Finite Group Databasebeta

← C2×#72221852 #72244136

Subgroups — C2×#72221852 #72244136

Order 5832.

664 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 664 conjugacy classes of subgroups — too many to draw. Only its 80 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×#92182⋊C3C211yesmaximal, characteristic, complemented
2C2×#13090945C211yesmaximal, characteristic, complemented
3C2×#92213⋊C3C211yesmaximal, characteristic, complemented
4#72221852C214yesmaximal, direct product factor
5C2×C3×#4207⋊C2—31nomaximal
6C2×D3×D27⋊C3—31nomaximal
7C2×#92181⋊C2C311yesmaximal, characteristic, complemented
8#92182⋊C3C2212yescomplemented
9#92213⋊C3C2212yescomplemented
10#13090945C2212yescomplemented
11C2×#10624702C2211yescharacteristic, complemented
12C2×C3×#4208D311yescharacteristic
13C2×C3×#4207—31no—
14C2×C3×#4207—31no—
15C3×#4207⋊C2—34no—
16C2×C3⋊(D27⋊C3)D311yescharacteristic, complemented
17C2×D3×C27⋊C3—31no—
18C2×C3×D27⋊C3—31no—
19C2×C3×D27⋊C3—61no—
20D3×D27⋊C3—34no—
21C2×#92213C2×C311yescharacteristic, complemented
22C2×#92182—21no—
23C2×#92182C2×C311yescharacteristic, complemented
24C2×#92181—21no—
25C2×#92181C2×C311yescharacteristic, complemented
26#92181⋊C2C2×C314yescomplemented
27#10624702C2311yescharacteristic, complemented, Sylow
28C2×#4207⋊C2—31no—
29C2×C3×((C3⋊D3)⋊C3)⋊C2—91no—
30C2×C3×D3×D9—91no—
31C2×C3×D3×D9—91no—
32C22×D27⋊C3—91no—
33C2×D3×D27—31no—
34C2×C3×#737C2×D311yescharacteristic
35C3×#4208C2×D312yes—
36C3×#4207—32no—
37C3×#4207—32no—
38C2×C3×C27⋊C3—21no—
39C2×C3×C27⋊C3C2×D311yescharacteristic, complemented
40C3⋊(D27⋊C3)C2×D312yescomplemented
41D3×C27⋊C3—32no—
42C3×D27⋊C3—32no—
43C3×D27⋊C3—62no—
44C2×#6872—21no—
45C2×#6872—21no—
46C2×#6872—41no—
47C2×#6872C22×C311yescharacteristic, complemented
48#92213C22×C312yescomplemented
49#92182—22no—
50#92182C22×C312yescomplemented
51#92181—22no—
52#92181C22×C312yescomplemented
53C2×#4208—21no—
54C2×#4208C3×D311yescharacteristic
55C2×#4207—31no—
56C2×#4207—31no—
57C2×#4207—61no—
58C2×#4207—61no—
59C2×C3×#272D911yescharacteristic
60C2×C3×C3⋊D9—31no—
61C2×C3×C3⋊D9—31no—
62C2×C3×AGL(1,ℤ/9)—181no—
63C2×C3×AGL(1,ℤ/9)—181no—
64C2×C3×(C3⋊D3)⋊C3—91no—
65C2×C3×(C3⋊D3)⋊C3—91no—
66C2×C3×D3×C9—31no—
67C2×C3×D3×C9—181no—
68C2×C32×D9—91no—
69C2×C32×D9—91no—
70#4207⋊C2—34no—
71C3×((C3⋊D3)⋊C3)⋊C2—94no—
72C3×D3×D9—94no—
73C3×D3×D9—94no—
74C22×C27⋊C3—91no—
75C2×C3⋊D27C3×D311yescharacteristic, complemented
76C2×D27⋊C3—31no—
77C2×D27⋊C3—61no—
78C2×D27⋊C3—61no—
79C2×D27⋊C3—61no—
80C2×D27⋊C3—61no—
81C2×D27⋊C3—94no—
82C2×D27⋊C3—181no—
83C2×D3×C27—31no—
84C2×D3×C27—61no—
85C2×C3×D27—31no—
86C2×C3×D27—61no—
87D3×D27—34no—
88C3×#737C22×D311yescharacteristic
89C3×C27⋊C3—21no—
90C3×C27⋊C3C22×D311yescharacteristic, complemented
91#6872—21no—
92#6872—21no—
93#6872—41no—
94#6872C23×C311yescharacteristic, complemented
95C2×C3×D32—271no—
96C2×C3×D32—271no—
97C22×C3×D9—271no—
98C2×((C3⋊D3)⋊C3)⋊C2—91no—
99C2×D3×D9—91no—
100C2×D3×D9—91no—
101C22×D27—91no—
102C2×#737—21no—
103C2×#737—21no—
104C2×#737—41no—
105C2×#737—121no—
106C2×#737—121no—
107C2×#737C2×C3×D311yescharacteristic
108C2×C3×3−1+2—21no—
109C2×C3×3−1+2—61no—
110C2×C3×H(3,𝔽3)C2×D911yescharacteristic
111C2×C32×C9—61no—
112C2×C32×C9D3211yescharacteristic
113#4208—22no—
114#4208C2×C3×D312yes—
115#4207—32no—
116#4207—32no—
117#4207—62no—
118#4207—62no—
119C3×#272C2×D912yes—
120C3×C3⋊D9—32no—
121C3×C3⋊D9—32no—
122C3×AGL(1,ℤ/9)—182no—
123C3×AGL(1,ℤ/9)—182no—
124C3×(C3⋊D3)⋊C3—92no—
125C3×(C3⋊D3)⋊C3—92no—
126C3×D3×C9—32no—
127C3×D3×C9—182no—
128C32×D9—92no—
129C32×D9—92no—
130C2×C27⋊C3—21no—
131C2×C27⋊C3—41no—
132C2×C27⋊C3—61no—
133C2×C27⋊C3—61no—
134C2×C27⋊C3—61no—
135C2×C27⋊C3—92no—
136C2×C27⋊C3—121no—
137C2×C3×C27—21no—
138C2×C3×C27—121no—
139C2×C3×C27C2×C3×D311yescharacteristic, complemented
140C3⋊D27C2×C3×D312yescomplemented
141D27⋊C3—32no—
142D27⋊C3—62no—
143D27⋊C3—62no—
144D27⋊C3—62no—
145D27⋊C3—62no—
146D27⋊C3—182no—
147D3×C27—32no—
148D3×C27—62no—
149C3×D27—32no—
150C3×D27—62no—
151C2×C3×C3⋊D3—91no—
152C2×C3×C3⋊D3—91no—
153C2×C32×D3—31no—
154C2×C32×D3—181no—
155C2×C32×D3—271no—
156C2×C32×D3—271no—
157C3×D32—274no—
158C3×D32—274no—
159C22×C3×C9—91no—
160C2×#272C3×D911yescharacteristic
161C2×C3⋊D9—31no—
162C2×C3⋊D9—31no—
163C2×AGL(1,ℤ/9)—181no—
164C2×AGL(1,ℤ/9)—181no—
165C2×AGL(1,ℤ/9)—181no—
166C2×AGL(1,ℤ/9)—181no—
167C2×AGL(1,ℤ/9)—181no—
168C2×AGL(1,ℤ/9)—181no—
169C2×(C3⋊D3)⋊C3—91no—
170C2×(C3⋊D3)⋊C3—91no—
171C2×(C3⋊D3)⋊C3—181no—
172C2×(C3⋊D3)⋊C3—181no—
173C2×D3×C9—31no—
174C2×D3×C9—61no—
175C2×D3×C9—91no—
176C2×D3×C9—181no—
177C2×C3×D9—91no—
178C2×C3×D9—91no—
179C2×C3×D9—91no—
180C2×C3×D9—91no—
181C2×C3×D9—181no—
182C2×C3×D9—181no—
183C2×C3×D9—181no—
184C2×C3×D9—181no—
185C2×C3×D9—274no—
186((C3⋊D3)⋊C3)⋊C2—94no—
187D3×D9—94no—
188D3×D9—94no—
189C22×C27—91no—
190C22×C27—181no—
191C2×D27—31no—
192C2×D27—61no—
193C2×D27—94no—
194C2×D27—181no—
195#737—21no—
196#737—21no—
197#737—41no—
198#737—121no—
199#737—121no—
200#737C22×C3×D311yescharacteristic
201C3×3−1+2—21no—
202C3×3−1+2—61no—
203C3×H(3,𝔽3)C22×D911yescharacteristic
204C32×C9—61no—
205C32×C9C2×D3211yescharacteristic
206C27⋊C3—21no—
207C27⋊C3—41no—
208C27⋊C3—61no—
209C27⋊C3—61no—
210C27⋊C3—61no—
211C27⋊C3—121no—
212C3×C27—21no—
213C3×C27—121no—
214C3×C27C22×C3×D311yescharacteristic, complemented
215C22×C3×D3—811no—
216C22×C3×D3—811no—
217C22×C3×D3—811no—
218C2×D32—271no—
219C2×D32—271no—
220C22×D9—271no—
221C2×C33—61no—
222C2×C33D3×D911yescharacteristic
223C3×C3⋊D3—92no—
224C3×C3⋊D3—92no—
225C32×D3—32no—
226C32×D3—182no—
227C32×D3—272no—
228C32×D3—272no—
229C2×3−1+2—21no—
230C2×3−1+2—41no—
231C2×3−1+2—61no—
232C2×3−1+2—61no—
233C2×3−1+2—121no—
234C2×3−1+2—121no—
235C2×3−1+2—122no—
236C2×3−1+2—121no—
237C2×3−1+2—121no—
238C2×H(3,𝔽3)—21no—
239C2×H(3,𝔽3)—121no—
240C2×H(3,𝔽3)C2×C3×D911yescharacteristic
241C2×C3×C9—21no—
242C2×C3×C9—21no—
243C2×C3×C9—21no—
244C2×C3×C9—31no—
245C2×C3×C9—41no—
246C2×C3×C9—61no—
247C2×C3×C9—92no—
248C2×C3×C9—121no—
249C2×C3×C9—121no—
250C2×C3×C9—121no—
251C2×C3×C9((C3⋊D3)⋊C3)⋊C211yescharacteristic
252C2×C3×C9C3×D3211yescharacteristic
253#272C2×C3×D912yes—
254C3⋊D9—32no—
255C3⋊D9—32no—
256AGL(1,ℤ/9)—182no—
257AGL(1,ℤ/9)—182no—
258AGL(1,ℤ/9)—182no—
259AGL(1,ℤ/9)—182no—
260AGL(1,ℤ/9)—182no—
261AGL(1,ℤ/9)—182no—
262(C3⋊D3)⋊C3—92no—
263(C3⋊D3)⋊C3—92no—
264(C3⋊D3)⋊C3—182no—
265(C3⋊D3)⋊C3—182no—
266D3×C9—32no—
267D3×C9—62no—
268D3×C9—92no—
269D3×C9—182no—
270C3×D9—92no—
271C3×D9—92no—
272C3×D9—92no—
273C3×D9—92no—
274C3×D9—182no—
275C3×D9—182no—
276C3×D9—182no—
277C3×D9—182no—
278C2×C27—61no—
279C2×C27—61no—
280C2×C27—61no—
281C2×C27—92no—
282C2×C27—121no—
283C2×C27—182no—
284D27—32no—
285D27—62no—
286D27—182no—
287C22×C32—91no—
288C22×C32—811no—
289C22×C32—811no—
290C2×C3⋊D3—91no—
291C2×C3⋊D3—91no—
292C2×C3×D3—31no—
293C2×C3×D3—91no—
294C2×C3×D3—181no—
295C2×C3×D3—181no—
296C2×C3×D3—181no—
297C2×C3×D3—271no—
298C2×C3×D3—271no—
299C2×C3×D3—271no—
300C2×C3×D3—271no—
301C2×C3×D3—541no—
302C2×C3×D3—541no—
303C2×C3×D3—814no—
304C2×C3×D3—814no—
305C2×C3×D3—814no—
306C2×C3×D3—1621no—
307C2×C3×D3—1621no—
308D32—274no—
309D32—274no—
310C22×C9—91no—
311C22×C9—181no—
312C2×D9—91no—
313C2×D9—91no—
314C2×D9—181no—
315C2×D9—181no—
316C2×D9—274no—
317C33—61no—
318C33C2×D3×D911yescharacteristic
3193−1+2—21no—
3203−1+2—41no—
3213−1+2—61no—
3223−1+2—61no—
3233−1+2—121no—
3243−1+2—121no—
3253−1+2—122no—
3263−1+2—121no—
3273−1+2—121no—
328H(3,𝔽3)—21no—
329H(3,𝔽3)—121no—
330H(3,𝔽3)C22×C3×D911yescharacteristic
331C3×C9—21no—
332C3×C9—21no—
333C3×C9—21no—
334C3×C9—31no—
335C3×C9—41no—
336C3×C9—61no—
337C3×C9—121no—
338C3×C9—121no—
339C3×C9—121no—
340C3×C9C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
341C3×C9C2×C3×D3211yescharacteristic
342C27—61no—
343C27—61no—
344C27—61no—
345C27—121no—
346C23×C3—2431no—
347C22×D3—811no—
348C22×D3—811no—
349C22×D3—811no—
350C2×C32—21no—
351C2×C32—61no—
352C2×C32—92no—
353C2×C32—121no—
354C2×C32—181no—
355C2×C32—361no—
356C2×C32—361no—
357C2×C32—812no—
358C2×C32—812no—
359C2×C32((C3⋊D3)⋊C9)⋊C211yescharacteristic
360C2×C32C3×D3×D911yescharacteristic
361C3⋊D3—92no—
362C3⋊D3—92no—
363C3×D3—32no—
364C3×D3—92no—
365C3×D3—182no—
366C3×D3—182no—
367C3×D3—182no—
368C3×D3—272no—
369C3×D3—272no—
370C3×D3—272no—
371C3×D3—272no—
372C3×D3—542no—
373C3×D3—542no—
374C3×D3—1622no—
375C3×D3—1622no—
376C2×C9—21no—
377C2×C9—21no—
378C2×C9—41no—
379C2×C9—61no—
380C2×C9—92no—
381C2×C9—121no—
382C2×C9—182no—
383C2×C9C3×((C3⋊D3)⋊C3)⋊C211yescharacteristic
384D9—92no—
385D9—92no—
386D9—182no—
387D9—182no—
388C22×C3—91no—
389C22×C3—271no—
390C22×C3—811no—
391C22×C3—811no—
392C22×C3—811no—
393C22×C3—811no—
394C22×C3—1621no—
395C22×C3—1621no—
396C22×C3—2434no—
397C2×D3—31no—
398C2×D3—181no—
399C2×D3—271no—
400C2×D3—271no—
401C2×D3—814no—
402C2×D3—814no—
403C2×D3—814no—
404C2×D3—1621no—
405C2×D3—1621no—
406C32—21no—
407C32—61no—
408C32—121no—
409C32—181no—
410C32—361no—
411C32—361no—
412C32C2×((C3⋊D3)⋊C9)⋊C211yescharacteristic
413C32C2×C3×D3×D911yescharacteristic
414C9—21no—
415C9—21no—
416C9—41no—
417C9—61no—
418C9—121no—
419C9C2×C3×((C3⋊D3)⋊C3)⋊C211yescharacteristic
420C23—2431noSylow
421C2×C3—31no—
422C2×C3—61no—
423C2×C3—92no—
424C2×C3—121no—
425C2×C3—181no—
426C2×C3—272no—
427C2×C3—361no—
428C2×C3—812no—
429C2×C3—812no—
430C2×C3—812no—
431C2×C3—812no—
432C2×C3—1622no—
433C2×C3—1622no—
434C2×C3C3×((C3⋊D3)⋊C9)⋊C211yescharacteristic
435D3—32no—
436D3—182no—
437D3—272no—
438D3—272no—
439D3—1622no—
440D3—1622no—
441C22—91no—
442C22—811no—
443C22—811no—
444C22—2434no—
445C3—31no—
446C3—61no—
447C3—121no—
448C3—181no—
449C3—361no—
450C3C2×C3×((C3⋊D3)⋊C9)⋊C211yescharacteristic
451C2—92no—
452C2—812no—
453C2—812no—
454C2#7222185211yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2#72221852
commutator, cosocle, hypocenter#6872C23×C3
FrattiniC3×C9C2×C3×D32
FittingC2×#10624702C22
socleC2×C3C3×((C3⋊D3)⋊C9)⋊C2
LCSG ▷ #6872
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. C2×#92182⋊C3C2semidirect
2. C2×#13090945C2semidirect
3. C2×#92213⋊C3C2semidirect
4. #72221852C2direct (N × H)
7. C2×#92181⋊C2C3semidirect
8. #92182⋊C3C22semidirect
9. #92213⋊C3C22semidirect
10. #13090945C22semidirect
11. C2×#10624702C22semidirect
16. C2×C3⋊(D27⋊C3)D3semidirect
21. C2×#92213C2×C3semidirect
23. C2×#92182C2×C3semidirect
25. C2×#92181C2×C3semidirect
26. #92181⋊C2C2×C3semidirect
27. #10624702C23semidirect
39. C2×C3×C27⋊C3C2×D3semidirect
40. C3⋊(D27⋊C3)C2×D3semidirect
47. C2×#6872C22×C3semidirect
48. #92213C22×C3semidirect
50. #92182C22×C3semidirect
52. #92181C22×C3semidirect
75. C2×C3⋊D27C3×D3semidirect
90. C3×C27⋊C3C22×D3semidirect
94. #6872C23×C3semidirect
139. C2×C3×C27C2×C3×D3semidirect
140. C3⋊D27C2×C3×D3semidirect
214. C3×C27C22×C3×D3semidirect
454. C2#72221852direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#92182⋊C32
C2×#130909452
C2×#92213⋊C32
#722218522
C2×C3×#4207⋊C23
C2×D3×D27⋊C33
C2×#92181⋊C23