Finite Group Databasebeta

← #13159345

Subgroups — #13159345

Order 1728.

1076 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 1076 conjugacy classes of subgroups — too many to draw. Only its 170 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
1C4⋊C4×C3⋊(C3⋊D3)C211yesmaximal, characteristic
2#11723413C211yesmaximal, characteristic
3C2×#88788C211yesmaximal, characteristic
4C3×#63634C211yesmaximal, characteristic, complemented
5C32×#808C211yesmaximal, characteristic, complemented
6#11722055C211yesmaximal, characteristic, complemented
7#11722054C211yesmaximal, characteristic, complemented
8#10592560—31nomaximal
9#10590583—34nomaximal
10C2×C4×C3⋊(C3⋊D3)—21no—
11C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
12C33×C4⋊C4C2211yescharacteristic
13C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
14#88857—21no—
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88804—21no—
20Dic3×C3⋊Dic3—21no—
21#88788C414yescomplemented
22C2×#3743—31no—
23#64101—31no—
24C4⋊C4×C3⋊D3—31no—
25C4⋊C4×C3⋊D3—34no—
26C4⋊C4×C3⋊D3—64no—
27#63849—34no—
28C2×#3712—34no—
29C32×#106—31no—
30#63645—31no—
31#63635—31no—
32#63634D311yescharacteristic, complemented
33C3×#808D314yescomplemented
34C3×#807—34no—
35#63575—34no—
36#63573—34no—
37C22×C3⋊(C3⋊D3)D411yescharacteristic
38C2×C33×C4—21no—
39C2×C33×C4C2311yescharacteristic
40C2×C3⋊(C3⋊Dic3)—21no—
41C2×C3⋊(C3⋊Dic3)D411yescharacteristic
42C4×C3⋊(C3⋊D3)—42no—
43C4×C3⋊(C3⋊D3)C2×C412yes—
44C4×C3⋊(C3⋊D3)Q812yes—
45C2×C3×C3⋊Dic3—21no—
46C2×C32×Dic3—21no—
47C3×C3⋊(C3⋊C8)C2×C412yescomplemented
48C32×C3⋊C8C2×C412yescomplemented
49#4881—91no—
50#4861—94no—
51C2×C4×C3⋊D3—31no—
52C2×C4×C3⋊D3—34no—
53C2×C4×C3⋊D3—64no—
54C2×C4×C3⋊D3—61no—
55C2×C4×C3⋊D3—64no—
56C2×C4×C3⋊D3—68no—
57C2×C32×C8—31no—
58C32×C4⋊C4—24no—
59C32×C4⋊C4—31no—
60C32×C4⋊C4C2×D314yes—
61C32×C4⋊C4C2×D311yescharacteristic
62#3752—61no—
63C4⋊(C3⋊Dic3)—31no—
64C4⋊(C3⋊Dic3)—34no—
65C4⋊(C3⋊Dic3)—64no—
66C4⋊(C3⋊Dic3)C2×D311yescharacteristic
67#3750—61no—
68#3750—64no—
69#3750—68no—
70C4×C3⋊Dic3—61no—
71C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
72#3743—34no—
73C3×C4⋊Dic3—34no—
74C3×C4⋊Dic3C2×D314yes—
75C2×C3×C3⋊C8—34no—
76C2×C3×C3⋊C8C2×D314yes—
77#3722—64no—
78Dic32—64no—
79#3712—316no—
80C22×C33C2×D411yescharacteristic
81C2×C3⋊(C3⋊D3)—21no—
82C2×C3⋊(C3⋊D3)—21no—
83C2×C3⋊(C3⋊D3)C4⋊C412yes—
84C33×C4—41no—
85C33×C4C22×C411yescharacteristic
86C33×C4C2×Q811yescharacteristic
87C3⋊(C3⋊Dic3)—41no—
88C3⋊(C3⋊Dic3)C4⋊C412yes—
89C3×C3⋊Dic3—41no—
90C32×Dic3—41no—
91C2×ΓL(1,𝔽25)—94no—
92C2×ΓL(1,𝔽25)—91no—
93#892—94no—
94#892—91no—
95D3×C4⋊C4—94no—
96D3×C4⋊C4—91no—
97D3×C4⋊C4—184no—
98C3×#107—91no—
99C3×#106—34no—
100#818—94no—
101#817—91no—
102#808—94no—
103#808C3⋊D311yescharacteristic, complemented
104#807—34no—
105#807—91no—
106C22×C3⋊D3—31no—
107C22×C3⋊D3—34no—
108C22×C3⋊D3—64no—
109C2×C32×C4—28no—
110C2×C32×C4—24no—
111C2×C32×C4—21no—
112C2×C32×C4—24no—
113C2×C32×C4—61no—
114C2×C32×C4C22×D314yes—
115C2×C32×C4C22×D311yescharacteristic
116C2×C3⋊Dic3—21no—
117C2×C3⋊Dic3—31no—
118C2×C3⋊Dic3—34no—
119C2×C3⋊Dic3—64no—
120C2×C3⋊Dic3—68no—
121C2×C3⋊Dic3—64no—
122C2×C3⋊Dic3—61no—
123C4×C3⋊D3—32no—
124C4×C3⋊D3—32no—
125C4×C3⋊D3—38no—
126C4×C3⋊D3—38no—
127C4×C3⋊D3—68no—
128C4×C3⋊D3—68no—
129C4×C3⋊D3—1216no—
130C4×C3⋊D3—122no—
131C4×C3⋊D3—128no—
132C2×C3×Dic3—24no—
133C2×C3×Dic3—64no—
134C32×C8—32no—
135C3⋊(C3⋊C8)C4×D312yescomplemented
136C3×C3⋊C8—38no—
137C3×C3⋊C8C4×D318yescomplemented
138#428—271noSylow
139C2×C33C2×C4⋊C411yescharacteristic
140C2×C33AGL(1,ℤ/8)11yescharacteristic
141C2×C33#13711yescharacteristic
142C3⋊(C3⋊D3)—22no—
143C2×C4×D3—94no—
144C2×C4×D3—91no—
145C2×C4×D3—184no—
146C2×C4×D3—184no—
147C2×C4×D3—181no—
148C2×C4×D3—188no—
149C2×C3×C8—34no—
150C2×C3×C8—91no—
151C3×C4⋊C4—24no—
152C3×C4⋊C4—34no—
153C3×C4⋊C4—91no—
154C3×C4⋊C4D3214yes—
155C3×C4⋊C4C2×C3⋊D311yescharacteristic
156(C2×D3)⋊C4—181no—
157(C2×D3)⋊C4—184no—
158C4⋊Dic3—34no—
159C4⋊Dic3—94no—
160C4⋊Dic3—91no—
161C4⋊Dic3—184no—
162C4⋊Dic3C2×C3⋊D311yescharacteristic
163Dic3⋊C4—184no—
164Dic3⋊C4—181no—
165Dic3⋊C4—188no—
166C4×Dic3—181no—
167C4×Dic3—184no—
168C2×C3⋊C8—34no—
169C2×C3⋊C8C2×C3⋊D311yescharacteristic
170ΓL(1,𝔽25)—916no—
171ΓL(1,𝔽25)—94no—
172C22×C32—24no—
173C22×C32D3×D414yes—
174C22×C32D3×D411yescharacteristic
175C2×C3⋊D3—32no—
176C2×C3⋊D3—38no—
177C2×C3⋊D3—68no—
178C2×C3⋊D3—68no—
179C2×C3⋊D3—68no—
180C2×C3⋊D3—61no—
181C2×C3⋊D3—64no—
182C2×C3⋊D3—64no—
183C2×C3⋊D3—61no—
184C32×C4—24no—
185C32×C4—24no—
186C32×C4—48no—
187C32×C4—41no—
188C32×C4—44no—
189C32×C4—121no—
190C32×C4C2×C4×D314yes—
191C32×C4C2×C4×D311yescharacteristic
192C32×C4D3×Q814yes—
193C32×C4D3×Q811yescharacteristic
194C3⋊Dic3—32no—
195C3⋊Dic3—38no—
196C3⋊Dic3—41no—
197C3⋊Dic3—68no—
198C3⋊Dic3—128no—
199C3⋊Dic3—121no—
200C3⋊Dic3—124no—
201C3×Dic3—44no—
202C3×Dic3—124no—
203C2×M16—271no—
204#117—271no—
205C2×C4⋊C4—271no—
206#107—91no—
207#107—271no—
208#106—31no—
209#106—271no—
210C33#42811yescharacteristic, complemented, Sylow
211C22×D3—94no—
212C22×D3—91no—
213C22×D3—184no—
214C2×C3×C4—28no—
215C2×C3×C4—24no—
216C2×C3×C4—21no—
217C2×C3×C4—24no—
218C2×C3×C4—64no—
219C2×C3×C4—181no—
220C2×C3×C4C2×D3214yes—
221C2×C3×C4C22×C3⋊D311yescharacteristic
222C2×Dic3—21no—
223C2×Dic3—64no—
224C2×Dic3—94no—
225C2×Dic3—91no—
226C2×Dic3—184no—
227C2×Dic3—188no—
228C2×Dic3—184no—
229C2×Dic3—181no—
230C4×D3—98no—
231C4×D3—98no—
232C4×D3—92no—
233C4×D3—92no—
234C4×D3—188no—
235C4×D3—188no—
236C4×D3—3616no—
237C4×D3—362no—
238C4×D3—368no—
239C3×C8—38no—
240C3×C8—92no—
241C3⋊C8—38no—
242C3⋊C8C4×C3⋊D312yescomplemented
243C2×C32—24no—
244C2×C32—24no—
245C2×C32—24no—
246C2×C32D3×C4⋊C414yes—
247C2×C32D3×C4⋊C411yescharacteristic
248C2×C32#91111yescharacteristic
249C2×C32#91414yes—
250C2×C32#91514yes—
251C2×C32#91811yescharacteristic
252C3⋊D3—616no—
253C3⋊D3—62no—
254C3⋊D3—68no—
255C22×C4—271no—
256C22×C4—541no—
257M16—274no—
258C2×C8—31no—
259C2×C8—91no—
260C4⋊C4—31no—
261C4⋊C4—91no—
262C4⋊C4—271no—
263C4⋊C4—541no—
264C4⋊C4D3×C3⋊D311yescharacteristic
265C22⋊C4—541no—
266C42—541no—
267C22×C3—24no—
268C22×C3#381114yes—
269C22×C3D4×C3⋊D311yescharacteristic
270C2×D3—92no—
271C2×D3—98no—
272C2×D3—188no—
273C2×D3—188no—
274C2×D3—188no—
275C2×D3—181no—
276C2×D3—181no—
277C2×D3—184no—
278C2×D3—184no—
279C3×C4—24no—
280C3×C4—24no—
281C3×C4—48no—
282C3×C4—41no—
283C3×C4—44no—
284C3×C4—124no—
285C3×C4—361no—
286C3×C4#379714yes—
287C3×C4C4×D3214yes—
288C3×C4C2×C4×C3⋊D311yescharacteristic
289C3×C4Q8×C3⋊D311yescharacteristic
290Dic3—41no—
291Dic3—92no—
292Dic3—98no—
293Dic3—124no—
294Dic3—188no—
295Dic3—368no—
296Dic3—361no—
297Dic3—364no—
298C32—24no—
299C32#486114yescomplemented
300C32#488111yescharacteristic, complemented
301C23—271no—
302C2×C4—21no—
303C2×C4—61no—
304C2×C4—181no—
305C2×C4—271no—
306C2×C4—272no—
307C2×C4—272no—
308C2×C4—541no—
309C2×C4—1082no—
310C2×C4C2×D3×C3⋊D311yescharacteristic
311C8—32no—
312C8—92no—
313C2×C3—24no—
314C2×C3—24no—
315C2×C3—24no—
316C2×C3#6388314yes—
317C2×C3#6393714yes—
318C2×C3#6395114yes—
319C2×C3C4⋊C4×C3⋊D311yescharacteristic
320C2×C3#6412311yescharacteristic
321C2×C3#6412411yescharacteristic
322D3—1816no—
323D3—182no—
324D3—188no—
325C22—272no—
326C22—541no—
327C22—541no—
328C22#8903911yescharacteristic
329C4—41no—
330C4—121no—
331C4—272no—
332C4—361no—
333C4—1081no—
334C4#8901911yescharacteristic
335C4C4×D3×C3⋊D311yescharacteristic
336C3—24no—
337C3#1059058314yescomplemented
338C3#1059256011yescharacteristic, complemented
339C2—542no—
340C2#1172347011yescharacteristic
341C2#1172356611yescharacteristic
342C2#1172358811yescharacteristic
subgroupG/H
centerC22#89039
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC22×C3⋊(C3⋊D3)D4
cosocleC2×C33×C4C23
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#428
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
4. C3×#63634C2semidirect
5. C32×#808C2semidirect
6. #11722055C2semidirect
7. #11722054C2semidirect
21. #88788C4semidirect
32. #63634D3semidirect
33. C3×#808D3semidirect
47. C3×C3⋊(C3⋊C8)C2×C4semidirect
48. C32×C3⋊C8C2×C4semidirect
103. #808C3⋊D3semidirect
135. C3⋊(C3⋊C8)C4×D3semidirect
137. C3×C3⋊C8C4×D3semidirect
210. C33#428semidirect
242. C3⋊C8C4×C3⋊D3semidirect
299. C32#4861semidirect
300. C32#4881semidirect
337. C3#10590583semidirect
338. C3#10592560semidirect

Maximal subgroups

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

subgroupindex
C4⋊C4×C3⋊(C3⋊D3)2
#117234132
C2×#887882
C3×#636342
C32×#8082
#117220552
#117220542
#105925603
#105905833