Finite Group Databasebeta

← C2×#72221840 #72244083

Subgroups — C2×#72221840 #72244083

Order 8748.

646 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 646 conjugacy classes of subgroups — too many to draw. Only its 106 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×#10624473⋊C2C311yesmaximal, characteristic, complemented
2C2×#10624473⋊C2C311yesmaximal, characteristic, complemented
3C2×#92184⋊C3C311yesmaximal, characteristic
4C2×#4207⋊C9—31nomaximal
5C2×#14550491C211yesmaximal, characteristic, complemented
6C2×C3⋊AGL(1,ℤ/27)C311yesmaximal, characteristic, complemented
7#72221840C212yesmaximal, direct product factor
8#14550491C2211yescharacteristic, complemented, Sylow
9#4207⋊C9—32no—
10C3⋊AGL(1,ℤ/27)C2×C312yescomplemented
11#92184⋊C3C2×C312yes—
12#10624473⋊C2C2×C312yescomplemented
13#10624473⋊C2C2×C312yescomplemented
14C2×#10624864D311yescharacteristic
15C2×C3×#6875—21no—
16C2×C3×#6875C2×C311yescharacteristic, complemented
17C2×#10624704—21no—
18C2×#10624704C2×C311yescharacteristic
19C2×#10624473—21no—
20C2×#10624473—21no—
21C2×#10624473C2×C311yescharacteristic, complemented
22C2×#10624473C2×C311yescharacteristic, complemented
23C2×C3×#4207—31no—
24C2×C3⋊(D27⋊C3)C3211yescharacteristic
25C2×#92294—31no—
26C2×#92294—31no—
27C2×C3⋊(D9⋊C9)—31no—
28C2×C9×AGL(1,ℤ/9)—91no—
29C2×C9×AGL(1,ℤ/9)—91no—
30C2×C9×(C3⋊D3)⋊C3—91no—
31C2×C3×D9⋊C9—91no—
32C2×AGL(1,ℤ/27)—91no—
33C2×#92184C911yescharacteristic, complemented
34C2×#92184C911yescharacteristic, complemented
35C2×#92184C911yescharacteristic, complemented
36#10624864C2×D311yescharacteristic
37C3×#6875—21no—
38C3×#6875C22×C311yescharacteristic, complemented
39#10624704—21no—
40#10624704C22×C311yescharacteristic
41#10624473—21no—
42#10624473—21no—
43#10624473C22×C311yescharacteristic, complemented
44#10624473C22×C311yescharacteristic, complemented
45C2×C3×#737C3×D311yescharacteristic
46C3×#4207—32no—
47C2×C3×C27⋊C3—21no—
48C2×C3×C27⋊C3—21no—
49C2×C3×C27⋊C3—21no—
50C2×C3×C27⋊C3—21no—
51C2×C3×C27⋊C3C2×C3211yescharacteristic
52C2×C9×3−1+2—61no—
53C2×C9×3−1+2—61no—
54C2×C9×3−1+2—61no—
55C2×C9×3−1+2—61no—
56C2×C9×3−1+2—61no—
57C2×C9×H(3,𝔽3)—61no—
58C2×#6887—61no—
59C2×#6887C3×D311yescharacteristic
60C2×#6887C3×D311yescharacteristic
61C2×C3×C9⋊C9—31no—
62C2×C3×C9⋊C9C3×D311yescharacteristic
63C3⋊(D27⋊C3)C2×C3212yes—
64#92294—32no—
65#92294—32no—
66C3⋊(D9⋊C9)—32no—
67C9×AGL(1,ℤ/9)—92no—
68C9×AGL(1,ℤ/9)—92no—
69C9×(C3⋊D3)⋊C3—92no—
70C3×D9⋊C9—92no—
71C2×#6875—31no—
72C2×#6875—61no—
73C2×#6875—62no—
74C2×#6875—61no—
75C2×#6873—21no—
76C2×#6873—21no—
77C2×#6873—21no—
78C2×#6873—61no—
79C2×#6873—61no—
80C2×#6873—61no—
81C2×#6873C2×C911yescharacteristic, complemented
82C2×#6873C2×C911yescharacteristic, complemented
83C2×#6873C2×C911yescharacteristic, complemented
84AGL(1,ℤ/27)—92no—
85#92184C2×C912yescomplemented
86#92184C2×C912yescomplemented
87#92184C2×C912yescomplemented
88C2×#4207—31no—
89C2×#4207—31no—
90C2×#4207—31no—
91C2×C9×C3⋊D3—91no—
92C2×C3×C3⋊D9—31no—
93C2×C3×AGL(1,ℤ/9)—91no—
94C2×C3×AGL(1,ℤ/9)—91no—
95C2×C3×(C3⋊D3)⋊C3—91no—
96C2×C3×D3×C9—271no—
97C2×C32×D9—91no—
98C2×C3⋊D27C3×C911yescharacteristic, complemented
99C2×D27⋊C3—91no—
100C2×D9⋊C9—91no—
101C2×D9⋊C9—91no—
102C2×D9⋊C9—91no—
103C2×D9⋊C9—91no—
104C2×D9⋊C9—91no—
105C2×D9⋊C9—91no—
106C2×D9⋊C9—91no—
107C2×(C3⋊D3)⋊C9—91no—
108C2×(C3⋊D3)⋊C9—91no—
109C2×C9×D9—91no—
110C2×C9×D9—91no—
111C3×#737C2×C3×D311yescharacteristic
112C3×C27⋊C3—21no—
113C3×C27⋊C3—21no—
114C3×C27⋊C3—21no—
115C3×C27⋊C3—21no—
116C3×C27⋊C3C22×C3211yescharacteristic
117C9×3−1+2—61no—
118C9×3−1+2—61no—
119C9×3−1+2—61no—
120C9×3−1+2—61no—
121C9×3−1+2—61no—
122C9×H(3,𝔽3)—61no—
123#6887—61no—
124#6887C2×C3×D311yescharacteristic
125#6887C2×C3×D311yescharacteristic
126C3×C9⋊C9—31no—
127C3×C9⋊C9C2×C3×D311yescharacteristic
128#6875—31no—
129#6875—61no—
130#6875—62no—
131#6875—61no—
132#6873—21no—
133#6873—21no—
134#6873—21no—
135#6873—61no—
136#6873—61no—
137#6873—61no—
138#6873C22×C911yescharacteristic, complemented
139#6873C22×C911yescharacteristic, complemented
140#6873C22×C911yescharacteristic, complemented
141C2×#737—21no—
142C2×#737—21no—
143C2×#737—21no—
144C2×#737—61no—
145C2×#737—61no—
146C2×#737—61no—
147C2×#737D3×C911yescharacteristic
148C2×#737D3×C911yescharacteristic
149C2×#737D3×C911yescharacteristic
150C2×C3×3−1+2—21no—
151C2×C3×3−1+2—61no—
152C2×C3×3−1+2—61no—
153C2×C3×H(3,𝔽3)AGL(1,ℤ/9)11yescharacteristic
154C2×C32×C9—31no—
155C2×C32×C9—61no—
156C2×C32×C9—61no—
157C2×C32×C9—61no—
158C2×C32×C9C32×D311yescharacteristic
159#4207—32no—
160#4207—32no—
161#4207—32no—
162C9×C3⋊D3—92no—
163C3×C3⋊D9—32no—
164C3×AGL(1,ℤ/9)—92no—
165C3×AGL(1,ℤ/9)—92no—
166C3×(C3⋊D3)⋊C3—92no—
167C3×D3×C9—272no—
168C32×D9—92no—
169C2×C27⋊C3—21no—
170C2×C27⋊C3—21no—
171C2×C27⋊C3—21no—
172C2×C27⋊C3—31no—
173C2×C27⋊C3—61no—
174C2×C27⋊C3—61no—
175C2×C27⋊C3—61no—
176C2×C27⋊C3—61no—
177C2×C27⋊C3—61no—
178C2×C27⋊C3—61no—
179C2×C27⋊C3—61no—
180C2×C27⋊C3—61no—
181C2×C27⋊C3—61no—
182C2×C27⋊C3—61no—
183C2×C27⋊C3—61no—
184C2×C27⋊C3—61no—
185C2×C3×C27—21no—
186C2×C3×C27—61no—
187C2×C3×C27—61no—
188C2×C3×C27—61no—
189C2×C3×C27—61no—
190C2×C3×C27C2×C3×C911yescharacteristic, complemented
191C2×C9⋊C9—21no—
192C2×C9⋊C9—21no—
193C2×C9⋊C9—21no—
194C2×C9⋊C9—31no—
195C2×C9⋊C9—61no—
196C2×C9⋊C9—61no—
197C2×C9⋊C9—61no—
198C2×C9⋊C9—61no—
199C2×C9⋊C9—61no—
200C2×C9⋊C9—61no—
201C2×C9⋊C9—61no—
202C2×C9⋊C9—61no—
203C2×C9⋊C9—61no—
204C2×C9⋊C9(C3⋊D3)⋊C311yescharacteristic
205C2×C9⋊C9AGL(1,ℤ/9)11yescharacteristic
206C2×C9⋊C9AGL(1,ℤ/9)11yescharacteristic
207C2×C32⋊C9—61no—
208C2×C32⋊C9—61no—
209C2×C32⋊C9—61no—
210C2×C32⋊C9—61no—
211C2×C32⋊C9—61no—
212C2×C92—31no—
213C2×C92—61no—
214C2×C92—62no—
215C2×C92—61no—
216C3⋊D27C2×C3×C912yescomplemented
217D27⋊C3—92no—
218D9⋊C9—92no—
219D9⋊C9—92no—
220D9⋊C9—92no—
221D9⋊C9—92no—
222D9⋊C9—92no—
223D9⋊C9—92no—
224D9⋊C9—92no—
225(C3⋊D3)⋊C9—92no—
226(C3⋊D3)⋊C9—92no—
227C9×D9—92no—
228C9×D9—92no—
229C2×C3×C3⋊D3—91no—
230C2×C32×D3—271no—
231C22×C3×C9—811no—
232C2×C3⋊D9—31no—
233C2×AGL(1,ℤ/9)—91no—
234C2×AGL(1,ℤ/9)—91no—
235C2×AGL(1,ℤ/9)—91no—
236C2×AGL(1,ℤ/9)—91no—
237C2×AGL(1,ℤ/9)—91no—
238C2×AGL(1,ℤ/9)—91no—
239C2×(C3⋊D3)⋊C3—91no—
240C2×(C3⋊D3)⋊C3—91no—
241C2×(C3⋊D3)⋊C3—91no—
242C2×D3×C9—271no—
243C2×D3×C9—271no—
244C2×D3×C9—271no—
245C2×D3×C9—811no—
246C2×C3×D9—91no—
247C2×C3×D9—91no—
248C2×C3×D9—91no—
249C2×C3×D9—91no—
250C2×C3×D9—91no—
251C2×C3×D9—91no—
252C2×D27—91no—
253#737—21no—
254#737—21no—
255#737—21no—
256#737—61no—
257#737—61no—
258#737—61no—
259#737C2×D3×C911yescharacteristic
260#737C2×D3×C911yescharacteristic
261#737C2×D3×C911yescharacteristic
262C3×3−1+2—21no—
263C3×3−1+2—61no—
264C3×3−1+2—61no—
265C3×H(3,𝔽3)C2×AGL(1,ℤ/9)11yescharacteristic
266C32×C9—31no—
267C32×C9—61no—
268C32×C9—61no—
269C32×C9—61no—
270C32×C9C2×C32×D311yescharacteristic
271C27⋊C3—21no—
272C27⋊C3—21no—
273C27⋊C3—21no—
274C27⋊C3—31no—
275C27⋊C3—61no—
276C27⋊C3—61no—
277C27⋊C3—61no—
278C27⋊C3—61no—
279C27⋊C3—61no—
280C27⋊C3—61no—
281C27⋊C3—61no—
282C27⋊C3—61no—
283C27⋊C3—61no—
284C27⋊C3—61no—
285C27⋊C3—61no—
286C27⋊C3—61no—
287C3×C27—21no—
288C3×C27—61no—
289C3×C27—61no—
290C3×C27—61no—
291C3×C27—61no—
292C3×C27C22×C3×C911yescharacteristic, complemented
293C9⋊C9—21no—
294C9⋊C9—21no—
295C9⋊C9—21no—
296C9⋊C9—31no—
297C9⋊C9—61no—
298C9⋊C9—61no—
299C9⋊C9—61no—
300C9⋊C9—61no—
301C9⋊C9—61no—
302C9⋊C9—61no—
303C9⋊C9—61no—
304C9⋊C9—61no—
305C9⋊C9—61no—
306C9⋊C9C2×(C3⋊D3)⋊C311yescharacteristic
307C9⋊C9C2×AGL(1,ℤ/9)11yescharacteristic
308C9⋊C9C2×AGL(1,ℤ/9)11yescharacteristic
309C32⋊C9—61no—
310C32⋊C9—61no—
311C32⋊C9—61no—
312C32⋊C9—61no—
313C32⋊C9—61no—
314C92—31no—
315C92—61no—
316C92—62no—
317C92—61no—
318C2×C33—31no—
319C2×C33C3×AGL(1,ℤ/9)11yescharacteristic
320C3×C3⋊D3—92no—
321C32×D3—272no—
322C2×3−1+2—21no—
323C2×3−1+2—21no—
324C2×3−1+2—21no—
325C2×3−1+2—31no—
326C2×3−1+2—31no—
327C2×3−1+2—31no—
328C2×3−1+2—61no—
329C2×3−1+2—61no—
330C2×3−1+2—61no—
331C2×3−1+2—61no—
332C2×3−1+2—61no—
333C2×3−1+2—61no—
334C2×3−1+2—62no—
335C2×3−1+2—61no—
336C2×3−1+2—61no—
337C2×3−1+2—61no—
338C2×H(3,𝔽3)—61no—
339C2×H(3,𝔽3)D9⋊C911yescharacteristic
340C2×H(3,𝔽3)D9⋊C911yescharacteristic
341C2×H(3,𝔽3)D9⋊C911yescharacteristic
342C2×C3×C9—21no—
343C2×C3×C9—21no—
344C2×C3×C9—21no—
345C2×C3×C9—21no—
346C2×C3×C9—31no—
347C2×C3×C9—31no—
348C2×C3×C9—31no—
349C2×C3×C9—61no—
350C2×C3×C9—61no—
351C2×C3×C9—61no—
352C2×C3×C9—61no—
353C2×C3×C9—61no—
354C2×C3×C9—61no—
355C2×C3×C9—61no—
356C2×C3×C9—61no—
357C2×C3×C9—61no—
358C2×C3×C9—61no—
359C2×C3×C9—541no—
360C2×C3×C9—541no—
361C2×C3×C9—541no—
362C2×C3×C9—812no—
363C2×C3×C9C3×D3×C911yescharacteristic
364C2×C3×C9C3×(C3⋊D3)⋊C311yescharacteristic
365C2×C3×C9C3×AGL(1,ℤ/9)11yescharacteristic
366C2×C3×C9C3×AGL(1,ℤ/9)11yescharacteristic
367C3⋊D9—32no—
368AGL(1,ℤ/9)—92no—
369AGL(1,ℤ/9)—92no—
370AGL(1,ℤ/9)—92no—
371AGL(1,ℤ/9)—92no—
372AGL(1,ℤ/9)—92no—
373AGL(1,ℤ/9)—92no—
374(C3⋊D3)⋊C3—92no—
375(C3⋊D3)⋊C3—92no—
376(C3⋊D3)⋊C3—92no—
377D3×C9—272no—
378D3×C9—272no—
379D3×C9—272no—
380D3×C9—812no—
381C3×D9—92no—
382C3×D9—92no—
383C3×D9—92no—
384C3×D9—92no—
385C3×D9—92no—
386C3×D9—92no—
387C2×C27—31no—
388C2×C27—61no—
389C2×C27—61no—
390C2×C27—61no—
391C2×C27—61no—
392C2×C27—181no—
393C2×C27—181no—
394C2×C27—181no—
395D27—92no—
396C22×C32—811no—
397C2×C3⋊D3—91no—
398C2×C3×D3—271no—
399C2×C3×D3—271no—
400C2×C3×D3—271no—
401C2×C3×D3—271no—
402C2×C3×D3—811no—
403C22×C9—811no—
404C22×C9—811no—
405C22×C9—811no—
406C2×D9—91no—
407C2×D9—91no—
408C2×D9—91no—
409C33—31no—
410C33C2×C3×AGL(1,ℤ/9)11yescharacteristic
4113−1+2—21no—
4123−1+2—21no—
4133−1+2—21no—
4143−1+2—31no—
4153−1+2—31no—
4163−1+2—31no—
4173−1+2—61no—
4183−1+2—61no—
4193−1+2—61no—
4203−1+2—61no—
4213−1+2—61no—
4223−1+2—61no—
4233−1+2—62no—
4243−1+2—61no—
4253−1+2—61no—
4263−1+2—61no—
427H(3,𝔽3)—61no—
428H(3,𝔽3)C2×D9⋊C911yescharacteristic
429H(3,𝔽3)C2×D9⋊C911yescharacteristic
430H(3,𝔽3)C2×D9⋊C911yescharacteristic
431C3×C9—21no—
432C3×C9—21no—
433C3×C9—21no—
434C3×C9—21no—
435C3×C9—31no—
436C3×C9—31no—
437C3×C9—31no—
438C3×C9—61no—
439C3×C9—61no—
440C3×C9—61no—
441C3×C9—61no—
442C3×C9—61no—
443C3×C9—61no—
444C3×C9—61no—
445C3×C9—61no—
446C3×C9—61no—
447C3×C9—61no—
448C3×C9—541no—
449C3×C9—541no—
450C3×C9—541no—
451C3×C9C2×C3×D3×C911yescharacteristic
452C3×C9C2×C3×(C3⋊D3)⋊C311yescharacteristic
453C3×C9C2×C3×AGL(1,ℤ/9)11yescharacteristic
454C3×C9C2×C3×AGL(1,ℤ/9)11yescharacteristic
455C27—31no—
456C27—61no—
457C27—61no—
458C27—61no—
459C27—61no—
460C27—181no—
461C27—181no—
462C27—181no—
463C2×C32—21no—
464C2×C32—31no—
465C2×C32—31no—
466C2×C32—61no—
467C2×C32—181no—
468C2×C32—181no—
469C2×C32—181no—
470C2×C32—812no—
471C2×C32C3×D9⋊C911yescharacteristic
472C2×C32#4169⋊C311yescharacteristic
473C3⋊D3—92no—
474C3×D3—272no—
475C3×D3—272no—
476C3×D3—272no—
477C3×D3—272no—
478C3×D3—812no—
479C2×C9—21no—
480C2×C9—21no—
481C2×C9—21no—
482C2×C9—61no—
483C2×C9—61no—
484C2×C9—61no—
485C2×C9—181no—
486C2×C9—181no—
487C2×C9—181no—
488C2×C9—181no—
489C2×C9—181no—
490C2×C9—181no—
491C2×C9—812no—
492C2×C9—812no—
493C2×C9—812no—
494C2×C9C9×(C3⋊D3)⋊C311yescharacteristic
495C2×C9C9×AGL(1,ℤ/9)11yescharacteristic
496C2×C9C9×AGL(1,ℤ/9)11yescharacteristic
497D9—92no—
498D9—92no—
499D9—92no—
500C22×C3—811no—
501C22×C3—811no—
502C22×C3—811no—
503C22×C3—811no—
504C2×D3—271no—
505C2×D3—811no—
506C32—21no—
507C32—31no—
508C32—31no—
509C32—61no—
510C32—181no—
511C32—181no—
512C32—181no—
513C32C2×C3×D9⋊C911yescharacteristic
514C32C2×#4169⋊C311yescharacteristic
515C9—21no—
516C9—21no—
517C9—21no—
518C9—61no—
519C9—61no—
520C9—61no—
521C9—181no—
522C9—181no—
523C9—181no—
524C9—181no—
525C9—181no—
526C9—181no—
527C9C2×C9×(C3⋊D3)⋊C311yescharacteristic
528C9C2×C9×AGL(1,ℤ/9)11yescharacteristic
529C9C2×C9×AGL(1,ℤ/9)11yescharacteristic
530C2×C3—31no—
531C2×C3—31no—
532C2×C3—61no—
533C2×C3—181no—
534C2×C3—181no—
535C2×C3—181no—
536C2×C3—812no—
537C2×C3—812no—
538C2×C3—812no—
539C2×C3—812no—
540C2×C3#4169⋊C911yescharacteristic
541D3—272no—
542D3—812no—
543C22—811noSylow
544C3—31no—
545C3—31no—
546C3—61no—
547C3—181no—
548C3—181no—
549C3—181no—
550C3C2×#4169⋊C911yescharacteristic
551C2—812no—
552C2#7222184011yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2#72221840
commutator, hypocenterC3×C27C22×C3×C9
FrattiniC32×C9C2×C32×D3
FittingC2×#14550491C2
socleC2×C3#4169⋊C9
prime radicalC2×#92184⋊C3C3
cosocleC3×C27⋊C3C22×C32
LCSG ▷ C3×C27
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. C2×#10624473⋊C2C3semidirect
2. C2×#10624473⋊C2C3semidirect
5. C2×#14550491C2semidirect
6. C2×C3⋊AGL(1,ℤ/27)C3semidirect
7. #72221840C2direct (N × H)
8. #14550491C22semidirect
10. C3⋊AGL(1,ℤ/27)C2×C3semidirect
12. #10624473⋊C2C2×C3semidirect
13. #10624473⋊C2C2×C3semidirect
16. C2×C3×#6875C2×C3semidirect
21. C2×#10624473C2×C3semidirect
22. C2×#10624473C2×C3semidirect
33. C2×#92184C9semidirect
34. C2×#92184C9semidirect
35. C2×#92184C9semidirect
38. C3×#6875C22×C3semidirect
43. #10624473C22×C3semidirect
44. #10624473C22×C3semidirect
81. C2×#6873C2×C9semidirect
82. C2×#6873C2×C9semidirect
83. C2×#6873C2×C9semidirect
85. #92184C2×C9semidirect
86. #92184C2×C9semidirect
87. #92184C2×C9semidirect
98. C2×C3⋊D27C3×C9semidirect
138. #6873C22×C9semidirect
139. #6873C22×C9semidirect
140. #6873C22×C9semidirect
190. C2×C3×C27C2×C3×C9semidirect
216. C3⋊D27C2×C3×C9semidirect
292. C3×C27C22×C3×C9semidirect
552. C2#72221840direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#10624473⋊C23
C2×#92184⋊C33
C2×#4207⋊C93
C2×#145504912
C2×C3⋊AGL(1,ℤ/27)3
#722218402