Finite Group Databasebeta

← #13138855

Subgroups — #13138855

Order 1728.

470 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 83 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
2C2×C8×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
3#11721141C211yesmaximal, characteristic
4#11721106C211yesmaximal, characteristic, complemented
5#11720550C211yesmaximal, characteristic, complemented
6#11720309C211yesmaximal, characteristic, complemented
7#11720308C211yesmaximal, characteristic, complemented
8#10590843—31nomaximal
9#10590830—31nomaximal
10C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
11C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
12C2×#634⋊C3C2211yescharacteristic
13C2×#6198—21no—
14C2×C8×H(3,𝔽3)C2211yescharacteristic
15C4⋊#1084C2211yescharacteristic
16C4⋊#1084C2211yescharacteristic
17C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
18#3744⋊C3—22no—
19C8×(C3⋊D3)⋊C3—22no—
20#88450—21no—
21#88450—21no—
22#88449—21no—
23C2×#3744D311yescharacteristic, complemented
24C2×C8×C3⋊D3D311yescharacteristic, complemented
25C2×C3×#204—31no—
26C2×C3×D3×C8—31no—
27#63916—31no—
28#63909—31no—
29#63871—31no—
30#63857—31no—
31C3×#817—31no—
32C3×#817—31no—
33C3×#817—61no—
34#63567—31no—
35#63565—31no—
36#63564—31no—
37#63562—31no—
38C22×(C3⋊D3)⋊C3—21no—
39C22×(C3⋊D3)⋊C3D411yescharacteristic
40C2×C4×H(3,𝔽3)C2311yescharacteristic
41C2×#1084—21no—
42C2×#1084—21no—
43C2×(C3⋊Dic3)⋊C3—21no—
44C2×(C3⋊Dic3)⋊C3D411yescharacteristic
45C4⋊((C3⋊D3)⋊C3)—21no—
46C4⋊((C3⋊D3)⋊C3)—21no—
47C4×(C3⋊D3)⋊C3—21no—
48C4×(C3⋊D3)⋊C3—21no—
49#634⋊C3—21no—
50#634⋊C3—21no—
51#6198—42no—
52C8×H(3,𝔽3)D412yescomplemented
53(C3⋊(C3⋊C8))⋊C3—21no—
54#5171—91no—
55#5114—91no—
56#4864—91no—
57C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
58C2×C4×C3⋊D3C2×D311yescharacteristic
59C2×#634C2×D311yescharacteristic
60C2×C3×D12—31no—
61C2×C3×C4×D3—31no—
62C2×C3×Dic6—31no—
63C2×#626—61no—
64C2×#626—61no—
65C2×C32×C8—21no—
66C2×C32×C8C2×D311yescharacteristic
67C2×C32×C8C2×D311yescharacteristic
68C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
69#3744—22no—
70C8×C3⋊D3—22no—
71C3×C4⋊Dic3—31no—
72C3×C4⋊Dic3—31no—
73C3×C4⋊Dic3—31no—
74C3×C4⋊Dic3—31no—
75C3×C4⋊Dic3—61no—
76C3×C4⋊Dic3—61no—
77C2×C3×C3⋊C8—31no—
78C3×#204—62no—
79C3×D3×C8—62no—
80Dic3⋊Dic3—61no—
81Dic3⋊Dic3—61no—
82#3722—61no—
83#3722—61no—
84(C2×D3)⋊Dic3—61no—
85(C2×D3)⋊Dic3—61no—
86C22×H(3,𝔽3)C2×D411yescharacteristic
87C2×(C3⋊D3)⋊C3—21no—
88C2×(C3⋊D3)⋊C3—21no—
89C2×(C3⋊D3)⋊C3—41no—
90C2×(C3⋊D3)⋊C3—41no—
91C2×(C3⋊D3)⋊C3—41no—
92C2×(C3⋊D3)⋊C3SD1612yescomplemented
93C4×H(3,𝔽3)C2×D411yescharacteristic
94C4×H(3,𝔽3)#4111yescharacteristic
95#1084—41no—
96#1084—41no—
97(C3⋊Dic3)⋊C3—21no—
98(C3⋊Dic3)⋊C3—41no—
99C2×C3×SD16—91no—
100C22×C3×C8—91no—
101#946—91no—
102#938—91no—
103#930—91no—
104#924—91no—
105C2×#204—31no—
106C2×#204—91no—
107C2×D3×C8—31no—
108C2×D3×C8—91no—
109#897—91no—
110#895—91no—
111C3×#106—91no—
112#835—91no—
113#832—91no—
114#821—91no—
115#817—31no—
116#817—31no—
117#817—61no—
118#816—91no—
119#810—91no—
120#809—91no—
121C22×C3⋊D3—21no—
122C22×C3⋊D3D1211yescharacteristic
123C22×C3×D3—31no—
124C22×C3×D3—61no—
125C2×C32×C4—21no—
126C2×C32×C4C22×D311yescharacteristic
127C2×C32×C4C22×D311yescharacteristic
128C2×C3⋊Dic3—21no—
129C2×C3⋊Dic3D1211yescharacteristic
130C4⋊(C3⋊D3)—21no—
131C4⋊(C3⋊D3)—21no—
132C4×C3⋊D3—21no—
133C4×C3⋊D3—21no—
134#634—21no—
135#634—21no—
136C2×C3×Dic3—31no—
137C2×C3×Dic3—61no—
138C2×C3×Dic3—61no—
139C2×C3×Dic3—61no—
140C2×C3×Dic3—61no—
141C2×C3×Dic3—61no—
142C2×C3×Dic3—121no—
143C2×C3×Dic3—121no—
144C3×D12—61no—
145C3×D12—61no—
146C3×C4×D3—61no—
147C3×C4×D3—61no—
148C3×Dic6—61no—
149C3×Dic6—61no—
150#626—122no—
151#626—122no—
152C32×C8—22no—
153C32×C8#6712yescomplemented
154C32×C8#6712yescomplemented
155C3⋊(C3⋊C8)—21no—
156C3×C3⋊C8—61no—
157#465—271noSylow
158C2×H(3,𝔽3)#12111yescharacteristic
159C2×H(3,𝔽3)C2×SD1611yescharacteristic
160C2×H(3,𝔽3)#13511yescharacteristic
161(C3⋊D3)⋊C3—22no—
162(C3⋊D3)⋊C3—81no—
163C2×C3×Q8—91no—
164C2×C3×D4—91no—
165C22×C3×C4—91no—
166C2×#67—181no—
167C2×#67—181no—
168C2×D12—31no—
169C2×D12—91no—
170C2×D12—181no—
171C2×C4×D3—31no—
172C2×C4×D3—91no—
173C2×Dic6—31no—
174C2×Dic6—91no—
175C3×SD16—182no—
176C2×C3×C8—31no—
177C2×C3×C8—31no—
178C2×C3×C8—61no—
179C2×C3×C8—91no—
180C2×C3×C8—182no—
181C2×C3×C8D3211yescharacteristic
182C3×C4⋊C4—91no—
183C3×C4⋊C4—91no—
184#217—181no—
185#217—181no—
186(C2×D3)⋊C4—181no—
187(C2×D3)⋊C4—181no—
188(C2×D3)⋊C4—181no—
189(C2×D3)⋊C4—181no—
190C4⋊Dic3—31no—
191C4⋊Dic3—31no—
192C4⋊Dic3—31no—
193C4⋊Dic3—31no—
194C4⋊Dic3—61no—
195C4⋊Dic3—61no—
196C4⋊Dic3—181no—
197Dic3⋊C4—181no—
198Dic3⋊C4—181no—
199C2×C3⋊C8—31no—
200C2×C3⋊C8—91no—
201#204—62no—
202#204—182no—
203D3×C8—62no—
204D3×C8—182no—
205C22×C32—21no—
206C22×C32C2×D1211yescharacteristic
207C22×C32D3×D411yescharacteristic
208C2×C3⋊D3—21no—
209C2×C3⋊D3—21no—
210C2×C3⋊D3—41no—
211C2×C3⋊D3—41no—
212C2×C3⋊D3—41no—
213C2×C3⋊D3#20412yescomplemented
214C2×C3×D3—32no—
215C2×C3×D3—61no—
216C2×C3×D3—61no—
217C2×C3×D3—121no—
218C2×C3×D3—121no—
219C2×C3×D3—121no—
220C32×C4—21no—
221C32×C4—21no—
222C32×C4#23511yescharacteristic
223C32×C4#23711yescharacteristic
224C32×C4C2×#6711yescharacteristic
225C32×C4C2×#6711yescharacteristic
226C3⋊Dic3—21no—
227C3⋊Dic3—41no—
228C3×Dic3—61no—
229C3×Dic3—121no—
230C3×Dic3—121no—
231C3×Dic3—121no—
232C3×Dic3—121no—
233C3×Dic3—121no—
234C3×Dic3—241no—
235C3×Dic3—241no—
236C2×SD16—91no—
237C22×C8—91no—
238C22⋊Q8—271no—
239#121—271no—
240#106—91no—
241#103—271no—
242#102—271no—
243H(3,𝔽3)#46511yescharacteristic, complemented, Sylow
244C23×C3—91no—
245C23×C3—181no—
246C22×D3—31no—
247C22×D3—61no—
248C22×D3—91no—
249C22×D3—181no—
250C3×Q8—181no—
251C3×Q8—181no—
252C3×D4—181no—
253C3×D4—181no—
254C2×C3×C4—31no—
255C2×C3×C4—31no—
256C2×C3×C4—61no—
257C2×C3×C4—91no—
258C2×C3×C4—181no—
259C2×C3×C4—181no—
260C2×C3×C4—181no—
261C2×C3×C4—181no—
262C2×C3×C4—181no—
263C2×C3×C4C2×D3211yescharacteristic
264#67—362no—
265#67—362no—
266C2×Dic3—31no—
267C2×Dic3—61no—
268C2×Dic3—61no—
269C2×Dic3—61no—
270C2×Dic3—61no—
271C2×Dic3—61no—
272C2×Dic3—91no—
273C2×Dic3—121no—
274C2×Dic3—121no—
275C2×Dic3—181no—
276D12—61no—
277D12—61no—
278D12—181no—
279D12—181no—
280D12—362no—
281C4×D3—61no—
282C4×D3—61no—
283C4×D3—181no—
284C4×D3—181no—
285Dic6—61no—
286Dic6—61no—
287Dic6—181no—
288Dic6—181no—
289C3×C8—32no—
290C3×C8—32no—
291C3×C8—62no—
292C3×C8—181no—
293C3×C8#62512yes—
294C3⋊C8—61no—
295C3⋊C8—181no—
296C2×C32—21no—
297C2×C32—21no—
298C2×C32—21no—
299C2×C32C2×#20411yescharacteristic
300C2×C32#90411yescharacteristic
301C2×C32D3×SD1611yescharacteristic
302C2×C32#91611yescharacteristic
303C2×C32#93011yescharacteristic
304C2×C32#93811yescharacteristic
305C3⋊D3—22no—
306C3⋊D3—81no—
307C3×D3—62no—
308C3×D3—241no—
309C2×Q8—91no—
310C2×D4—91no—
311C2×D4—541no—
312C22×C4—91no—
313SD16—182no—
314C2×C8—91no—
315C2×C8—182no—
316C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
317C4⋊C4—91no—
318C4⋊C4—91no—
319C4⋊C4—541no—
320C22⋊C4—541no—
321C22⋊C4—541no—
322C22×C3—31no—
323C22×C3—31no—
324C22×C3—61no—
325C22×C3—92no—
326C22×C3—181no—
327C22×C3—181no—
328C22×C3—361no—
329C22×C3—361no—
330C22×C3—361no—
331C22×C3D3×D1211yescharacteristic
332C2×D3—32no—
333C2×D3—61no—
334C2×D3—61no—
335C2×D3—92no—
336C2×D3—121no—
337C2×D3—121no—
338C2×D3—121no—
339C2×D3—181no—
340C2×D3—181no—
341C2×D3—361no—
342C2×D3—361no—
343C2×D3—361no—
344C3×C4—31no—
345C3×C4—31no—
346C3×C4—31no—
347C3×C4—31no—
348C3×C4—61no—
349C3×C4—61no—
350C3×C4—181no—
351C3×C4—361no—
352C3×C4—361no—
353C3×C4—361no—
354C3×C4#379511yescharacteristic
355C3×C4C2×#62511yescharacteristic
356Dic3—61no—
357Dic3—121no—
358Dic3—121no—
359Dic3—121no—
360Dic3—121no—
361Dic3—121no—
362Dic3—181no—
363Dic3—241no—
364Dic3—241no—
365Dic3—361no—
366C32—21no—
367C32#511411yescharacteristic, complemented
368C32#517111yescharacteristic, complemented
369C23—91no—
370C23—181no—
371Q8—181no—
372Q8—181no—
373D4—181no—
374D4—181no—
375D4—1082no—
376C2×C4—91no—
377C2×C4—181no—
378C2×C4—181no—
379C2×C4—181no—
380C2×C4—181no—
381C2×C4—181no—
382C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
383C8—181no—
384C8#619812yescomplemented
385C2×C3—31no—
386C2×C3—31no—
387C2×C3—31no—
388C2×C3—31no—
389C2×C3—31no—
390C2×C3—31no—
391C2×C3—61no—
392C2×C3—61no—
393C2×C3—61no—
394C2×C3—182no—
395C2×C3—721no—
396C2×C3D3×#20411yescharacteristic
397C2×C3#6380611yescharacteristic
398C2×C3#6390811yescharacteristic
399D3—62no—
400D3—182no—
401D3—241no—
402D3—721no—
403C22—92no—
404C22—181no—
405C22—181no—
406C22—361no—
407C22—361no—
408C22—361no—
409C22#8865411yescharacteristic
410C4—181no—
411C4—361no—
412C4—361no—
413C4—361no—
414C4#8864811yescharacteristic
415C4C2×#619811yescharacteristic
416C3—31no—
417C3—31no—
418C3—61no—
419C3#1059082911yescharacteristic
420C2—182no—
421C2—721no—
422C2#1172084611yescharacteristic
423C2#1172085411yescharacteristic
424C2#1172114111yescharacteristic
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#11721141C2
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#465
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
2. C2×C8×(C3⋊D3)⋊C3C2semidirect
4. #11721106C2semidirect
5. #11720550C2semidirect
6. #11720309C2semidirect
7. #11720308C2semidirect
23. C2×#3744D3semidirect
24. C2×C8×C3⋊D3D3semidirect
52. C8×H(3,𝔽3)D4semidirect
92. C2×(C3⋊D3)⋊C3SD16semidirect
153. C32×C8#67semidirect
154. C32×C8#67semidirect
213. C2×C3⋊D3#204semidirect
243. H(3,𝔽3)#465semidirect
367. C32#5114semidirect
368. C32#5171semidirect
384. C8#6198semidirect

Maximal subgroups

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

subgroupindex
C2×#3744⋊C32
C2×C8×(C3⋊D3)⋊C32
#117211412
#117211062
#117205502
#117203092
#117203082
#105908433
#105908303