Finite Group Databasebeta

← #10592544⋊C3 #13142052

Subgroups — #10592544⋊C3 #13142052

Order 1728.

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

normal subgroups only

This group has 430 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
2C2×#3746⋊C3C211yesmaximal, characteristic, complemented
3#64084⋊C3C211yesmaximal, characteristic
4C4×#634⋊C3C211yesmaximal, characteristic, complemented
5H(3,𝔽3)×C4⋊C8C211yesmaximal, characteristic, complemented
6#63648⋊C3C211yesmaximal, characteristic
7#63643⋊C3C211yesmaximal, characteristic, complemented
8#10592544C311yesmaximal, characteristic, complemented
9C3×#4845—31nomaximal
10C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
11C2×#634⋊C3C2211yescharacteristic
12C2×#634⋊C3C2211yescharacteristic
13C2×C8×H(3,𝔽3)C2211yescharacteristic
14C2×C8×H(3,𝔽3)C2211yescharacteristic
15C42×H(3,𝔽3)C2211yescharacteristic
16#3752⋊C3—21no—
17#3752⋊C3—21no—
18C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
19#3750⋊C3—21no—
20C4×(C3⋊Dic3)⋊C3—21no—
21#3744⋊C3—22no—
22#3746⋊C3—22no—
23C2×#3746C2×C311yescharacteristic, complemented
24C2×#3744C2×C311yescharacteristic
25#64084C2×C311yescharacteristic
26C4×#634C2×C311yescharacteristic, complemented
27C2×C3×Dic12—31no—
28C2×C3×#204—31no—
29C3×#875—31no—
30C3×C4×Dic6—31no—
31C32×C4⋊C8—21no—
32C32×C4⋊C8D311yescharacteristic, complemented
33C32×C4⋊C8C2×C311yescharacteristic, complemented
34#63648C2×C311yescharacteristic
35#63643C2×C311yescharacteristic, complemented
36C3×#821—31no—
37C3×#816—31no—
38C22×(C3⋊D3)⋊C3—21no—
39C2×C4×H(3,𝔽3)D411yescharacteristic
40C2×C4×H(3,𝔽3)D411yescharacteristic
41C2×C4×H(3,𝔽3)C2311yescharacteristic
42C2×(C3⋊Dic3)⋊C3—21no—
43C2×(C3⋊Dic3)⋊C3—21no—
44C2×(C3⋊Dic3)⋊C3—21no—
45C4⋊((C3⋊D3)⋊C3)—21no—
46C4⋊((C3⋊D3)⋊C3)—21no—
47#634⋊C3—21no—
48#634⋊C3—21no—
49#634⋊C3—21no—
50#634⋊C3D412yes—
51C8×H(3,𝔽3)—21no—
52C8×H(3,𝔽3)—21no—
53C3×#464—91no—
54#4845—31no—
55#4845—91no—
56C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
57C2×#634C22×C311yescharacteristic
58C2×#634C22×C311yescharacteristic
59C2×C3×D12—31no—
60C2×C3×Dic6—31no—
61C2×C3×Dic6—31no—
62C2×C32×C8—21no—
63C2×C32×C8—21no—
64C2×C32×C8C2×D311yescharacteristic
65C2×C32×C8C2×D311yescharacteristic
66C2×C32×C8C22×C311yescharacteristic
67C2×C32×C8C22×C311yescharacteristic
68C32×C42—21no—
69C32×C42C2×D311yescharacteristic
70C32×C42C22×C311yescharacteristic
71#3752—21no—
72#3752—21no—
73C4⋊(C3⋊Dic3)C22×C311yescharacteristic
74#3750—21no—
75C4×C3⋊Dic3—21no—
76#3746—22no—
77#3744—22no—
78C3×(C2×D3)⋊C4—61no—
79C3×(C2×D3)⋊C4—61no—
80C3×C4⋊Dic3—31no—
81C3×Dic3⋊C4—61no—
82C3×C4×Dic3—61no—
83C3×Dic12—62no—
84C3×#204—62no—
85C22×H(3,𝔽3)C2×D411yescharacteristic
86C2×(C3⋊D3)⋊C3—41no—
87C2×(C3⋊D3)⋊C3—41no—
88C2×(C3⋊D3)⋊C3—41no—
89C4×H(3,𝔽3)—21no—
90C4×H(3,𝔽3)—21no—
91C4×H(3,𝔽3)C2×D411yescharacteristic
92C4×H(3,𝔽3)#4111yescharacteristic
93(C3⋊Dic3)⋊C3—22no—
94(C3⋊Dic3)⋊C3—41no—
95(C3⋊Dic3)⋊C3—41no—
96C2×C3×Q16—91no—
97C2×C3×SD16—91no—
98C3×#124—91no—
99C3×C4×Q8—91no—
100C2×Dic12—31no—
101C2×Dic12—91no—
102C2×#204—31no—
103C2×#204—91no—
104#875—31no—
105#875—91no—
106C4×Dic6—31no—
107C4×Dic6—91no—
108C3×C4⋊C8—31no—
109C3×C4⋊C8—31no—
110C3×C4⋊C8—61no—
111C3×C4⋊C8C3×D311yescharacteristic
112C3×#103—91no—
113C3×#102—91no—
114#821—31no—
115#821—91no—
116#816—31no—
117#816—91no—
118C22×C3⋊D3—21no—
119C22×C3×D3—61no—
120C2×C32×C4—21no—
121C2×C32×C4—21no—
122C2×C32×C4—21no—
123C2×C32×C4D1211yescharacteristic
124C2×C32×C4D1211yescharacteristic
125C2×C32×C4C3×D411yescharacteristic
126C2×C32×C4C3×D411yescharacteristic
127C2×C32×C4C22×D311yescharacteristic
128C2×C32×C4C23×C311yescharacteristic
129C2×C3⋊Dic3—21no—
130C2×C3⋊Dic3—21no—
131C2×C3⋊Dic3—21no—
132C4⋊(C3⋊D3)—21no—
133C4⋊(C3⋊D3)—21no—
134#634—21no—
135#634—21no—
136#634—21no—
137#634C3×D412yes—
138C2×C3×Dic3—61no—
139C2×C3×Dic3—61no—
140C2×C3×Dic3—61no—
141C3×D12—61no—
142C3×D12—61no—
143C3×Dic6—32no—
144C3×Dic6—61no—
145C3×Dic6—61no—
146C3×Dic6—61no—
147C32×C8—21no—
148C32×C8—21no—
149C32×C8—21no—
150C32×C8—21no—
151C32×C8—41no—
152C32×C8—41no—
153#464—91noSylow
154C2×H(3,𝔽3)#12111yescharacteristic
155C2×H(3,𝔽3)#13511yescharacteristic
156C2×H(3,𝔽3)#13711yescharacteristic
157(C3⋊D3)⋊C3—81no—
158C2×C3×Q8—91no—
159C2×C3×Q8—91no—
160C2×C3×D4—91no—
161C2×D12—31no—
162C2×D12—91no—
163C2×Dic6—31no—
164C2×Dic6—31no—
165C2×Dic6—91no—
166C2×Dic6—91no—
167C3×Q16—182no—
168C3×SD16—182no—
169C2×C3×C8—31no—
170C2×C3×C8—31no—
171C2×C3×C8—31no—
172C2×C3×C8—31no—
173C2×C3×C8—61no—
174C2×C3×C8—61no—
175C2×C3×C8C2×C3×D311yescharacteristic
176C2×C3×C8C2×C3×D311yescharacteristic
177C3×C4⋊C4—91no—
178C3×C4⋊C4—181no—
179C3×C22⋊C4—181no—
180C3×C22⋊C4—181no—
181C3×C42—31no—
182C3×C42—31no—
183C3×C42—61no—
184C3×C42—181no—
185C3×C42C2×C3×D311yescharacteristic
186(C2×D3)⋊C4—61no—
187(C2×D3)⋊C4—61no—
188(C2×D3)⋊C4—181no—
189(C2×D3)⋊C4—181no—
190C4⋊Dic3—31no—
191C4⋊Dic3—91no—
192Dic3⋊C4—61no—
193Dic3⋊C4—181no—
194C4×Dic3—61no—
195C4×Dic3—181no—
196Dic12—62no—
197Dic12—182no—
198#204—62no—
199#204—182no—
200C22×C32—21no—
201C22×C32C2×D1211yescharacteristic
202C22×C32C2×C3×D411yescharacteristic
203C2×C3⋊D3—41no—
204C2×C3⋊D3—41no—
205C2×C3⋊D3—41no—
206C2×C3×D3—121no—
207C2×C3×D3—121no—
208C2×C3×D3—121no—
209C32×C4—22no—
210C32×C4—21no—
211C32×C4—21no—
212C32×C4—21no—
213C32×C4—21no—
214C32×C4—21no—
215C32×C4—21no—
216C32×C4—41no—
217C32×C4D3×D411yescharacteristic
218C32×C4#23911yescharacteristic
219C32×C4C2×C3×D411yescharacteristic
220C32×C4C3×#4111yescharacteristic
221C3⋊Dic3—22no—
222C3⋊Dic3—41no—
223C3⋊Dic3—41no—
224C3×Dic3—62no—
225C3×Dic3—121no—
226C3×Dic3—121no—
227C2×Q16—91no—
228C2×SD16—91no—
229#124—91no—
230C4×Q8—91no—
231C4⋊C8(C3⋊D3)⋊C311yescharacteristic, complemented
232#103—91no—
233#102—91no—
234H(3,𝔽3)#46411yescharacteristic, complemented, Sylow
235C23×C3—181no—
236C22×D3—61no—
237C22×D3—181no—
238C3×Q8—92no—
239C3×Q8—181no—
240C3×Q8—181no—
241C3×Q8—181no—
242C3×D4—181no—
243C3×D4—181no—
244C2×C3×C4—31no—
245C2×C3×C4—31no—
246C2×C3×C4—31no—
247C2×C3×C4—31no—
248C2×C3×C4—31no—
249C2×C3×C4—31no—
250C2×C3×C4—61no—
251C2×C3×C4—61no—
252C2×C3×C4—61no—
253C2×C3×C4—181no—
254C2×C3×C4—181no—
255C2×C3×C4—181no—
256C2×C3×C4C3×D1211yescharacteristic
257C2×C3×C4C3×D1211yescharacteristic
258C2×C3×C4C22×C3×D311yescharacteristic
259C2×Dic3—61no—
260C2×Dic3—61no—
261C2×Dic3—61no—
262C2×Dic3—181no—
263C2×Dic3—181no—
264C2×Dic3—181no—
265D12—61no—
266D12—61no—
267D12—181no—
268D12—181no—
269Dic6—32no—
270Dic6—61no—
271Dic6—61no—
272Dic6—61no—
273Dic6—92no—
274Dic6—181no—
275Dic6—181no—
276Dic6—181no—
277C3×C8—21no—
278C3×C8—21no—
279C3×C8—61no—
280C3×C8—61no—
281C3×C8—61no—
282C3×C8—61no—
283C3×C8—121no—
284C3×C8—121no—
285C2×C32—21no—
286C2×C32—21no—
287C2×C32—21no—
288C2×C32#89511yescharacteristic
289C2×C32#90411yescharacteristic
290C2×C32#90911yescharacteristic
291C2×C32C3×#12111yescharacteristic
292C2×C32C3×#13511yescharacteristic
293C2×C32C3×#13711yescharacteristic
294C3⋊D3—81no—
295C3×D3—241no—
296C2×Q8—91no—
297C2×Q8—91no—
298C2×D4—91no—
299Q16—182no—
300SD16—182no—
301C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
302C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
303C4⋊C4—91no—
304C4⋊C4—181no—
305C22⋊C4—181no—
306C22⋊C4—181no—
307C42—181no—
308C42C2×(C3⋊D3)⋊C311yescharacteristic
309C22×C3—31no—
310C22×C3—31no—
311C22×C3—61no—
312C22×C3—361no—
313C22×C3—361no—
314C22×C3—361no—
315C22×C3C2×C3×D1211yescharacteristic
316C2×D3—121no—
317C2×D3—121no—
318C2×D3—121no—
319C2×D3—361no—
320C2×D3—361no—
321C2×D3—361no—
322C3×C4—21no—
323C3×C4—21no—
324C3×C4—31no—
325C3×C4—31no—
326C3×C4—31no—
327C3×C4—31no—
328C3×C4—61no—
329C3×C4—61no—
330C3×C4—61no—
331C3×C4—61no—
332C3×C4—62no—
333C3×C4—61no—
334C3×C4—61no—
335C3×C4—121no—
336C3×C4—182no—
337C3×C4—361no—
338C3×C4—361no—
339C3×C4C3×D3×D411yescharacteristic
340C3×C4C3×#23911yescharacteristic
341Dic3—62no—
342Dic3—121no—
343Dic3—121no—
344Dic3—182no—
345Dic3—361no—
346Dic3—361no—
347C32—21no—
348C32#484511yescharacteristic, complemented
349C32C3×#46411yescharacteristic, complemented
350C23—181no—
351Q8—92no—
352Q8—181no—
353Q8—181no—
354Q8—181no—
355D4—181no—
356D4—181no—
357C2×C4—181no—
358C2×C4—181no—
359C2×C4—181no—
360C2×C4C4⋊((C3⋊D3)⋊C3)11yescharacteristic
361C2×C4C4⋊((C3⋊D3)⋊C3)11yescharacteristic
362C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
363C8—21no—
364C8—21no—
365C2×C3—31no—
366C2×C3—31no—
367C2×C3—31no—
368C2×C3—31no—
369C2×C3—31no—
370C2×C3—31no—
371C2×C3—61no—
372C2×C3—61no—
373C2×C3—61no—
374C2×C3—721no—
375C2×C3C3×#89511yescharacteristic
376C2×C3C3×#90411yescharacteristic
377C2×C3C3×#90911yescharacteristic
378D3—241no—
379D3—721no—
380C22—361no—
381C22—361no—
382C22—361no—
383C22C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
384C4—21no—
385C4—21no—
386C4—182no—
387C4—361no—
388C4—361no—
389C4D4×(C3⋊D3)⋊C311yescharacteristic
390C4#8872211yescharacteristic
391C3—31no—
392C3—31no—
393C3—61no—
394C3C3×#484511yescharacteristic
395C2—721no—
396C2#64104⋊C311yescharacteristic
397C2#64113⋊C311yescharacteristic
398C2#64118⋊C311yescharacteristic
subgroupG/H
centerC22C2×C4⋊((C3⋊D3)⋊C3)
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×C4⋊C8C2
socleC22×C3C2×C3×D12
prime radicalC2×C4⋊((C3⋊D3)⋊C3)C22
hypercenterC4⋊C8(C3⋊D3)⋊C3
hypocenterC32C3×#464
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×#3746⋊C3C2semidirect
4. C4×#634⋊C3C2semidirect
5. H(3,𝔽3)×C4⋊C8C2semidirect
7. #63643⋊C3C2semidirect
8. #10592544C3semidirect
23. C2×#3746C2×C3semidirect
26. C4×#634C2×C3semidirect
32. C32×C4⋊C8D3semidirect
33. C32×C4⋊C8C2×C3semidirect
35. #63643C2×C3semidirect
231. C4⋊C8(C3⋊D3)⋊C3semidirect
234. H(3,𝔽3)#464semidirect
348. C32#4845semidirect
349. C32C3×#464semidirect

Maximal subgroups

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

subgroupindex
C2×#3744⋊C32
C2×#3746⋊C32
#64084⋊C32
C4×#634⋊C32
H(3,𝔽3)×C4⋊C82
#63648⋊C32
#63643⋊C32
#105925443
C3×#48453