Finite Group Databasebeta

← #13138781

Subgroups — #13138781

Order 1728.

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

normal subgroups only

This group has 374 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#11721105C211yesmaximal, characteristic
2#11720547C211yesmaximal, characteristic, complemented
3#63647⋊C3C211yesmaximal, characteristic
4#63642⋊C3C211yesmaximal, characteristic, complemented
5#11720317C211yesmaximal, characteristic
6#11720310C211yesmaximal, characteristic, complemented
7#11720304C211yesmaximal, characteristic, complemented
8#10590699—31nomaximal
9#10590647—31nomaximal
10C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
11C2×C8×H(3,𝔽3)C2211yescharacteristic
12C4×#1084C2211yescharacteristic
13C2×#6180C2211yescharacteristic
14C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
15C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
16C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
17#88450C411yescharacteristic
18#88450C411yescharacteristic
19#88448C411yescharacteristic
20#88448C411yescharacteristic
21#63874—31no—
22#63855—31no—
23#63647D311yescharacteristic
24#63642D311yescharacteristic, complemented
25C3×(C2×D3)⋊C8—31no—
26C3×#815—31no—
27C3×Dic3⋊C8—31no—
28C3×Dic3⋊C8—31no—
29C3×Dic3⋊C8—61no—
30Dic3⋊(C3⋊C8)—31no—
31Dic3⋊(C3⋊C8)—31no—
32#63557—31no—
33(C2×D3)⋊(C3⋊C8)—31no—
34#63553—31no—
35Dic3×C3⋊C8—31no—
36C22×(C3⋊D3)⋊C3C2×C411yescharacteristic
37C2×C4×H(3,𝔽3)C2311yescharacteristic
38C2×#1084C2×C411yescharacteristic
39C2×#1084C2×C411yescharacteristic
40C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
41C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
42C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
43C4×(C3⋊D3)⋊C3—21no—
44C4×(C3⋊D3)⋊C3—21no—
45C8×H(3,𝔽3)—21no—
46#6180—21no—
47(C3⋊(C3⋊C8))⋊C3—21no—
48(C3⋊(C3⋊C8))⋊C3—21no—
49#4840—91no—
50#4720—91no—
51#4709—91no—
52C2×C4×C3⋊D3C2×D311yescharacteristic
53C2×C3×C4×D3—31no—
54C2×C32×C8—21no—
55C2×C32×C8C2×D311yescharacteristic
56C2×C32×C8C2×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—61no—
64C2×C3×C3⋊C8—31no—
65C2×C3×C3⋊C8—31no—
66C2×C3×C3⋊C8—31no—
67C2×C3×C3⋊C8—31no—
68C2×C3×C3⋊C8—61no—
69#3724—31no—
70#3724—31no—
71Dic3⋊Dic3—31no—
72Dic3⋊Dic3—31no—
73#3722—31no—
74#3722—31no—
75(C2×D3)⋊Dic3—31no—
76(C2×D3)⋊Dic3—31no—
77C22×H(3,𝔽3)C22×C411yescharacteristic
78C2×(C3⋊D3)⋊C3—21no—
79C2×(C3⋊D3)⋊C3—21no—
80C2×(C3⋊D3)⋊C3—21no—
81C4×H(3,𝔽3)#4111yescharacteristic
82C4×H(3,𝔽3)#4111yescharacteristic
83#1084—21no—
84#1084—21no—
85(C3⋊Dic3)⋊C3—21no—
86(C3⋊Dic3)⋊C3—21no—
87(C3⋊Dic3)⋊C3—21no—
88#926—91no—
89#872—91no—
90#872—91no—
91C3×C4⋊C8—91no—
92C3×C22⋊C8—91no—
93C3×#97—91no—
94#830—91no—
95(C2×D3)⋊C8—31no—
96(C2×D3)⋊C8—91no—
97(C2×D3)⋊C8—91no—
98(C2×D3)⋊C8—91no—
99#815—31no—
100#815—91no—
101Dic3⋊C8—31no—
102Dic3⋊C8—31no—
103Dic3⋊C8—61no—
104Dic3⋊C8—91no—
105Dic3⋊C8—91no—
106C8×Dic3—91no—
107C8×Dic3—91no—
108C4⋊(C3⋊C8)—91no—
109C4×C3⋊C8—91no—
110C22×C3⋊D3C4×D311yescharacteristic
111C22×C3×D3—31no—
112C2×C32×C4—21no—
113C2×C32×C4C22×D311yescharacteristic
114C2×C32×C4C22×D311yescharacteristic
115C2×C3⋊Dic3C4×D311yescharacteristic
116C2×C3⋊Dic3C4×D311yescharacteristic
117C2×C3⋊Dic3C4×D311yescharacteristic
118C4×C3⋊D3—21no—
119C4×C3⋊D3—21no—
120C2×C3×Dic3—31no—
121C2×C3×Dic3—31no—
122C2×C3×Dic3—31no—
123C2×C3×Dic3—31no—
124C2×C3×Dic3—31no—
125C2×C3×Dic3—31no—
126C2×C3×Dic3—31no—
127C2×C3×Dic3—61no—
128C2×C3×Dic3—61no—
129C3×C4×D3—61no—
130C3×C4×D3—61no—
131C32×C8—21no—
132C32×C8—21no—
133C32×C8—41no—
134C3⋊(C3⋊C8)—21no—
135C3⋊(C3⋊C8)—21no—
136C3×C3⋊C8—61no—
137C3×C3⋊C8—61no—
138C3×C3⋊C8—61no—
139C3×C3⋊C8—61no—
140C3×C3⋊C8—121no—
141#433—271noSylow
142C2×H(3,𝔽3)#11711yescharacteristic
143C2×H(3,𝔽3)#13111yescharacteristic
144C2×H(3,𝔽3)#13111yescharacteristic
145(C3⋊D3)⋊C3—41no—
146C22×C3×C4—91no—
147C2×C4×D3—31no—
148C2×C4×D3—91no—
149C2×C3×C8—31no—
150C2×C3×C8—31no—
151C2×C3×C8—61no—
152C2×C3×C8—91no—
153C2×C3×C8—91no—
154C2×C3×C8—91no—
155C2×C3×C8D3211yescharacteristic
156C3×C42—91no—
157C3×C42—91no—
158#217—91no—
159#217—91no—
160(C2×D3)⋊C4—91no—
161(C2×D3)⋊C4—91no—
162(C2×D3)⋊C4—91no—
163(C2×D3)⋊C4—91no—
164C4⋊Dic3—91no—
165C4⋊Dic3—91no—
166Dic3⋊C4—91no—
167Dic3⋊C4—91no—
168Dic3⋊C4—91no—
169Dic3⋊C4—91no—
170C4×Dic3—31no—
171C4×Dic3—31no—
172C4×Dic3—31no—
173C4×Dic3—61no—
174C4×Dic3—91no—
175C2×C3⋊C8—31no—
176C2×C3⋊C8—31no—
177C2×C3⋊C8—31no—
178C2×C3⋊C8—31no—
179C2×C3⋊C8—61no—
180C2×C3⋊C8—91no—
181C2×C3⋊C8—91no—
182C22×C32—21no—
183C22×C32C2×C4×D311yescharacteristic
184C22×C32C2×C4×D311yescharacteristic
185C2×C3⋊D3—21no—
186C2×C3⋊D3—21no—
187C2×C3⋊D3—21no—
188C2×C3×D3—61no—
189C2×C3×D3—61no—
190C2×C3×D3—61no—
191C32×C4—21no—
192C32×C4—21no—
193C32×C4#23511yescharacteristic
194C32×C4#23511yescharacteristic
195C32×C4#23711yescharacteristic
196C32×C4#23711yescharacteristic
197C3⋊Dic3—21no—
198C3⋊Dic3—21no—
199C3⋊Dic3—21no—
200C3×Dic3—62no—
201C3×Dic3—61no—
202C3×Dic3—61no—
203C3×Dic3—61no—
204C3×Dic3—61no—
205C3×Dic3—61no—
206C3×Dic3—61no—
207C3×Dic3—61no—
208C3×Dic3—121no—
209#117—271no—
210C4⋊C8—91no—
211C4⋊C8—271no—
212C22⋊C8—91no—
213C22⋊C8—271no—
214#97—91no—
215C4×C8—271no—
216H(3,𝔽3)#43311yescharacteristic, complemented, Sylow
217C23×C3—91no—
218C22×D3—31no—
219C22×D3—91no—
220C2×C3×C4—31no—
221C2×C3×C4—31no—
222C2×C3×C4—61no—
223C2×C3×C4—91no—
224C2×C3×C4—91no—
225C2×C3×C4—91no—
226C2×C3×C4—91no—
227C2×C3×C4—91no—
228C2×C3×C4—181no—
229C2×C3×C4—181no—
230C2×C3×C4C2×D3211yescharacteristic
231C2×Dic3—31no—
232C2×Dic3—31no—
233C2×Dic3—31no—
234C2×Dic3—31no—
235C2×Dic3—31no—
236C2×Dic3—31no—
237C2×Dic3—31no—
238C2×Dic3—61no—
239C2×Dic3—61no—
240C2×Dic3—91no—
241C2×Dic3—91no—
242C2×Dic3—91no—
243C4×D3—61no—
244C4×D3—61no—
245C4×D3—181no—
246C4×D3—181no—
247C3×C8—21no—
248C3×C8—61no—
249C3×C8—61no—
250C3×C8—121no—
251C3×C8—181no—
252C3×C8—181no—
253C3×C8—181no—
254C3⋊C8—61no—
255C3⋊C8—61no—
256C3⋊C8—61no—
257C3⋊C8—61no—
258C3⋊C8—121no—
259C3⋊C8—181no—
260C3⋊C8—181no—
261C2×C32—21no—
262C2×C32—21no—
263C2×C32—21no—
264C2×C32#87211yescharacteristic
265C2×C32#87711yescharacteristic
266C2×C32#90111yescharacteristic
267C2×C32#90711yescharacteristic
268C2×C32#90711yescharacteristic
269C2×C32#90711yescharacteristic
270C3⋊D3—41no—
271C3×D3—121no—
272C22×C4—91no—
273C2×C8—91no—
274C2×C8—91no—
275C2×C8—91no—
276C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
277C4⋊C4—271no—
278C4⋊C4—271no—
279C22⋊C4—271no—
280C22⋊C4—271no—
281C42—91no—
282C42—91no—
283C22×C3—31no—
284C22×C3—31no—
285C22×C3—61no—
286C22×C3—181no—
287C22×C3—181no—
288C22×C3—181no—
289C22×C3C4×D3211yescharacteristic
290C2×D3—61no—
291C2×D3—61no—
292C2×D3—61no—
293C2×D3—181no—
294C2×D3—181no—
295C2×D3—181no—
296C3×C4—31no—
297C3×C4—31no—
298C3×C4—31no—
299C3×C4—31no—
300C3×C4—61no—
301C3×C4—61no—
302C3×C4—181no—
303C3×C4—181no—
304C3×C4—181no—
305C3×C4—181no—
306C3×C4—181no—
307C3×C4#379511yescharacteristic
308C3×C4#380411yescharacteristic
309Dic3—61no—
310Dic3—61no—
311Dic3—61no—
312Dic3—61no—
313Dic3—62no—
314Dic3—61no—
315Dic3—61no—
316Dic3—61no—
317Dic3—121no—
318Dic3—181no—
319Dic3—181no—
320Dic3—181no—
321C32—21no—
322C32#470911yescharacteristic, complemented
323C32#472011yescharacteristic, complemented
324C23—91no—
325C2×C4—91no—
326C2×C4—91no—
327C2×C4—91no—
328C2×C4—91no—
329C2×C4—91no—
330C2×C4—181no—
331C2×C4—181no—
332C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
333C8—21no—
334C8—181no—
335C8—181no—
336C8—181no—
337C2×C3—31no—
338C2×C3—31no—
339C2×C3—31no—
340C2×C3—31no—
341C2×C3—31no—
342C2×C3—31no—
343C2×C3—61no—
344C2×C3—61no—
345C2×C3—61no—
346C2×C3—361no—
347C2×C3#6380411yescharacteristic
348C2×C3#6380511yescharacteristic
349C2×C3#6387711yescharacteristic
350D3—121no—
351D3—361no—
352C22—181no—
353C22—181no—
354C22—181no—
355C22C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
356C4—181no—
357C4—181no—
358C4—181no—
359C4—181no—
360C4—181no—
361C4#8864811yescharacteristic
362C4#8867111yescharacteristic
363C3—31no—
364C3—31no—
365C3—61no—
366C3#1059069711yescharacteristic
367C2—361no—
368C2#1172084111yescharacteristic
369C2#1172084211yescharacteristic
370C2#1172112011yescharacteristic
subgroupG/H
centerC2×C4C2×((C3⋊D3)⋊C3)⋊C2
commutatorC22×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3C4×D32
prime radicalC22×(C3⋊D3)⋊C3C2×C4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#433
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
2. #11720547C2semidirect
4. #63642⋊C3C2semidirect
6. #11720310C2semidirect
7. #11720304C2semidirect
24. #63642D3semidirect
216. H(3,𝔽3)#433semidirect
322. C32#4709semidirect
323. C32#4720semidirect

Maximal subgroups

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

subgroupindex
#117211052
#117205472
#63647⋊C32
#63642⋊C32
#117203172
#117203102
#117203042
#105906993
#105906473