Finite Group Databasebeta

← #13137392

Subgroups — #13137392

Order 1728.

472 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 79 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#64084⋊C3C211yesmaximal, characteristic, complemented
2C4×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic, complemented
3C2×#88432C211yesmaximal, characteristic
4C2×#88431C211yesmaximal, characteristic, complemented
5C4⋊#6180C211yesmaximal, characteristic, complemented
6#11720318C211yesmaximal, characteristic, complemented
7#11720313C211yesmaximal, characteristic, complemented
8#10590366—31nomaximal
9#10590356—31nomaximal
10C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
11C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
12C2×C4×(C3⋊D3)⋊C3—21no—
13C2×#634⋊C3C2211yescharacteristic
14C42×H(3,𝔽3)C2211yescharacteristic
15C2×#6180C2211yescharacteristic
16C2×#6180C2211yescharacteristic
17#3752⋊C3—21no—
18#3752⋊C3—21no—
19#3752⋊C3—21no—
20C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
21#88432—22no—
22#88431—22no—
23#64084D311yescharacteristic, complemented
24C4×C4⋊(C3⋊D3)D311yescharacteristic, complemented
25C3×#875—31no—
26C3×C4×D12—31no—
27C2×#3717—31no—
28C2×#3716—31no—
29C2×#3714—31no—
30C2×#3714—31no—
31C3×C4⋊(C3⋊C8)—31no—
32C3×C4⋊(C3⋊C8)—31no—
33C3×C4⋊(C3⋊C8)—61no—
34#63567—31no—
35#63566—31no—
36#63564—31no—
37#63563—31no—
38C22×(C3⋊D3)⋊C3—21no—
39C22×(C3⋊D3)⋊C3—21no—
40C2×C4×H(3,𝔽3)D411yescharacteristic
41C2×C4×H(3,𝔽3)D411yescharacteristic
42C2×C4×H(3,𝔽3)C2311yescharacteristic
43C2×(C3⋊Dic3)⋊C3—21no—
44C2×(C3⋊Dic3)⋊C3—21no—
45C4⋊((C3⋊D3)⋊C3)—21no—
46C4⋊((C3⋊D3)⋊C3)—21no—
47C4⋊((C3⋊D3)⋊C3)—21no—
48C4⋊((C3⋊D3)⋊C3)D412yes—
49C4×(C3⋊D3)⋊C3—42no—
50#634⋊C3—21no—
51#634⋊C3—21no—
52#6180—21no—
53#6180—21no—
54#5057—91no—
55#5016—91no—
56#4844—91no—
57C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
58C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
59C2×C4×C3⋊D3—21no—
60C2×#634C2×D311yescharacteristic
61C2×C3×D12—31no—
62C2×C3×D12—31no—
63C2×C3×C4×D3—61no—
64C2×C3×Dic6—31no—
65C32×C42—21no—
66C32×C42C2×D311yescharacteristic
67C32×C42C2×D311yescharacteristic
68#3752—21no—
69#3752—21no—
70#3752—21no—
71C4⋊(C3⋊Dic3)C2×D311yescharacteristic
72C3×(C2×D3)⋊C4—61no—
73C3×(C2×D3)⋊C4—61no—
74C3×(C2×D3)⋊C4—61no—
75C3×C4⋊Dic3—31no—
76C2×C3×C3⋊C8—31no—
77C2×C3×C3⋊C8—31no—
78C2×C3×C3⋊C8—31no—
79C2×C3×C3⋊C8—31no—
80C2×C3×C3⋊C8—61no—
81C2×C3×C3⋊C8—61no—
82#3717—62no—
83#3716—62no—
84#3714—62no—
85#3714—62no—
86C22×H(3,𝔽3)C2×D411yescharacteristic
87C2×(C3⋊D3)⋊C3—22no—
88C2×(C3⋊D3)⋊C3—41no—
89C2×(C3⋊D3)⋊C3—41no—
90C2×(C3⋊D3)⋊C3—41no—
91C2×(C3⋊D3)⋊C3—41no—
92C2×(C3⋊D3)⋊C3—41no—
93C4×H(3,𝔽3)—21no—
94C4×H(3,𝔽3)—21no—
95C4×H(3,𝔽3)C2×D411yescharacteristic
96C4×H(3,𝔽3)#4111yescharacteristic
97(C3⋊Dic3)⋊C3—41no—
98(C3⋊Dic3)⋊C3—41no—
99C3×#124—91no—
100C3×C4×D4—91no—
101C2×#215—91no—
102C2×#214—91no—
103C2×#213—91no—
104C2×#213—91no—
105C2×D24—91no—
106C2×#204—91no—
107#875—31no—
108#875—91no—
109C4×D12—31no—
110C4×D12—91no—
111C3×C4⋊C8—91no—
112#835—91no—
113#832—91no—
114#821—91no—
115#816—91no—
116#810—91no—
117#809—91no—
118C4⋊(C3⋊C8)—31no—
119C4⋊(C3⋊C8)—31no—
120C4⋊(C3⋊C8)—61no—
121C22×C3⋊D3—21no—
122C22×C3⋊D3—21no—
123C22×C3×D3—61no—
124C22×C3×D3—61no—
125C2×C32×C4—21no—
126C2×C32×C4—21no—
127C2×C32×C4—21no—
128C2×C32×C4#6711yescharacteristic
129C2×C32×C4#6711yescharacteristic
130C2×C32×C4#6711yescharacteristic
131C2×C32×C4#6711yescharacteristic
132C2×C32×C4C22×D311yescharacteristic
133C2×C32×C4C22×D311yescharacteristic
134C2×C3⋊Dic3—21no—
135C2×C3⋊Dic3—21no—
136C4⋊(C3⋊D3)—21no—
137C4⋊(C3⋊D3)—21no—
138C4⋊(C3⋊D3)—21no—
139C4⋊(C3⋊D3)D1212yes—
140C4×C3⋊D3—42no—
141#634—21no—
142#634—21no—
143C2×C3×Dic3—61no—
144C2×C3×Dic3—61no—
145C3×D12—32no—
146C3×D12—61no—
147C3×D12—61no—
148C3×D12—61no—
149C3×C4×D3—122no—
150C3×Dic6—61no—
151C3×Dic6—61no—
152C3×C3⋊C8—61no—
153C3×C3⋊C8—61no—
154C3×C3⋊C8—61no—
155C3×C3⋊C8—61no—
156C3×C3⋊C8—121no—
157C3×C3⋊C8—121no—
158#463—271noSylow
159C2×H(3,𝔽3)#12111yescharacteristic
160C2×H(3,𝔽3)#13511yescharacteristic
161C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
162(C3⋊D3)⋊C3—42no—
163(C3⋊D3)⋊C3—81no—
164C2×C3×Q8—91no—
165C2×C3×D4—91no—
166C2×C3×D4—91no—
167C22×C3×C4—181no—
168C2×D12—31no—
169C2×D12—31no—
170C2×D12—91no—
171C2×D12—91no—
172C2×C4×D3—61no—
173C2×C4×D3—181no—
174C2×Dic6—31no—
175C2×Dic6—91no—
176C2×C3×C8—91no—
177C2×C3×C8—91no—
178C3×C4⋊C4—91no—
179C3×C22⋊C4—181no—
180C3×C22⋊C4—181no—
181C3×C22⋊C4—181no—
182C3×C42—31no—
183C3×C42—31no—
184C3×C42—61no—
185C3×C42D3211yescharacteristic
186#215—182no—
187#214—182no—
188#213—182no—
189#213—182no—
190(C2×D3)⋊C4—61no—
191(C2×D3)⋊C4—61no—
192(C2×D3)⋊C4—61no—
193(C2×D3)⋊C4—181no—
194(C2×D3)⋊C4—181no—
195(C2×D3)⋊C4—181no—
196C4⋊Dic3—31no—
197C4⋊Dic3—91no—
198C2×C3⋊C8—31no—
199C2×C3⋊C8—31no—
200C2×C3⋊C8—31no—
201C2×C3⋊C8—31no—
202C2×C3⋊C8—61no—
203C2×C3⋊C8—61no—
204D24—182no—
205#204—182no—
206C22×C32—21no—
207C22×C32C2×#6711yescharacteristic
208C22×C32C2×#6711yescharacteristic
209C2×C3⋊D3—22no—
210C2×C3⋊D3—41no—
211C2×C3⋊D3—41no—
212C2×C3⋊D3—41no—
213C2×C3⋊D3—41no—
214C2×C3⋊D3—41no—
215C2×C3×D3—62no—
216C2×C3×D3—121no—
217C2×C3×D3—121no—
218C2×C3×D3—121no—
219C2×C3×D3—121no—
220C2×C3×D3—121no—
221C32×C4—22no—
222C32×C4—21no—
223C32×C4—21no—
224C32×C4—21no—
225C32×C4—21no—
226C32×C4—21no—
227C32×C4—21no—
228C32×C4—41no—
229C32×C4C2×D1211yescharacteristic
230C32×C4#23511yescharacteristic
231C32×C4D3×D411yescharacteristic
232C32×C4#23711yescharacteristic
233C3⋊Dic3—41no—
234C3⋊Dic3—41no—
235C3×Dic3—121no—
236C3×Dic3—121no—
237C2×SD16—271no—
238C2×D8—271no—
239#124—91no—
240C4×D4—91no—
241C4⋊C8—91no—
242#103—271no—
243#102—271no—
244H(3,𝔽3)#46311yescharacteristic, complemented, Sylow
245C23×C3—181no—
246C23×C3—181no—
247C22×D3—61no—
248C22×D3—61no—
249C22×D3—181no—
250C22×D3—181no—
251C3×Q8—181no—
252C3×Q8—181no—
253C3×D4—92no—
254C3×D4—181no—
255C3×D4—181no—
256C3×D4—181no—
257C2×C3×C4—31no—
258C2×C3×C4—31no—
259C2×C3×C4—31no—
260C2×C3×C4—31no—
261C2×C3×C4—31no—
262C2×C3×C4—31no—
263C2×C3×C4—61no—
264C2×C3×C4—61no—
265C2×C3×C4—61no—
266C2×C3×C4—181no—
267C2×C3×C4—181no—
268C2×C3×C4—362no—
269C2×C3×C4#62511yescharacteristic
270C2×C3×C4#62511yescharacteristic
271C2×C3×C4C2×D3211yescharacteristic
272C2×Dic3—61no—
273C2×Dic3—61no—
274C2×Dic3—181no—
275C2×Dic3—181no—
276D12—32no—
277D12—61no—
278D12—61no—
279D12—61no—
280D12—92no—
281D12—181no—
282D12—181no—
283D12—181no—
284C4×D3—122no—
285C4×D3—362no—
286Dic6—61no—
287Dic6—61no—
288Dic6—181no—
289Dic6—181no—
290C3×C8—181no—
291C3×C8—181no—
292C3⋊C8—61no—
293C3⋊C8—61no—
294C3⋊C8—61no—
295C3⋊C8—61no—
296C3⋊C8—121no—
297C3⋊C8—121no—
298C2×C32—21no—
299C2×C32—21no—
300C2×C32—21no—
301C2×C32#93011yescharacteristic
302C2×C32#93211yescharacteristic
303C2×C32#93811yescharacteristic
304C2×C32#94911yescharacteristic
305C2×C32#95011yescharacteristic
306C2×C32#95011yescharacteristic
307C3⋊D3—42no—
308C3⋊D3—81no—
309C3×D3—122no—
310C3×D3—241no—
311C2×Q8—91no—
312C2×D4—91no—
313C2×D4—91no—
314C22×C4—181no—
315SD16—542no—
316D8—542no—
317C2×C8—91no—
318C2×C8—91no—
319C4⋊C4—91no—
320C22⋊C4—181no—
321C22⋊C4—181no—
322C22⋊C4—181no—
323C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
324C22×C3—31no—
325C22×C3—31no—
326C22×C3—61no—
327C22×C3—182no—
328C22×C3—361no—
329C22×C3—361no—
330C22×C3—361no—
331C22×C3—361no—
332C22×C3—361no—
333C22×C3C2×#62511yescharacteristic
334C2×D3—62no—
335C2×D3—121no—
336C2×D3—121no—
337C2×D3—121no—
338C2×D3—121no—
339C2×D3—121no—
340C2×D3—182no—
341C2×D3—361no—
342C2×D3—361no—
343C2×D3—361no—
344C2×D3—361no—
345C2×D3—361no—
346C3×C4—21no—
347C3×C4—21no—
348C3×C4—31no—
349C3×C4—31no—
350C3×C4—31no—
351C3×C4—31no—
352C3×C4—61no—
353C3×C4—61no—
354C3×C4—61no—
355C3×C4—61no—
356C3×C4—62no—
357C3×C4—61no—
358C3×C4—61no—
359C3×C4—121no—
360C3×C4—361no—
361C3×C4—361no—
362C3×C4#379511yescharacteristic
363C3×C4D3×D1211yescharacteristic
364Dic3—121no—
365Dic3—121no—
366Dic3—361no—
367Dic3—361no—
368C32—21no—
369C32#501611yescharacteristic, complemented
370C32#505711yescharacteristic, complemented
371C23—181no—
372C23—181no—
373Q8—181no—
374Q8—181no—
375D4—92no—
376D4—181no—
377D4—181no—
378D4—181no—
379C2×C4—181no—
380C2×C4—181no—
381C2×C4—362no—
382C2×C4#619811yescharacteristic
383C2×C4#619811yescharacteristic
384C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
385C8—181no—
386C8—181no—
387C2×C3—31no—
388C2×C3—31no—
389C2×C3—31no—
390C2×C3—31no—
391C2×C3—31no—
392C2×C3—31no—
393C2×C3—61no—
394C2×C3—61no—
395C2×C3—61no—
396C2×C3—362no—
397C2×C3—721no—
398C2×C3#6382211yescharacteristic
399C2×C3#6382311yescharacteristic
400C2×C3#6390811yescharacteristic
401D3—122no—
402D3—241no—
403D3—362no—
404D3—721no—
405C22—182no—
406C22—361no—
407C22—361no—
408C22—361no—
409C22—361no—
410C22—361no—
411C22C2×#619811yescharacteristic
412C4—21no—
413C4—21no—
414C4—361no—
415C4—361no—
416C4#8864811yescharacteristic
417C4#8865411yescharacteristic
418C3—31no—
419C3—31no—
420C3—61no—
421C3#1059034811yescharacteristic
422C2—362no—
423C2—721no—
424C2#1172106011yescharacteristic
425C2#1172106111yescharacteristic
426C2#1172114111yescharacteristic
subgroupG/H
centerC22C2×#6198
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC42×H(3,𝔽3)C22
socleC22×C3C2×#625
prime radicalC2×#88432C2
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#463
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. #64084⋊C3C2semidirect
2. C4×C4⋊((C3⋊D3)⋊C3)C2semidirect
4. C2×#88431C2semidirect
5. C4⋊#6180C2semidirect
6. #11720318C2semidirect
7. #11720313C2semidirect
23. #64084D3semidirect
24. C4×C4⋊(C3⋊D3)D3semidirect
244. H(3,𝔽3)#463semidirect
369. C32#5016semidirect
370. C32#5057semidirect

Maximal subgroups

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

subgroupindex
#64084⋊C32
C4×C4⋊((C3⋊D3)⋊C3)2
C2×#884322
C2×#884312
C4⋊#61802
#117203182
#117203132
#105903663
#105903563