Finite Group Databasebeta

← #13137373

Subgroups — #13137373

Order 1728.

474 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 97 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
1C4×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic
2C2×#88430C211yesmaximal, characteristic, complemented
3C2×((C3⋊D3)⋊C3)⋊C8C211yesmaximal, characteristic, complemented
4#11720520C211yesmaximal, characteristic, complemented
5C4⋊((C3⋊(C3⋊C8))⋊C3)C211yesmaximal, characteristic, complemented
6#11720317C211yesmaximal, characteristic, complemented
7#11720317C211yesmaximal, characteristic, complemented
8#10590328—31nomaximal
9#10590326—31nomaximal
10C2×C4⋊((C3⋊D3)⋊C3)C411yescharacteristic
11C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
12C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
13C42×H(3,𝔽3)C2211yescharacteristic
14C2×#6180C2211yescharacteristic
15C2×#6180C2211yescharacteristic
16#3752⋊C3C411yescharacteristic
17#3752⋊C3C411yescharacteristic
18C4⋊((C3⋊Dic3)⋊C3)C411yescharacteristic
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21#88430—22no—
22((C3⋊D3)⋊C3)⋊C8—22no—
23C4×C4⋊(C3⋊D3)D311yescharacteristic
24C3×C4×D12—31no—
25C2×#3712—31no—
26C2×#3711—31no—
27C2×#3710—31no—
28C2×D3×C3⋊C8—31no—
29C4⋊(C3⋊(C3⋊C8))D311yescharacteristic, complemented
30C3×C4⋊(C3⋊C8)—31no—
31C3×#803—31no—
32C3×#803—31no—
33C3×#803—61no—
34#63557—31no—
35#63557—31no—
36(C2×D3)⋊(C3⋊C8)—31no—
37(C2×D3)⋊(C3⋊C8)—31no—
38C22×(C3⋊D3)⋊C3C2×C411yescharacteristic
39C22×(C3⋊D3)⋊C3C2×C411yescharacteristic
40C2×C4×H(3,𝔽3)C2×C411yescharacteristic
41C2×C4×H(3,𝔽3)C2×C411yescharacteristic
42C2×C4×H(3,𝔽3)C2311yescharacteristic
43C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
44C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
45C4⋊((C3⋊D3)⋊C3)—22no—
46C4×(C3⋊D3)⋊C3—21no—
47C4×(C3⋊D3)⋊C3—21no—
48C4×(C3⋊D3)⋊C3—21no—
49C4×(C3⋊D3)⋊C3—21no—
50#6180—21no—
51#6180D412yescomplemented
52(C3⋊(C3⋊C8))⋊C3—21no—
53(C3⋊(C3⋊C8))⋊C3—21no—
54#5011—91no—
55#4836—91no—
56#4706—91no—
57C2×C4⋊(C3⋊D3)Dic311yescharacteristic
58C2×C4×C3⋊D3C2×D311yescharacteristic
59C2×C4×C3⋊D3C2×D311yescharacteristic
60C2×C3×D12—31no—
61C2×C3×C4×D3—31no—
62C2×C3×C4×D3—31no—
63C32×C42—21no—
64C32×C42C2×D311yescharacteristic
65C32×C42C2×D311yescharacteristic
66#3752Dic311yescharacteristic
67#3752Dic311yescharacteristic
68C4⋊(C3⋊Dic3)Dic311yescharacteristic
69C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
70C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
71C3×(C2×D3)⋊C4—31no—
72C3×(C2×D3)⋊C4—31no—
73C3×C4⋊Dic3—31no—
74C2×C3×C3⋊C8—31no—
75C2×C3×C3⋊C8—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#3712—62no—
83#3711—62no—
84#3710—62no—
85D3×C3⋊C8—62no—
86C22×H(3,𝔽3)C22×C411yescharacteristic
87C2×(C3⋊D3)⋊C3—21no—
88C2×(C3⋊D3)⋊C3—21no—
89C2×(C3⋊D3)⋊C3—21no—
90C2×(C3⋊D3)⋊C3—21no—
91C2×(C3⋊D3)⋊C3—21no—
92C2×(C3⋊D3)⋊C3M1612yescomplemented
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—21no—
98(C3⋊Dic3)⋊C3—21no—
99C3×C4×D4—91no—
100C2×#208—91no—
101C22×C3⋊C8—91no—
102C2×ΓL(1,𝔽25)—91no—
103C2×ΓL(1,𝔽25)—91no—
104C2×D3×C8—91no—
105C2×D3×C8—91no—
106C4×D12—31no—
107C4×D12—91no—
108C3×C4⋊C8—91no—
109C3×#97—91no—
110#830—91no—
111#830—91no—
112(C2×D3)⋊C8—91no—
113(C2×D3)⋊C8—91no—
114(C2×D3)⋊C8—91no—
115(C2×D3)⋊C8—91no—
116C4⋊(C3⋊C8)—31no—
117C4⋊(C3⋊C8)—91no—
118#803—31no—
119#803—31no—
120#803—61no—
121C22×C3⋊D3C2×Dic311yescharacteristic
122C22×C3⋊D3C2×Dic311yescharacteristic
123C22×C3×D3—31no—
124C22×C3×D3—31no—
125C2×C32×C4—21no—
126C2×C32×C4—21no—
127C2×C32×C4—21no—
128C2×C32×C4C4×D311yescharacteristic
129C2×C32×C4C4×D311yescharacteristic
130C2×C32×C4C2×Dic311yescharacteristic
131C2×C32×C4C2×Dic311yescharacteristic
132C2×C32×C4C22×D311yescharacteristic
133C2×C32×C4C22×D311yescharacteristic
134C2×C3⋊Dic3C2×Dic311yescharacteristic
135C2×C3⋊Dic3C2×Dic311yescharacteristic
136C4⋊(C3⋊D3)—22no—
137C4×C3⋊D3—21no—
138C4×C3⋊D3—21no—
139C4×C3⋊D3—21no—
140C4×C3⋊D3—21no—
141C2×C3×Dic3—31no—
142C2×C3×Dic3—31no—
143C3×D12—62no—
144C3×C4×D3—61no—
145C3×C4×D3—61no—
146C3×C4×D3—61no—
147C3×C4×D3—61no—
148C3⋊(C3⋊C8)—21no—
149C3⋊(C3⋊C8)—21no—
150C3×C3⋊C8—32no—
151C3×C3⋊C8—32no—
152C3×C3⋊C8—62no—
153C3×C3⋊C8—62no—
154C3×C3⋊C8—61no—
155C3×C3⋊C8—61no—
156C3×C3⋊C8—61no—
157C3×C3⋊C8—61no—
158#435—271noSylow
159C2×H(3,𝔽3)C4×D411yescharacteristic
160C2×H(3,𝔽3)C2×M1611yescharacteristic
161C2×H(3,𝔽3)#13111yescharacteristic
162(C3⋊D3)⋊C3—22no—
163(C3⋊D3)⋊C3—41no—
164C2×C3×D4—91no—
165C22×C3×C4—91no—
166C22×C3×C4—91no—
167C2×D12—31no—
168C2×D12—91no—
169C2×C4×D3—31no—
170C2×C4×D3—31no—
171C2×C4×D3—91no—
172C2×C4×D3—91no—
173C2×C3×C8—91no—
174C2×C3×C8—91no—
175C2×C3×C8—91no—
176C2×C3×C8—91no—
177C3×C4⋊C4—91no—
178C3×C22⋊C4—91no—
179C3×C22⋊C4—91no—
180C3×C42—31no—
181C3×C42—31no—
182C3×C42—61no—
183C3×C42D3211yescharacteristic
184(C2×D3)⋊C4—31no—
185(C2×D3)⋊C4—31no—
186(C2×D3)⋊C4—91no—
187(C2×D3)⋊C4—91no—
188C4⋊Dic3—31no—
189C4⋊Dic3—91no—
190#208—182no—
191C2×C3⋊C8—31no—
192C2×C3⋊C8—31no—
193C2×C3⋊C8—31no—
194C2×C3⋊C8—31no—
195C2×C3⋊C8—31no—
196C2×C3⋊C8—31no—
197C2×C3⋊C8—61no—
198C2×C3⋊C8—61no—
199C2×C3⋊C8—91no—
200C2×C3⋊C8—91no—
201C2×C3⋊C8—182no—
202ΓL(1,𝔽25)—182no—
203ΓL(1,𝔽25)—182no—
204D3×C8—182no—
205D3×C8—182no—
206C22×C32—21no—
207C22×C32C2×C4×D311yescharacteristic
208C22×C32C22×Dic311yescharacteristic
209C2×C3⋊D3—21no—
210C2×C3⋊D3—21no—
211C2×C3⋊D3—21no—
212C2×C3⋊D3—21no—
213C2×C3⋊D3—21no—
214C2×C3⋊D3#20812yescomplemented
215C2×C3×D3—32no—
216C2×C3×D3—61no—
217C2×C3×D3—61no—
218C2×C3×D3—61no—
219C2×C3×D3—61no—
220C2×C3×D3—61no—
221C32×C4—21no—
222C32×C4—21no—
223C32×C4—21no—
224C32×C4—21no—
225C32×C4—21no—
226C32×C4—21no—
227C32×C4—41no—
228C32×C4—41no—
229C32×C4D3×D411yescharacteristic
230C32×C4D3×D411yescharacteristic
231C32×C4#23711yescharacteristic
232C32×C4#23911yescharacteristic
233C3⋊Dic3—21no—
234C3⋊Dic3—21no—
235C3×Dic3—61no—
236C3×Dic3—61no—
237C2×M16—271no—
238C22×C8—271no—
239C4×D4—91no—
240C4⋊C8—91no—
241C22⋊C8—271no—
242C22⋊C8—271no—
243#97—91no—
244H(3,𝔽3)#43511yescharacteristic, complemented, Sylow
245C23×C3—91no—
246C23×C3—91no—
247C22×D3—31no—
248C22×D3—31no—
249C22×D3—91no—
250C22×D3—91no—
251C3×D4—182no—
252C2×C3×C4—31no—
253C2×C3×C4—31no—
254C2×C3×C4—31no—
255C2×C3×C4—31no—
256C2×C3×C4—31no—
257C2×C3×C4—31no—
258C2×C3×C4—61no—
259C2×C3×C4—61no—
260C2×C3×C4—61no—
261C2×C3×C4—91no—
262C2×C3×C4—91no—
263C2×C3×C4—181no—
264C2×C3×C4—181no—
265C2×C3×C4—181no—
266C2×C3×C4—181no—
267C2×C3×C4D3×Dic311yescharacteristic
268C2×C3×C4D3×Dic311yescharacteristic
269C2×C3×C4C2×D3211yescharacteristic
270C2×Dic3—31no—
271C2×Dic3—31no—
272C2×Dic3—91no—
273C2×Dic3—91no—
274D12—62no—
275D12—182no—
276C4×D3—61no—
277C4×D3—61no—
278C4×D3—61no—
279C4×D3—61no—
280C4×D3—181no—
281C4×D3—181no—
282C4×D3—181no—
283C4×D3—181no—
284C3×C8—92no—
285C3×C8—181no—
286C3×C8—181no—
287C3×C8—181no—
288C3⋊C8—32no—
289C3⋊C8—32no—
290C3⋊C8—62no—
291C3⋊C8—61no—
292C3⋊C8—61no—
293C3⋊C8—62no—
294C3⋊C8—61no—
295C3⋊C8—61no—
296C3⋊C8—181no—
297C3⋊C8—181no—
298C2×C32—21no—
299C2×C32—21no—
300C2×C32—21no—
301C2×C32#89311yescharacteristic
302C2×C32#90111yescharacteristic
303C2×C32D3×M1611yescharacteristic
304C2×C32C2×#20811yescharacteristic
305C2×C32D4×Dic311yescharacteristic
306C2×C32#94811yescharacteristic
307C3⋊D3—22no—
308C3⋊D3—41no—
309C3×D3—62no—
310C3×D3—121no—
311C2×D4—91no—
312C22×C4—91no—
313C22×C4—91no—
314M16—542no—
315C2×C8—91no—
316C2×C8—91no—
317C2×C8—91no—
318C2×C8—91no—
319C2×C8—542no—
320C4⋊C4—91no—
321C22⋊C4—91no—
322C22⋊C4—91no—
323C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
324C22×C3—31no—
325C22×C3—31no—
326C22×C3—61no—
327C22×C3—92no—
328C22×C3—181no—
329C22×C3—181no—
330C22×C3—181no—
331C22×C3—181no—
332C22×C3—181no—
333C22×C3C2×D3×Dic311yescharacteristic
334C2×D3—32no—
335C2×D3—61no—
336C2×D3—61no—
337C2×D3—61no—
338C2×D3—61no—
339C2×D3—61no—
340C2×D3—92no—
341C2×D3—181no—
342C2×D3—181no—
343C2×D3—181no—
344C2×D3—181no—
345C2×D3—181no—
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—61no—
357C3×C4—61no—
358C3×C4—121no—
359C3×C4—121no—
360C3×C4—181no—
361C3×C4—181no—
362C3×C4#379611yescharacteristic
363C3×C4#380211yescharacteristic
364Dic3—61no—
365Dic3—61no—
366Dic3—181no—
367Dic3—181no—
368C32—21no—
369C32#483611yescharacteristic, complemented
370C32#501111yescharacteristic, complemented
371C23—91no—
372C23—91no—
373D4—182no—
374C2×C4—91no—
375C2×C4—91no—
376C2×C4—181no—
377C2×C4—181no—
378C2×C4—181no—
379C2×C4—181no—
380C2×C4((C3⋊D3)⋊C3)⋊C411yescharacteristic
381C2×C4((C3⋊D3)⋊C3)⋊C411yescharacteristic
382C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
383C8—92no—
384C8—181no—
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—182no—
397C2×C3—361no—
398C2×C3D3×#20811yescharacteristic
399C2×C3#6381411yescharacteristic
400C2×C3#6389811yescharacteristic
401D3—62no—
402D3—121no—
403D3—182no—
404D3—361no—
405C22—92no—
406C22—181no—
407C22—181no—
408C22—181no—
409C22—181no—
410C22—181no—
411C22C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
412C4—21no—
413C4—21no—
414C4—181no—
415C4—181no—
416C4#8864911yescharacteristic
417C4#8865511yescharacteristic
418C3—31no—
419C3—31no—
420C3—61no—
421C3#1059033011yescharacteristic
422C2—182no—
423C2—361no—
424C2#1172105311yescharacteristic
425C2#1172105411yescharacteristic
426C2#1172113611yescharacteristic
subgroupG/H
centerC2×C4C2×((C3⋊D3)⋊C3)⋊C2
commutatorC22×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC42×H(3,𝔽3)C22
socleC22×C3C2×D3×Dic3
prime radicalC2×C4⋊((C3⋊D3)⋊C3)C4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#435
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C2×#88430C2semidirect
3. C2×((C3⋊D3)⋊C3)⋊C8C2semidirect
4. #11720520C2semidirect
5. C4⋊((C3⋊(C3⋊C8))⋊C3)C2semidirect
6. #11720317C2semidirect
7. #11720317C2semidirect
29. C4⋊(C3⋊(C3⋊C8))D3semidirect
51. #6180D4semidirect
92. C2×(C3⋊D3)⋊C3M16semidirect
214. C2×C3⋊D3#208semidirect
244. H(3,𝔽3)#435semidirect
369. C32#4836semidirect
370. C32#5011semidirect

Maximal subgroups

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

subgroupindex
C4×C4⋊((C3⋊D3)⋊C3)2
C2×#884302
C2×((C3⋊D3)⋊C3)⋊C82
#117205202
C4⋊((C3⋊(C3⋊C8))⋊C3)2
#117203172
#105903283
#105903263