Finite Group Databasebeta

← #13138953

Subgroups — #13138953

Order 1728.

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

normal subgroups only

This group has 498 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×#3744⋊C3C211yesmaximal, characteristic, complemented
2#11721158C211yesmaximal, characteristic
3#11721082C211yesmaximal, characteristic, complemented
4C2×#88437C211yesmaximal, characteristic, complemented
5#11720554C211yesmaximal, characteristic, complemented
6#63641⋊C3C211yesmaximal, characteristic, complemented
7#11720308C211yesmaximal, characteristic, complemented
8#10590879—31nomaximal
9#10590876—31nomaximal
10C2×C4⋊#272C2211yescharacteristic
11C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
12C2×#634⋊C3C2211yescharacteristic
13C2×#6200—21no—
14C2×#6200—21no—
15C2×C8×H(3,𝔽3)C2211yescharacteristic
16C4⋊#1084C2211yescharacteristic
17C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
18C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
19#3744⋊C3—22no—
20#88448—21no—
21#88447—21no—
22#88437—22no—
23C2×#3744D311yescharacteristic, complemented
24C2×C3×#204—31no—
25#63914—31no—
26#63911—31no—
27#63844—31no—
28#63843—31no—
29C2×#3716—31no—
30C2×#3715—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—
40C2×C4×H(3,𝔽3)C2311yescharacteristic
41C2×#1084—21no—
42C4⋊#272—21no—
43C4⋊#272—21no—
44C2×(C3⋊Dic3)⋊C3—21no—
45C2×(C3⋊Dic3)⋊C3D411yescharacteristic
46C2×(C3⋊Dic3)⋊C3D411yescharacteristic
47C4⋊((C3⋊D3)⋊C3)—21no—
48C4⋊((C3⋊D3)⋊C3)—21no—
49#634⋊C3—21no—
50#634⋊C3D412yescomplemented
51#6200—42no—
52#6200—42no—
53C8×H(3,𝔽3)—21no—
54(C3⋊(C3⋊C8))⋊C3—21no—
55#5163—91no—
56#4848—91no—
57#4762—91no—
58C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
59C2×#634C2×D311yescharacteristic
60C2×C3×D12—31no—
61C2×C3×D12—31no—
62C2×C3×D12—31no—
63C2×C3×D12—61no—
64C2×C3×Dic6—31no—
65C2×#626—61no—
66C2×#626—61no—
67C2×#625—61no—
68C2×#625—61no—
69C2×C32×C8—21no—
70C2×C32×C8C2×D311yescharacteristic
71C2×C32×C8C2×D311yescharacteristic
72C4×C3⋊Dic3C2×D311yescharacteristic
73C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
74#3744—22no—
75C3×C4⋊Dic3—31no—
76C3×C4⋊Dic3—31no—
77C3×C4⋊Dic3—61no—
78C3×C4×Dic3—31no—
79C2×C3×C3⋊C8—31no—
80C3×#204—62no—
81#3724—61no—
82Dic3⋊Dic3—61no—
83Dic32—61no—
84Dic32—61no—
85#3716—62no—
86#3715—62no—
87C22×H(3,𝔽3)C2×D411yescharacteristic
88C2×#272—41no—
89C2×#272—41no—
90C2×#272—41no—
91C2×(C3⋊D3)⋊C3—41no—
92C2×(C3⋊D3)⋊C3—41no—
93C2×(C3⋊D3)⋊C3—41no—
94C4×H(3,𝔽3)C2×D411yescharacteristic
95C4×H(3,𝔽3)#4111yescharacteristic
96#1084—41no—
97(C3⋊Dic3)⋊C3—22no—
98(C3⋊Dic3)⋊C3—21no—
99(C3⋊Dic3)⋊C3SD1612yescomplemented
100C2×C3×SD16—91no—
101Q8×Dic3—91no—
102C2×#215—91no—
103#940—91no—
104#940—91no—
105C2×#214—91no—
106C2×#204—31no—
107C2×#204—91no—
108C2×#204—91no—
109#887—91no—
110#874—91no—
111C4×Dic6—91no—
112C3×C4⋊C8—91no—
113C3×#102—91no—
114#832—91no—
115#821—31no—
116#821—31no—
117#821—61no—
118#821—91no—
119Dic3⋊C8—31no—
120Dic3⋊C8—91no—
121#809—91no—
122C22×C3⋊D3—21no—
123C22×C3×D3—61no—
124C22×C3×D3—61no—
125C22×C3×D3—61no—
126C22×C3×D3—121no—
127C2×C32×C4—21no—
128C2×C32×C4C22×D311yescharacteristic
129C2×C32×C4C22×D311yescharacteristic
130C2×C3⋊Dic3—21no—
131C2×C3⋊Dic3D1211yescharacteristic
132C2×C3⋊Dic3D1211yescharacteristic
133C4⋊(C3⋊D3)—21no—
134C4⋊(C3⋊D3)—21no—
135#634—21no—
136#634#6712yescomplemented
137C2×C3×Dic3—31no—
138C2×C3×Dic3—31no—
139C2×C3×Dic3—61no—
140C2×C3×Dic3—61no—
141C2×C3×Dic3—61no—
142C2×C3×Dic3—121no—
143C3×D12—62no—
144C3×D12—61no—
145C3×D12—61no—
146C3×D12—61no—
147C3×D12—61no—
148C3×D12—61no—
149C3×D12—61no—
150C3×D12—121no—
151C3×Dic6—32no—
152C3×Dic6—61no—
153#626—122no—
154#626—122no—
155#625—122no—
156#625—122no—
157C32×C8—21no—
158C32×C8—21no—
159C32×C8—41no—
160C3⋊(C3⋊C8)—21no—
161C3×C3⋊C8—61no—
162#460—271noSylow
163C2×H(3,𝔽3)#12111yescharacteristic
164C2×H(3,𝔽3)C2×SD1611yescharacteristic
165C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
166#272—81no—
167(C3⋊D3)⋊C3—81no—
168C2×C3×Q8—91no—
169C2×C3×D4—91no—
170C2×C3×D4—91no—
171C2×#67—181no—
172C2×#67—181no—
173C2×#67—181no—
174C2×#67—181no—
175C2×D12—31no—
176C2×D12—31no—
177C2×D12—31no—
178C2×D12—61no—
179C2×D12—91no—
180C2×D12—181no—
181C2×D12—181no—
182C2×Dic6—31no—
183C2×Dic6—91no—
184C3×SD16—182no—
185C2×C3×C8—31no—
186C2×C3×C8—31no—
187C2×C3×C8—61no—
188C2×C3×C8—91no—
189C2×C3×C8D3211yescharacteristic
190C3×C4⋊C4—91no—
191C3×C42—91no—
192#215—182no—
193#214—182no—
194C4⋊Dic3—31no—
195C4⋊Dic3—31no—
196C4⋊Dic3—61no—
197C4⋊Dic3—181no—
198Dic3⋊C4—181no—
199Dic3⋊C4—181no—
200C4×Dic3—31no—
201C4×Dic3—91no—
202C4×Dic3—181no—
203C4×Dic3—181no—
204C4×Dic3—181no—
205C2×C3⋊C8—31no—
206C2×C3⋊C8—91no—
207#204—62no—
208#204—182no—
209#204—182no—
210C22×C32—21no—
211C22×C32C2×D1211yescharacteristic
212C22×C32D3×D411yescharacteristic
213C2×C3⋊D3—41no—
214C2×C3⋊D3—41no—
215C2×C3⋊D3—41no—
216C2×C3×D3—122no—
217C2×C3×D3—121no—
218C2×C3×D3—121no—
219C2×C3×D3—121no—
220C2×C3×D3—121no—
221C2×C3×D3—121no—
222C2×C3×D3—121no—
223C2×C3×D3—121no—
224C2×C3×D3—121no—
225C2×C3×D3—121no—
226C2×C3×D3—241no—
227C2×C3×D3—241no—
228C32×C4—21no—
229C32×C4—21no—
230C32×C4D3×D411yescharacteristic
231C32×C4#23711yescharacteristic
232C32×C4#23911yescharacteristic
233C32×C4C2×#6711yescharacteristic
234C3⋊Dic3—21no—
235C3⋊Dic3—22no—
236C3⋊Dic3#20412yescomplemented
237C3×Dic3—32no—
238C3×Dic3—62no—
239C3×Dic3—61no—
240C3×Dic3—121no—
241C3×Dic3—121no—
242C3×Dic3—241no—
243C2×SD16—91no—
244C2×SD16—271no—
245#127—271no—
246C4×Q8—271no—
247C4⋊C8—91no—
248#102—91no—
249#102—271no—
250H(3,𝔽3)#46011yescharacteristic, complemented, Sylow
251C23×C3—181no—
252C23×C3—181no—
253C22×D3—61no—
254C22×D3—61no—
255C22×D3—61no—
256C22×D3—121no—
257C22×D3—181no—
258C3×Q8—92no—
259C3×Q8—181no—
260C3×D4—181no—
261C3×D4—181no—
262C3×D4—181no—
263C3×D4—181no—
264C2×C3×C4—31no—
265C2×C3×C4—31no—
266C2×C3×C4—61no—
267C2×C3×C4—91no—
268C2×C3×C4—91no—
269C2×C3×C4—181no—
270C2×C3×C4—181no—
271C2×C3×C4C2×D3211yescharacteristic
272#67—362no—
273#67—362no—
274#67—362no—
275#67—362no—
276C2×Dic3—31no—
277C2×Dic3—31no—
278C2×Dic3—61no—
279C2×Dic3—61no—
280C2×Dic3—61no—
281C2×Dic3—91no—
282C2×Dic3—91no—
283C2×Dic3—121no—
284C2×Dic3—181no—
285D12—61no—
286D12—61no—
287D12—61no—
288D12—61no—
289D12—62no—
290D12—61no—
291D12—61no—
292D12—121no—
293D12—181no—
294D12—181no—
295D12—362no—
296D12—362no—
297Dic6—32no—
298Dic6—61no—
299Dic6—92no—
300Dic6—181no—
301C3×C8—21no—
302C3×C8—61no—
303C3×C8—61no—
304C3×C8—121no—
305C3×C8—181no—
306C3⋊C8—61no—
307C3⋊C8—181no—
308C2×C32—21no—
309C2×C32—21no—
310C2×C32—21no—
311C2×C32#89511yescharacteristic
312C2×C32C2×#20411yescharacteristic
313C2×C32#90811yescharacteristic
314C2×C32D3×SD1611yescharacteristic
315C2×C32#91411yescharacteristic
316C2×C32#93911yescharacteristic
317C3⋊D3—81no—
318C3×D3—242no—
319C3×D3—241no—
320C3×D3—241no—
321C3×D3—241no—
322C2×Q8—91no—
323C2×D4—91no—
324C2×D4—91no—
325C2×D4—541no—
326C2×D4—541no—
327SD16—182no—
328SD16—542no—
329C2×C8—91no—
330C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
331C4⋊C4—91no—
332C4⋊C4—541no—
333C42—91no—
334C42—541no—
335C22×C3—31no—
336C22×C3—31no—
337C22×C3—61no—
338C22×C3—361no—
339C22×C3—361no—
340C22×C3—361no—
341C22×C3—361no—
342C22×C3—361no—
343C22×C3—361no—
344C22×C3D3×D1211yescharacteristic
345C2×D3—121no—
346C2×D3—121no—
347C2×D3—121no—
348C2×D3—121no—
349C2×D3—121no—
350C2×D3—121no—
351C2×D3—122no—
352C2×D3—121no—
353C2×D3—121no—
354C2×D3—121no—
355C2×D3—241no—
356C2×D3—241no—
357C2×D3—361no—
358C2×D3—361no—
359C2×D3—361no—
360C3×C4—31no—
361C3×C4—31no—
362C3×C4—31no—
363C3×C4—31no—
364C3×C4—61no—
365C3×C4—61no—
366C3×C4—92no—
367C3×C4—181no—
368C3×C4—182no—
369C3×C4—361no—
370C3×C4#379611yescharacteristic
371C3×C4D3×#6711yescharacteristic
372Dic3—32no—
373Dic3—61no—
374Dic3—62no—
375Dic3—92no—
376Dic3—121no—
377Dic3—121no—
378Dic3—182no—
379Dic3—181no—
380Dic3—241no—
381C32—21no—
382C32#484811yescharacteristic, complemented
383C32#516311yescharacteristic, complemented
384C23—181no—
385C23—181no—
386Q8—92no—
387Q8—181no—
388D4—181no—
389D4—181no—
390D4—181no—
391D4—181no—
392D4—1082no—
393D4—1082no—
394C2×C4—91no—
395C2×C4—91no—
396C2×C4—181no—
397C2×C4—181no—
398C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
399C8—21no—
400C8—181no—
401C2×C3—31no—
402C2×C3—31no—
403C2×C3—31no—
404C2×C3—31no—
405C2×C3—31no—
406C2×C3—31no—
407C2×C3—61no—
408C2×C3—61no—
409C2×C3—61no—
410C2×C3—721no—
411C2×C3—721no—
412C2×C3D3×#20411yescharacteristic
413C2×C3#6379411yescharacteristic
414C2×C3#6388611yescharacteristic
415D3—241no—
416D3—241no—
417D3—242no—
418D3—241no—
419D3—721no—
420C22—361no—
421C22—361no—
422C22—361no—
423C22—361no—
424C22—361no—
425C22—361no—
426C22#8865411yescharacteristic
427C4—92no—
428C4—181no—
429C4—182no—
430C4—361no—
431C4#8864911yescharacteristic
432C4#8867611yescharacteristic
433C3—31no—
434C3—31no—
435C3—61no—
436C3#1059079911yescharacteristic
437C2—721no—
438C2—721no—
439C2#1172084611yescharacteristic
440C2#1172084811yescharacteristic
441C2#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
prime radical#11721158C2
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#460
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×#3744⋊C3C2semidirect
3. #11721082C2semidirect
4. C2×#88437C2semidirect
5. #11720554C2semidirect
6. #63641⋊C3C2semidirect
7. #11720308C2semidirect
23. C2×#3744D3semidirect
31. #63641D3semidirect
50. #634⋊C3D4semidirect
99. (C3⋊Dic3)⋊C3SD16semidirect
136. #634#67semidirect
236. C3⋊Dic3#204semidirect
250. H(3,𝔽3)#460semidirect
382. C32#4848semidirect
383. C32#5163semidirect

Maximal subgroups

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

subgroupindex
C2×#3744⋊C32
#117211582
#117210822
C2×#884372
#117205542
#63641⋊C32
#117203082
#105908793
#105908763