Finite Group Databasebeta

← #13137955

Subgroups — #13137955

Order 1728.

414 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 83 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#64096⋊C3C211yesmaximal, characteristic
2#11721082C211yesmaximal, characteristic
3C2×#88433C211yesmaximal, characteristic
4#11720530C211yesmaximal, characteristic
5#63637⋊C3C211yesmaximal, characteristic
6#11720312C211yesmaximal, characteristic
7#11720309C211yesmaximal, characteristic
8#10590485—31nomaximal
9#10590480—31nomaximal
10C2×#634⋊C3C2211yescharacteristic
11C2×#634⋊C3C2211yescharacteristic
12C4⋊C4×H(3,𝔽3)C2211yescharacteristic
13C4⋊#1084C2211yescharacteristic
14C2×#6180C2211yescharacteristic
15#3750⋊C3—21no—
16C4×(C3⋊Dic3)⋊C3—21no—
17C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
18C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
19#88448—21no—
20#88447—21no—
21#88433C414yescomplemented
22#64096D311yescharacteristic
23C3×#887—31no—
24#63844—31no—
25#63843—31no—
26C2×#3719—31no—
27C2×#3719—31no—
28#63637D311yescharacteristic
29C3×#810—31no—
30C3×#808—31no—
31C3×#808—31no—
32C3×#808—61no—
33#63567—31no—
34#63565—31no—
35#63555—31no—
36#63554—31no—
37C2×C4×H(3,𝔽3)—21no—
38C2×C4×H(3,𝔽3)C2311yescharacteristic
39C2×#1084—21no—
40C2×(C3⋊Dic3)⋊C3—21no—
41C2×(C3⋊Dic3)⋊C3—21no—
42C2×(C3⋊Dic3)⋊C3D411yescharacteristic
43C2×(C3⋊Dic3)⋊C3D411yescharacteristic
44#634⋊C3—21no—
45#634⋊C3—21no—
46#634⋊C3C2×C412yes—
47#634⋊C3C2×C412yes—
48#6180C2×C412yes—
49(C3⋊(C3⋊C8))⋊C3—21no—
50#5030—91no—
51#4853—91no—
52#4789—91no—
53C2×#634C2×D311yescharacteristic
54C2×#634C2×D311yescharacteristic
55C2×C3×Dic6—31no—
56C2×C3×Dic6—31no—
57C32×C4⋊C4—21no—
58C32×C4⋊C4C2×D311yescharacteristic
59C32×C4⋊C4C2×D311yescharacteristic
60#3750—21no—
61C4×C3⋊Dic3—21no—
62C4×C3⋊Dic3C2×D311yescharacteristic
63C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
64C3×C4⋊Dic3—31no—
65C3×C4⋊Dic3—31no—
66C3×C4⋊Dic3—61no—
67C3×Dic3⋊C4—61no—
68C3×C4×Dic3—31no—
69C3×C4×Dic3—61no—
70C2×C3×C3⋊C8—31no—
71C2×C3×C3⋊C8—31no—
72C2×C3×C3⋊C8—31no—
73C2×C3×C3⋊C8—61no—
74#3724—61no—
75Dic3⋊Dic3—61no—
76Dic32—61no—
77Dic32—61no—
78#3719—34no—
79#3719—34no—
80C22×H(3,𝔽3)C2×D411yescharacteristic
81C4×H(3,𝔽3)—41no—
82C4×H(3,𝔽3)C22×C411yescharacteristic
83C4×H(3,𝔽3)#4111yescharacteristic
84#1084—41no—
85(C3⋊Dic3)⋊C3—22no—
86(C3⋊Dic3)⋊C3—22no—
87(C3⋊Dic3)⋊C3—21no—
88(C3⋊Dic3)⋊C3—21no—
89C3×C4×Q8—91no—
90Q8×Dic3—91no—
91C2×#216—91no—
92C2×#216—91no—
93C2×Dic12—91no—
94#887—31no—
95#887—91no—
96#887—91no—
97C4×Dic6—91no—
98C3×#106—91no—
99C3×#103—91no—
100#835—91no—
101#816—91no—
102#815—91no—
103#815—91no—
104#810—31no—
105#810—91no—
106#810—91no—
107#808—31no—
108#808—31no—
109#808—61no—
110#803—91no—
111C2×C32×C4—21no—
112C2×C32×C4—21no—
113C2×C32×C4—21no—
114C2×C32×C4—41no—
115C2×C32×C4C22×D311yescharacteristic
116C2×C32×C4C22×D311yescharacteristic
117C2×C3⋊Dic3—21no—
118C2×C3⋊Dic3—21no—
119C2×C3⋊Dic3#6711yescharacteristic
120C2×C3⋊Dic3#6711yescharacteristic
121#634—21no—
122#634—21no—
123#634C4×D312yes—
124#634C4×D312yes—
125C2×C3×Dic3—31no—
126C2×C3×Dic3—31no—
127C2×C3×Dic3—61no—
128C2×C3×Dic3—61no—
129C2×C3×Dic3—61no—
130C2×C3×Dic3—61no—
131C2×C3×Dic3—121no—
132C3×Dic6—32no—
133C3×Dic6—32no—
134C3×Dic6—61no—
135C3×Dic6—61no—
136C3⋊(C3⋊C8)—21no—
137C3×C3⋊C8—32no—
138C3×C3⋊C8—32no—
139C3×C3⋊C8—62no—
140C3×C3⋊C8—61no—
141#441—271noSylow
142C2×H(3,𝔽3)C4×D411yescharacteristic
143C2×H(3,𝔽3)#13711yescharacteristic
144C2×H(3,𝔽3)#13711yescharacteristic
145C2×C3×Q8—91no—
146C2×C3×Q8—91no—
147C2×Dic6—31no—
148C2×Dic6—31no—
149C2×Dic6—91no—
150C2×Dic6—91no—
151C2×C3×C8—91no—
152C2×C3×C8—91no—
153C3×C4⋊C4—31no—
154C3×C4⋊C4—31no—
155C3×C4⋊C4—61no—
156C3×C4⋊C4—91no—
157C3×C4⋊C4—181no—
158C3×C4⋊C4D3211yescharacteristic
159C3×C42—91no—
160C3×C42—181no—
161#216—94no—
162#216—94no—
163C4⋊Dic3—31no—
164C4⋊Dic3—31no—
165C4⋊Dic3—61no—
166C4⋊Dic3—181no—
167Dic3⋊C4—61no—
168Dic3⋊C4—181no—
169Dic3⋊C4—181no—
170Dic3⋊C4—181no—
171C4×Dic3—31no—
172C4×Dic3—61no—
173C4×Dic3—91no—
174C4×Dic3—181no—
175C4×Dic3—181no—
176C4×Dic3—181no—
177C4×Dic3—181no—
178C2×C3⋊C8—31no—
179C2×C3⋊C8—31no—
180C2×C3⋊C8—31no—
181C2×C3⋊C8—61no—
182C2×C3⋊C8—91no—
183Dic12—94no—
184C22×C32—21no—
185C22×C32D3×D411yescharacteristic
186C22×C32C2×#6711yescharacteristic
187C32×C4—21no—
188C32×C4—21no—
189C32×C4—41no—
190C32×C4—41no—
191C32×C4—81no—
192C32×C4C2×C4×D311yescharacteristic
193C32×C4C2×C4×D311yescharacteristic
194C32×C4#23511yescharacteristic
195C32×C4#23711yescharacteristic
196C3⋊Dic3—22no—
197C3⋊Dic3—22no—
198C3⋊Dic3—21no—
199C3⋊Dic3—21no—
200C3×Dic3—62no—
201C3×Dic3—61no—
202C3×Dic3—61no—
203C3×Dic3—62no—
204C3×Dic3—121no—
205C3×Dic3—121no—
206C3×Dic3—241no—
207C2×Q16—271no—
208C4×Q8—91no—
209C4×Q8—271no—
210#106—91no—
211#103—91no—
212#103—271no—
213#97—271no—
214H(3,𝔽3)#44111yescharacteristic, complemented, Sylow
215C3×Q8—92no—
216C3×Q8—92no—
217C3×Q8—181no—
218C3×Q8—181no—
219C2×C3×C4—21no—
220C2×C3×C4—31no—
221C2×C3×C4—31no—
222C2×C3×C4—61no—
223C2×C3×C4—61no—
224C2×C3×C4—61no—
225C2×C3×C4—91no—
226C2×C3×C4—91no—
227C2×C3×C4—121no—
228C2×C3×C4—181no—
229C2×C3×C4—181no—
230C2×C3×C4—181no—
231C2×C3×C4C2×D3211yescharacteristic
232C2×Dic3—31no—
233C2×Dic3—31no—
234C2×Dic3—61no—
235C2×Dic3—61no—
236C2×Dic3—61no—
237C2×Dic3—61no—
238C2×Dic3—91no—
239C2×Dic3—91no—
240C2×Dic3—121no—
241C2×Dic3—181no—
242C2×Dic3—181no—
243Dic6—32no—
244Dic6—32no—
245Dic6—61no—
246Dic6—61no—
247Dic6—92no—
248Dic6—92no—
249Dic6—181no—
250Dic6—181no—
251C3×C8—92no—
252C3×C8—181no—
253C3⋊C8—32no—
254C3⋊C8—32no—
255C3⋊C8—62no—
256C3⋊C8—61no—
257C3⋊C8—181no—
258C2×C32—21no—
259C2×C32—21no—
260C2×C32—21no—
261C2×C32#67⋊C411yescharacteristic
262C2×C32#91511yescharacteristic
263C2×C32#91811yescharacteristic
264C2×C32C4×#6711yescharacteristic
265C2×C32#94211yescharacteristic
266C2×C32#95111yescharacteristic
267C2×Q8—91no—
268C2×Q8—91no—
269Q16—274no—
270C2×C8—91no—
271C2×C8—91no—
272C4⋊C4—91no—
273C4⋊C4—181no—
274C4⋊C4—541no—
275C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
276C42—91no—
277C42—181no—
278C42—541no—
279C22×C3—31no—
280C22×C3—31no—
281C22×C3—61no—
282C22×C3D3×#6711yescharacteristic
283C3×C4—31no—
284C3×C4—31no—
285C3×C4—31no—
286C3×C4—31no—
287C3×C4—41no—
288C3×C4—61no—
289C3×C4—61no—
290C3×C4—121no—
291C3×C4—121no—
292C3×C4—181no—
293C3×C4—181no—
294C3×C4—182no—
295C3×C4—182no—
296C3×C4—241no—
297C3×C4—361no—
298C3×C4#379511yescharacteristic
299C3×C4C4×D3211yescharacteristic
300Dic3—62no—
301Dic3—62no—
302Dic3—61no—
303Dic3—61no—
304Dic3—121no—
305Dic3—121no—
306Dic3—182no—
307Dic3—182no—
308Dic3—181no—
309Dic3—181no—
310Dic3—241no—
311C32—21no—
312C32#478911yescharacteristic, complemented
313C32#503011yescharacteristic, complemented
314Q8—92no—
315Q8—92no—
316Q8—181no—
317Q8—181no—
318C2×C4—21no—
319C2×C4—91no—
320C2×C4—91no—
321C2×C4—181no—
322C2×C4—181no—
323C2×C4—181no—
324C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
325C8—92no—
326C8—181no—
327C2×C3—31no—
328C2×C3—31no—
329C2×C3—31no—
330C2×C3—31no—
331C2×C3—31no—
332C2×C3—31no—
333C2×C3—61no—
334C2×C3—61no—
335C2×C3—61no—
336C2×C3#6387911yescharacteristic
337C2×C3#6392911yescharacteristic
338C2×C3#6394811yescharacteristic
339C22#8867611yescharacteristic
340C4—41no—
341C4—181no—
342C4—182no—
343C4—182no—
344C4—181no—
345C4—361no—
346C4#8864811yescharacteristic
347C4C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
348C3—31no—
349C3—31no—
350C3—61no—
351C3#1059047711yescharacteristic
352C2#1172112211yescharacteristic
353C2#1172119911yescharacteristic
354C2#1172123411yescharacteristic
subgroupG/H
centerC22#88676
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC4⋊C4×H(3,𝔽3)C22
socleC22×C3D3×#67
prime radicalC22×H(3,𝔽3)C2×D4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#441
LCSG ▷ 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
21. #88433C4semidirect
214. H(3,𝔽3)#441semidirect
312. C32#4789semidirect
313. C32#5030semidirect

Maximal subgroups

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

subgroupindex
#64096⋊C32
#117210822
C2×#884332
#117205302
#63637⋊C32
#117203122
#117203092
#105904853
#105904803