Finite Group Databasebeta

← #13137933

Subgroups — #13137933

Order 1728.

435 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 435 conjugacy classes of subgroups — too many to draw. Only its 75 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#64106⋊C3C211yesmaximal, characteristic
2#11721106C211yesmaximal, characteristic
3C2×#88433C211yesmaximal, characteristic, complemented
4C2×((C3⋊D3)⋊C3)⋊C8C211yesmaximal, characteristic
5#11720529C211yesmaximal, characteristic, complemented
6#63637⋊C3C211yesmaximal, characteristic, complemented
7#11720309C211yesmaximal, characteristic, complemented
8#10590579—31nomaximal
9#10590525—31nomaximal
10C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
11C2×#634⋊C3C2211yescharacteristic
12C2×#634⋊C3C2211yescharacteristic
13C4⋊C4×H(3,𝔽3)C2211yescharacteristic
14C4⋊#1084C2211yescharacteristic
15C2×#6180C2211yescharacteristic
16#3752⋊C3—21no—
17C4⋊((C3⋊Dic3)⋊C3)—21no—
18C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
19#88450—21no—
20#88449—21no—
21#88433—22no—
22((C3⋊D3)⋊C3)⋊C8—22no—
23#64106D311yescharacteristic
24C3×#897—31no—
25#63871—31no—
26#63857—31no—
27C2×#3719—31no—
28C2×#3719—31no—
29C2×#3710—31no—
30C2×D3×C3⋊C8—31no—
31#63637D311yescharacteristic, complemented
32C3×#810—31no—
33C3×#807—31no—
34C3×#807—31no—
35C3×#807—61no—
36#63567—31no—
37#63565—31no—
38C22×(C3⋊D3)⋊C3D411yescharacteristic
39C2×C4×H(3,𝔽3)—21no—
40C2×C4×H(3,𝔽3)C2311yescharacteristic
41C2×#1084—21no—
42C2×(C3⋊Dic3)⋊C3—21no—
43C2×(C3⋊Dic3)⋊C3—21no—
44C2×(C3⋊Dic3)⋊C3D411yescharacteristic
45C4×(C3⋊D3)⋊C3—21no—
46C4×(C3⋊D3)⋊C3—21no—
47#634⋊C3—21no—
48#634⋊C3—21no—
49#634⋊C3—21no—
50#634⋊C3—21no—
51#6180D412yes—
52(C3⋊(C3⋊C8))⋊C3—21no—
53#5051—91no—
54#4887—91no—
55#4813—91no—
56C2×C4×C3⋊D3C2×D311yescharacteristic
57C2×#634C2×D311yescharacteristic
58C2×#634C2×D311yescharacteristic
59C2×C3×C4×D3—31no—
60C2×C3×Dic6—31no—
61C2×C3×Dic6—31no—
62C32×C4⋊C4—21no—
63C32×C4⋊C4C2×D311yescharacteristic
64C32×C4⋊C4C2×D311yescharacteristic
65#3752—21no—
66C4⋊(C3⋊Dic3)—21no—
67C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
68C3×(C2×D3)⋊C4—61no—
69C3×C4⋊Dic3—31no—
70C3×C4⋊Dic3—31no—
71C3×C4⋊Dic3—61no—
72C3×C4⋊Dic3—61no—
73C2×C3×C3⋊C8—31no—
74C2×C3×C3⋊C8—31no—
75C2×C3×C3⋊C8—31no—
76C2×C3×C3⋊C8—61no—
77Dic3⋊Dic3—61no—
78Dic3⋊Dic3—61no—
79#3722—61no—
80(C2×D3)⋊Dic3—61no—
81#3719—62no—
82#3719—62no—
83#3710—62no—
84D3×C3⋊C8—62no—
85C22×H(3,𝔽3)C2×D411yescharacteristic
86C2×(C3⋊D3)⋊C3—21no—
87C2×(C3⋊D3)⋊C3—21no—
88C2×(C3⋊D3)⋊C3Q1612yescomplemented
89C4×H(3,𝔽3)—41no—
90C4×H(3,𝔽3)C2×D411yescharacteristic
91C4×H(3,𝔽3)#4111yescharacteristic
92#1084—41no—
93(C3⋊Dic3)⋊C3—21no—
94(C3⋊Dic3)⋊C3—41no—
95(C3⋊Dic3)⋊C3—41no—
96C3×C22⋊Q8—91no—
97#946—91no—
98C2×#216—91no—
99C2×#216—91no—
100#924—91no—
101C22×C3⋊C8—91no—
102C2×Dic12—91no—
103C2×D3×C8—91no—
104C2×D3×C8—91no—
105#897—31no—
106#897—91no—
107#897—91no—
108C3×#107—91no—
109C3×#103—91no—
110#835—91no—
111#816—91no—
112#810—31no—
113#810—91no—
114#810—91no—
115#807—31no—
116#807—31no—
117#807—61no—
118C22×C3⋊D3#6711yescharacteristic
119C22×C3×D3—31no—
120C2×C32×C4—21no—
121C2×C32×C4—21no—
122C2×C32×C4—21no—
123C2×C32×C4—41no—
124C2×C32×C4C22×D311yescharacteristic
125C2×C32×C4C22×D311yescharacteristic
126C2×C3⋊Dic3—21no—
127C2×C3⋊Dic3—21no—
128C2×C3⋊Dic3#6711yescharacteristic
129C4×C3⋊D3—21no—
130C4×C3⋊D3—21no—
131#634—21no—
132#634—21no—
133#634—21no—
134#634—21no—
135C2×C3×Dic3—31no—
136C2×C3×Dic3—61no—
137C2×C3×Dic3—61no—
138C2×C3×Dic3—61no—
139C2×C3×Dic3—61no—
140C2×C3×Dic3—121no—
141C3×C4×D3—61no—
142C3×C4×D3—61no—
143C3×Dic6—61no—
144C3×Dic6—61no—
145C3×Dic6—61no—
146C3×Dic6—61no—
147C3⋊(C3⋊C8)—21no—
148C3×C3⋊C8—32no—
149C3×C3⋊C8—32no—
150C3×C3⋊C8—62no—
151C3×C3⋊C8—61no—
152#467—271noSylow
153C2×H(3,𝔽3)#12111yescharacteristic
154C2×H(3,𝔽3)C2×Q1611yescharacteristic
155C2×H(3,𝔽3)#13511yescharacteristic
156(C3⋊D3)⋊C3—22no—
157C2×C3×Q8—91no—
158C2×C3×Q8—91no—
159C22×C3×C4—91no—
160C2×C4×D3—31no—
161C2×C4×D3—91no—
162C2×Dic6—31no—
163C2×Dic6—31no—
164C2×Dic6—91no—
165C2×Dic6—91no—
166C2×C3×C8—91no—
167C2×C3×C8—91no—
168C3×C4⋊C4—31no—
169C3×C4⋊C4—31no—
170C3×C4⋊C4—61no—
171C3×C4⋊C4—91no—
172C3×C4⋊C4—181no—
173C3×C4⋊C4D3211yescharacteristic
174C3×C22⋊C4—181no—
175#217—181no—
176#216—182no—
177#216—182no—
178(C2×D3)⋊C4—61no—
179(C2×D3)⋊C4—181no—
180(C2×D3)⋊C4—181no—
181(C2×D3)⋊C4—181no—
182C4⋊Dic3—31no—
183C4⋊Dic3—31no—
184C4⋊Dic3—61no—
185C4⋊Dic3—61no—
186C4⋊Dic3—181no—
187C4⋊Dic3—181no—
188Dic3⋊C4—181no—
189Dic3⋊C4—181no—
190C2×C3⋊C8—31no—
191C2×C3⋊C8—31no—
192C2×C3⋊C8—31no—
193C2×C3⋊C8—61no—
194C2×C3⋊C8—91no—
195C2×C3⋊C8—182no—
196Dic12—182no—
197D3×C8—182no—
198D3×C8—182no—
199C22×C32—21no—
200C22×C32D3×D411yescharacteristic
201C22×C32C2×#6711yescharacteristic
202C2×C3⋊D3—21no—
203C2×C3⋊D3—21no—
204C2×C3⋊D3#21612yescomplemented
205C2×C3×D3—32no—
206C2×C3×D3—61no—
207C2×C3×D3—61no—
208C32×C4—21no—
209C32×C4—21no—
210C32×C4—41no—
211C32×C4—41no—
212C32×C4—81no—
213C32×C4#23511yescharacteristic
214C32×C4D3×D411yescharacteristic
215C32×C4D3×D411yescharacteristic
216C32×C4#23711yescharacteristic
217C3⋊Dic3—21no—
218C3⋊Dic3—41no—
219C3⋊Dic3—41no—
220C3×Dic3—61no—
221C3×Dic3—121no—
222C3×Dic3—121no—
223C3×Dic3—121no—
224C3×Dic3—121no—
225C3×Dic3—241no—
226C2×Q16—271no—
227C22×C8—271no—
228C22⋊Q8—91no—
229C22⋊Q8—271no—
230#107—91no—
231#103—91no—
232#103—271no—
233H(3,𝔽3)#46711yescharacteristic, complemented, Sylow
234C23×C3—91no—
235C22×D3—31no—
236C22×D3—91no—
237C3×Q8—181no—
238C3×Q8—181no—
239C3×Q8—181no—
240C3×Q8—181no—
241C2×C3×C4—21no—
242C2×C3×C4—31no—
243C2×C3×C4—31no—
244C2×C3×C4—61no—
245C2×C3×C4—61no—
246C2×C3×C4—61no—
247C2×C3×C4—91no—
248C2×C3×C4—121no—
249C2×C3×C4—181no—
250C2×C3×C4—181no—
251C2×C3×C4—181no—
252C2×C3×C4—181no—
253C2×C3×C4—181no—
254C2×C3×C4C2×D3211yescharacteristic
255C2×Dic3—31no—
256C2×Dic3—61no—
257C2×Dic3—61no—
258C2×Dic3—61no—
259C2×Dic3—61no—
260C2×Dic3—91no—
261C2×Dic3—121no—
262C2×Dic3—181no—
263C2×Dic3—181no—
264C4×D3—61no—
265C4×D3—61no—
266C4×D3—181no—
267C4×D3—181no—
268Dic6—61no—
269Dic6—61no—
270Dic6—61no—
271Dic6—61no—
272Dic6—181no—
273Dic6—181no—
274Dic6—181no—
275Dic6—181no—
276C3×C8—92no—
277C3×C8—181no—
278C3⋊C8—32no—
279C3⋊C8—32no—
280C3⋊C8—62no—
281C3⋊C8—61no—
282C3⋊C8—181no—
283C2×C32—21no—
284C2×C32—21no—
285C2×C32—21no—
286C2×C32#88411yescharacteristic
287C2×C32#91611yescharacteristic
288C2×C32D3×Q1611yescharacteristic
289C2×C32#93911yescharacteristic
290C2×C32C2×#21611yescharacteristic
291C2×C32#95011yescharacteristic
292C3⋊D3—22no—
293C3×D3—62no—
294C2×Q8—91no—
295C2×Q8—91no—
296C22×C4—91no—
297Q16—542no—
298C2×C8—91no—
299C2×C8—91no—
300C2×C8—542no—
301C4⋊C4—91no—
302C4⋊C4—181no—
303C4⋊C4—541no—
304C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
305C22⋊C4—181no—
306C22⋊C4—541no—
307C22×C3—31no—
308C22×C3—31no—
309C22×C3—61no—
310C22×C3—92no—
311C22×C3—181no—
312C22×C3—181no—
313C22×C3D3×#6711yescharacteristic
314C2×D3—32no—
315C2×D3—61no—
316C2×D3—61no—
317C2×D3—92no—
318C2×D3—181no—
319C2×D3—181no—
320C3×C4—31no—
321C3×C4—31no—
322C3×C4—31no—
323C3×C4—31no—
324C3×C4—41no—
325C3×C4—61no—
326C3×C4—61no—
327C3×C4—121no—
328C3×C4—121no—
329C3×C4—181no—
330C3×C4—241no—
331C3×C4—361no—
332C3×C4—361no—
333C3×C4—361no—
334C3×C4#379511yescharacteristic
335C3×C4#380211yescharacteristic
336Dic3—61no—
337Dic3—121no—
338Dic3—121no—
339Dic3—121no—
340Dic3—121no—
341Dic3—181no—
342Dic3—241no—
343Dic3—361no—
344Dic3—361no—
345C32—21no—
346C32#481311yescharacteristic, complemented
347C32#505111yescharacteristic, complemented
348C23—91no—
349Q8—181no—
350Q8—181no—
351Q8—181no—
352Q8—181no—
353C2×C4—21no—
354C2×C4—91no—
355C2×C4—181no—
356C2×C4—181no—
357C2×C4—181no—
358C2×C4—181no—
359C2×C4—181no—
360C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
361C8—92no—
362C8—181no—
363C2×C3—31no—
364C2×C3—31no—
365C2×C3—31no—
366C2×C3—31no—
367C2×C3—31no—
368C2×C3—31no—
369C2×C3—61no—
370C2×C3—61no—
371C2×C3—61no—
372C2×C3—182no—
373C2×C3#6388511yescharacteristic
374C2×C3#6393511yescharacteristic
375C2×C3D3×#21611yescharacteristic
376D3—62no—
377D3—182no—
378C22—92no—
379C22—181no—
380C22—181no—
381C22#8867611yescharacteristic
382C4—41no—
383C4—181no—
384C4—361no—
385C4—361no—
386C4—361no—
387C4#8864811yescharacteristic
388C4#8865511yescharacteristic
389C3—31no—
390C3—31no—
391C3—61no—
392C3#1059051811yescharacteristic
393C2—182no—
394C2#1172112611yescharacteristic
395C2#1172119711yescharacteristic
396C2#1172123611yescharacteristic
subgroupG/H
centerC22#88676
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC4⋊C4×H(3,𝔽3)C22
socleC22×C3D3×#67
prime radicalC22×(C3⋊D3)⋊C3D4
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#467
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
3. C2×#88433C2semidirect
5. #11720529C2semidirect
6. #63637⋊C3C2semidirect
7. #11720309C2semidirect
31. #63637D3semidirect
88. C2×(C3⋊D3)⋊C3Q16semidirect
204. C2×C3⋊D3#216semidirect
233. H(3,𝔽3)#467semidirect
346. C32#4813semidirect
347. C32#5051semidirect

Maximal subgroups

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

subgroupindex
#64106⋊C32
#117211062
C2×#884332
C2×((C3⋊D3)⋊C3)⋊C82
#117205292
#63637⋊C32
#117203092
#105905793
#105905253