Finite Group Databasebeta

← #13181146

Subgroups — #13181146

Order 1728.

1028 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 1028 conjugacy classes of subgroups — too many to draw. Only its 131 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#11724618C212yesmaximal, complemented
2C2×C3×C4×#157C211yesmaximal, characteristic, complemented
3#11724355C214yesmaximal, complemented
4#11724344C211yesmaximal, characteristic, complemented
5C4×#157⋊D3C214yesmaximal, complemented
6#11724335C211yesmaximal, characteristic, complemented
7#11724334C212yesmaximal, complemented
8#10596511—31nomaximal
9C22×C3×#157C2212yes—
10C2×#157⋊D3C2214yes—
11C3×#3828—22no—
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13#89049C2214yes—
14#89046—22no—
15C4⋊#1113C2212yes—
16C4×#1113C2214yes—
17#89042—22no—
18#89041—22no—
19#89040C2212yes—
20C3×C4×#157C2211yescharacteristic
21C3×C4×#157C2211yescharacteristic
22C3×C4×#157C2212yes—
23#88940C2214yes—
24#88939C2218yes—
25#64306—32no—
26D3×#235—62no—
27C2×C4×#157D311yescharacteristic, complemented
28#64242—34no—
29#64231—31no—
30C4×GO+(4,𝔽2)—34no—
31#64222—31no—
32#64221—32no—
33C22×C3×C3⋊D3C2311yescharacteristic
34C2×#1113C2314yes—
35C2×C3×#157—24no—
36C2×C3×#157C2312yes—
37C2×C3×#157C2312yes—
38#157⋊D3—28no—
39C2×C33×C4D411yescharacteristic, complemented
40C3×#638—24no—
41C2×C3×C3⋊Dic3D411yescharacteristic, complemented
42C3×C4⋊(C3⋊D3)—22no—
43C3×C4×C3⋊D3—24no—
44C3×C4×C3⋊D3D411yescharacteristic, complemented
45C3×C4×C3⋊D3D411yescharacteristic, complemented
46C3×C4×C3⋊D3C2311yescharacteristic
47C3×C4×C3⋊D3C2311yescharacteristic
48C3×#634—22no—
49#6296—22no—
50#6295—24no—
51#6295—22no—
52#6294—24no—
53#6294C2314yes—
54#5477—91nomaximal
55C22×#157C2×D312yes—
56C2×GO+(4,𝔽2)—34no—
57#3828—22no—
58C2×C4×C3⋊D3C2×D311yescharacteristic
59C3×#235—62no—
60C3×#235—62no—
61C3×#235—122no—
62C2×C3×C4×D3—62no—
63#3811—34no—
64D3×#67—64no—
65#3805—62no—
66#3804—64no—
67#3802—32no—
68D3×D12—62no—
69C4×D32—34no—
70C4×D32—64no—
71#3799—62no—
72#3798—64no—
73#3798—62no—
74#3797—32no—
75#3796—62no—
76#3795—62no—
77D3×Dic6—62no—
78C4×#157C2×D311yescharacteristic
79C4×#157C2×D311yescharacteristic
80C4×#157C2×D312yes—
81#3773—34no—
82D32⋊C4—38no—
83C22×C33C2×D411yescharacteristic
84C2×C3×C3⋊D3—22no—
85C2×C3×C3⋊D3—24no—
86C2×C3×C3⋊D3C2×D411yescharacteristic
87C2×C3×C3⋊D3C2411yescharacteristic
88#1113—28no—
89C3×#157—22no—
90C3×#157#4114yescomplemented
91C33×C4C2×D411yescharacteristic
92C33×C4C2×D411yescharacteristic
93C3×C3⋊Dic3—24no—
94C3×C3⋊Dic3C2×D411yescharacteristic
95C3×C3⋊Dic3C2×D411yescharacteristic
96D3×#41—182no—
97C2×#235—92no—
98C2×C3×C42—91no—
99#930—94no—
100#875—92no—
101#874—91no—
102C4×D12—94no—
103#869—91no—
104C22×C3⋊D3C22×D311yescharacteristic
105C22×C3×D3—62no—
106C2×D32—34no—
107C2×D32—64no—
108C2×#157—24no—
109C2×#157C22×D312yes—
110C2×#157C22×D312yes—
111GO+(4,𝔽2)—68no—
112C2×C32×C4—22no—
113C2×C32×C4—42no—
114C2×C32×C4D1211yescharacteristic, complemented
115#638—24no—
116C2×C3⋊Dic3D1211yescharacteristic, complemented
117C4⋊(C3⋊D3)—22no—
118C4×C3⋊D3—24no—
119C4×C3⋊D3D1211yescharacteristic, complemented
120C4×C3⋊D3D1211yescharacteristic, complemented
121C4×C3⋊D3C22×D311yescharacteristic
122C4×C3⋊D3C22×D311yescharacteristic
123#634—22no—
124C3×#67—64no—
125C3×#67—64no—
126C3×#67—124no—
127C2×C3×Dic3—62no—
128C3×D12—62no—
129C3×D12—62no—
130C3×D12—122no—
131C3×C4×D3—62no—
132C3×C4×D3—62no—
133C3×C4×D3—62no—
134C3×C4×D3—62no—
135C3×C4×D3—64no—
136C3×C4×D3—64no—
137C3×C4×D3—124no—
138C3×Dic6—62no—
139C3×Dic6—62no—
140C3×Dic6—122no—
141#627—64no—
142#627—62no—
143#626—64no—
144#626—64no—
145#626—64no—
146#625—64no—
147#625—62no—
148#624—34no—
149#624—64no—
150D3×Dic3—64no—
151D3×Dic3—64no—
152D3×Dic3—64no—
153#532—271noSylow
154C2×C33—21no—
155C2×C33C22×D411yescharacteristic
156C3×C3⋊D3—21no—
157C3×C3⋊D3—44no—
158C3×C3⋊D3C2×#4112yes—
159C3×#41—182no—
160C22×C3×C4—91no—
161C22×C3×C4—92no—
162C2×#67—94no—
163#239—182no—
164D3×Q8—182no—
165#237—182no—
166#237—184no—
167D3×D4—182no—
168D3×D4—184no—
169#235—62no—
170#235—62no—
171#235—122no—
172#235—182no—
173#235—182no—
174#235—184no—
175#235—182no—
176C2×D12—92no—
177C2×D12—94no—
178C2×C4×D3—62no—
179C2×C4×D3—94no—
180C2×C4×D3—184no—
181C2×Dic6—92no—
182C3×C42—91no—
183C3×C42—92no—
184C3×C42—91no—
185(C2×D3)⋊C4—98no—
186C4⋊Dic3—94no—
187C22×C32—22no—
188C22×C32—42no—
189C22×C32C2×D1211yescharacteristic
190C2×C3⋊D3—24no—
191C2×C3⋊D3—22no—
192C2×C3⋊D3C2×D1211yescharacteristic
193C2×C3⋊D3C23×D311yescharacteristic
194C2×C3×D3—62no—
195C2×C3×D3—62no—
196C2×C3×D3—64no—
197C2×C3×D3—64no—
198C2×C3×D3—124no—
199C2×C3×D3—124no—
200D32—68no—
201D32—128no—
202#157—22no—
203#157#23514yescomplemented
204C32×C4—22no—
205C32×C4—22no—
206C32×C4—42no—
207C32×C4—42no—
208C32×C4C2×D1211yescharacteristic
209C32×C4C2×D1211yescharacteristic
210C3⋊Dic3—24no—
211C3⋊Dic3C2×D1211yescharacteristic
212C3⋊Dic3C2×D1211yescharacteristic
213C3×Dic3—62no—
214C3×Dic3—62no—
215C3×Dic3—64no—
216C3×Dic3—64no—
217C3×Dic3—124no—
218C2×#41—272no—
219C4⋊Q8—271no—
220#127—271no—
221#124—272no—
222#121—274no—
223C4×D4—274no—
224C2×C42—91no—
225C33#53211yescharacteristic, complemented, Sylow
226C23×C3—91no—
227C22×D3—62no—
228C22×D3—94no—
229C22×D3—184no—
230C3×Q8—182no—
231C3×D4—182no—
232C3×D4—184no—
233C2×C3×C4—22no—
234C2×C3×C4—42no—
235C2×C3×C4—92no—
236C2×C3×C4—91no—
237C2×C3×C4—91no—
238C2×C3×C4—91no—
239C2×C3×C4—91no—
240C2×C3×C4—91no—
241C2×C3×C4—92no—
242C2×C3×C4—184no—
243C2×C3×C4—184no—
244C2×C3×C4GO+(4,𝔽2)11yescharacteristic, complemented
245#67—64no—
246#67—64no—
247#67—124no—
248#67—184no—
249#67—184no—
250#67—184no—
251C2×Dic3—62no—
252C2×Dic3—94no—
253C2×Dic3—184no—
254D12—62no—
255D12—62no—
256D12—122no—
257D12—182no—
258D12—188no—
259D12—184no—
260C4×D3—62no—
261C4×D3—62no—
262C4×D3—62no—
263C4×D3—62no—
264C4×D3—64no—
265C4×D3—64no—
266C4×D3—124no—
267C4×D3—184no—
268C4×D3—184no—
269C4×D3—184no—
270Dic6—62no—
271Dic6—62no—
272Dic6—122no—
273Dic6—182no—
274Dic6—184no—
275C2×C32—22no—
276C2×C32—21no—
277C2×C32—42no—
278C2×C32—42no—
279C2×C32—82no—
280C2×C32C22×D1211yescharacteristic
281C3⋊D3—21no—
282C3⋊D3—44no—
283C3⋊D3C2×#23512yes—
284C3×D3—64no—
285C3×D3—122no—
286C3×D3—124no—
287C3×D3—124no—
288C3×D3—244no—
289#41—182no—
290#41—542no—
291#41—542no—
292#41—542no—
293C2×Q8—272no—
294C2×D4—272no—
295C2×D4—274no—
296C2×D4—274no—
297C22×C4—91no—
298C22×C4—92no—
299C22×C4—274no—
300C4⋊C4—274no—
301C22⋊C4—278no—
302C42—92no—
303C42—91no—
304C42—91no—
305C22×C3—22no—
306C22×C3—42no—
307C22×C3—91no—
308C22×C3—91no—
309C22×C3—182no—
310C22×C3—184no—
311C22×C3C2×GO+(4,𝔽2)11yescharacteristic
312C2×D3—62no—
313C2×D3—62no—
314C2×D3—64no—
315C2×D3—64no—
316C2×D3—124no—
317C2×D3—124no—
318C2×D3—188no—
319C2×D3—368no—
320C3×C4—22no—
321C3×C4—22no—
322C3×C4—42no—
323C3×C4—42no—
324C3×C4—94no—
325C3×C4—91no—
326C3×C4—91no—
327C3×C4—184no—
328C3×C4—182no—
329C3×C4C2×GO+(4,𝔽2)11yescharacteristic
330C3×C4C2×GO+(4,𝔽2)11yescharacteristic
331Dic3—62no—
332Dic3—62no—
333Dic3—64no—
334Dic3—64no—
335Dic3—124no—
336C32—22no—
337C32—42no—
338C32#547711yescharacteristic, complemented
339C23—91no—
340C23—274no—
341Q8—182no—
342Q8—542no—
343D4—184no—
344D4—182no—
345D4—542no—
346D4—548no—
347D4—544no—
348C2×C4—92no—
349C2×C4—91no—
350C2×C4—91no—
351C2×C4—91no—
352C2×C4—91no—
353C2×C4—91no—
354C2×C4—92no—
355C2×C4—184no—
356C2×C4—184no—
357C2×C4—274no—
358C2×C4—544no—
359C2×C4#157⋊D311yescharacteristic, complemented
360C2×C3—21no—
361C2×C3—22no—
362C2×C3—42no—
363C2×C3—42no—
364C2×C3—82no—
365C2×C3—92no—
366C2×C3—181no—
367C2×C3—364no—
368C2×C3C22×GO+(4,𝔽2)11yescharacteristic
369D3—64no—
370D3—122no—
371D3—124no—
372D3—124no—
373D3—244no—
374C22—91no—
375C22—91no—
376C22—182no—
377C22—184no—
378C22—548no—
379C22C2×#157⋊D311yescharacteristic
380C4—94no—
381C4—91no—
382C4—91no—
383C4—182no—
384C4—184no—
385C4C2×#157⋊D311yescharacteristic
386C4C2×#157⋊D311yescharacteristic
387C3—22no—
388C3—42no—
389C3#1059651111yescharacteristic, complemented
390C2—21no—
391C2—92no—
392C2—181no—
393C2—364no—
394C2C22×#157⋊D311yescharacteristic
subgroupG/H
centerC4C2×#157⋊D3
commutator, cosocleC2×C3×C3⋊D3C24
FrattiniC2C22×#157⋊D3
FittingC2×C33×C4D4
socleC2×C33C22×D4
hypercenterC2×C4#157⋊D3
hypocenterC33#532
LCSG ▷ C2×C3×C3⋊D3 ▷ C33
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. #11724618C2semidirect
2. C2×C3×C4×#157C2semidirect
3. #11724355C2semidirect
4. #11724344C2semidirect
5. C4×#157⋊D3C2semidirect
6. #11724335C2semidirect
7. #11724334C2semidirect
27. C2×C4×#157D3semidirect
39. C2×C33×C4D4semidirect
41. C2×C3×C3⋊Dic3D4semidirect
44. C3×C4×C3⋊D3D4semidirect
45. C3×C4×C3⋊D3D4semidirect
90. C3×#157#41semidirect
114. C2×C32×C4D12semidirect
116. C2×C3⋊Dic3D12semidirect
119. C4×C3⋊D3D12semidirect
120. C4×C3⋊D3D12semidirect
203. #157#235semidirect
225. C33#532semidirect
244. C2×C3×C4GO+(4,𝔽2)semidirect
338. C32#5477semidirect
359. C2×C4#157⋊D3semidirect
389. C3#10596511semidirect

Maximal subgroups

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

subgroupindex
#117246182
C2×C3×C4×#1572
#117243552
#117243442
C4×#157⋊D32
#117243352
#117243342
#105965113
#54779