Finite Group Databasebeta

← #13140984

Subgroups — #13140984

Order 1728.

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

normal subgroups only

This group has 459 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
1C2×#88550C211yesmaximal, characteristic, complemented
2#64155⋊C3C211yesmaximal, characteristic
3#11721107C211yesmaximal, characteristic
4C2×((C3⋊D3)⋊C3)⋊C8C211yesmaximal, characteristic
5#63662⋊C3C211yesmaximal, characteristic, complemented
6#11720326C211yesmaximal, characteristic, complemented
7#11720315C211yesmaximal, characteristic, complemented
8#10591635—31nomaximal
9#10591601—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C2×#6229C2211yescharacteristic
12C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
13#88550—22no—
14C2×#6180C2211yescharacteristic
15#3752⋊C3—21no—
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
18#3750⋊C3—21no—
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20#88450—21no—
21#88448—21no—
22((C3⋊D3)⋊C3)⋊C8—22no—
23#64155D311yescharacteristic
24C3×#946—31no—
25C2×C3×#216—31no—
26C2×C3×#216—31no—
27C2×C3×#216—61no—
28#63871—31no—
29#63856—31no—
30C2×#3710—31no—
31C2×D3×C3⋊C8—31no—
32#63662D311yescharacteristic, complemented
33C3×#835—31no—
34#63575—31no—
35#63575—31no—
36#63566—31no—
37#63565—31no—
38C22×(C3⋊D3)⋊C3D411yescharacteristic
39Q8×H(3,𝔽3)—21no—
40Q8×H(3,𝔽3)—21no—
41C2×C4×H(3,𝔽3)—21no—
42C2×C4×H(3,𝔽3)C2311yescharacteristic
43C2×#1084—21no—
44#6229—21no—
45#6229—21no—
46C2×(C3⋊Dic3)⋊C3—21no—
47C2×(C3⋊Dic3)⋊C3—21no—
48C2×(C3⋊Dic3)⋊C3D411yescharacteristic
49C4×(C3⋊D3)⋊C3—21no—
50C4×(C3⋊D3)⋊C3—21no—
51#6180D412yes—
52(C3⋊(C3⋊C8))⋊C3—21no—
53#5188—91no—
54#5051—91no—
55#4813—91no—
56C2×C32×Q8—21no—
57C2×C32×Q8C2×D311yescharacteristic
58C2×C32×Q8C2×D311yescharacteristic
59C2×C4×C3⋊D3C2×D311yescharacteristic
60C2×C3×C4×D3—31no—
61C2×C3×Dic6—31no—
62C2×C3×Dic6—31no—
63C2×C3×Dic6—61no—
64#3752—21no—
65C4⋊(C3⋊Dic3)C2×D311yescharacteristic
66C4⋊(C3⋊Dic3)C2×D311yescharacteristic
67#3750—21no—
68C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
69C3×#216—62no—
70C3×#216—62no—
71C3×#216—64no—
72C3×(C2×D3)⋊C4—61no—
73C3×C4⋊Dic3—31no—
74C3×C4⋊Dic3—31no—
75C3×Dic3⋊C4—61no—
76C2×C3×C3⋊C8—31no—
77C2×C3×C3⋊C8—31no—
78C2×C3×C3⋊C8—31no—
79C2×C3×C3⋊C8—61no—
80#3724—61no—
81Dic3⋊Dic3—61no—
82#3722—61no—
83(C2×D3)⋊Dic3—61no—
84#3710—62no—
85D3×C3⋊C8—62no—
86C22×H(3,𝔽3)C2×D411yescharacteristic
87C2×(C3⋊D3)⋊C3—21no—
88C2×(C3⋊D3)⋊C3—21no—
89C2×(C3⋊D3)⋊C3Q1612yescomplemented
90C4×H(3,𝔽3)—41no—
91C4×H(3,𝔽3)C2×D411yescharacteristic
92C4×H(3,𝔽3)#4111yescharacteristic
93#1084—41no—
94(C3⋊Dic3)⋊C3—21no—
95(C3⋊Dic3)⋊C3—41no—
96(C3⋊Dic3)⋊C3—41no—
97C2×C3×Q16—91no—
98C3×C22⋊Q8—91no—
99#946—31no—
100#946—91no—
101#946—91no—
102C2×#216—31no—
103C2×#216—31no—
104C2×#216—61no—
105#924—91no—
106C22×C3⋊C8—91no—
107C2×D3×C8—91no—
108C2×D3×C8—91no—
109#897—91no—
110C3×#103—91no—
111#835—31no—
112#835—91no—
113#835—91no—
114#818—91no—
115#816—91no—
116#810—91no—
117#807—91no—
118#807—91no—
119C22×C3⋊D3#6711yescharacteristic
120C22×C3×D3—31no—
121C32×Q8—22no—
122C32×Q8—21no—
123C32×Q8—21no—
124C32×Q8—21no—
125C32×Q8—21no—
126C32×Q8—41no—
127C2×C32×C4—21no—
128C2×C32×C4—21no—
129C2×C32×C4—21no—
130C2×C32×C4—41no—
131C2×C32×C4C22×D311yescharacteristic
132C2×C32×C4C22×D311yescharacteristic
133C2×C3⋊Dic3—21no—
134C2×C3⋊Dic3—21no—
135C2×C3⋊Dic3#6711yescharacteristic
136C4×C3⋊D3—21no—
137C4×C3⋊D3—21no—
138C2×C3×Dic3—31no—
139C2×C3×Dic3—61no—
140C2×C3×Dic3—61no—
141C2×C3×Dic3—61no—
142C2×C3×Dic3—61no—
143C2×C3×Dic3—121no—
144C3×C4×D3—61no—
145C3×C4×D3—61no—
146C3×Dic6—62no—
147C3×Dic6—61no—
148C3×Dic6—61no—
149C3×Dic6—61no—
150C3×Dic6—61no—
151C3×Dic6—121no—
152C3⋊(C3⋊C8)—21no—
153C3×C3⋊C8—32no—
154C3×C3⋊C8—32no—
155C3×C3⋊C8—62no—
156C3×C3⋊C8—61no—
157#467—271noSylow
158C2×H(3,𝔽3)#12111yescharacteristic
159C2×H(3,𝔽3)C2×Q1611yescharacteristic
160C2×H(3,𝔽3)#13511yescharacteristic
161(C3⋊D3)⋊C3—22no—
162C2×C3×Q8—31no—
163C2×C3×Q8—31no—
164C2×C3×Q8—61no—
165C2×C3×Q8—91no—
166C2×C3×Q8D3211yescharacteristic
167C22×C3×C4—91no—
168C2×C4×D3—31no—
169C2×C4×D3—91no—
170C2×Dic6—31no—
171C2×Dic6—31no—
172C2×Dic6—61no—
173C3×Q16—182no—
174C2×C3×C8—91no—
175C2×C3×C8—91no—
176C3×C4⋊C4—91no—
177C3×C4⋊C4—91no—
178C3×C4⋊C4—181no—
179C3×C22⋊C4—181no—
180#217—181no—
181#216—62no—
182#216—62no—
183#216—64no—
184(C2×D3)⋊C4—61no—
185(C2×D3)⋊C4—181no—
186(C2×D3)⋊C4—181no—
187(C2×D3)⋊C4—181no—
188C4⋊Dic3—31no—
189C4⋊Dic3—31no—
190C4⋊Dic3—91no—
191C4⋊Dic3—91no—
192C4⋊Dic3—181no—
193Dic3⋊C4—61no—
194Dic3⋊C4—181no—
195Dic3⋊C4—181no—
196Dic3⋊C4—181no—
197C2×C3⋊C8—31no—
198C2×C3⋊C8—31no—
199C2×C3⋊C8—31no—
200C2×C3⋊C8—61no—
201C2×C3⋊C8—91no—
202C2×C3⋊C8—182no—
203D3×C8—182no—
204D3×C8—182no—
205C22×C32—21no—
206C22×C32D3×D411yescharacteristic
207C22×C32C2×#6711yescharacteristic
208C2×C3⋊D3—21no—
209C2×C3⋊D3—21no—
210C2×C3⋊D3#21612yescomplemented
211C2×C3×D3—32no—
212C2×C3×D3—61no—
213C2×C3×D3—61no—
214C32×C4—21no—
215C32×C4—21no—
216C32×C4—42no—
217C32×C4—41no—
218C32×C4—41no—
219C32×C4#23511yescharacteristic
220C32×C4D3×D411yescharacteristic
221C32×C4D3×D411yescharacteristic
222C32×C4#23711yescharacteristic
223C3⋊Dic3—21no—
224C3⋊Dic3—41no—
225C3⋊Dic3—41no—
226C3×Dic3—61no—
227C3×Dic3—122no—
228C3×Dic3—121no—
229C3×Dic3—121no—
230C3×Dic3—121no—
231C3×Dic3—121no—
232C2×Q16—91no—
233C22×C8—271no—
234C22⋊Q8—91no—
235C22⋊Q8—271no—
236#107—271no—
237#103—91no—
238#103—271no—
239H(3,𝔽3)#46711yescharacteristic, complemented, Sylow
240C23×C3—91no—
241C22×D3—31no—
242C22×D3—91no—
243C3×Q8—21no—
244C3×Q8—21no—
245C3×Q8—61no—
246C3×Q8—61no—
247C3×Q8—61no—
248C3×Q8—61no—
249C3×Q8—62no—
250C3×Q8—121no—
251C3×Q8—181no—
252C3×Q8—181no—
253C2×C3×C4—21no—
254C2×C3×C4—31no—
255C2×C3×C4—31no—
256C2×C3×C4—61no—
257C2×C3×C4—61no—
258C2×C3×C4—61no—
259C2×C3×C4—91no—
260C2×C3×C4—121no—
261C2×C3×C4—181no—
262C2×C3×C4—181no—
263C2×C3×C4—181no—
264C2×C3×C4—181no—
265C2×C3×C4—181no—
266C2×C3×C4C2×D3211yescharacteristic
267C2×Dic3—31no—
268C2×Dic3—61no—
269C2×Dic3—61no—
270C2×Dic3—61no—
271C2×Dic3—61no—
272C2×Dic3—91no—
273C2×Dic3—121no—
274C2×Dic3—181no—
275C2×Dic3—181no—
276C4×D3—61no—
277C4×D3—61no—
278C4×D3—181no—
279C4×D3—181no—
280Dic6—61no—
281Dic6—61no—
282Dic6—61no—
283Dic6—61no—
284Dic6—62no—
285Dic6—121no—
286C3×C8—92no—
287C3×C8—181no—
288C3⋊C8—32no—
289C3⋊C8—32no—
290C3⋊C8—62no—
291C3⋊C8—61no—
292C3⋊C8—181no—
293C2×C32—21no—
294C2×C32—21no—
295C2×C32—21no—
296C2×C32#88411yescharacteristic
297C2×C32#91611yescharacteristic
298C2×C32D3×Q1611yescharacteristic
299C2×C32#93911yescharacteristic
300C2×C32C2×#21611yescharacteristic
301C2×C32#95011yescharacteristic
302C3⋊D3—22no—
303C3×D3—62no—
304C2×Q8—91no—
305C2×Q8((C3⋊D3)⋊C3)⋊C211yescharacteristic
306C22×C4—91no—
307Q16—182no—
308C2×C8—91no—
309C2×C8—91no—
310C2×C8—542no—
311C4⋊C4—91no—
312C4⋊C4—91no—
313C4⋊C4—181no—
314C4⋊C4—541no—
315C22⋊C4—181no—
316C22⋊C4—541no—
317C22×C3—31no—
318C22×C3—31no—
319C22×C3—61no—
320C22×C3—92no—
321C22×C3—181no—
322C22×C3—181no—
323C22×C3D3×#6711yescharacteristic
324C2×D3—32no—
325C2×D3—61no—
326C2×D3—61no—
327C2×D3—92no—
328C2×D3—181no—
329C2×D3—181no—
330C3×C4—31no—
331C3×C4—31no—
332C3×C4—31no—
333C3×C4—31no—
334C3×C4—41no—
335C3×C4—61no—
336C3×C4—61no—
337C3×C4—121no—
338C3×C4—121no—
339C3×C4—122no—
340C3×C4—181no—
341C3×C4—361no—
342C3×C4—361no—
343C3×C4—361no—
344C3×C4#380411yescharacteristic
345C3×C4#381111yescharacteristic
346Dic3—61no—
347Dic3—121no—
348Dic3—121no—
349Dic3—122no—
350Dic3—121no—
351Dic3—121no—
352Dic3—181no—
353Dic3—361no—
354Dic3—361no—
355C32—21no—
356C32#481311yescharacteristic, complemented
357C32#505111yescharacteristic, complemented
358C23—91no—
359Q8—21no—
360Q8—21no—
361Q8—181no—
362Q8—181no—
363C2×C4—21no—
364C2×C4—91no—
365C2×C4—181no—
366C2×C4—181no—
367C2×C4—181no—
368C2×C4—181no—
369C2×C4—181no—
370C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
371C8—92no—
372C8—181no—
373C2×C3—31no—
374C2×C3—31no—
375C2×C3—31no—
376C2×C3—31no—
377C2×C3—31no—
378C2×C3—31no—
379C2×C3—61no—
380C2×C3—61no—
381C2×C3—61no—
382C2×C3—182no—
383C2×C3D3×#21611yescharacteristic
384C2×C3#6394711yescharacteristic
385C2×C3#6397111yescharacteristic
386D3—62no—
387D3—182no—
388C22—92no—
389C22—181no—
390C22—181no—
391C22#8867611yescharacteristic
392C4—41no—
393C4—181no—
394C4—361no—
395C4—361no—
396C4—361no—
397C4#8867111yescharacteristic
398C4#8867711yescharacteristic
399C3—31no—
400C3—31no—
401C3—61no—
402C3#1059159611yescharacteristic
403C2—182no—
404C2#1172123611yescharacteristic
405C2#1172123911yescharacteristic
406C2#1172131211yescharacteristic
subgroupG/H
centerC22#88676
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×Q8×H(3,𝔽3)C22
socleC22×C3D3×#67
prime radicalC22×(C3⋊D3)⋊C3D4
hypercenterC2×Q8((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 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#88550C2semidirect
5. #63662⋊C3C2semidirect
6. #11720326C2semidirect
7. #11720315C2semidirect
32. #63662D3semidirect
89. C2×(C3⋊D3)⋊C3Q16semidirect
210. C2×C3⋊D3#216semidirect
239. H(3,𝔽3)#467semidirect
356. C32#4813semidirect
357. C32#5051semidirect

Maximal subgroups

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

subgroupindex
C2×#885502
#64155⋊C32
#117211072
C2×((C3⋊D3)⋊C3)⋊C82
#63662⋊C32
#117203262
#117203152
#105916353
#105916013