Finite Group Databasebeta

← #13137951

Subgroups — #13137951

Order 1728.

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

normal subgroups only

This group has 456 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#64096⋊C3C211yesmaximal, characteristic, complemented
2#11721132C211yesmaximal, characteristic
3C2×#88431C211yesmaximal, characteristic
4#11720529C211yesmaximal, characteristic, complemented
5#63636⋊C3C211yesmaximal, characteristic
6#11720312C211yesmaximal, characteristic, complemented
7#11720309C211yesmaximal, characteristic, complemented
8#10590461—31nomaximal
9#10590457—31nomaximal
10C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
11C2×#634⋊C3C2211yescharacteristic
12C2×((C3⋊D3)⋊C3)⋊C4—21no—
13C4⋊C4×H(3,𝔽3)C2211yescharacteristic
14C4⋊#1084C2211yescharacteristic
15C2×#6180C2211yescharacteristic
16#3750⋊C3—21no—
17C4×(C3⋊Dic3)⋊C3—21no—
18C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20#88450—21no—
21#88431C414yescomplemented
22#64096D311yescharacteristic, complemented
23C3×#887—31no—
24#63898—31no—
25#63894—31no—
26C2×#3717—31no—
27C2×#3716—31no—
28#63636D311yescharacteristic
29C3×#809—31no—
30C3×#807—31no—
31C3×#807—31no—
32C3×#807—61no—
33#63567—31no—
34#63565—31no—
35#63555—31no—
36#63554—31no—
37C22×(C3⋊D3)⋊C3—21no—
38C2×C4×H(3,𝔽3)—21no—
39C2×C4×H(3,𝔽3)C2311yescharacteristic
40C2×#1084—21no—
41C2×(C3⋊Dic3)⋊C3—21no—
42C2×(C3⋊Dic3)⋊C3D411yescharacteristic
43C2×(C3⋊Dic3)⋊C3D411yescharacteristic
44C4⋊((C3⋊D3)⋊C3)—21no—
45C4⋊((C3⋊D3)⋊C3)C2×C412yes—
46#634⋊C3—21no—
47#634⋊C3C2×C412yescomplemented
48((C3⋊D3)⋊C3)⋊C4—42no—
49#6180C2×C412yescomplemented
50(C3⋊(C3⋊C8))⋊C3—21no—
51#5026—91no—
52#4889—91no—
53#4757—91no—
54C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
55C2×#634C2×D311yescharacteristic
56C2×C3×D12—31no—
57C2×C3×Dic6—31no—
58C2×#624—61no—
59C2×D3×Dic3—61no—
60C32×C4⋊C4—21no—
61C32×C4⋊C4C2×D311yescharacteristic
62C32×C4⋊C4C2×D311yescharacteristic
63#3750—21no—
64C4×C3⋊Dic3—21no—
65C4×C3⋊Dic3C2×D311yescharacteristic
66C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
67C3×C4⋊Dic3—31no—
68C3×C4⋊Dic3—31no—
69C3×C4⋊Dic3—61no—
70C3×Dic3⋊C4—61no—
71C3×C4×Dic3—31no—
72C3×C4×Dic3—61no—
73C2×C3×C3⋊C8—31no—
74C2×C3×C3⋊C8—31no—
75C2×C3×C3⋊C8—31no—
76C2×C3×C3⋊C8—61no—
77#3722—61no—
78(C2×D3)⋊Dic3—61no—
79#3717—34no—
80#3716—34no—
81C22×H(3,𝔽3)C2×D411yescharacteristic
82C2×(C3⋊D3)⋊C3—22no—
83C2×(C3⋊D3)⋊C3—41no—
84C2×(C3⋊D3)⋊C3—41no—
85C4×H(3,𝔽3)—41no—
86C4×H(3,𝔽3)C22×C411yescharacteristic
87C4×H(3,𝔽3)#4111yescharacteristic
88#1084—41no—
89(C3⋊Dic3)⋊C3—22no—
90(C3⋊Dic3)⋊C3—21no—
91(C3⋊Dic3)⋊C3—21no—
92C3×C4×Q8—91no—
93C2×#215—91no—
94D4×Dic3—91no—
95C2×#214—91no—
96C2×#204—91no—
97#893—91no—
98#887—31no—
99#887—91no—
100C4×D12—91no—
101C3×#107—91no—
102C3×#102—91no—
103#835—91no—
104#816—91no—
105#815—91no—
106#815—91no—
107#810—91no—
108#809—31no—
109#809—91no—
110#807—31no—
111#807—31no—
112#807—61no—
113#803—91no—
114C22×C3⋊D3—21no—
115C22×C3×D3—61no—
116C2×C32×C4—21no—
117C2×C32×C4—21no—
118C2×C32×C4—21no—
119C2×C32×C4—41no—
120C2×C32×C4C22×D311yescharacteristic
121C2×C32×C4C22×D311yescharacteristic
122C2×C3⋊Dic3—21no—
123C2×C3⋊Dic3#6711yescharacteristic
124C2×C3⋊Dic3#6711yescharacteristic
125C4⋊(C3⋊D3)—21no—
126C4⋊(C3⋊D3)C4×D312yes—
127#634—21no—
128#634C4×D312yescomplemented
129C2×C3×Dic3—31no—
130C2×C3×Dic3—31no—
131C2×C3×Dic3—61no—
132C2×C3×Dic3—61no—
133C2×C3×Dic3—61no—
134C2×C3×Dic3—121no—
135C3×D12—32no—
136C3×D12—61no—
137C3×Dic6—32no—
138C3×Dic6—61no—
139#624—122no—
140D3×Dic3—122no—
141C3⋊(C3⋊C8)—21no—
142C3×C3⋊C8—32no—
143C3×C3⋊C8—32no—
144C3×C3⋊C8—62no—
145C3×C3⋊C8—61no—
146#440—271noSylow
147C2×H(3,𝔽3)C4×D411yescharacteristic
148C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
149C2×H(3,𝔽3)#13711yescharacteristic
150(C3⋊D3)⋊C3—42no—
151C2×C3×Q8—91no—
152C2×C3×D4—91no—
153C22×Dic3—181no—
154C2×D12—31no—
155C2×D12—91no—
156C2×C4×D3—181no—
157C2×C4×D3—181no—
158C2×Dic6—31no—
159C2×Dic6—91no—
160C2×C3×C8—91no—
161C2×C3×C8—91no—
162C3×C4⋊C4—31no—
163C3×C4⋊C4—31no—
164C3×C4⋊C4—61no—
165C3×C4⋊C4—91no—
166C3×C4⋊C4—181no—
167C3×C4⋊C4D3211yescharacteristic
168C3×C42—91no—
169C3×C42—181no—
170#217—181no—
171#215—94no—
172#214—94no—
173(C2×D3)⋊C4—181no—
174(C2×D3)⋊C4—181no—
175C4⋊Dic3—31no—
176C4⋊Dic3—31no—
177C4⋊Dic3—61no—
178Dic3⋊C4—61no—
179Dic3⋊C4—181no—
180C4×Dic3—31no—
181C4×Dic3—61no—
182C4×Dic3—91no—
183C4×Dic3—181no—
184C2×C3⋊C8—31no—
185C2×C3⋊C8—31no—
186C2×C3⋊C8—31no—
187C2×C3⋊C8—61no—
188C2×C3⋊C8—91no—
189#204—94no—
190C22×C32—21no—
191C22×C32D3×D411yescharacteristic
192C22×C32C2×#6711yescharacteristic
193C2×C3⋊D3—22no—
194C2×C3⋊D3—41no—
195C2×C3⋊D3—41no—
196C2×C3×D3—62no—
197C2×C3×D3—121no—
198C2×C3×D3—121no—
199C32×C4—21no—
200C32×C4—21no—
201C32×C4—41no—
202C32×C4—41no—
203C32×C4—81no—
204C32×C4C2×C4×D311yescharacteristic
205C32×C4C2×C4×D311yescharacteristic
206C32×C4#23511yescharacteristic
207C32×C4#23711yescharacteristic
208C3⋊Dic3—22no—
209C3⋊Dic3—21no—
210C3⋊Dic3—21no—
211C3×Dic3—62no—
212C3×Dic3—61no—
213C3×Dic3—61no—
214C3×Dic3—121no—
215C3×Dic3—121no—
216C3×Dic3—241no—
217C2×SD16—271no—
218C4×Q8—91no—
219C4×D4—271no—
220#107—91no—
221#103—271no—
222#102—91no—
223#97—271no—
224H(3,𝔽3)#44011yescharacteristic, complemented, Sylow
225C23×C3—181no—
226C22×D3—61no—
227C22×D3—181no—
228C3×Q8—92no—
229C3×Q8—181no—
230C3×D4—92no—
231C3×D4—181no—
232C2×C3×C4—21no—
233C2×C3×C4—31no—
234C2×C3×C4—31no—
235C2×C3×C4—61no—
236C2×C3×C4—61no—
237C2×C3×C4—61no—
238C2×C3×C4—91no—
239C2×C3×C4—91no—
240C2×C3×C4—121no—
241C2×C3×C4—181no—
242C2×C3×C4—181no—
243C2×C3×C4C2×D3211yescharacteristic
244C2×Dic3—31no—
245C2×Dic3—31no—
246C2×Dic3—61no—
247C2×Dic3—61no—
248C2×Dic3—61no—
249C2×Dic3—91no—
250C2×Dic3—91no—
251C2×Dic3—121no—
252C2×Dic3—181no—
253C2×Dic3—362no—
254D12—32no—
255D12—61no—
256D12—92no—
257D12—181no—
258C4×D3—362no—
259C4×D3—362no—
260Dic6—32no—
261Dic6—61no—
262Dic6—92no—
263Dic6—181no—
264C3×C8—92no—
265C3×C8—181no—
266C3⋊C8—32no—
267C3⋊C8—32no—
268C3⋊C8—62no—
269C3⋊C8—61no—
270C3⋊C8—181no—
271C2×C32—21no—
272C2×C32—21no—
273C2×C32—21no—
274C2×C32#67⋊C411yescharacteristic
275C2×C32#91111yescharacteristic
276C2×C32#91511yescharacteristic
277C2×C32C4×#6711yescharacteristic
278C2×C32#94211yescharacteristic
279C2×C32#94911yescharacteristic
280C3⋊D3—42no—
281C3×D3—122no—
282C2×Q8—91no—
283C2×D4—91no—
284C22×C4—541no—
285SD16—274no—
286C2×C8—91no—
287C2×C8—91no—
288C4⋊C4—91no—
289C4⋊C4—181no—
290C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
291C22⋊C4—541no—
292C42—91no—
293C42—181no—
294C22×C3—31no—
295C22×C3—31no—
296C22×C3—61no—
297C22×C3—182no—
298C22×C3—361no—
299C22×C3—361no—
300C22×C3D3×#6711yescharacteristic
301C2×D3—62no—
302C2×D3—121no—
303C2×D3—121no—
304C2×D3—182no—
305C2×D3—361no—
306C2×D3—361no—
307C3×C4—31no—
308C3×C4—31no—
309C3×C4—31no—
310C3×C4—31no—
311C3×C4—41no—
312C3×C4—61no—
313C3×C4—61no—
314C3×C4—121no—
315C3×C4—121no—
316C3×C4—181no—
317C3×C4—181no—
318C3×C4—182no—
319C3×C4—241no—
320C3×C4—361no—
321C3×C4#379511yescharacteristic
322C3×C4C4×D3211yescharacteristic
323Dic3—62no—
324Dic3—61no—
325Dic3—61no—
326Dic3—121no—
327Dic3—121no—
328Dic3—182no—
329Dic3—181no—
330Dic3—181no—
331Dic3—241no—
332C32—21no—
333C32#475711yescharacteristic, complemented
334C32#502611yescharacteristic, complemented
335C23—181no—
336Q8—92no—
337Q8—181no—
338D4—92no—
339D4—181no—
340C2×C4—21no—
341C2×C4—91no—
342C2×C4—91no—
343C2×C4—181no—
344C2×C4—181no—
345C2×C4—1082no—
346C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
347C8—92no—
348C8—181no—
349C2×C3—31no—
350C2×C3—31no—
351C2×C3—31no—
352C2×C3—31no—
353C2×C3—31no—
354C2×C3—31no—
355C2×C3—61no—
356C2×C3—61no—
357C2×C3—61no—
358C2×C3—362no—
359C2×C3#6387911yescharacteristic
360C2×C3#6392511yescharacteristic
361C2×C3#6394611yescharacteristic
362D3—122no—
363D3—362no—
364C22—182no—
365C22—361no—
366C22—361no—
367C22#8867611yescharacteristic
368C4—41no—
369C4—181no—
370C4—182no—
371C4—181no—
372C4—361no—
373C4#8864811yescharacteristic
374C4C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
375C3—31no—
376C3—31no—
377C3—61no—
378C3#1059045411yescharacteristic
379C2—362no—
380C2#1172112211yescharacteristic
381C2#1172119511yescharacteristic
382C2#1172123811yescharacteristic
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×C4⋊((C3⋊D3)⋊C3)C22
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#440
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. #64096⋊C3C2semidirect
4. #11720529C2semidirect
6. #11720312C2semidirect
7. #11720309C2semidirect
21. #88431C4semidirect
22. #64096D3semidirect
47. #634⋊C3C2×C4semidirect
49. #6180C2×C4semidirect
128. #634C4×D3semidirect
224. H(3,𝔽3)#440semidirect
333. C32#4757semidirect
334. C32#5026semidirect

Maximal subgroups

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

subgroupindex
#64096⋊C32
#117211322
C2×#884312
#117205292
#63636⋊C32
#117203122
#117203092
#105904613
#105904573