Finite Group Databasebeta

← #10592585⋊C3 #13142227

Subgroups — #10592585⋊C3 #13142227

Order 1728.

456 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 96 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×#3744⋊C3C211yesmaximal, characteristic, complemented
2C2×#3743⋊C3C211yesmaximal, characteristic, complemented
3#64106⋊C3C211yesmaximal, characteristic, complemented
4#64104⋊C3C211yesmaximal, characteristic
5H(3,𝔽3)×#107C211yesmaximal, characteristic, complemented
6#63636⋊C3C211yesmaximal, characteristic, complemented
7#63637⋊C3C211yesmaximal, characteristic, complemented
8#10592585C311yesmaximal, characteristic, complemented
9C3×#4886—31nomaximal
10C2×C4⋊((C3⋊D3)⋊C3)—21no—
11C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
12C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
13C2×#634⋊C3C2211yescharacteristic
14C2×C8×H(3,𝔽3)C2211yescharacteristic
15C4⋊C4×H(3,𝔽3)C2211yescharacteristic
16C4⋊C4×H(3,𝔽3)C2211yescharacteristic
17#3752⋊C3—21no—
18#3752⋊C3—21no—
19C4⋊((C3⋊Dic3)⋊C3)—21no—
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21#3744⋊C3—22no—
22#3743⋊C3—22no—
23C2×#3744C2×C311yescharacteristic, complemented
24C2×#3743C2×C311yescharacteristic, complemented
25#64106C2×C311yescharacteristic, complemented
26#64104C2×C311yescharacteristic
27C2×C3×#204—31no—
28C2×C3×ΓL(1,𝔽25)—31no—
29C3×#897—31no—
30C3×#895—31no—
31C32×#107—21no—
32C32×#107D311yescharacteristic, complemented
33C32×#107C2×C311yescharacteristic, complemented
34#63637C2×C311yescharacteristic, complemented
35#63636C2×C311yescharacteristic, complemented
36C3×#810—31no—
37C3×#809—31no—
38C22×(C3⋊D3)⋊C3—21no—
39C22×(C3⋊D3)⋊C3D411yescharacteristic
40C2×C4×H(3,𝔽3)—21no—
41C2×C4×H(3,𝔽3)—21no—
42C2×C4×H(3,𝔽3)C2311yescharacteristic
43C2×(C3⋊Dic3)⋊C3—21no—
44C2×(C3⋊Dic3)⋊C3D411yescharacteristic
45C4⋊((C3⋊D3)⋊C3)—21no—
46C4⋊((C3⋊D3)⋊C3)—21no—
47C4⋊((C3⋊D3)⋊C3)—42no—
48C4×(C3⋊D3)⋊C3—21no—
49C4×(C3⋊D3)⋊C3—21no—
50#634⋊C3—21no—
51#634⋊C3—21no—
52C8×H(3,𝔽3)D412yescomplemented
53(C3⋊(C3⋊C8))⋊C3—21no—
54C3×#468—91no—
55#4886—31no—
56#4886—91no—
57C2×C4⋊(C3⋊D3)—21no—
58C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
59C2×C4×C3⋊D3C22×C311yescharacteristic
60C2×#634C22×C311yescharacteristic
61C2×C3×D12—31no—
62C2×C3×D12—61no—
63C2×C3×C4×D3—31no—
64C2×C3×Dic6—31no—
65C2×C32×C8—21no—
66C2×C32×C8C2×D311yescharacteristic
67C2×C32×C8C22×C311yescharacteristic
68C32×C4⋊C4—21no—
69C32×C4⋊C4—21no—
70C32×C4⋊C4C2×D311yescharacteristic
71C32×C4⋊C4C2×D311yescharacteristic
72C32×C4⋊C4C22×C311yescharacteristic
73C32×C4⋊C4C22×C311yescharacteristic
74#3752—21no—
75#3752—21no—
76C4⋊(C3⋊Dic3)—21no—
77C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
78#3744—22no—
79#3743—22no—
80C3×(C2×D3)⋊C4—61no—
81C3×(C2×D3)⋊C4—61no—
82C3×C4⋊Dic3—61no—
83C2×C3×C3⋊C8—31no—
84C3×#204—62no—
85C3×ΓL(1,𝔽25)—62no—
86C22×H(3,𝔽3)C2×D411yescharacteristic
87C2×(C3⋊D3)⋊C3—21no—
88C2×(C3⋊D3)⋊C3—21no—
89C2×(C3⋊D3)⋊C3—21no—
90C2×(C3⋊D3)⋊C3—41no—
91C2×(C3⋊D3)⋊C3—41no—
92C2×(C3⋊D3)⋊C3—41no—
93C4×H(3,𝔽3)—41no—
94C4×H(3,𝔽3)—41no—
95C4×H(3,𝔽3)C2×D411yescharacteristic
96C4×H(3,𝔽3)#4111yescharacteristic
97(C3⋊Dic3)⋊C3—21no—
98(C3⋊Dic3)⋊C3—41no—
99C2×C3×SD16—91no—
100C2×C3×M16—91no—
101C3×C22⋊Q8—91no—
102C3×#121—91no—
103C2×#204—31no—
104C2×#204—91no—
105C2×ΓL(1,𝔽25)—31no—
106C2×ΓL(1,𝔽25)—91no—
107#897—31no—
108#897—91no—
109#895—31no—
110#895—91no—
111C3×#107—31no—
112C3×#107—31no—
113C3×#107—61no—
114C3×#107C3×D311yescharacteristic
115C3×#103—91no—
116C3×#102—91no—
117#810—31no—
118#810—91no—
119#809—31no—
120#809—91no—
121C22×C3⋊D3—21no—
122C22×C3⋊D3C3×D411yescharacteristic
123C22×C3×D3—31no—
124C22×C3×D3—61no—
125C2×C32×C4—21no—
126C2×C32×C4—21no—
127C2×C32×C4—21no—
128C2×C32×C4—21no—
129C2×C32×C4—21no—
130C2×C32×C4—41no—
131C2×C32×C4—41no—
132C2×C32×C4C22×D311yescharacteristic
133C2×C32×C4C23×C311yescharacteristic
134C2×C3⋊Dic3—21no—
135C2×C3⋊Dic3C3×D411yescharacteristic
136C4⋊(C3⋊D3)—21no—
137C4⋊(C3⋊D3)—21no—
138C4⋊(C3⋊D3)—42no—
139C4×C3⋊D3—21no—
140C4×C3⋊D3—21no—
141#634—21no—
142#634—21no—
143C2×C3×Dic3—31no—
144C2×C3×Dic3—61no—
145C3×D12—61no—
146C3×D12—61no—
147C3×D12—122no—
148C3×C4×D3—61no—
149C3×C4×D3—61no—
150C3×Dic6—61no—
151C3×Dic6—61no—
152C32×C8—22no—
153C32×C8D1212yescomplemented
154C32×C8C3×D412yescomplemented
155C3⋊(C3⋊C8)—21no—
156C3×C3⋊C8—61no—
157#468—91noSylow
158C2×H(3,𝔽3)#12111yescharacteristic
159C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
160C2×H(3,𝔽3)#13711yescharacteristic
161(C3⋊D3)⋊C3—41no—
162(C3⋊D3)⋊C3—81no—
163C2×C3×Q8—91no—
164C2×C3×D4—91no—
165C2×C3×D4—181no—
166C22×C3×C4—91no—
167C2×D12—31no—
168C2×D12—61no—
169C2×D12—91no—
170C2×D12—181no—
171C2×C4×D3—31no—
172C2×C4×D3—91no—
173C2×Dic6—31no—
174C2×Dic6—91no—
175C3×SD16—182no—
176C3×M16—182no—
177C2×C3×C8—31no—
178C2×C3×C8—31no—
179C2×C3×C8—61no—
180C2×C3×C8—91no—
181C2×C3×C8C2×C3×D311yescharacteristic
182C3×C4⋊C4—31no—
183C3×C4⋊C4—31no—
184C3×C4⋊C4—31no—
185C3×C4⋊C4—31no—
186C3×C4⋊C4—61no—
187C3×C4⋊C4—61no—
188C3×C4⋊C4—181no—
189C3×C4⋊C4C2×C3×D311yescharacteristic
190C3×C4⋊C4C2×C3×D311yescharacteristic
191C3×C22⋊C4—181no—
192C3×C22⋊C4—181no—
193(C2×D3)⋊C4—61no—
194(C2×D3)⋊C4—61no—
195(C2×D3)⋊C4—181no—
196(C2×D3)⋊C4—181no—
197C4⋊Dic3—61no—
198C4⋊Dic3—181no—
199C2×C3⋊C8—31no—
200C2×C3⋊C8—91no—
201#204—62no—
202#204—182no—
203ΓL(1,𝔽25)—62no—
204ΓL(1,𝔽25)—182no—
205C22×C32—21no—
206C22×C32D3×D411yescharacteristic
207C22×C32C2×C3×D411yescharacteristic
208C2×C3⋊D3—21no—
209C2×C3⋊D3—21no—
210C2×C3⋊D3—21no—
211C2×C3⋊D3—41no—
212C2×C3⋊D3—41no—
213C2×C3⋊D3—41no—
214C2×C3×D3—61no—
215C2×C3×D3—61no—
216C2×C3×D3—61no—
217C2×C3×D3—121no—
218C2×C3×D3—121no—
219C2×C3×D3—121no—
220C32×C4—21no—
221C32×C4—21no—
222C32×C4—41no—
223C32×C4—41no—
224C32×C4—41no—
225C32×C4—41no—
226C32×C4—81no—
227C32×C4—81no—
228C32×C4C2×D1211yescharacteristic
229C32×C4#23911yescharacteristic
230C32×C4C2×C3×D411yescharacteristic
231C32×C4C3×#4111yescharacteristic
232C3⋊Dic3—21no—
233C3⋊Dic3—41no—
234C3×Dic3—61no—
235C3×Dic3—121no—
236C2×SD16—91no—
237C2×M16—91no—
238C22⋊Q8—91no—
239#121—91no—
240#107(C3⋊D3)⋊C311yescharacteristic, complemented
241#103—91no—
242#102—91no—
243H(3,𝔽3)#46811yescharacteristic, complemented, Sylow
244C23×C3—91no—
245C23×C3—181no—
246C22×D3—31no—
247C22×D3—61no—
248C22×D3—91no—
249C22×D3—181no—
250C3×Q8—181no—
251C3×Q8—181no—
252C3×D4—181no—
253C3×D4—181no—
254C3×D4—362no—
255C2×C3×C4—21no—
256C2×C3×C4—21no—
257C2×C3×C4—31no—
258C2×C3×C4—31no—
259C2×C3×C4—61no—
260C2×C3×C4—61no—
261C2×C3×C4—61no—
262C2×C3×C4—61no—
263C2×C3×C4—61no—
264C2×C3×C4—91no—
265C2×C3×C4—121no—
266C2×C3×C4—121no—
267C2×C3×C4—181no—
268C2×C3×C4—181no—
269C2×C3×C4—181no—
270C2×C3×C4C22×C3×D311yescharacteristic
271C2×Dic3—31no—
272C2×Dic3—61no—
273C2×Dic3—91no—
274C2×Dic3—181no—
275D12—61no—
276D12—61no—
277D12—122no—
278D12—181no—
279D12—181no—
280D12—362no—
281C4×D3—61no—
282C4×D3—61no—
283C4×D3—181no—
284C4×D3—181no—
285Dic6—61no—
286Dic6—61no—
287Dic6—181no—
288Dic6—181no—
289C3×C8—32no—
290C3×C8—32no—
291C3×C8—62no—
292C3×C8—181no—
293C3×C8C3×D1212yes—
294C3⋊C8—61no—
295C3⋊C8—181no—
296C2×C32—21no—
297C2×C32—21no—
298C2×C32—21no—
299C2×C32#89511yescharacteristic
300C2×C32#91111yescharacteristic
301C2×C32#91811yescharacteristic
302C2×C32C3×#12111yescharacteristic
303C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
304C2×C32C3×#13711yescharacteristic
305C3⋊D3—41no—
306C3⋊D3—81no—
307C3×D3—121no—
308C3×D3—241no—
309C2×Q8—91no—
310C2×D4—91no—
311C2×D4—181no—
312C22×C4—91no—
313SD16—182no—
314M16—182no—
315C2×C8—91no—
316C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
317C4⋊C4—181no—
318C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
319C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
320C22⋊C4—181no—
321C22⋊C4—181no—
322C22×C3—31no—
323C22×C3—31no—
324C22×C3—61no—
325C22×C3—181no—
326C22×C3—181no—
327C22×C3—181no—
328C22×C3—361no—
329C22×C3—361no—
330C22×C3—361no—
331C22×C3C3×D3×D411yescharacteristic
332C2×D3—61no—
333C2×D3—61no—
334C2×D3—61no—
335C2×D3—121no—
336C2×D3—121no—
337C2×D3—121no—
338C2×D3—181no—
339C2×D3—181no—
340C2×D3—181no—
341C2×D3—361no—
342C2×D3—361no—
343C2×D3—361no—
344C3×C4—31no—
345C3×C4—31no—
346C3×C4—31no—
347C3×C4—31no—
348C3×C4—41no—
349C3×C4—41no—
350C3×C4—61no—
351C3×C4—61no—
352C3×C4—121no—
353C3×C4—121no—
354C3×C4—121no—
355C3×C4—121no—
356C3×C4—181no—
357C3×C4—241no—
358C3×C4—241no—
359C3×C4—361no—
360C3×C4C2×C3×D1211yescharacteristic
361C3×C4C3×#23911yescharacteristic
362Dic3—61no—
363Dic3—121no—
364Dic3—181no—
365Dic3—361no—
366C32—21no—
367C32#488611yescharacteristic, complemented
368C32C3×#46811yescharacteristic, complemented
369C23—91no—
370C23—181no—
371Q8—181no—
372Q8—181no—
373D4—181no—
374D4—181no—
375D4—362no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—91no—
379C2×C4—181no—
380C2×C4—181no—
381C2×C4—181no—
382C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
383C8—181no—
384C8C4⋊((C3⋊D3)⋊C3)12yescomplemented
385C2×C3—31no—
386C2×C3—31no—
387C2×C3—31no—
388C2×C3—31no—
389C2×C3—31no—
390C2×C3—31no—
391C2×C3—61no—
392C2×C3—61no—
393C2×C3—61no—
394C2×C3—361no—
395C2×C3—721no—
396C2×C3C3×#89511yescharacteristic
397C2×C3C3×#91111yescharacteristic
398C2×C3C3×#91811yescharacteristic
399D3—121no—
400D3—241no—
401D3—361no—
402D3—721no—
403C22—181no—
404C22—181no—
405C22—181no—
406C22—361no—
407C22—361no—
408C22—361no—
409C22D4×(C3⋊D3)⋊C311yescharacteristic
410C4—41no—
411C4—41no—
412C4—181no—
413C4—361no—
414C4C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
415C4#8872211yescharacteristic
416C3—31no—
417C3—31no—
418C3—61no—
419C3C3×#488611yescharacteristic
420C2—361no—
421C2—721no—
422C2#64104⋊C311yescharacteristic
423C2#64120⋊C311yescharacteristic
424C2#64127⋊C311yescharacteristic
subgroupG/H
centerC22D4×(C3⋊D3)⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#107C2
socleC22×C3C3×D3×D4
prime radical#64104⋊C3C2
hypercenter#107(C3⋊D3)⋊C3
hypocenterC32C3×#468
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ #107

Complements

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

NH ≅ G/Ntype
1. C2×#3744⋊C3C2semidirect
2. C2×#3743⋊C3C2semidirect
3. #64106⋊C3C2semidirect
5. H(3,𝔽3)×#107C2semidirect
6. #63636⋊C3C2semidirect
7. #63637⋊C3C2semidirect
8. #10592585C3semidirect
23. C2×#3744C2×C3semidirect
24. C2×#3743C2×C3semidirect
25. #64106C2×C3semidirect
32. C32×#107D3semidirect
33. C32×#107C2×C3semidirect
34. #63637C2×C3semidirect
35. #63636C2×C3semidirect
52. C8×H(3,𝔽3)D4semidirect
153. C32×C8D12semidirect
154. C32×C8C3×D4semidirect
240. #107(C3⋊D3)⋊C3semidirect
243. H(3,𝔽3)#468semidirect
367. C32#4886semidirect
368. C32C3×#468semidirect
384. C8C4⋊((C3⋊D3)⋊C3)semidirect

Maximal subgroups

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

subgroupindex
C2×#3744⋊C32
C2×#3743⋊C32
#64106⋊C32
#64104⋊C32
H(3,𝔽3)×#1072
#63636⋊C32
#63637⋊C32
#105925853
C3×#48863