Finite Group Databasebeta

← #13137985

Subgroups — #13137985

Order 1728.

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

normal subgroups only

This group has 512 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
1#64102⋊C3C211yesmaximal, characteristic, complemented
2#11721149C211yesmaximal, characteristic, complemented
3C2×#88436C211yesmaximal, characteristic
4#11720531C211yesmaximal, characteristic, complemented
5#63635⋊C3C211yesmaximal, characteristic, complemented
6#11720318C211yesmaximal, characteristic, complemented
7#11720312C211yesmaximal, characteristic, complemented
8#10590462—31nomaximal
9#10590455—31nomaximal
10C2×C4⋊#272C2211yescharacteristic
11C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
12C2×C4×(C3⋊D3)⋊C3—21no—
13C2×#6199—21no—
14C4⋊C4×H(3,𝔽3)C2211yescharacteristic
15C2×#6180C2211yescharacteristic
16#3752⋊C3—21no—
17C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
18C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20#88451—21no—
21#88436C414yescomplemented
22#64102D311yescharacteristic, complemented
23C3×#893—31no—
24#63898—31no—
25Dic3×D12—31no—
26C2×#3714—31no—
27C2×#3713—31no—
28#63635D311yescharacteristic, complemented
29C3×#809—31no—
30C3×#809—31no—
31C3×#809—61no—
32C3×#808—31no—
33#63564—31no—
34#63563—31no—
35#63555—31no—
36#63554—31no—
37C22×#272—21no—
38C22×(C3⋊D3)⋊C3—21no—
39C2×C4×H(3,𝔽3)—21no—
40C2×C4×H(3,𝔽3)C2311yescharacteristic
41C4⋊#272—21no—
42C4⋊#272C2×C412yescomplemented
43C2×(C3⋊Dic3)⋊C3—21no—
44C2×(C3⋊Dic3)⋊C3D411yescharacteristic
45C2×(C3⋊Dic3)⋊C3D411yescharacteristic
46C4⋊((C3⋊D3)⋊C3)—21no—
47C4⋊((C3⋊D3)⋊C3)C2×C412yescomplemented
48C4×(C3⋊D3)⋊C3—42no—
49#6199—42no—
50#6180—21no—
51(C3⋊(C3⋊C8))⋊C3C2×C412yescomplemented
52#5014—91no—
53#4869—91no—
54#4785—91no—
55C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
56C2×C4×C3⋊D3—21no—
57C2×C3×D12—31no—
58C2×C3×D12—31no—
59C2×C3×D12—31no—
60C2×C3×D12—61no—
61C2×C3×C4×D3—61no—
62C2×D3×Dic3—61no—
63C2×D3×Dic3—61no—
64C32×C4⋊C4—21no—
65C32×C4⋊C4C2×D311yescharacteristic
66C32×C4⋊C4C2×D311yescharacteristic
67#3752—21no—
68C4⋊(C3⋊Dic3)C2×D311yescharacteristic
69C4×C3⋊Dic3C2×D311yescharacteristic
70C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
71C3×(C2×D3)⋊C4—61no—
72C3×C4⋊Dic3—31no—
73C3×C4×Dic3—31no—
74C2×C3×C3⋊C8—31no—
75C2×C3×C3⋊C8—31no—
76C2×C3×C3⋊C8—31no—
77C2×C3×C3⋊C8—61no—
78(C2×D3)⋊Dic3—61no—
79(C2×D3)⋊Dic3—61no—
80#3714—34no—
81#3713—34no—
82C22×H(3,𝔽3)C2×D411yescharacteristic
83C2×#272—22no—
84C2×#272—41no—
85C2×#272—41no—
86C2×(C3⋊D3)⋊C3—22no—
87C2×(C3⋊D3)⋊C3—41no—
88C2×(C3⋊D3)⋊C3—41no—
89C4×H(3,𝔽3)—41no—
90C4×H(3,𝔽3)C22×C411yescharacteristic
91C4×H(3,𝔽3)#4111yescharacteristic
92(C3⋊Dic3)⋊C3—21no—
93(C3⋊Dic3)⋊C3—21no—
94(C3⋊Dic3)⋊C3—41no—
95C3×C4×D4—91no—
96D4×Dic3—91no—
97C2×#213—91no—
98C2×#213—91no—
99C2×D24—91no—
100#893—31no—
101#893—91no—
102#893—91no—
103C4×D12—91no—
104C3×#106—91no—
105C3×#102—91no—
106#832—91no—
107#821—91no—
108#815—91no—
109#815—91no—
110#809—31no—
111#809—31no—
112#809—61no—
113#809—91no—
114#808—31no—
115#808—91no—
116#803—91no—
117C22×C3⋊D3—21no—
118C22×C3×D3—61no—
119C22×C3×D3—61no—
120C22×C3×D3—61no—
121C22×C3×D3—121no—
122C2×C32×C4—21no—
123C2×C32×C4—21no—
124C2×C32×C4—21no—
125C2×C32×C4—41no—
126C2×C32×C4C22×D311yescharacteristic
127C2×C32×C4C22×D311yescharacteristic
128C2×C3⋊Dic3—21no—
129C2×C3⋊Dic3#6711yescharacteristic
130C2×C3⋊Dic3#6711yescharacteristic
131C4⋊(C3⋊D3)—21no—
132C4⋊(C3⋊D3)C4×D312yescomplemented
133C4×C3⋊D3—42no—
134C2×C3×Dic3—31no—
135C2×C3×Dic3—31no—
136C2×C3×Dic3—61no—
137C3×D12—32no—
138C3×D12—32no—
139C3×D12—32no—
140C3×D12—62no—
141C3×D12—61no—
142C3×D12—61no—
143C3×D12—61no—
144C3×D12—121no—
145C3×C4×D3—122no—
146D3×Dic3—122no—
147D3×Dic3—122no—
148C3⋊(C3⋊C8)C4×D312yescomplemented
149C3×C3⋊C8—32no—
150C3×C3⋊C8—61no—
151C3×C3⋊C8—61no—
152C3×C3⋊C8—121no—
153#442—271noSylow
154C2×H(3,𝔽3)C4×D411yescharacteristic
155C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
156C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
157#272—42no—
158(C3⋊D3)⋊C3—42no—
159C2×C3×D4—91no—
160C2×C3×D4—91no—
161C22×C3×C4—181no—
162C22×Dic3—181no—
163C2×D12—31no—
164C2×D12—31no—
165C2×D12—31no—
166C2×D12—61no—
167C2×D12—91no—
168C2×C4×D3—61no—
169C2×C4×D3—181no—
170C2×C4×D3—181no—
171C2×C4×D3—181no—
172C2×C3×C8—91no—
173C2×C3×C8—91no—
174C3×C4⋊C4—31no—
175C3×C4⋊C4—31no—
176C3×C4⋊C4—61no—
177C3×C4⋊C4—91no—
178C3×C4⋊C4D3211yescharacteristic
179C3×C22⋊C4—181no—
180C3×C42—91no—
181#217—181no—
182#213—94no—
183#213—94no—
184(C2×D3)⋊C4—61no—
185(C2×D3)⋊C4—181no—
186(C2×D3)⋊C4—181no—
187(C2×D3)⋊C4—181no—
188C4⋊Dic3—31no—
189C4⋊Dic3—91no—
190C4×Dic3—31no—
191C4×Dic3—91no—
192C2×C3⋊C8—31no—
193C2×C3⋊C8—31no—
194C2×C3⋊C8—31no—
195C2×C3⋊C8—61no—
196C2×C3⋊C8—91no—
197D24—94no—
198C22×C32—21no—
199C22×C32D3×D411yescharacteristic
200C22×C32C2×#6711yescharacteristic
201C2×C3⋊D3—22no—
202C2×C3⋊D3—41no—
203C2×C3⋊D3—41no—
204C2×C3×D3—62no—
205C2×C3×D3—62no—
206C2×C3×D3—62no—
207C2×C3×D3—122no—
208C2×C3×D3—121no—
209C2×C3×D3—121no—
210C2×C3×D3—121no—
211C2×C3×D3—121no—
212C2×C3×D3—121no—
213C2×C3×D3—121no—
214C2×C3×D3—241no—
215C2×C3×D3—241no—
216C32×C4—21no—
217C32×C4—21no—
218C32×C4—41no—
219C32×C4—41no—
220C32×C4—81no—
221C32×C4C2×C4×D311yescharacteristic
222C32×C4C2×C4×D311yescharacteristic
223C32×C4#23511yescharacteristic
224C32×C4#23911yescharacteristic
225C3⋊Dic3—21no—
226C3⋊Dic3—21no—
227C3⋊Dic3—41no—
228C3×Dic3—61no—
229C3×Dic3—61no—
230C3×Dic3—121no—
231C2×D8—271no—
232C4×D4—91no—
233C4×D4—271no—
234#106—91no—
235#102—91no—
236#102—271no—
237#97—271no—
238H(3,𝔽3)#44211yescharacteristic, complemented, Sylow
239C23×C3—181no—
240C23×C3—181no—
241C22×D3—61no—
242C22×D3—61no—
243C22×D3—61no—
244C22×D3—121no—
245C22×D3—181no—
246C3×D4—92no—
247C3×D4—92no—
248C3×D4—181no—
249C3×D4—181no—
250C2×C3×C4—21no—
251C2×C3×C4—31no—
252C2×C3×C4—31no—
253C2×C3×C4—61no—
254C2×C3×C4—61no—
255C2×C3×C4—61no—
256C2×C3×C4—91no—
257C2×C3×C4—91no—
258C2×C3×C4—121no—
259C2×C3×C4—181no—
260C2×C3×C4—362no—
261C2×C3×C4C2×D3211yescharacteristic
262C2×Dic3—31no—
263C2×Dic3—31no—
264C2×Dic3—61no—
265C2×Dic3—91no—
266C2×Dic3—91no—
267C2×Dic3—181no—
268C2×Dic3—362no—
269D12—32no—
270D12—32no—
271D12—32no—
272D12—62no—
273D12—61no—
274D12—61no—
275D12—61no—
276D12—92no—
277D12—121no—
278D12—181no—
279C4×D3—122no—
280C4×D3—362no—
281C4×D3—362no—
282C4×D3—362no—
283C3×C8—92no—
284C3×C8—181no—
285C3⋊C8—32no—
286C3⋊C8—61no—
287C3⋊C8—61no—
288C3⋊C8—92no—
289C3⋊C8—121no—
290C2×C32—21no—
291C2×C32—21no—
292C2×C32—21no—
293C2×C32#89311yescharacteristic
294C2×C32#91411yescharacteristic
295C2×C32#91411yescharacteristic
296C2×C32C4×#6711yescharacteristic
297C2×C32#93211yescharacteristic
298C2×C32#94911yescharacteristic
299C3⋊D3—42no—
300C3×D3—122no—
301C3×D3—122no—
302C3×D3—122no—
303C3×D3—242no—
304C2×D4—91no—
305C2×D4—91no—
306C22×C4—181no—
307C22×C4—541no—
308D8—274no—
309C2×C8—91no—
310C2×C8—91no—
311C4⋊C4—91no—
312C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
313C22⋊C4—181no—
314C22⋊C4—541no—
315C42—91no—
316C22×C3—31no—
317C22×C3—31no—
318C22×C3—61no—
319C22×C3—182no—
320C22×C3—182no—
321C22×C3—361no—
322C22×C3—361no—
323C22×C3—361no—
324C22×C3—361no—
325C22×C3D3×#6711yescharacteristic
326C2×D3—62no—
327C2×D3—62no—
328C2×D3—62no—
329C2×D3—122no—
330C2×D3—121no—
331C2×D3—121no—
332C2×D3—121no—
333C2×D3—121no—
334C2×D3—121no—
335C2×D3—121no—
336C2×D3—182no—
337C2×D3—241no—
338C2×D3—241no—
339C2×D3—361no—
340C2×D3—361no—
341C3×C4—31no—
342C3×C4—31no—
343C3×C4—31no—
344C3×C4—31no—
345C3×C4—41no—
346C3×C4—61no—
347C3×C4—61no—
348C3×C4—121no—
349C3×C4—121no—
350C3×C4—181no—
351C3×C4—181no—
352C3×C4—241no—
353C3×C4—361no—
354C3×C4#379911yescharacteristic
355C3×C4C4×D3211yescharacteristic
356Dic3—61no—
357Dic3—61no—
358Dic3—121no—
359Dic3—181no—
360Dic3—181no—
361Dic3—361no—
362C32—21no—
363C32#486911yescharacteristic, complemented
364C32#501411yescharacteristic, complemented
365C23—181no—
366C23—181no—
367D4—92no—
368D4—92no—
369D4—181no—
370D4—181no—
371C2×C4—21no—
372C2×C4—91no—
373C2×C4—91no—
374C2×C4—181no—
375C2×C4—362no—
376C2×C4—1082no—
377C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
378C8—92no—
379C8—181no—
380C2×C3—31no—
381C2×C3—31no—
382C2×C3—31no—
383C2×C3—31no—
384C2×C3—31no—
385C2×C3—31no—
386C2×C3—61no—
387C2×C3—61no—
388C2×C3—61no—
389C2×C3—362no—
390C2×C3—362no—
391C2×C3#6390411yescharacteristic
392C2×C3#6393411yescharacteristic
393C2×C3#6393911yescharacteristic
394D3—122no—
395D3—122no—
396D3—122no—
397D3—242no—
398D3—362no—
399C22—182no—
400C22—182no—
401C22—361no—
402C22—361no—
403C22—361no—
404C22—361no—
405C22#8867611yescharacteristic
406C4—41no—
407C4—181no—
408C4—181no—
409C4—361no—
410C4#8865211yescharacteristic
411C4C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
412C3—31no—
413C3—31no—
414C3—61no—
415C3#1059058911yescharacteristic
416C2—362no—
417C2—362no—
418C2#1172115411yescharacteristic
419C2#1172119611yescharacteristic
420C2#1172123311yescharacteristic
subgroupG/H
centerC22#88676
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC4⋊C4×H(3,𝔽3)C22
socleC22×C3D3×#67
prime radicalC2×#88436C2
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#442
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #64102⋊C3C2semidirect
2. #11721149C2semidirect
4. #11720531C2semidirect
5. #63635⋊C3C2semidirect
6. #11720318C2semidirect
7. #11720312C2semidirect
21. #88436C4semidirect
22. #64102D3semidirect
28. #63635D3semidirect
42. C4⋊#272C2×C4semidirect
47. C4⋊((C3⋊D3)⋊C3)C2×C4semidirect
51. (C3⋊(C3⋊C8))⋊C3C2×C4semidirect
132. C4⋊(C3⋊D3)C4×D3semidirect
148. C3⋊(C3⋊C8)C4×D3semidirect
238. H(3,𝔽3)#442semidirect
363. C32#4869semidirect
364. C32#5014semidirect

Maximal subgroups

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

subgroupindex
#64102⋊C32
#117211492
C2×#884362
#117205312
#63635⋊C32
#117203182
#117203122
#105904623
#105904553