Finite Group Databasebeta

← C2×#72221825 #72243987

Subgroups — C2×#72221825 #72243987

Order 8748.

726 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 726 conjugacy classes of subgroups — too many to draw. Only its 85 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×#4207⋊C9—31nomaximal
2C2×#92214⋊C3C311yesmaximal, characteristic
3C2×#14550491C211yesmaximal, characteristic, complemented
4#72221825C212yesmaximal, direct product factor
5C2×C3×AGL(1,ℤ/27)—31nomaximal
6C2×C3×AGL(1,ℤ/27)—31nomaximal
7C2×C3×AGL(1,ℤ/27)—31nomaximal
8#14550491C2211yescharacteristic, complemented, Sylow
9#4207⋊C9—32no—
10C3×AGL(1,ℤ/27)—32no—
11C3×AGL(1,ℤ/27)—32no—
12C3×AGL(1,ℤ/27)—32no—
13#92214⋊C3C2×C312yes—
14C2×#10624864D311yescharacteristic
15C2×C3×#6875D311yescharacteristic, complemented
16C2×C3×#6875D311yescharacteristic, complemented
17C2×C3×#6875D311yescharacteristic, complemented
18C2×#10624704—21no—
19C2×#10624704C2×C311yescharacteristic
20C2×#10624473—21no—
21C2×#10624473—21no—
22C2×#10624473—21no—
23C2×C3×#4207—31no—
24C2×#92294—31no—
25C2×#92294—31no—
26C2×C3⋊(D9⋊C9)—31no—
27C2×C3×D27⋊C3—31no—
28C2×C3×D27⋊C3—31no—
29C2×C3×D27⋊C3—31no—
30C2×C9×AGL(1,ℤ/9)—91no—
31C2×C9×AGL(1,ℤ/9)—91no—
32C2×C9×(C3⋊D3)⋊C3—91no—
33C2×C3×D9⋊C9—91no—
34C2×#92214C911yescharacteristic, complemented
35C2×AGL(1,ℤ/27)—31no—
36C2×AGL(1,ℤ/27)—31no—
37C2×AGL(1,ℤ/27)—31no—
38C2×AGL(1,ℤ/27)—31no—
39C2×AGL(1,ℤ/27)—31no—
40C2×AGL(1,ℤ/27)—31no—
41C2×AGL(1,ℤ/27)—31no—
42C2×AGL(1,ℤ/27)—31no—
43C2×AGL(1,ℤ/27)—31no—
44#10624864C2×D311yescharacteristic
45C3×#6875C2×D311yescharacteristic, complemented
46C3×#6875C2×D311yescharacteristic, complemented
47C3×#6875C2×D311yescharacteristic, complemented
48#10624704—21no—
49#10624704C22×C311yescharacteristic
50#10624473—21no—
51#10624473—21no—
52#10624473—21no—
53C2×C3×#737C3×D311yescharacteristic
54C3×#4207—32no—
55C2×C3×C27⋊C3—21no—
56C2×C3×C27⋊C3—21no—
57C2×C3×C27⋊C3—21no—
58C2×C3×C27⋊C3C3×D311yescharacteristic
59C2×C3×C27⋊C3C3×D311yescharacteristic
60C2×C3×C27⋊C3C3×D311yescharacteristic
61C2×C9×3−1+2—61no—
62C2×C9×3−1+2—61no—
63C2×C9×3−1+2—61no—
64C2×C9×3−1+2—61no—
65C2×C9×3−1+2—61no—
66C2×C9×H(3,𝔽3)—31no—
67C2×#6887—21no—
68C2×#6887—31no—
69C2×#6887—61no—
70C2×C3×C9⋊C9—31no—
71C2×C3×C9⋊C9C3⋊D311yescharacteristic
72#92294—32no—
73#92294—32no—
74C3⋊(D9⋊C9)—32no—
75C3×D27⋊C3—32no—
76C3×D27⋊C3—32no—
77C3×D27⋊C3—32no—
78C9×AGL(1,ℤ/9)—92no—
79C9×AGL(1,ℤ/9)—92no—
80C9×(C3⋊D3)⋊C3—92no—
81C3×D9⋊C9—92no—
82C2×#6875—31no—
83C2×#6875—31no—
84C2×#6875—31no—
85C2×#6875—61no—
86C2×#6875—61no—
87C2×#6875—61no—
88C2×#6875—61no—
89C2×#6875—61no—
90C2×#6875—61no—
91C2×#6873—21no—
92C2×#6873—21no—
93C2×#6873—21no—
94C2×#6873—21no—
95C2×#6873—61no—
96C2×#6873—61no—
97C2×#6873—61no—
98C2×#6873C2×C911yescharacteristic, complemented
99#92214C2×C912yescomplemented
100AGL(1,ℤ/27)—32no—
101AGL(1,ℤ/27)—32no—
102AGL(1,ℤ/27)—32no—
103AGL(1,ℤ/27)—32no—
104AGL(1,ℤ/27)—32no—
105AGL(1,ℤ/27)—32no—
106AGL(1,ℤ/27)—32no—
107AGL(1,ℤ/27)—32no—
108AGL(1,ℤ/27)—32no—
109C2×#4207—31no—
110C2×#4207—31no—
111C2×#4207—31no—
112C2×C9×C3⋊D3—91no—
113C2×C3×C3⋊D9—31no—
114C2×C3×AGL(1,ℤ/9)—91no—
115C2×C3×AGL(1,ℤ/9)—91no—
116C2×C3×(C3⋊D3)⋊C3—91no—
117C2×C3×D3×C9—271no—
118C2×C32×D9—91no—
119C2×D27⋊C3—31no—
120C2×D27⋊C3—31no—
121C2×D27⋊C3—31no—
122C2×D27⋊C3—31no—
123C2×D27⋊C3—31no—
124C2×D27⋊C3—31no—
125C2×D27⋊C3—31no—
126C2×D27⋊C3—31no—
127C2×D27⋊C3—31no—
128C2×C3×D27—31no—
129C2×C3×D27—31no—
130C2×C3×D27—31no—
131C2×D9⋊C9—91no—
132C2×D9⋊C9—91no—
133C2×D9⋊C9—91no—
134C2×D9⋊C9—91no—
135C2×D9⋊C9—91no—
136C2×D9⋊C9—91no—
137C2×D9⋊C9—91no—
138C2×(C3⋊D3)⋊C9—91no—
139C2×(C3⋊D3)⋊C9—91no—
140C2×C9×D9—91no—
141C2×C9×D9—91no—
142C3×#737C2×C3×D311yescharacteristic
143C3×C27⋊C3—21no—
144C3×C27⋊C3—21no—
145C3×C27⋊C3—21no—
146C3×C27⋊C3C2×C3×D311yescharacteristic
147C3×C27⋊C3C2×C3×D311yescharacteristic
148C3×C27⋊C3C2×C3×D311yescharacteristic
149C9×3−1+2—61no—
150C9×3−1+2—61no—
151C9×3−1+2—61no—
152C9×3−1+2—61no—
153C9×3−1+2—61no—
154C9×H(3,𝔽3)—31no—
155#6887—21no—
156#6887—31no—
157#6887—61no—
158C3×C9⋊C9—31no—
159C3×C9⋊C9C2×C3⋊D311yescharacteristic
160#6875—31no—
161#6875—31no—
162#6875—31no—
163#6875—61no—
164#6875—61no—
165#6875—61no—
166#6875—61no—
167#6875—61no—
168#6875—61no—
169#6873—21no—
170#6873—21no—
171#6873—21no—
172#6873—21no—
173#6873—61no—
174#6873—61no—
175#6873—61no—
176#6873C22×C911yescharacteristic, complemented
177C2×#737—21no—
178C2×#737—21no—
179C2×#737—21no—
180C2×#737—21no—
181C2×#737—31no—
182C2×#737—31no—
183C2×#737—61no—
184C2×#737—61no—
185C2×#737D3×C911yescharacteristic
186C2×C3×3−1+2—61no—
187C2×C3×3−1+2AGL(1,ℤ/9)11yescharacteristic
188C2×C3×3−1+2AGL(1,ℤ/9)11yescharacteristic
189C2×C3×H(3,𝔽3)AGL(1,ℤ/9)11yescharacteristic
190C2×C32×C9—31no—
191C2×C32×C9—31no—
192C2×C32×C9—61no—
193C2×C32×C9—61no—
194C2×C32×C9C3×C3⋊D311yescharacteristic
195#4207—32no—
196#4207—32no—
197#4207—32no—
198C9×C3⋊D3—92no—
199C3×C3⋊D9—32no—
200C3×AGL(1,ℤ/9)—92no—
201C3×AGL(1,ℤ/9)—92no—
202C3×(C3⋊D3)⋊C3—92no—
203C3×D3×C9—272no—
204C32×D9—92no—
205C2×C27⋊C3—21no—
206C2×C27⋊C3—21no—
207C2×C27⋊C3—31no—
208C2×C27⋊C3—31no—
209C2×C27⋊C3—61no—
210C2×C27⋊C3—61no—
211C2×C27⋊C3—61no—
212C2×C27⋊C3—61no—
213C2×C27⋊C3—61no—
214C2×C27⋊C3—61no—
215C2×C27⋊C3—61no—
216C2×C27⋊C3—61no—
217C2×C27⋊C3—61no—
218C2×C27⋊C3—61no—
219C2×C27⋊C3—61no—
220C2×C27⋊C3—61no—
221C2×C27⋊C3—61no—
222C2×C27⋊C3—61no—
223C2×C27⋊C3D3×C911yescharacteristic, complemented
224C2×C27⋊C3D3×C911yescharacteristic, complemented
225C2×C3×C27—21no—
226C2×C3×C27—31no—
227C2×C3×C27—31no—
228C2×C3×C27—61no—
229C2×C3×C27—61no—
230C2×C3×C27—61no—
231C2×C3×C27D3×C911yescharacteristic, complemented
232C2×C9⋊C9—21no—
233C2×C9⋊C9—21no—
234C2×C9⋊C9—21no—
235C2×C9⋊C9—61no—
236C2×C9⋊C9—61no—
237C2×C9⋊C9—61no—
238C2×C9⋊C9—61no—
239C2×C9⋊C9—61no—
240C2×C9⋊C9—61no—
241C2×C9⋊C9—61no—
242C2×C9⋊C9—61no—
243C2×C9⋊C9—61no—
244C2×C9⋊C9—61no—
245C2×C9⋊C9#27211yescharacteristic
246C2×C9⋊C9#27211yescharacteristic
247C2×C9⋊C9#27211yescharacteristic
248C2×C32⋊C9—31no—
249C2×C32⋊C9—61no—
250C2×C32⋊C9—61no—
251C2×C32⋊C9—61no—
252C2×C32⋊C9—61no—
253C2×C92—61no—
254C2×C92—61no—
255C2×C92—61no—
256C2×C92—61no—
257C2×C92—61no—
258D27⋊C3—32no—
259D27⋊C3—32no—
260D27⋊C3—32no—
261D27⋊C3—32no—
262D27⋊C3—32no—
263D27⋊C3—32no—
264D27⋊C3—32no—
265D27⋊C3—32no—
266D27⋊C3—32no—
267C3×D27—32no—
268C3×D27—32no—
269C3×D27—32no—
270D9⋊C9—92no—
271D9⋊C9—92no—
272D9⋊C9—92no—
273D9⋊C9—92no—
274D9⋊C9—92no—
275D9⋊C9—92no—
276D9⋊C9—92no—
277(C3⋊D3)⋊C9—92no—
278(C3⋊D3)⋊C9—92no—
279C9×D9—92no—
280C9×D9—92no—
281C2×C3×C3⋊D3—91no—
282C2×C32×D3—271no—
283C22×C3×C9—811no—
284C2×C3⋊D9—31no—
285C2×AGL(1,ℤ/9)—91no—
286C2×AGL(1,ℤ/9)—91no—
287C2×AGL(1,ℤ/9)—91no—
288C2×AGL(1,ℤ/9)—91no—
289C2×AGL(1,ℤ/9)—91no—
290C2×AGL(1,ℤ/9)—91no—
291C2×(C3⋊D3)⋊C3—91no—
292C2×(C3⋊D3)⋊C3—91no—
293C2×(C3⋊D3)⋊C3—91no—
294C2×D3×C9—271no—
295C2×D3×C9—271no—
296C2×D3×C9—271no—
297C2×D3×C9—811no—
298C2×C3×D9—91no—
299C2×C3×D9—91no—
300C2×C3×D9—91no—
301C2×C3×D9—91no—
302C2×C3×D9—91no—
303C2×C3×D9—91no—
304C2×D27—31no—
305C2×D27—31no—
306C2×D27—31no—
307#737—21no—
308#737—21no—
309#737—21no—
310#737—21no—
311#737—31no—
312#737—31no—
313#737—61no—
314#737—61no—
315#737C2×D3×C911yescharacteristic
316C3×3−1+2—61no—
317C3×3−1+2C2×AGL(1,ℤ/9)11yescharacteristic
318C3×3−1+2C2×AGL(1,ℤ/9)11yescharacteristic
319C3×H(3,𝔽3)C2×AGL(1,ℤ/9)11yescharacteristic
320C32×C9—31no—
321C32×C9—31no—
322C32×C9—61no—
323C32×C9—61no—
324C32×C9C2×C3×C3⋊D311yescharacteristic
325C27⋊C3—21no—
326C27⋊C3—21no—
327C27⋊C3—31no—
328C27⋊C3—31no—
329C27⋊C3—61no—
330C27⋊C3—61no—
331C27⋊C3—61no—
332C27⋊C3—61no—
333C27⋊C3—61no—
334C27⋊C3—61no—
335C27⋊C3—61no—
336C27⋊C3—61no—
337C27⋊C3—61no—
338C27⋊C3—61no—
339C27⋊C3—61no—
340C27⋊C3—61no—
341C27⋊C3—61no—
342C27⋊C3—61no—
343C27⋊C3C2×D3×C911yescharacteristic, complemented
344C27⋊C3C2×D3×C911yescharacteristic, complemented
345C3×C27—21no—
346C3×C27—31no—
347C3×C27—31no—
348C3×C27—61no—
349C3×C27—61no—
350C3×C27—61no—
351C3×C27C2×D3×C911yescharacteristic, complemented
352C9⋊C9—21no—
353C9⋊C9—21no—
354C9⋊C9—21no—
355C9⋊C9—61no—
356C9⋊C9—61no—
357C9⋊C9—61no—
358C9⋊C9—61no—
359C9⋊C9—61no—
360C9⋊C9—61no—
361C9⋊C9—61no—
362C9⋊C9—61no—
363C9⋊C9—61no—
364C9⋊C9—61no—
365C9⋊C9C2×#27211yescharacteristic
366C9⋊C9C2×#27211yescharacteristic
367C9⋊C9C2×#27211yescharacteristic
368C32⋊C9—31no—
369C32⋊C9—61no—
370C32⋊C9—61no—
371C32⋊C9—61no—
372C32⋊C9—61no—
373C92—61no—
374C92—61no—
375C92—61no—
376C92—61no—
377C92—61no—
378C2×C33—61no—
379C2×C33C3⋊AGL(1,ℤ/9)11yescharacteristic
380C3×C3⋊D3—92no—
381C32×D3—272no—
382C2×3−1+2—21no—
383C2×3−1+2—21no—
384C2×3−1+2—31no—
385C2×3−1+2—61no—
386C2×3−1+2—61no—
387C2×3−1+2—61no—
388C2×3−1+2—61no—
389C2×3−1+2—61no—
390C2×3−1+2—61no—
391C2×3−1+2—61no—
392C2×3−1+2—62no—
393C2×3−1+2—61no—
394C2×3−1+2—61no—
395C2×3−1+2—61no—
396C2×3−1+2D9⋊C911yescharacteristic
397C2×3−1+2D9⋊C911yescharacteristic
398C2×H(3,𝔽3)—21no—
399C2×H(3,𝔽3)—31no—
400C2×H(3,𝔽3)—61no—
401C2×H(3,𝔽3)D9⋊C911yescharacteristic
402C2×C3×C9—21no—
403C2×C3×C9—21no—
404C2×C3×C9—21no—
405C2×C3×C9—21no—
406C2×C3×C9—31no—
407C2×C3×C9—31no—
408C2×C3×C9—31no—
409C2×C3×C9—61no—
410C2×C3×C9—61no—
411C2×C3×C9—61no—
412C2×C3×C9—61no—
413C2×C3×C9—61no—
414C2×C3×C9—61no—
415C2×C3×C9—61no—
416C2×C3×C9—61no—
417C2×C3×C9—61no—
418C2×C3×C9—61no—
419C2×C3×C9—541no—
420C2×C3×C9—541no—
421C2×C3×C9—541no—
422C2×C3×C9—812no—
423C2×C3×C9C9×C3⋊D311yescharacteristic
424C2×C3×C9C3×#27211yescharacteristic
425C2×C3×C9#420811yescharacteristic
426C2×C3×C9#420811yescharacteristic
427C3⋊D9—32no—
428AGL(1,ℤ/9)—92no—
429AGL(1,ℤ/9)—92no—
430AGL(1,ℤ/9)—92no—
431AGL(1,ℤ/9)—92no—
432AGL(1,ℤ/9)—92no—
433AGL(1,ℤ/9)—92no—
434(C3⋊D3)⋊C3—92no—
435(C3⋊D3)⋊C3—92no—
436(C3⋊D3)⋊C3—92no—
437D3×C9—272no—
438D3×C9—272no—
439D3×C9—272no—
440D3×C9—812no—
441C3×D9—92no—
442C3×D9—92no—
443C3×D9—92no—
444C3×D9—92no—
445C3×D9—92no—
446C3×D9—92no—
447C2×C27—31no—
448C2×C27—61no—
449C2×C27—61no—
450C2×C27—61no—
451C2×C27—61no—
452C2×C27—61no—
453C2×C27—181no—
454C2×C27—181no—
455C2×C27—181no—
456D27—32no—
457D27—32no—
458D27—32no—
459C22×C32—811no—
460C2×C3⋊D3—91no—
461C2×C3×D3—271no—
462C2×C3×D3—271no—
463C2×C3×D3—271no—
464C2×C3×D3—271no—
465C2×C3×D3—811no—
466C22×C9—811no—
467C22×C9—811no—
468C22×C9—811no—
469C2×D9—91no—
470C2×D9—91no—
471C2×D9—91no—
472C33—61no—
473C33C2×C3⋊AGL(1,ℤ/9)11yescharacteristic
4743−1+2—21no—
4753−1+2—21no—
4763−1+2—31no—
4773−1+2—61no—
4783−1+2—61no—
4793−1+2—61no—
4803−1+2—61no—
4813−1+2—61no—
4823−1+2—61no—
4833−1+2—61no—
4843−1+2—62no—
4853−1+2—61no—
4863−1+2—61no—
4873−1+2—61no—
4883−1+2C2×D9⋊C911yescharacteristic
4893−1+2C2×D9⋊C911yescharacteristic
490H(3,𝔽3)—21no—
491H(3,𝔽3)—31no—
492H(3,𝔽3)—61no—
493H(3,𝔽3)C2×D9⋊C911yescharacteristic
494C3×C9—21no—
495C3×C9—21no—
496C3×C9—21no—
497C3×C9—21no—
498C3×C9—31no—
499C3×C9—31no—
500C3×C9—31no—
501C3×C9—61no—
502C3×C9—61no—
503C3×C9—61no—
504C3×C9—61no—
505C3×C9—61no—
506C3×C9—61no—
507C3×C9—61no—
508C3×C9—61no—
509C3×C9—61no—
510C3×C9—61no—
511C3×C9—541no—
512C3×C9—541no—
513C3×C9—541no—
514C3×C9C2×C9×C3⋊D311yescharacteristic
515C3×C9C2×C3×#27211yescharacteristic
516C3×C9C2×#420811yescharacteristic
517C3×C9C2×#420811yescharacteristic
518C27—31no—
519C27—61no—
520C27—61no—
521C27—61no—
522C27—61no—
523C27—61no—
524C27—181no—
525C27—181no—
526C27—181no—
527C2×C32—61no—
528C2×C32—61no—
529C2×C32—91no—
530C2×C32—181no—
531C2×C32—181no—
532C2×C32—812no—
533C2×C32C3⋊(D9⋊C9)11yescharacteristic
534C2×C32#9230211yescharacteristic
535C2×C32#6895⋊C211yescharacteristic
536C2×C32#6895⋊C211yescharacteristic
537C3⋊D3—92no—
538C3×D3—272no—
539C3×D3—272no—
540C3×D3—272no—
541C3×D3—272no—
542C3×D3—812no—
543C2×C9—21no—
544C2×C9—21no—
545C2×C9—21no—
546C2×C9—21no—
547C2×C9—31no—
548C2×C9—31no—
549C2×C9—61no—
550C2×C9—61no—
551C2×C9—181no—
552C2×C9—181no—
553C2×C9—181no—
554C2×C9—181no—
555C2×C9—181no—
556C2×C9—181no—
557C2×C9—812no—
558C2×C9—812no—
559C2×C9—812no—
560C2×C9C9×#27211yescharacteristic
561D9—92no—
562D9—92no—
563D9—92no—
564C22×C3—811no—
565C22×C3—811no—
566C22×C3—811no—
567C22×C3—811no—
568C2×D3—271no—
569C2×D3—811no—
570C32—61no—
571C32—61no—
572C32—91no—
573C32—181no—
574C32—181no—
575C32C2×C3⋊(D9⋊C9)11yescharacteristic
576C32C2×#9230211yescharacteristic
577C32C2×#6895⋊C211yescharacteristic
578C32C2×#6895⋊C211yescharacteristic
579C9—21no—
580C9—21no—
581C9—21no—
582C9—21no—
583C9—31no—
584C9—31no—
585C9—61no—
586C9—61no—
587C9—181no—
588C9—181no—
589C9—181no—
590C9—181no—
591C9—181no—
592C9—181no—
593C9C2×C9×#27211yescharacteristic
594C2×C3—31no—
595C2×C3—61no—
596C2×C3—61no—
597C2×C3—61no—
598C2×C3—91no—
599C2×C3—181no—
600C2×C3—812no—
601C2×C3—812no—
602C2×C3—812no—
603C2×C3—812no—
604C2×C3#1309064611yescharacteristic
605D3—272no—
606D3—812no—
607C22—811noSylow
608C3—31no—
609C3—61no—
610C3—61no—
611C3—61no—
612C3—91no—
613C3—181no—
614C3C2×#1309064611yescharacteristic
615C2—812no—
616C2#7222182511yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2#72221825
commutator, hypocenter#6873C22×C9
FrattiniC32×C9C2×C3×C3⋊D3
FittingC2×#14550491C2
socleC2×C3#13090646
prime radicalC2×#92214⋊C3C3
cosocle#10624704C22×C3
LCSG ▷ #6873
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
3. C2×#14550491C2semidirect
4. #72221825C2direct (N × H)
8. #14550491C22semidirect
15. C2×C3×#6875D3semidirect
16. C2×C3×#6875D3semidirect
17. C2×C3×#6875D3semidirect
34. C2×#92214C9semidirect
45. C3×#6875C2×D3semidirect
46. C3×#6875C2×D3semidirect
47. C3×#6875C2×D3semidirect
98. C2×#6873C2×C9semidirect
99. #92214C2×C9semidirect
176. #6873C22×C9semidirect
223. C2×C27⋊C3D3×C9semidirect
224. C2×C27⋊C3D3×C9semidirect
231. C2×C3×C27D3×C9semidirect
343. C27⋊C3C2×D3×C9semidirect
344. C27⋊C3C2×D3×C9semidirect
351. C3×C27C2×D3×C9semidirect
616. C2#72221825direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#4207⋊C93
C2×#92214⋊C33
C2×#145504912
#722218252
C2×C3×AGL(1,ℤ/27)3