#127—271no—
256C4×D4—271no—
257C4⋊C8—91no—
258#102—91no—
259#102—271no—
260H(3,𝔽3)#45911yescharacteristic, complemented, Sylow
261C23×C3—181no—
262C23×C3—181no—
263C23×C3—181no—
264C22×D3—61no—
265C22×D3—61no—
266C22×D3—61no—
267C22×D3—61no—
268C22×D3—121no—
269C22×D3—181no—
270C22×D3—181no—
271C3×D4—92no—
272C3×D4—181no—
273C3×D4—181no—
274C3×D4—181no—
275C3×D4—181no—
276C3×D4—181no—
277C2×C3×C4—31no—
278C2×C3×C4—31no—
279C2×C3×C4—61no—
280C2×C3×C4—91no—
281C2×C3×C4—91no—
282C2×C3×C4—181no—
283C2×C3×C4C2×D3211yescharacteristic
284#67—362no—
285#67—362no—
286#67—362no—
287#67—362no—
288C2×Dic3—31no—
289C2×Dic3—31no—
290C2×Dic3—61no—
291C2×Dic3—61no—
292C2×Dic3—91no—
293C2×Dic3—91no—
294C2×Dic3—121no—
295C2×Dic3—362no—
296D12—32no—
297D12—61no—
298D12—61no—
299D12—61no—
300D12—61no—
301D12—62no—
302D12—61no—
303D12—61no—
304D12—61no—
305D12—92no—
306D12—121no—
307D12—181no—
308D12—181no—
309D12—181no—
310D12—362no—
311D12—362no—
312C4×D3—362no—
313C4×D3—362no—
314C3×C8—21no—
315C3×C8—61no—
316C3×C8—61no—
317C3×C8—121no—
318C3×C8—181no—
319C3⋊C8—61no—
320C3⋊C8—181no—
321C2×C32—21no—
322C2×C32—21no—
323C2×C32—21no—
324C2×C32#89511yescharacteristic
325C2×C32C2×D2411yescharacteristic
326C2×C32#90811yescharacteristic
327C2×C32D3×D811yescharacteristic
328C2×C32#91111yescharacteristic
329C2×C32#93911yescharacteristic
330C3⋊D3—42no—
331C3⋊D3—81no—
332C3×D3—122no—
333C3×D3—242no—
334C3×D3—241no—
335C3×D3—241no—
336C3×D3—241no—
337C2×D4—91no—
338C2×D4—91no—
339C2×D4—91no—
340C2×D4—541no—
341C2×D4—541no—
342C22×C4—541no—
343D8—182no—
344D8—542no—
345C2×C8—91no—
346C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
347C4⋊C4—91no—
348C22⋊C4—541no—
349C42—91no—
350C22×C3—31no—
351C22×C3—31no—
352C22×C3—61no—
353C22×C3—182no—
354C22×C3—361no—
355C22×C3—361no—
356C22×C3—361no—
357C22×C3—361no—
358C22×C3—361no—
359C22×C3—361no—
360C22×C3—361no—
361C22×C3—361no—
362C22×C3D3×D1211yescharacteristic
363C2×D3—62no—
364C2×D3—121no—
365C2×D3—121no—
366C2×D3—121no—
367C2×D3—121no—
368C2×D3—121no—
369C2×D3—121no—
370C2×D3—121no—
371C2×D3—121no—
372C2×D3—122no—
373C2×D3—121no—
374C2×D3—121no—
375C2×D3—121no—
376C2×D3—182no—
377C2×D3—241no—
378C2×D3—241no—
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#379611yescharacteristic
394C3×C4D3×#6711yescharacteristic
395Dic3—32no—
396Dic3—61no—
397Dic3—92no—
398Dic3—121no—
399Dic3—121no—
400Dic3—181no—
401Dic3—241no—
402C32—21no—
403C32#484311yescharacteristic, complemented
404C32#515011yescharacteristic, complemented
405C23—181no—
406C23—181no—
407C23—181no—
408D4—92no—
409D4—181no—
410D4—181no—
411D4—181no—
412D4—181no—
413D4—181no—
414D4—1082no—
415D4—1082no—
416C2×C4—91no—
417C2×C4—91no—
418C2×C4—181no—
419C2×C4—1082no—
420C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
421C8—21no—
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—362no—
433C2×C3—721no—
434C2×C3—721no—
435C2×C3D3×D2411yescharacteristic
436C2×C3#6379511yescharacteristic
437C2×C3#6388611yescharacteristic
438D3—122no—
439D3—241no—
440D3—241no—
441D3—242no—
442D3—241no—
443D3—362no—
444D3—721no—
445C22—182no—
446C22—361no—
447C22—361no—
448C22—361no—
449C22—361no—
450C22—361no—
451C22—361no—
452C22—361no—
453C22—361no—
454C22#8865411yescharacteristic
455C4—92no—
456C4—181no—
457C4—361no—
458C4#8864911yescharacteristic
459C4#8867611yescharacteristic
460C3—31no—
461C3—31no—
462C3—61no—
463C3#1059075311yescharacteristic
464C2—362no—
465C2—721no—
466C2—721no—
467C2#1172084711yescharacteristic
468C2#1172084911yescharacteristic
469C2#1172112711yescharacteristic
subgroupG/H
centerC22#88654
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3D3×D12
hypercenterC2×C8((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×C8

Complements

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

NH ≅ G/Ntype
1. C2×#3745⋊C3C2semidirect
2. #11721158C2semidirect
3. #11721132C2semidirect
4. C2×#88436C2semidirect
5. #11720554C2semidirect
6. #63641⋊C3C2semidirect
7. #11720308C2semidirect
23. C2×#3745D3semidirect
31. #63641D3semidirect
50. C4⋊((C3⋊D3)⋊C3)D4semidirect
102. (C3⋊Dic3)⋊C3D8semidirect
140. C4⋊(C3⋊D3)#67semidirect
247. C3⋊Dic3D24semidirect
260. H(3,𝔽3)#459semidirect
403. C32#4843semidirect
404. C32#5150semidirect

Maximal subgroups

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

subgroupindex
C2×#3745⋊C32
#117211582
#117211322
C2×#884362
#117205542
#63641⋊C32
#117203082
#105907653
#105907493