Finite Group Databasebeta

← #13137880

Subgroups — #13137880

Order 1728.

383 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 383 conjugacy classes of subgroups — too many to draw. Only its 75 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#11721756C211yesmaximal, characteristic
2#11721095C211yesmaximal, characteristic
3#63635⋊C3C211yesmaximal, characteristic
4#63634⋊C3C211yesmaximal, characteristic
5#11720307C211yesmaximal, characteristic
6#11720306C211yesmaximal, characteristic
7#11720303C211yesmaximal, characteristic
8#10590421—31nomaximal
9#10590420—31nomaximal
10C2×#6229—21no—
11C4⋊C4×H(3,𝔽3)C2211yescharacteristic
12C4⋊#1084C2211yescharacteristic
13#88540—21no—
14C4×#1084C2211yescharacteristic
15C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
18C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
19#88448—21no—
20#88448—21no—
21C3×#888—31no—
22C3×#888—31no—
23C3×#888—61no—
24#63869—31no—
25#63869—31no—
26#63635D311yescharacteristic
27#63634D311yescharacteristic
28C3×#808—31no—
29C3×#807—31no—
30#63576—31no—
31#63575—31no—
32#63574—31no—
33#63573—31no—
34#63554—31no—
35#63554—31no—
36C2×C4×H(3,𝔽3)—21no—
37C2×C4×H(3,𝔽3)C2311yescharacteristic
38C2×#1084—21no—
39C2×#1084D411yescharacteristic
40C2×#1084D411yescharacteristic
41#6229—42no—
42C2×(C3⋊Dic3)⋊C3—21no—
43C2×(C3⋊Dic3)⋊C3—21no—
44(C3⋊(C3⋊C8))⋊C3Q812yescomplemented
45(C3⋊(C3⋊C8))⋊C3Q812yescomplemented
46#5081—91no—
47#4876—91no—
48#4856—91no—
49C2×C3×Dic6—61no—
50C2×C3×Dic6—61no—
51C2×C3×Dic6—62no—
52C32×C4⋊C4—21no—
53C32×C4⋊C4C2×D311yescharacteristic
54C32×C4⋊C4C2×D311yescharacteristic
55C4⋊(C3⋊Dic3)C2×D311yescharacteristic
56C4⋊(C3⋊Dic3)C2×D311yescharacteristic
57C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
58C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
59C3×C4⋊Dic3—31no—
60C3×C4⋊Dic3—31no—
61C3×C4⋊Dic3—31no—
62C3×C4⋊Dic3—31no—
63C3×C4⋊Dic3—61no—
64C3×Dic3⋊C4—61no—
65C3×Dic3⋊C4—61no—
66C3×Dic3⋊C4—62no—
67C3×C4×Dic3—31no—
68C3×C4×Dic3—31no—
69C3×C4×Dic3—61no—
70C2×C3×C3⋊C8—31no—
71C2×C3×C3⋊C8—31no—
72#3724—61no—
73#3724—61no—
74Dic3⋊Dic3—61no—
75Dic3⋊Dic3—61no—
76C22×H(3,𝔽3)C2×D411yescharacteristic
77C4×H(3,𝔽3)—41no—
78C4×H(3,𝔽3)C2×Q811yescharacteristic
79C4×H(3,𝔽3)C2×Q811yescharacteristic
80#1084—21no—
81#1084—21no—
82#1084—41no—
83(C3⋊Dic3)⋊C3—41no—
84(C3⋊Dic3)⋊C3—41no—
85C3×C4⋊Q8—91no—
86#890—91no—
87#890—91no—
88#888—31no—
89#888—31no—
90#888—61no—
91#870—91no—
92C3×#107—91no—
93C3×#106—91no—
94#818—91no—
95#817—91no—
96#815—91no—
97#815—91no—
98#808—31no—
99#808—91no—
100#808—91no—
101#808—91no—
102#807—31no—
103#807—91no—
104#807—91no—
105#807—91no—
106#803—91no—
107C2×C32×C4—22no—
108C2×C32×C4—21no—
109C2×C32×C4—21no—
110C2×C32×C4—21no—
111C2×C32×C4C22×D311yescharacteristic
112C2×C32×C4C22×D311yescharacteristic
113C2×C3⋊Dic3—21no—
114C2×C3⋊Dic3—21no—
115C2×C3×Dic3—31no—
116C2×C3×Dic3—31no—
117C2×C3×Dic3—31no—
118C2×C3×Dic3—31no—
119C2×C3×Dic3—61no—
120C2×C3×Dic3—61no—
121C2×C3×Dic3—62no—
122C2×C3×Dic3—61no—
123C2×C3×Dic3—61no—
124C2×C3×Dic3—61no—
125C2×C3×Dic3—61no—
126C3×Dic6—124no—
127C3×Dic6—122no—
128C3×Dic6—122no—
129C3⋊(C3⋊C8)Dic612yescomplemented
130C3⋊(C3⋊C8)Dic612yescomplemented
131C3×C3⋊C8—32no—
132C3×C3⋊C8—32no—
133#501—271noSylow
134C2×H(3,𝔽3)C4⋊Q811yescharacteristic
135C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
136C2×H(3,𝔽3)#13711yescharacteristic
137C2×C3×Q8—181no—
138C2×Dic6—61no—
139C2×Dic6—61no—
140C2×Dic6—62no—
141C2×C3×C8—91no—
142C2×C3×C8—91no—
143C3×C4⋊C4—31no—
144C3×C4⋊C4—31no—
145C3×C4⋊C4—61no—
146C3×C4⋊C4—91no—
147C3×C4⋊C4—91no—
148C3×C4⋊C4—91no—
149C3×C4⋊C4—181no—
150C3×C4⋊C4D3211yescharacteristic
151C3×C42—91no—
152C4⋊Dic3—31no—
153C4⋊Dic3—31no—
154C4⋊Dic3—31no—
155C4⋊Dic3—31no—
156C4⋊Dic3—61no—
157C4⋊Dic3—91no—
158C4⋊Dic3—91no—
159C4⋊Dic3—181no—
160C4⋊Dic3—181no—
161Dic3⋊C4—61no—
162Dic3⋊C4—61no—
163Dic3⋊C4—62no—
164Dic3⋊C4—181no—
165Dic3⋊C4—181no—
166Dic3⋊C4—181no—
167Dic3⋊C4—181no—
168C4×Dic3—31no—
169C4×Dic3—31no—
170C4×Dic3—61no—
171C2×C3⋊C8—31no—
172C2×C3⋊C8—31no—
173C2×C3⋊C8—91no—
174C2×C3⋊C8—91no—
175C22×C32—21no—
176C22×C32D3×D411yescharacteristic
177C22×C32D3×D411yescharacteristic
178C32×C4—21no—
179C32×C4—21no—
180C32×C4—42no—
181C32×C4—41no—
182C32×C4—41no—
183C32×C4C2×Dic611yescharacteristic
184C32×C4C2×Dic611yescharacteristic
185C32×C4D3×Q811yescharacteristic
186C32×C4D3×Q811yescharacteristic
187C3⋊Dic3—41no—
188C3⋊Dic3—41no—
189C3×Dic3—62no—
190C3×Dic3—62no—
191C3×Dic3—61no—
192C3×Dic3—61no—
193C3×Dic3—61no—
194C3×Dic3—61no—
195C3×Dic3—122no—
196C3×Dic3—121no—
197C3×Dic3—121no—
198C3×Dic3—121no—
199C3×Dic3—121no—
200C4⋊Q8—91no—
201#125—271no—
202#107—91no—
203#107—271no—
204#106—91no—
205#106—271no—
206#97—271no—
207H(3,𝔽3)#50111yescharacteristic, complemented, Sylow
208C3×Q8—362no—
209C2×C3×C4—21no—
210C2×C3×C4—31no—
211C2×C3×C4—31no—
212C2×C3×C4—61no—
213C2×C3×C4—61no—
214C2×C3×C4—62no—
215C2×C3×C4—61no—
216C2×C3×C4—91no—
217C2×C3×C4—91no—
218C2×C3×C4—181no—
219C2×C3×C4—181no—
220C2×C3×C4—181no—
221C2×C3×C4C2×D3211yescharacteristic
222C2×Dic3—31no—
223C2×Dic3—31no—
224C2×Dic3—31no—
225C2×Dic3—31no—
226C2×Dic3—61no—
227C2×Dic3—61no—
228C2×Dic3—61no—
229C2×Dic3—61no—
230C2×Dic3—62no—
231C2×Dic3—61no—
232C2×Dic3—61no—
233C2×Dic3—181no—
234C2×Dic3—181no—
235Dic6—122no—
236Dic6—122no—
237Dic6—124no—
238C3×C8—92no—
239C3×C8—92no—
240C3⋊C8—32no—
241C3⋊C8—32no—
242C3⋊C8—92no—
243C3⋊C8—92no—
244C2×C32—21no—
245C2×C32—21no—
246C2×C32—21no—
247C2×C32#88811yescharacteristic
248C2×C32#88811yescharacteristic
249C2×C32#91111yescharacteristic
250C2×C32#91411yescharacteristic
251C2×C32#91511yescharacteristic
252C2×C32#91811yescharacteristic
253C2×Q8—181no—
254C2×C8—91no—
255C2×C8—91no—
256C4⋊C4—91no—
257C4⋊C4—91no—
258C4⋊C4—91no—
259C4⋊C4—181no—
260C4⋊C4—541no—
261C4⋊C4—541no—
262C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
263C42—91no—
264C22×C3—31no—
265C22×C3—31no—
266C22×C3—61no—
267C22×C3#381111yescharacteristic
268C3×C4—31no—
269C3×C4—31no—
270C3×C4—31no—
271C3×C4—31no—
272C3×C4—41no—
273C3×C4—61no—
274C3×C4—61no—
275C3×C4—121no—
276C3×C4—121no—
277C3×C4—122no—
278C3×C4—181no—
279C3×C4—181no—
280C3×C4—361no—
281C3×C4—361no—
282C3×C4—361no—
283C3×C4D3×Dic611yescharacteristic
284C3×C4D3×Dic611yescharacteristic
285Dic3—61no—
286Dic3—61no—
287Dic3—61no—
288Dic3—61no—
289Dic3—62no—
290Dic3—62no—
291Dic3—121no—
292Dic3—121no—
293Dic3—122no—
294Dic3—121no—
295Dic3—121no—
296Dic3—361no—
297Dic3—361no—
298C32—21no—
299C32#485611yescharacteristic, complemented
300C32#487611yescharacteristic, complemented
301Q8—362no—
302C2×C4—21no—
303C2×C4—91no—
304C2×C4—91no—
305C2×C4—181no—
306C2×C4—181no—
307C2×C4—181no—
308C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
309C8—92no—
310C8—92no—
311C2×C3—31no—
312C2×C3—31no—
313C2×C3—31no—
314C2×C3—31no—
315C2×C3—31no—
316C2×C3—31no—
317C2×C3—61no—
318C2×C3—61no—
319C2×C3—61no—
320C2×C3#6384011yescharacteristic
321C2×C3#6393711yescharacteristic
322C2×C3#6395111yescharacteristic
323C22#8867711yescharacteristic
324C4—41no—
325C4—181no—
326C4—181no—
327C4—361no—
328C4—361no—
329C4—361no—
330C4((C3⋊D3)⋊C3)⋊Q811yescharacteristic
331C4((C3⋊D3)⋊C3)⋊Q811yescharacteristic
332C3—31no—
333C3—31no—
334C3—61no—
335C3#1059054611yescharacteristic
336C2#1172108011yescharacteristic
337C2#1172119311yescharacteristic
338C2#1172123111yescharacteristic
subgroupG/H
centerC22#88677
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC4⋊C4×H(3,𝔽3)C22
socleC22×C3#3811
prime radicalC22×H(3,𝔽3)C2×D4
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#501
LCSG ▷ C2×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
44. (C3⋊(C3⋊C8))⋊C3Q8semidirect
45. (C3⋊(C3⋊C8))⋊C3Q8semidirect
129. C3⋊(C3⋊C8)Dic6semidirect
130. C3⋊(C3⋊C8)Dic6semidirect
207. H(3,𝔽3)#501semidirect
299. C32#4856semidirect
300. C32#4876semidirect

Maximal subgroups

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

subgroupindex
#117217562
#117210952
#63635⋊C32
#63634⋊C32
#117203072
#117203062
#117203032
#105904213
#105904203