Finite Group Databasebeta

← #13137329

Subgroups — #13137329

Order 1728.

371 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 371 conjugacy classes of subgroups — too many to draw. Only its 79 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#11721716C211yesmaximal, characteristic
2C4×#634⋊C3C211yesmaximal, characteristic
3C4⋊((C3⋊(C3⋊C8))⋊C3)C211yesmaximal, characteristic
4#11720309C211yesmaximal, characteristic
5#11720309C211yesmaximal, characteristic
6#11720307C211yesmaximal, characteristic
7#11720306C211yesmaximal, characteristic
8#10590292—31nomaximal
9#10590286—31nomaximal
10C2×#634⋊C3C2211yescharacteristic
11C42×H(3,𝔽3)C2211yescharacteristic
12C4⋊#1084C2211yescharacteristic
13C4⋊#1084C2211yescharacteristic
14#88540—21no—
15#88540—21no—
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17#3750⋊C3—21no—
18C4×(C3⋊Dic3)⋊C3—21no—
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21C4×#634D311yescharacteristic
22C3×#870—31no—
23C3×#870—31no—
24C3×#870—61no—
25C3×C4×Dic6—31no—
26C4⋊(C3⋊(C3⋊C8))D311yescharacteristic
27C3×C4⋊(C3⋊C8)—31no—
28#63576—31no—
29#63575—31no—
30#63574—31no—
31#63573—31no—
32#63567—31no—
33#63567—31no—
34#63565—31no—
35#63565—31no—
36C2×C4×H(3,𝔽3)D411yescharacteristic
37C2×C4×H(3,𝔽3)D411yescharacteristic
38C2×C4×H(3,𝔽3)C2311yescharacteristic
39C2×#1084—21no—
40C2×#1084—21no—
41C2×(C3⋊Dic3)⋊C3—21no—
42C2×(C3⋊Dic3)⋊C3—21no—
43#634⋊C3—21no—
44#634⋊C3Q812yes—
45(C3⋊(C3⋊C8))⋊C3—21no—
46(C3⋊(C3⋊C8))⋊C3—21no—
47#5063—91no—
48#5022—91no—
49#4829—91no—
50C2×#634C2×D311yescharacteristic
51C2×C3×Dic6—31no—
52C32×C42—21no—
53C32×C42C2×D311yescharacteristic
54C32×C42C2×D311yescharacteristic
55C4⋊(C3⋊Dic3)C2×D311yescharacteristic
56#3750—21no—
57C4×C3⋊Dic3—21no—
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—31no—
65C3×C4⋊Dic3—61no—
66C3×C4⋊Dic3—61no—
67C3×Dic3⋊C4—62no—
68C3×Dic3⋊C4—62no—
69C3×Dic3⋊C4—61no—
70C3×Dic3⋊C4—61no—
71C3×Dic3⋊C4—61no—
72C3×Dic3⋊C4—61no—
73C3×Dic3⋊C4—61no—
74C3×C4×Dic3—61no—
75C2×C3×C3⋊C8—31no—
76C2×C3×C3⋊C8—31no—
77C22×H(3,𝔽3)C2×D411yescharacteristic
78C4×H(3,𝔽3)—21no—
79C4×H(3,𝔽3)—21no—
80C4×H(3,𝔽3)C2×Q811yescharacteristic
81C4×H(3,𝔽3)#4111yescharacteristic
82#1084—41no—
83#1084—41no—
84(C3⋊Dic3)⋊C3—22no—
85(C3⋊Dic3)⋊C3—41no—
86C3×#125—91no—
87C3×C4×Q8—91no—
88#870—31no—
89#870—31no—
90#870—61no—
91C4×Dic6—31no—
92C4×Dic6—91no—
93C3×C4⋊C8—91no—
94#835—91no—
95#835—91no—
96#818—91no—
97#817—91no—
98#816—91no—
99#816—91no—
100#810—91no—
101#810—91no—
102#808—91no—
103#808—91no—
104#807—91no—
105#807—91no—
106C4⋊(C3⋊C8)—31no—
107C4⋊(C3⋊C8)—91no—
108C2×C32×C4—21no—
109C2×C32×C4—21no—
110C2×C32×C4—21no—
111C2×C32×C4D1211yescharacteristic
112C2×C32×C4D1211yescharacteristic
113C2×C32×C4#6711yescharacteristic
114C2×C32×C4#6711yescharacteristic
115C2×C32×C4C22×D311yescharacteristic
116C2×C32×C4C22×D311yescharacteristic
117C2×C3⋊Dic3—21no—
118C2×C3⋊Dic3—21no—
119#634—21no—
120#634Dic612yes—
121C2×C3×Dic3—61no—
122C2×C3×Dic3—62no—
123C2×C3×Dic3—62no—
124C2×C3×Dic3—61no—
125C2×C3×Dic3—61no—
126C2×C3×Dic3—61no—
127C2×C3×Dic3—61no—
128C2×C3×Dic3—61no—
129C3×Dic6—32no—
130C3×Dic6—61no—
131C3⋊(C3⋊C8)—21no—
132C3⋊(C3⋊C8)—21no—
133C3×C3⋊C8—61no—
134C3×C3⋊C8—61no—
135#479—271noSylow
136C2×H(3,𝔽3)C22⋊Q811yescharacteristic
137C2×H(3,𝔽3)#13511yescharacteristic
138C2×H(3,𝔽3)#13711yescharacteristic
139C2×C3×Q8—91no—
140C2×Dic6—31no—
141C2×Dic6—91no—
142C2×C3×C8—91no—
143C2×C3×C8—91no—
144C3×C4⋊C4—91no—
145C3×C4⋊C4—91no—
146C3×C4⋊C4—91no—
147C3×C4⋊C4—181no—
148C3×C4⋊C4—181no—
149C3×C4⋊C4—181no—
150C3×C42—31no—
151C3×C42—31no—
152C3×C42—61no—
153C3×C42—181no—
154C3×C42D3211yescharacteristic
155C4⋊Dic3—31no—
156C4⋊Dic3—31no—
157C4⋊Dic3—31no—
158C4⋊Dic3—31no—
159C4⋊Dic3—31no—
160C4⋊Dic3—61no—
161C4⋊Dic3—61no—
162C4⋊Dic3—91no—
163Dic3⋊C4—61no—
164Dic3⋊C4—61no—
165Dic3⋊C4—61no—
166Dic3⋊C4—61no—
167Dic3⋊C4—62no—
168Dic3⋊C4—62no—
169Dic3⋊C4—61no—
170Dic3⋊C4—181no—
171C4×Dic3—61no—
172C4×Dic3—181no—
173C2×C3⋊C8—31no—
174C2×C3⋊C8—31no—
175C2×C3⋊C8—91no—
176C2×C3⋊C8—91no—
177C22×C32—21no—
178C22×C32C2×D1211yescharacteristic
179C22×C32C2×#6711yescharacteristic
180C32×C4—21no—
181C32×C4—21no—
182C32×C4—21no—
183C32×C4—21no—
184C32×C4—21no—
185C32×C4—21no—
186C32×C4—41no—
187C32×C4—41no—
188C32×C4C2×Dic611yescharacteristic
189C32×C4#23511yescharacteristic
190C32×C4#23711yescharacteristic
191C32×C4D3×Q811yescharacteristic
192C3⋊Dic3—22no—
193C3⋊Dic3—41no—
194C3×Dic3—62no—
195C3×Dic3—122no—
196C3×Dic3—122no—
197C3×Dic3—121no—
198C3×Dic3—121no—
199C3×Dic3—121no—
200C3×Dic3—121no—
201C3×Dic3—121no—
202#125—91no—
203C4×Q8—91no—
204#107—271no—
205#106—271no—
206C4⋊C8—91no—
207#103—271no—
208#103—271no—
209H(3,𝔽3)#47911yescharacteristic, complemented, Sylow
210C3×Q8—92no—
211C3×Q8—181no—
212C2×C3×C4—31no—
213C2×C3×C4—31no—
214C2×C3×C4—31no—
215C2×C3×C4—31no—
216C2×C3×C4—31no—
217C2×C3×C4—31no—
218C2×C3×C4—61no—
219C2×C3×C4—61no—
220C2×C3×C4—61no—
221C2×C3×C4—181no—
222C2×C3×C4—181no—
223C2×C3×C4—181no—
224C2×C3×C4—181no—
225C2×C3×C4#62611yescharacteristic
226C2×C3×C4#62611yescharacteristic
227C2×C3×C4C2×D3211yescharacteristic
228C2×Dic3—61no—
229C2×Dic3—61no—
230C2×Dic3—61no—
231C2×Dic3—61no—
232C2×Dic3—61no—
233C2×Dic3—62no—
234C2×Dic3—62no—
235C2×Dic3—61no—
236C2×Dic3—181no—
237C2×Dic3—181no—
238Dic6—32no—
239Dic6—61no—
240Dic6—92no—
241Dic6—181no—
242C3×C8—181no—
243C3×C8—181no—
244C3⋊C8—61no—
245C3⋊C8—61no—
246C3⋊C8—181no—
247C3⋊C8—181no—
248C2×C32—21no—
249C2×C32—21no—
250C2×C32—21no—
251C2×C32#89711yescharacteristic
252C2×C32#90411yescharacteristic
253C2×C32#90911yescharacteristic
254C2×C32#92411yescharacteristic
255C2×C32#94211yescharacteristic
256C2×C32#95011yescharacteristic
257C2×Q8—91no—
258C2×C8—91no—
259C2×C8—91no—
260C4⋊C4—91no—
261C4⋊C4—91no—
262C4⋊C4—91no—
263C4⋊C4—181no—
264C4⋊C4—181no—
265C4⋊C4—181no—
266C42—181no—
267C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
268C22×C3—31no—
269C22×C3—31no—
270C22×C3—61no—
271C22×C3C2×#62611yescharacteristic
272C3×C4—21no—
273C3×C4—21no—
274C3×C4—31no—
275C3×C4—31no—
276C3×C4—31no—
277C3×C4—31no—
278C3×C4—61no—
279C3×C4—61no—
280C3×C4—61no—
281C3×C4—61no—
282C3×C4—61no—
283C3×C4—61no—
284C3×C4—121no—
285C3×C4—121no—
286C3×C4—182no—
287C3×C4—361no—
288C3×C4—361no—
289C3×C4—361no—
290C3×C4D3×Dic611yescharacteristic
291C3×C4#379511yescharacteristic
292Dic3—62no—
293Dic3—121no—
294Dic3—121no—
295Dic3—121no—
296Dic3—121no—
297Dic3—122no—
298Dic3—122no—
299Dic3—121no—
300Dic3—182no—
301Dic3—361no—
302C32—21no—
303C32#482911yescharacteristic, complemented
304C32#502211yescharacteristic, complemented
305Q8—92no—
306Q8—181no—
307C2×C4—181no—
308C2×C4—181no—
309C2×C4—181no—
310C2×C4—181no—
311C2×C4#620011yescharacteristic
312C2×C4#620011yescharacteristic
313C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
314C8—181no—
315C8—181no—
316C2×C3—31no—
317C2×C3—31no—
318C2×C3—31no—
319C2×C3—31no—
320C2×C3—31no—
321C2×C3—31no—
322C2×C3—61no—
323C2×C3—61no—
324C2×C3—61no—
325C2×C3#6382911yescharacteristic
326C2×C3#6383311yescharacteristic
327C2×C3#6385711yescharacteristic
328C22C2×#620011yescharacteristic
329C4—21no—
330C4—21no—
331C4—182no—
332C4—361no—
333C4—361no—
334C4—361no—
335C4((C3⋊D3)⋊C3)⋊Q811yescharacteristic
336C4#8864811yescharacteristic
337C3—31no—
338C3—31no—
339C3—61no—
340C3#1059030211yescharacteristic
341C2#1172106911yescharacteristic
342C2#1172107311yescharacteristic
343C2#1172110711yescharacteristic
subgroupG/H
centerC22C2×#6200
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC42×H(3,𝔽3)C22
socleC22×C3C2×#626
prime radicalC22×H(3,𝔽3)C2×D4
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#479
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
209. H(3,𝔽3)#479semidirect
303. C32#4829semidirect
304. C32#5022semidirect

Maximal subgroups

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

subgroupindex
#117217162
C4×#634⋊C32
C4⋊((C3⋊(C3⋊C8))⋊C3)2
#117203092
#117203072
#117203062
#105902923
#105902863