Finite Group Databasebeta

← #13138751

Subgroups — #13138751

Order 1728.

367 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 367 conjugacy classes of subgroups — too many to draw. Only its 87 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×#6196C211yesmaximal, characteristic
2#11720547C211yesmaximal, characteristic
3#63642⋊C3C211yesmaximal, characteristic
4#63641⋊C3C211yesmaximal, characteristic
5#11720311C211yesmaximal, characteristic
6#11720310C211yesmaximal, characteristic
7#11720303C211yesmaximal, characteristic
8#10590789—31nomaximal
9#10590646—31nomaximal
10C2×#6196C411yescharacteristic
11C2×C8×H(3,𝔽3)C2211yescharacteristic
12C4×#1084C2211yescharacteristic
13C2×#6180C2211yescharacteristic
14C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
15C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
16C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
17C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
18#88449C411yescharacteristic
19#88448C411yescharacteristic
20#88448C411yescharacteristic
21C4×#627—31no—
22C4×#627—31no—
23#63642D311yescharacteristic
24#63641D311yescharacteristic
25C3×#815—31no—
26C3×Dic3⋊C8—31no—
27C3×Dic3⋊C8—31no—
28C3×Dic3⋊C8—31no—
29C3×Dic3⋊C8—61no—
30#63572—31no—
31Dic3⋊(C3⋊C8)—31no—
32#63554—31no—
33#63554—31no—
34#63553—31no—
35Dic3×C3⋊C8—31no—
36C2×C4×H(3,𝔽3)C2311yescharacteristic
37C2×#1084C2×C411yescharacteristic
38C2×#1084C2×C411yescharacteristic
39C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
40C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
41C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
42C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
43#6196—22no—
44C8×H(3,𝔽3)—21no—
45#6180—21no—
46(C3⋊(C3⋊C8))⋊C3—21no—
47(C3⋊(C3⋊C8))⋊C3Q812yescomplemented
48#4831—91no—
49#4828—91no—
50#4701—91no—
51C2×#627—31no—
52C2×#627—31no—
53C2×C32×C8—21no—
54C2×C32×C8C2×D311yescharacteristic
55C2×C32×C8C2×D311yescharacteristic
56C4×C3⋊Dic3C2×D311yescharacteristic
57C4×C3⋊Dic3C2×D311yescharacteristic
58C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
59C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
60C3×C4×Dic3—31no—
61C3×C4×Dic3—31no—
62C3×C4×Dic3—31no—
63C3×C4×Dic3—31no—
64C3×C4×Dic3—61no—
65C2×C3×C3⋊C8—31no—
66C2×C3×C3⋊C8—31no—
67C2×C3×C3⋊C8—31no—
68C2×C3×C3⋊C8—31no—
69C2×C3×C3⋊C8—61no—
70#3724—31no—
71#3724—31no—
72Dic3⋊Dic3—31no—
73Dic3⋊Dic3—31no—
74Dic3⋊Dic3—31no—
75Dic3⋊Dic3—31no—
76C22×H(3,𝔽3)C22×C411yescharacteristic
77C4×H(3,𝔽3)C2×Q811yescharacteristic
78C4×H(3,𝔽3)#4111yescharacteristic
79#1084—21no—
80#1084—21no—
81(C3⋊Dic3)⋊C3—21no—
82(C3⋊Dic3)⋊C3—21no—
83(C3⋊Dic3)⋊C3—21no—
84(C3⋊Dic3)⋊C3M1612yes—
85C4×Dic6—91no—
86C4×Dic6—91no—
87C4×Dic6—91no—
88C3×C4⋊C8—91no—
89C3×C4⋊C8—91no—
90C3×#97—91no—
91#815—31no—
92#815—91no—
93#815—91no—
94#815—91no—
95Dic3⋊C8—31no—
96Dic3⋊C8—31no—
97Dic3⋊C8—31no—
98Dic3⋊C8—61no—
99Dic3⋊C8—91no—
100Dic3⋊C8—91no—
101Dic3⋊C8—91no—
102C8×Dic3—91no—
103C8×Dic3—91no—
104C4⋊(C3⋊C8)—91no—
105#803—91no—
106C4×C3⋊C8—91no—
107C2×C32×C4—21no—
108C2×C32×C4C22×D311yescharacteristic
109C2×C32×C4C22×D311yescharacteristic
110C2×C3⋊Dic3C4×D311yescharacteristic
111C2×C3⋊Dic3C4×D311yescharacteristic
112C2×C3⋊Dic3C4×D311yescharacteristic
113C2×C3⋊Dic3C4×D311yescharacteristic
114C2×C3×Dic3—31no—
115C2×C3×Dic3—31no—
116C2×C3×Dic3—31no—
117C2×C3×Dic3—31no—
118C2×C3×Dic3—31no—
119C2×C3×Dic3—31no—
120C2×C3×Dic3—31no—
121C2×C3×Dic3—31no—
122C2×C3×Dic3—61no—
123C2×C3×Dic3—61no—
124#627—62no—
125#627—62no—
126C32×C8—21no—
127C32×C8—21no—
128C32×C8—41no—
129C3⋊(C3⋊C8)—21no—
130C3⋊(C3⋊C8)Dic612yescomplemented
131C3×C3⋊C8—32no—
132C3×C3⋊C8—61no—
133C3×C3⋊C8—61no—
134C3×C3⋊C8—61no—
135C3×C3⋊C8—121no—
136#446—271noSylow
137C2×H(3,𝔽3)C4×Q811yescharacteristic
138C2×H(3,𝔽3)C2×M1611yescharacteristic
139C2×H(3,𝔽3)#13111yescharacteristic
140C2×Dic6—91no—
141C2×Dic6—91no—
142C2×Dic6—91no—
143C2×C3×C8—31no—
144C2×C3×C8—31no—
145C2×C3×C8—61no—
146C2×C3×C8—91no—
147C2×C3×C8—91no—
148C2×C3×C8—91no—
149C2×C3×C8D3211yescharacteristic
150C3×C42—91no—
151C3×C42—91no—
152C3×C42—91no—
153C4⋊Dic3—91no—
154C4⋊Dic3—91no—
155C4⋊Dic3—91no—
156Dic3⋊C4—91no—
157Dic3⋊C4—91no—
158Dic3⋊C4—91no—
159Dic3⋊C4—91no—
160Dic3⋊C4—91no—
161Dic3⋊C4—91no—
162C4×Dic3—31no—
163C4×Dic3—31no—
164C4×Dic3—31no—
165C4×Dic3—31no—
166C4×Dic3—61no—
167C4×Dic3—91no—
168C4×Dic3—91no—
169C2×C3⋊C8—31no—
170C2×C3⋊C8—31no—
171C2×C3⋊C8—31no—
172C2×C3⋊C8—31no—
173C2×C3⋊C8—61no—
174C2×C3⋊C8—91no—
175C2×C3⋊C8—91no—
176C22×C32—21no—
177C22×C32C2×C4×D311yescharacteristic
178C22×C32C2×C4×D311yescharacteristic
179C32×C4—21no—
180C32×C4—21no—
181C32×C4C2×Dic611yescharacteristic
182C32×C4#23511yescharacteristic
183C32×C4#23711yescharacteristic
184C32×C4D3×Q811yescharacteristic
185C3⋊Dic3—21no—
186C3⋊Dic3—21no—
187C3⋊Dic3—21no—
188C3⋊Dic3ΓL(1,𝔽25)12yes—
189C3×Dic3—32no—
190C3×Dic3—62no—
191C3×Dic3—61no—
192C3×Dic3—61no—
193C3×Dic3—61no—
194C3×Dic3—61no—
195C3×Dic3—61no—
196C3×Dic3—61no—
197C3×Dic3—61no—
198C3×Dic3—121no—
199C4×Q8—271no—
200C4⋊C8—91no—
201C4⋊C8—91no—
202C4⋊C8—271no—
203#97—91no—
204#97—271no—
205C4×C8—271no—
206H(3,𝔽3)#44611yescharacteristic, complemented, Sylow
207C2×C3×C4—31no—
208C2×C3×C4—31no—
209C2×C3×C4—61no—
210C2×C3×C4—91no—
211C2×C3×C4—91no—
212C2×C3×C4—91no—
213C2×C3×C4—91no—
214C2×C3×C4—91no—
215C2×C3×C4—91no—
216C2×C3×C4C2×D3211yescharacteristic
217C2×Dic3—31no—
218C2×Dic3—31no—
219C2×Dic3—31no—
220C2×Dic3—31no—
221C2×Dic3—31no—
222C2×Dic3—31no—
223C2×Dic3—31no—
224C2×Dic3—31no—
225C2×Dic3—61no—
226C2×Dic3—61no—
227C2×Dic3—91no—
228C2×Dic3—91no—
229C2×Dic3—91no—
230C2×Dic3—91no—
231Dic6—182no—
232Dic6—182no—
233Dic6—182no—
234C3×C8—21no—
235C3×C8—61no—
236C3×C8—61no—
237C3×C8—92no—
238C3×C8—121no—
239C3×C8—181no—
240C3×C8—181no—
241C3⋊C8—32no—
242C3⋊C8—61no—
243C3⋊C8—61no—
244C3⋊C8—61no—
245C3⋊C8—92no—
246C3⋊C8—121no—
247C3⋊C8—181no—
248C2×C32—21no—
249C2×C32—21no—
250C2×C32—21no—
251C2×C32C4×Dic611yescharacteristic
252C2×C32#88711yescharacteristic
253C2×C32C2×ΓL(1,𝔽25)11yescharacteristic
254C2×C32D3×M1611yescharacteristic
255C2×C32#90711yescharacteristic
256C2×C32#90711yescharacteristic
257C2×Q8—271no—
258C2×C8—91no—
259C2×C8—91no—
260C2×C8—91no—
261C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
262C4⋊C4—271no—
263C4⋊C4—271no—
264C4⋊C4—271no—
265C42—91no—
266C42—91no—
267C42—91no—
268C22×C3—31no—
269C22×C3—31no—
270C22×C3—61no—
271C22×C3C4×D3211yescharacteristic
272C3×C4—31no—
273C3×C4—31no—
274C3×C4—31no—
275C3×C4—31no—
276C3×C4—61no—
277C3×C4—61no—
278C3×C4—92no—
279C3×C4—181no—
280C3×C4—181no—
281C3×C4—181no—
282C3×C4—181no—
283C3×C4—181no—
284C3×C4D3×Dic611yescharacteristic
285C3×C4#380411yescharacteristic
286Dic3—32no—
287Dic3—61no—
288Dic3—61no—
289Dic3—61no—
290Dic3—61no—
291Dic3—62no—
292Dic3—61no—
293Dic3—61no—
294Dic3—61no—
295Dic3—92no—
296Dic3—121no—
297Dic3—181no—
298Dic3—181no—
299Dic3—181no—
300C32—21no—
301C32#470111yescharacteristic, complemented
302C32#483111yescharacteristic, complemented
303Q8—542no—
304C2×C4—91no—
305C2×C4—91no—
306C2×C4—91no—
307C2×C4—91no—
308C2×C4—91no—
309C2×C4—91no—
310C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
311C8—21no—
312C8—92no—
313C8—181no—
314C8—181no—
315C2×C3—31no—
316C2×C3—31no—
317C2×C3—31no—
318C2×C3—31no—
319C2×C3—31no—
320C2×C3—31no—
321C2×C3—61no—
322C2×C3—61no—
323C2×C3—61no—
324C2×C3D3×ΓL(1,𝔽25)11yescharacteristic
325C2×C3#6380511yescharacteristic
326C2×C3#6383711yescharacteristic
327C22C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
328C4—92no—
329C4—181no—
330C4—181no—
331C4—181no—
332C4—181no—
333C4—181no—
334C4((C3⋊D3)⋊C3)⋊Q811yescharacteristic
335C4#8867111yescharacteristic
336C3—31no—
337C3—31no—
338C3—61no—
339C3#1059078611yescharacteristic
340C2#1172083511yescharacteristic
341C2#1172084211yescharacteristic
342C2#1172107711yescharacteristic
subgroupG/H
centerC2×C4C2×((C3⋊D3)⋊C3)⋊C2
commutator, prime radicalC22×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3C4×D32
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#446
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
47. (C3⋊(C3⋊C8))⋊C3Q8semidirect
130. C3⋊(C3⋊C8)Dic6semidirect
206. H(3,𝔽3)#446semidirect
301. C32#4701semidirect
302. C32#4831semidirect

Maximal subgroups

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

subgroupindex
C4×#61962
#117205472
#63642⋊C32
#63641⋊C32
#117203112
#117203102
#117203032
#105907893
#105906463