Finite Group Databasebeta

← #13137864

Subgroups — #13137864

Order 1728.

404 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 73 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#11721084C211yesmaximal, characteristic
3#63637⋊C3C211yesmaximal, characteristic
4#63635⋊C3C211yesmaximal, characteristic
5#11720309C211yesmaximal, characteristic
6#11720307C211yesmaximal, characteristic
7#11720304C211yesmaximal, characteristic
8#10590487—31nomaximal
9#10590486—31nomaximal
10C2×#6229—21no—
11C2×#634⋊C3C2211yescharacteristic
12C4⋊C4×H(3,𝔽3)C2211yescharacteristic
13C4⋊#1084C2211yescharacteristic
14#88540—21no—
15C4×#1084C2211yescharacteristic
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
18C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
19#88448—21no—
20#88447—21no—
21C3×#888—31no—
22C3×#888—31no—
23C3×#888—61no—
24#63844—31no—
25Dic3×Dic6—31no—
26#63637D311yescharacteristic
27#63635D311yescharacteristic
28C3×#810—31no—
29C3×#808—31no—
30#63574—31no—
31#63573—31no—
32Dic3⋊(C3⋊C8)—31no—
33Dic3⋊(C3⋊C8)—31no—
34#63567—31no—
35#63565—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#634⋊C3—21no—
45#634⋊C3Q812yescomplemented
46(C3⋊(C3⋊C8))⋊C3—21no—
47(C3⋊(C3⋊C8))⋊C3—21no—
48#5094—91no—
49#4854—91no—
50#4791—91no—
51C2×#634C2×D311yescharacteristic
52C2×C3×Dic6—31no—
53C2×C3×Dic6—61no—
54C2×C3×Dic6—61no—
55C2×C3×Dic6—62no—
56C32×C4⋊C4—21no—
57C32×C4⋊C4C2×D311yescharacteristic
58C32×C4⋊C4C2×D311yescharacteristic
59C4⋊(C3⋊Dic3)C2×D311yescharacteristic
60C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
61C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
62C3×C4⋊Dic3—31no—
63C3×C4⋊Dic3—31no—
64C3×C4⋊Dic3—31no—
65C3×C4⋊Dic3—61no—
66C3×Dic3⋊C4—61no—
67C3×Dic3⋊C4—61no—
68C3×Dic3⋊C4—62no—
69C3×C4×Dic3—31no—
70C3×C4×Dic3—31no—
71C3×C4×Dic3—61no—
72C2×C3×C3⋊C8—31no—
73C2×C3×C3⋊C8—31no—
74#3724—61no—
75Dic3⋊Dic3—61no—
76Dic32—61no—
77Dic32—61no—
78C22×H(3,𝔽3)C2×D411yescharacteristic
79C4×H(3,𝔽3)—41no—
80C4×H(3,𝔽3)C2×Q811yescharacteristic
81C4×H(3,𝔽3)#4111yescharacteristic
82#1084—21no—
83#1084—41no—
84#1084SD1612yes—
85(C3⋊Dic3)⋊C3—22no—
86(C3⋊Dic3)⋊C3—41no—
87C3×C4⋊Q8—91no—
88Q8×Dic3—91no—
89#888—31no—
90#888—31no—
91#888—61no—
92#887—91no—
93C4×Dic6—91no—
94C3×#106—91no—
95C3×#103—91no—
96#835—91no—
97#817—91no—
98#816—91no—
99Dic3⋊C8—91no—
100Dic3⋊C8—91no—
101#810—31no—
102#810—91no—
103#810—91no—
104#808—31no—
105#808—91no—
106#808—91no—
107#808—91no—
108C4⋊(C3⋊C8)—91no—
109C2×C32×C4—22no—
110C2×C32×C4—21no—
111C2×C32×C4—21no—
112C2×C32×C4—21no—
113C2×C32×C4C22×D311yescharacteristic
114C2×C32×C4C22×D311yescharacteristic
115C2×C3⋊Dic3—21no—
116C2×C3⋊Dic3—21no—
117#634—21no—
118#634Dic612yescomplemented
119C2×C3×Dic3—31no—
120C2×C3×Dic3—31no—
121C2×C3×Dic3—31no—
122C2×C3×Dic3—31no—
123C2×C3×Dic3—61no—
124C2×C3×Dic3—61no—
125C2×C3×Dic3—61no—
126C2×C3×Dic3—62no—
127C2×C3×Dic3—61no—
128C2×C3×Dic3—61no—
129C2×C3×Dic3—61no—
130C3×Dic6—32no—
131C3×Dic6—61no—
132C3×Dic6—122no—
133C3×Dic6—122no—
134C3×Dic6—124no—
135C3⋊(C3⋊C8)—21no—
136C3⋊(C3⋊C8)—21no—
137C3×C3⋊C8—61no—
138C3×C3⋊C8—61no—
139#475—271noSylow
140C2×H(3,𝔽3)C22⋊Q811yescharacteristic
141C2×H(3,𝔽3)C2×SD1611yescharacteristic
142C2×H(3,𝔽3)#13711yescharacteristic
143C2×C3×Q8—91no—
144C2×C3×Q8—181no—
145C2×Dic6—31no—
146C2×Dic6—61no—
147C2×Dic6—61no—
148C2×Dic6—62no—
149C2×Dic6—91no—
150C2×C3×C8—91no—
151C2×C3×C8—91no—
152C3×C4⋊C4—31no—
153C3×C4⋊C4—31no—
154C3×C4⋊C4—61no—
155C3×C4⋊C4—91no—
156C3×C4⋊C4—91no—
157C3×C4⋊C4—181no—
158C3×C4⋊C4D3211yescharacteristic
159C3×C42—91no—
160C4⋊Dic3—31no—
161C4⋊Dic3—31no—
162C4⋊Dic3—31no—
163C4⋊Dic3—61no—
164C4⋊Dic3—91no—
165C4⋊Dic3—181no—
166Dic3⋊C4—61no—
167Dic3⋊C4—61no—
168Dic3⋊C4—62no—
169Dic3⋊C4—181no—
170Dic3⋊C4—181no—
171C4×Dic3—31no—
172C4×Dic3—31no—
173C4×Dic3—61no—
174C4×Dic3—181no—
175C4×Dic3—181no—
176C4×Dic3—181no—
177C2×C3⋊C8—31no—
178C2×C3⋊C8—31no—
179C2×C3⋊C8—91no—
180C2×C3⋊C8—91no—
181C22×C32—21no—
182C22×C32D3×D411yescharacteristic
183C22×C32D3×D411yescharacteristic
184C32×C4—21no—
185C32×C4—21no—
186C32×C4—42no—
187C32×C4—41no—
188C32×C4—41no—
189C32×C4C2×Dic611yescharacteristic
190C32×C4#23511yescharacteristic
191C32×C4#23711yescharacteristic
192C32×C4D3×Q811yescharacteristic
193C3⋊Dic3—22no—
194C3⋊Dic3—41no—
195C3×Dic3—32no—
196C3×Dic3—32no—
197C3×Dic3—62no—
198C3×Dic3—62no—
199C3×Dic3—62no—
200C3×Dic3—61no—
201C3×Dic3—61no—
202C3×Dic3—122no—
203C3×Dic3—121no—
204C3×Dic3—121no—
205C3×Dic3—121no—
206C4⋊Q8—91no—
207C4×Q8—271no—
208#106—91no—
209#106—271no—
210C4⋊C8—271no—
211#103—91no—
212#103—271no—
213H(3,𝔽3)#47511yescharacteristic, complemented, Sylow
214C3×Q8—92no—
215C3×Q8—181no—
216C3×Q8—362no—
217C2×C3×C4—21no—
218C2×C3×C4—31no—
219C2×C3×C4—31no—
220C2×C3×C4—61no—
221C2×C3×C4—61no—
222C2×C3×C4—62no—
223C2×C3×C4—61no—
224C2×C3×C4—91no—
225C2×C3×C4—91no—
226C2×C3×C4—181no—
227C2×C3×C4—181no—
228C2×C3×C4—181no—
229C2×C3×C4C2×D3211yescharacteristic
230C2×Dic3—31no—
231C2×Dic3—31no—
232C2×Dic3—31no—
233C2×Dic3—31no—
234C2×Dic3—61no—
235C2×Dic3—61no—
236C2×Dic3—61no—
237C2×Dic3—61no—
238C2×Dic3—61no—
239C2×Dic3—62no—
240C2×Dic3—61no—
241C2×Dic3—181no—
242C2×Dic3—181no—
243Dic6—32no—
244Dic6—61no—
245Dic6—92no—
246Dic6—122no—
247Dic6—122no—
248Dic6—124no—
249Dic6—181no—
250C3×C8—181no—
251C3×C8—181no—
252C3⋊C8—61no—
253C3⋊C8—61no—
254C3⋊C8—181no—
255C3⋊C8—181no—
256C2×C32—21no—
257C2×C32—21no—
258C2×C32—21no—
259C2×C32#87811yescharacteristic
260C2×C32#89611yescharacteristic
261C2×C32D3×SD1611yescharacteristic
262C2×C32D3×SD1611yescharacteristic
263C2×C32#91511yescharacteristic
264C2×C32#91811yescharacteristic
265C2×Q8—91no—
266C2×Q8—181no—
267C2×C8—91no—
268C2×C8—91no—
269C4⋊C4—91no—
270C4⋊C4—91no—
271C4⋊C4—181no—
272C4⋊C4—541no—
273C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
274C42—91no—
275C42—541no—
276C22×C3—31no—
277C22×C3—31no—
278C22×C3—61no—
279C22×C3#381111yescharacteristic
280C3×C4—31no—
281C3×C4—31no—
282C3×C4—31no—
283C3×C4—31no—
284C3×C4—41no—
285C3×C4—61no—
286C3×C4—61no—
287C3×C4—92no—
288C3×C4—121no—
289C3×C4—121no—
290C3×C4—122no—
291C3×C4—182no—
292C3×C4—181no—
293C3×C4—361no—
294C3×C4—361no—
295C3×C4D3×Dic611yescharacteristic
296C3×C4#379511yescharacteristic
297Dic3—32no—
298Dic3—32no—
299Dic3—62no—
300Dic3—61no—
301Dic3—61no—
302Dic3—62no—
303Dic3—62no—
304Dic3—121no—
305Dic3—121no—
306Dic3—122no—
307Dic3—121no—
308Dic3—182no—
309Dic3—361no—
310C32—21no—
311C32#479111yescharacteristic, complemented
312C32#485411yescharacteristic, complemented
313Q8—92no—
314Q8—181no—
315Q8—362no—
316C2×C4—21no—
317C2×C4—91no—
318C2×C4—91no—
319C2×C4—181no—
320C2×C4—181no—
321C2×C4—181no—
322C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
323C8—181no—
324C8—181no—
325C2×C3—31no—
326C2×C3—31no—
327C2×C3—31no—
328C2×C3—31no—
329C2×C3—31no—
330C2×C3—31no—
331C2×C3—61no—
332C2×C3—61no—
333C2×C3—61no—
334C2×C3#6389611yescharacteristic
335C2×C3#6393011yescharacteristic
336C2×C3#6395111yescharacteristic
337C22#8867711yescharacteristic
338C4—41no—
339C4—92no—
340C4—181no—
341C4—182no—
342C4—361no—
343C4—361no—
344C4((C3⋊D3)⋊C3)⋊Q811yescharacteristic
345C4#8864811yescharacteristic
346C3—31no—
347C3—31no—
348C3—61no—
349C3#1059049711yescharacteristic
350C2#1172113511yescharacteristic
351C2#1172118811yescharacteristic
352C2#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)#475
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
45. #634⋊C3Q8semidirect
118. #634Dic6semidirect
213. H(3,𝔽3)#475semidirect
311. C32#4791semidirect
312. C32#4854semidirect

Maximal subgroups

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

subgroupindex
#117217562
#117210842
#63637⋊C32
#63635⋊C32
#117203092
#117203072
#117203042
#105904873
#105904863