Finite Group Databasebeta

← #13138768

Subgroups — #13138768

Order 1728.

479 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 479 conjugacy classes of subgroups — too many to draw. Only its 101 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×#3743⋊C3C211yesmaximal, characteristic
2C4×((C3⋊D3)⋊C3)⋊C4C211yesmaximal, characteristic
3C2×#88430C211yesmaximal, characteristic
4#11720547C211yesmaximal, characteristic, complemented
5#63641⋊C3C211yesmaximal, characteristic, complemented
6#11720311C211yesmaximal, characteristic, complemented
7#11720304C211yesmaximal, characteristic, complemented
8#10590666—31nomaximal
9#10590663—31nomaximal
10C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
11C2×((C3⋊D3)⋊C3)⋊C4C411yescharacteristic
12C2×((C3⋊D3)⋊C3)⋊C4C411yescharacteristic
13C2×C8×H(3,𝔽3)C2211yescharacteristic
14C4×#1084C2211yescharacteristic
15C2×#6180C2211yescharacteristic
16C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
17C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
18C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
19#3743⋊C3—22no—
20#88447C411yescharacteristic
21#88447C411yescharacteristic
22#88430—22no—
23C2×#3743D311yescharacteristic
24C2×C3×ΓL(1,𝔽25)—31no—
25C4×#624—31no—
26C4×D3×Dic3—31no—
27C2×#3712—31no—
28C2×#3711—31no—
29#63641D311yescharacteristic, complemented
30C3×Dic3⋊C8—31no—
31C3×Dic3⋊C8—31no—
32C3×Dic3⋊C8—31no—
33C3×Dic3⋊C8—61no—
34#63572—31no—
35Dic3⋊(C3⋊C8)—31no—
36Dic3⋊(C3⋊C8)—31no—
37Dic3⋊(C3⋊C8)—31no—
38C22×(C3⋊D3)⋊C3C2×C411yescharacteristic
39C2×C4×H(3,𝔽3)C2311yescharacteristic
40C2×#1084C2×C411yescharacteristic
41C2×#1084C2×C411yescharacteristic
42C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
43C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
44C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
45C4×(C3⋊D3)⋊C3D412yes—
46C4×(C3⋊D3)⋊C3Q812yes—
47((C3⋊D3)⋊C3)⋊C4—22no—
48((C3⋊D3)⋊C3)⋊C4—22no—
49C8×H(3,𝔽3)—21no—
50#6180—21no—
51(C3⋊(C3⋊C8))⋊C3—21no—
52(C3⋊(C3⋊C8))⋊C3—21no—
53#5118—91no—
54#4837—91no—
55#4837—91no—
56C2×C4×C3⋊D3C2×D311yescharacteristic
57C2×C3×C4×D3—31no—
58C2×#624—31no—
59C2×#624—31no—
60C2×D3×Dic3—31no—
61C2×D3×Dic3—31no—
62C2×C32×C8—21no—
63C2×C32×C8C2×D311yescharacteristic
64C2×C32×C8C2×D311yescharacteristic
65C4×C3⋊Dic3C2×D311yescharacteristic
66C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
67C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
68#3743—22no—
69C3×C4×Dic3—31no—
70C3×C4×Dic3—31no—
71C3×C4×Dic3—31no—
72C3×C4×Dic3—61no—
73C2×C3×C3⋊C8—31no—
74C2×C3×C3⋊C8—31no—
75C2×C3×C3⋊C8—31no—
76C2×C3×C3⋊C8—31no—
77C2×C3×C3⋊C8—61no—
78C3×ΓL(1,𝔽25)—62no—
79Dic32—31no—
80Dic32—31no—
81Dic32—31no—
82Dic32—31no—
83#3712—62no—
84#3711—62no—
85C22×H(3,𝔽3)C22×C411yescharacteristic
86C2×(C3⋊D3)⋊C3—21no—
87C2×(C3⋊D3)⋊C3—21no—
88C2×(C3⋊D3)⋊C3C4⋊C412yes—
89C4×H(3,𝔽3)C2×D411yescharacteristic
90C4×H(3,𝔽3)C2×Q811yescharacteristic
91#1084—21no—
92#1084M1612yescomplemented
93(C3⋊Dic3)⋊C3—21no—
94(C3⋊Dic3)⋊C3C4⋊C412yes—
95(C3⋊Dic3)⋊C3M1612yescomplemented
96C2×C3×M16—91no—
97C2×C4×Dic3—91no—
98C2×#208—91no—
99C2×ΓL(1,𝔽25)—31no—
100C2×ΓL(1,𝔽25)—91no—
101C2×ΓL(1,𝔽25)—91no—
102C2×ΓL(1,𝔽25)—91no—
103C42×D3—91no—
104C42×D3—91no—
105C3×C4⋊C8—91no—
106C3×C4⋊C8—91no—
107Dic3⋊C8—31no—
108Dic3⋊C8—31no—
109Dic3⋊C8—31no—
110Dic3⋊C8—61no—
111Dic3⋊C8—91no—
112Dic3⋊C8—91no—
113Dic3⋊C8—91no—
114Dic3⋊C8—91no—
115Dic3⋊C8—91no—
116C4⋊(C3⋊C8)—91no—
117C4⋊(C3⋊C8)—91no—
118C22×C3⋊D3C4×D311yescharacteristic
119C22×C3×D3—31no—
120C2×C32×C4—21no—
121C2×C32×C4C22×D311yescharacteristic
122C2×C32×C4C22×D311yescharacteristic
123C2×C3⋊Dic3C4×D311yescharacteristic
124C2×C3⋊Dic3C4×D311yescharacteristic
125C2×C3⋊Dic3C4×D311yescharacteristic
126C4×C3⋊D3Dic612yes—
127C4×C3⋊D3#6712yes—
128C2×C3×Dic3—31no—
129C2×C3×Dic3—31no—
130C2×C3×Dic3—31no—
131C2×C3×Dic3—31no—
132C2×C3×Dic3—31no—
133C2×C3×Dic3—31no—
134C2×C3×Dic3—31no—
135C2×C3×Dic3—61no—
136C2×C3×Dic3—61no—
137C3×C4×D3—32no—
138C3×C4×D3—32no—
139#624—62no—
140#624—62no—
141D3×Dic3—62no—
142D3×Dic3—62no—
143C32×C8—21no—
144C32×C8—21no—
145C32×C8—41no—
146C3⋊(C3⋊C8)—21no—
147C3⋊(C3⋊C8)—21no—
148C3×C3⋊C8—61no—
149C3×C3⋊C8—61no—
150C3×C3⋊C8—61no—
151C3×C3⋊C8—61no—
152C3×C3⋊C8—121no—
153#423—271noSylow
154C2×H(3,𝔽3)C2×C4⋊C411yescharacteristic
155C2×H(3,𝔽3)C2×M1611yescharacteristic
156C2×H(3,𝔽3)C2×M1611yescharacteristic
157(C3⋊D3)⋊C3—22no—
158C22×C3×C4—91no—
159C22×Dic3—91no—
160C22×Dic3—91no—
161C2×C4×D3—31no—
162C2×C4×D3—91no—
163C2×C4×D3—91no—
164C2×C4×D3—91no—
165C2×C4×D3—91no—
166C2×C4×D3—91no—
167C3×M16—182no—
168C2×C3×C8—31no—
169C2×C3×C8—31no—
170C2×C3×C8—61no—
171C2×C3×C8—91no—
172C2×C3×C8—91no—
173C2×C3×C8—91no—
174C2×C3×C8D3211yescharacteristic
175C3×C42—91no—
176C3×C42—91no—
177C4×Dic3—31no—
178C4×Dic3—31no—
179C4×Dic3—31no—
180C4×Dic3—61no—
181C4×Dic3—91no—
182C4×Dic3—91no—
183C4×Dic3—91no—
184C4×Dic3—91no—
185C4×Dic3—91no—
186C4×Dic3—91no—
187C4×Dic3—91no—
188#208—182no—
189C2×C3⋊C8—31no—
190C2×C3⋊C8—31no—
191C2×C3⋊C8—31no—
192C2×C3⋊C8—31no—
193C2×C3⋊C8—61no—
194C2×C3⋊C8—91no—
195C2×C3⋊C8—91no—
196ΓL(1,𝔽25)—62no—
197ΓL(1,𝔽25)—182no—
198ΓL(1,𝔽25)—182no—
199ΓL(1,𝔽25)—182no—
200C22×C32—21no—
201C22×C32C2×C4×D311yescharacteristic
202C22×C32C2×C4×D311yescharacteristic
203C2×C3⋊D3—21no—
204C2×C3⋊D3—21no—
205C2×C3⋊D3Dic3⋊C412yes—
206C2×C3×D3—32no—
207C2×C3×D3—61no—
208C2×C3×D3—61no—
209C32×C4—21no—
210C32×C4—21no—
211C32×C4C2×Dic611yescharacteristic
212C32×C4D3×D411yescharacteristic
213C32×C4D3×Q811yescharacteristic
214C32×C4C2×#6711yescharacteristic
215C3⋊Dic3—21no—
216C3⋊Dic3ΓL(1,𝔽25)12yescomplemented
217C3⋊Dic3Dic3⋊C412yes—
218C3×Dic3—32no—
219C3×Dic3—32no—
220C3×Dic3—32no—
221C3×Dic3—32no—
222C3×Dic3—62no—
223C3×Dic3—61no—
224C3×Dic3—61no—
225C3×Dic3—61no—
226C3×Dic3—121no—
227C2×M16—91no—
228C2×M16—271no—
229C2×C42—271no—
230C4⋊C8—91no—
231C4⋊C8—91no—
232C4⋊C8—271no—
233C4⋊C8—271no—
234H(3,𝔽3)#42311yescharacteristic, complemented, Sylow
235C23×C3—91no—
236C22×D3—31no—
237C22×D3—91no—
238C2×C3×C4—31no—
239C2×C3×C4—31no—
240C2×C3×C4—61no—
241C2×C3×C4—92no—
242C2×C3×C4—92no—
243C2×C3×C4—91no—
244C2×C3×C4—91no—
245C2×C3×C4—91no—
246C2×C3×C4—91no—
247C2×C3×C4—91no—
248C2×C3×C4C2×D3211yescharacteristic
249C2×Dic3—31no—
250C2×Dic3—31no—
251C2×Dic3—31no—
252C2×Dic3—31no—
253C2×Dic3—31no—
254C2×Dic3—31no—
255C2×Dic3—31no—
256C2×Dic3—61no—
257C2×Dic3—61no—
258C2×Dic3—91no—
259C2×Dic3—91no—
260C2×Dic3—91no—
261C2×Dic3—182no—
262C2×Dic3—182no—
263C4×D3—32no—
264C4×D3—32no—
265C4×D3—92no—
266C4×D3—92no—
267C4×D3—182no—
268C4×D3—182no—
269C4×D3—182no—
270C4×D3—182no—
271C3×C8—21no—
272C3×C8—61no—
273C3×C8—61no—
274C3×C8—121no—
275C3×C8—181no—
276C3×C8—181no—
277C3×C8—181no—
278C3⋊C8—61no—
279C3⋊C8—61no—
280C3⋊C8—61no—
281C3⋊C8—61no—
282C3⋊C8—121no—
283C3⋊C8—181no—
284C3⋊C8—181no—
285C2×C32—21no—
286C2×C32—21no—
287C2×C32—21no—
288C2×C32D3×C4⋊C411yescharacteristic
289C2×C32C2×ΓL(1,𝔽25)11yescharacteristic
290C2×C32D3×M1611yescharacteristic
291C2×C32D3×M1611yescharacteristic
292C2×C32D3×M1611yescharacteristic
293C2×C32C2×Dic3⋊C411yescharacteristic
294C3⋊D3—22no—
295C3×D3—62no—
296C22×C4—91no—
297C22×C4—271no—
298C22×C4—271no—
299M16—182no—
300M16—542no—
301C2×C8—91no—
302C2×C8—91no—
303C2×C8—91no—
304C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
305C42—91no—
306C42—91no—
307C42—271no—
308C42—271no—
309C22×C3—31no—
310C22×C3—31no—
311C22×C3—61no—
312C22×C3—92no—
313C22×C3—181no—
314C22×C3—181no—
315C22×C3C4×D3211yescharacteristic
316C2×D3—32no—
317C2×D3—61no—
318C2×D3—61no—
319C2×D3—92no—
320C2×D3—181no—
321C2×D3—181no—
322C3×C4—31no—
323C3×C4—31no—
324C3×C4—31no—
325C3×C4—31no—
326C3×C4—61no—
327C3×C4—61no—
328C3×C4—92no—
329C3×C4—92no—
330C3×C4—92no—
331C3×C4—181no—
332C3×C4—181no—
333C3×C4D3×Dic611yescharacteristic
334C3×C4D3×#6711yescharacteristic
335Dic3—32no—
336Dic3—32no—
337Dic3—32no—
338Dic3—32no—
339Dic3—62no—
340Dic3—61no—
341Dic3—61no—
342Dic3—61no—
343Dic3—92no—
344Dic3—92no—
345Dic3—121no—
346Dic3—181no—
347C32—21no—
348C32#483711yescharacteristic, complemented
349C32#511811yescharacteristic, complemented
350C23—91no—
351C2×C4—91no—
352C2×C4—91no—
353C2×C4—91no—
354C2×C4—92no—
355C2×C4—92no—
356C2×C4—91no—
357C2×C4—91no—
358C2×C4—542no—
359C2×C4—542no—
360C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
361C8—21no—
362C8—181no—
363C8—181no—
364C8—181no—
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—182no—
375C2×C3D3×ΓL(1,𝔽25)11yescharacteristic
376C2×C3#6379111yescharacteristic
377C2×C3D3×Dic3⋊C411yescharacteristic
378D3—62no—
379D3—182no—
380C22—92no—
381C22—181no—
382C22—181no—
383C22C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
384C4—92no—
385C4—92no—
386C4—92no—
387C4—181no—
388C4—181no—
389C4((C3⋊D3)⋊C3)⋊Q811yescharacteristic
390C4#8867611yescharacteristic
391C3—31no—
392C3—31no—
393C3—61no—
394C3#1059066311yescharacteristic
395C2—182no—
396C2#1172083511yescharacteristic
397C2#1172083611yescharacteristic
398C2#1172111911yescharacteristic
subgroupG/H
centerC2×C4C2×((C3⋊D3)⋊C3)⋊C2
commutatorC22×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3C4×D32
prime radicalC22×(C3⋊D3)⋊C3C2×C4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#423
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
4. #11720547C2semidirect
5. #63641⋊C3C2semidirect
6. #11720311C2semidirect
7. #11720304C2semidirect
29. #63641D3semidirect
92. #1084M16semidirect
95. (C3⋊Dic3)⋊C3M16semidirect
216. C3⋊Dic3ΓL(1,𝔽25)semidirect
234. H(3,𝔽3)#423semidirect
348. C32#4837semidirect
349. C32#5118semidirect

Maximal subgroups

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

subgroupindex
C2×#3743⋊C32
C4×((C3⋊D3)⋊C3)⋊C42
C2×#884302
#117205472
#63641⋊C32
#117203112
#117203042
#105906663
#105906633