Finite Group Databasebeta

← C2×C3×C3⋊#4175 #72244331

Subgroups — C2×C3×C3⋊#4175 #72244331

Order 2916.

2200 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 2200 conjugacy classes of subgroups — too many to draw. Only its 122 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
1C3×C3⋊#4175C212yesmaximal, direct product factor
2C2×C32×AΣL(1,𝔽27)C211yesmaximal, characteristic, complemented
3C2×C3×C3⋊(C3⋊(C3⋊D3))C311yesmaximal, characteristic, complemented
4C2×C3×C3⋊((C3⋊D3)⋊C3)—31nomaximal
5C2×C3⋊#4175C312yesmaximal, direct product factor
6C2×C3⋊#4175C311yesmaximal, characteristic, direct product factor
7C2×C3×#4175—33nomaximal
8C32×AΣL(1,𝔽27)C2211yescharacteristic, complemented, Sylow
9C2×C35C2×C311yescharacteristic, complemented
10C3×C3⋊(C3⋊(C3⋊D3))C2×C312yescomplemented
11C2×C32×3−1+2—21no—
12C2×C32×H(3,𝔽3)D311yescharacteristic, complemented
13C3×C3⋊((C3⋊D3)⋊C3)—32no—
14C2×C3×AΣL(1,𝔽27)—23no—
15C2×C3×AΣL(1,𝔽27)—22no—
16C2×C3×AΣL(1,𝔽27)—21no—
17C2×C3×AΣL(1,𝔽27)—23no—
18C2×C3×AΣL(1,𝔽27)—26no—
19C2×C3×AΣL(1,𝔽27)D313yescomplemented
20C2×C3×AΣL(1,𝔽27)C2×C312yescomplemented
21C2×C3×AΣL(1,𝔽27)C2×C311yescharacteristic, complemented
22C3⋊#4175C2×C314yesdirect product factor
23C3⋊#4175C2×C312yesdirect product factor
24C3×#4175—36no—
25C2×C3⋊(C3⋊(C3⋊D3))C3211yescharacteristic, complemented
26C2×C3×C3⋊(C3⋊D3)—31no—
27C2×C3×C3⋊(C3⋊D3)—33no—
28C2×C3×C3⋊(C3⋊D3)—91no—
29C2×C3×C3⋊(C3⋊D3)—92no—
30C2×C3×C3⋊(C3⋊D3)—93no—
31C2×C3×C3⋊(C3⋊D3)—96no—
32C2×C32×C3⋊D3—91no—
33C2×C3⋊((C3⋊D3)⋊C3)—32no—
34C2×C3⋊((C3⋊D3)⋊C3)—31no—
35C2×C3×(C3⋊D3)⋊C3—92no—
36C2×C3×(C3⋊D3)⋊C3—91no—
37C2×#4175—36no—
38C2×#4175—33no—
39C35C22×C311yescharacteristic, complemented
40C32×3−1+2—21no—
41C32×H(3,𝔽3)C2×D311yescharacteristic, complemented
42C3×AΣL(1,𝔽27)—23no—
43C3×AΣL(1,𝔽27)—22no—
44C3×AΣL(1,𝔽27)—21no—
45C3×AΣL(1,𝔽27)—23no—
46C3×AΣL(1,𝔽27)—26no—
47C3×AΣL(1,𝔽27)C2×D313yescomplemented
48C3×AΣL(1,𝔽27)C22×C312yescomplemented
49C3×AΣL(1,𝔽27)C22×C311yescharacteristic, complemented
50C2×C34—21no—
51C2×C34—23no—
52C2×C34—32no—
53C2×C34—31no—
54C2×C34—33no—
55C2×C34—36no—
56C2×C34—31no—
57C2×C34—62no—
58C2×C34—61no—
59C2×C34—63no—
60C2×C34—66no—
61C2×C34C3×D311yescharacteristic
62C2×C34C3×D313yescomplemented
63C2×C34C2×C3211yescharacteristic, complemented
64C3⋊(C3⋊(C3⋊D3))C2×C3212yescomplemented
65C3×C3⋊(C3⋊D3)—32no—
66C3×C3⋊(C3⋊D3)—36no—
67C3×C3⋊(C3⋊D3)—92no—
68C3×C3⋊(C3⋊D3)—94no—
69C3×C3⋊(C3⋊D3)—96no—
70C3×C3⋊(C3⋊D3)—912no—
71C32×C3⋊D3—92no—
72C2×C3×3−1+2—26no—
73C2×C3×3−1+2—21no—
74C2×C3×3−1+2—22no—
75C2×C3×3−1+2—23no—
76C2×C3×3−1+2—64no—
77C2×C3×3−1+2—62no—
78C2×C3×3−1+2—62no—
79C2×C3×H(3,𝔽3)—21no—
80C2×C3×H(3,𝔽3)—21no—
81C2×C3×H(3,𝔽3)—22no—
82C2×C3×H(3,𝔽3)—32no—
83C2×C3×H(3,𝔽3)—62no—
84C2×C3×H(3,𝔽3)—61no—
85C2×C3×H(3,𝔽3)C3×D312yescomplemented
86C2×C3×H(3,𝔽3)C3×D311yescharacteristic, complemented
87C2×C3×H(3,𝔽3)C3⋊D311yescharacteristic, complemented
88C2×C32×C9—61no—
89C3⋊((C3⋊D3)⋊C3)—34no—
90C3⋊((C3⋊D3)⋊C3)—32no—
91C3×(C3⋊D3)⋊C3—94no—
92C3×(C3⋊D3)⋊C3—92no—
93C2×AΣL(1,𝔽27)—212no—
94C2×AΣL(1,𝔽27)—26no—
95C2×AΣL(1,𝔽27)—26no—
96C2×AΣL(1,𝔽27)—23no—
97C2×AΣL(1,𝔽27)—26no—
98C2×AΣL(1,𝔽27)—23no—
99C2×AΣL(1,𝔽27)C3×D316yescomplemented
100C2×AΣL(1,𝔽27)C3×D313yescomplemented
101#4175—312no—
102#4175—36no—
103C2×C3⋊(C3⋊D3)—31no—
104C2×C3⋊(C3⋊D3)—33no—
105C2×C3⋊(C3⋊D3)—91no—
106C2×C3⋊(C3⋊D3)—92no—
107C2×C3⋊(C3⋊D3)—93no—
108C2×C3⋊(C3⋊D3)—96no—
109C2×C3×C3⋊D3—91no—
110C2×C3×C3⋊D3—92no—
111C2×C3×C3⋊D3—91no—
112C2×C3×C3⋊D3—92no—
113C2×C3×C3⋊D3—91no—
114C2×C3×C3⋊D3—271no—
115C2×C3×C3⋊D3—272no—
116C2×C3×C3⋊D3—273no—
117C2×C3×C3⋊D3—273no—
118C2×C3×C3⋊D3—276no—
119C2×C3×C3⋊D3—2718no—
120C2×C3×C3⋊D3—279no—
121C2×C32×D3—271no—
122C2×C32×D3—271no—
123C2×C32×D3—272no—
124C2×(C3⋊D3)⋊C3—92no—
125C2×(C3⋊D3)⋊C3—92no—
126C2×(C3⋊D3)⋊C3—92no—
127C2×(C3⋊D3)⋊C3—92no—
128C2×(C3⋊D3)⋊C3—91no—
129C34—21no—
130C34—23no—
131C34—33no—
132C34—36no—
133C34—32no—
134C34—31no—
135C34—31no—
136C34—61no—
137C34—62no—
138C34—63no—
139C34—66no—
140C34C2×C3×D311yescharacteristic
141C34C2×C3×D313yescomplemented
142C34C22×C3211yescharacteristic, complemented
143C3×3−1+2—26no—
144C3×3−1+2—21no—
145C3×3−1+2—22no—
146C3×3−1+2—23no—
147C3×3−1+2—64no—
148C3×3−1+2—62no—
149C3×3−1+2—62no—
150C3×H(3,𝔽3)—21no—
151C3×H(3,𝔽3)—21no—
152C3×H(3,𝔽3)—22no—
153C3×H(3,𝔽3)—32no—
154C3×H(3,𝔽3)—62no—
155C3×H(3,𝔽3)—61no—
156C3×H(3,𝔽3)C2×C3×D312yescomplemented
157C3×H(3,𝔽3)C2×C3×D311yescharacteristic, complemented
158C3×H(3,𝔽3)C2×C3⋊D311yescharacteristic, complemented
159C32×C9—61no—
160AΣL(1,𝔽27)—212no—
161AΣL(1,𝔽27)—26no—
162AΣL(1,𝔽27)—26no—
163AΣL(1,𝔽27)—23no—
164AΣL(1,𝔽27)—26no—
165AΣL(1,𝔽27)—23no—
166AΣL(1,𝔽27)C2×C3×D316yescomplemented
167AΣL(1,𝔽27)C2×C3×D313yescomplemented
168C2×C33—22no—
169C2×C33—21no—
170C2×C33—21no—
171C2×C33—23no—
172C2×C33—32no—
173C2×C33—31no—
174C2×C33—33no—
175C2×C33—36no—
176C2×C33—318no—
177C2×C33—33no—
178C2×C33—36no—
179C2×C33—31no—
180C2×C33—32no—
181C2×C33—33no—
182C2×C33—39no—
183C2×C33—31no—
184C2×C33—63no—
185C2×C33—66no—
186C2×C33—63no—
187C2×C33—66no—
188C2×C33—61no—
189C2×C33—62no—
190C2×C33—63no—
191C2×C33—61no—
192C2×C33—62no—
193C2×C33—636no—
194C2×C33—618no—
195C2×C33—618no—
196C2×C33—66no—
197C2×C33—612no—
198C2×C33—63no—
199C2×C33—66no—
200C2×C33—618no—
201C2×C33—69no—
202C2×C33—63no—
203C2×C33—66no—
204C2×C33—69no—
205C2×C33—62no—
206C2×C33—61no—
207C2×C33—61no—
208C2×C33—61no—
209C2×C33—62no—
210C2×C33—181no—
211C2×C33—182no—
212C2×C33—181no—
213C2×C33—182no—
214C2×C33(C3⋊D3)⋊C311yescharacteristic
215C2×C33(C3⋊D3)⋊C312yes—
216C2×C33C32×D311yescharacteristic
217C2×C33C32×D313yescomplemented
218C2×C33C3×C3⋊D311yescharacteristic
219C3⋊(C3⋊D3)—32no—
220C3⋊(C3⋊D3)—36no—
221C3⋊(C3⋊D3)—92no—
222C3⋊(C3⋊D3)—94no—
223C3⋊(C3⋊D3)—96no—
224C3⋊(C3⋊D3)—912no—
225C3×C3⋊D3—92no—
226C3×C3⋊D3—94no—
227C3×C3⋊D3—92no—
228C3×C3⋊D3—94no—
229C3×C3⋊D3—92no—
230C3×C3⋊D3—272no—
231C3×C3⋊D3—274no—
232C3×C3⋊D3—276no—
233C3×C3⋊D3—276no—
234C3×C3⋊D3—2712no—
235C3×C3⋊D3—2736no—
236C3×C3⋊D3—2718no—
237C32×D3—272no—
238C32×D3—272no—
239C32×D3—274no—
240C2×3−1+2—23no—
241C2×3−1+2—26no—
242C2×3−1+2—612no—
243C2×3−1+2—66no—
244C2×3−1+2—62no—
245C2×3−1+2—64no—
246C2×H(3,𝔽3)—22no—
247C2×H(3,𝔽3)—21no—
248C2×H(3,𝔽3)—32no—
249C2×H(3,𝔽3)—62no—
250C2×H(3,𝔽3)—64no—
251C2×H(3,𝔽3)—62no—
252C2×H(3,𝔽3)—61no—
253C2×H(3,𝔽3)—62no—
254C2×H(3,𝔽3)—62no—
255C2×H(3,𝔽3)C3×C3⋊D312yescomplemented
256C2×H(3,𝔽3)C3×C3⋊D311yescharacteristic, complemented
257C2×C3×C9—66no—
258C2×C3×C9—61no—
259C2×C3×C9—62no—
260C2×C3×C9—63no—
261(C3⋊D3)⋊C3—94no—
262(C3⋊D3)⋊C3—94no—
263(C3⋊D3)⋊C3—94no—
264(C3⋊D3)⋊C3—94no—
265(C3⋊D3)⋊C3—92no—
266C22×C32—811no—
267C2×C3⋊D3—91no—
268C2×C3⋊D3—92no—
269C2×C3⋊D3—91no—
270C2×C3⋊D3—271no—
271C2×C3⋊D3—272no—
272C2×C3⋊D3—273no—
273C2×C3⋊D3—273no—
274C2×C3⋊D3—276no—
275C2×C3⋊D3—2718no—
276C2×C3⋊D3—279no—
277C2×C3×D3—272no—
278C2×C3×D3—271no—
279C2×C3×D3—271no—
280C2×C3×D3—272no—
281C2×C3×D3—272no—
282C2×C3×D3—274no—
283C2×C3×D3—271no—
284C2×C3×D3—272no—
285C2×C3×D3—271no—
286C2×C3×D3—812no—
287C2×C3×D3—811no—
288C2×C3×D3—819no—
289C33—22no—
290C33—21no—
291C33—21no—
292C33—23no—
293C33—33no—
294C33—36no—
295C33—32no—
296C33—31no—
297C33—318no—
298C33—39no—
299C33—33no—
300C33—36no—
301C33—31no—
302C33—32no—
303C33—33no—
304C33—31no—
305C33—61no—
306C33—62no—
307C33—63no—
308C33—63no—
309C33—66no—
310C33—63no—
311C33—66no—
312C33—62no—
313C33—61no—
314C33—69no—
315C33—618no—
316C33—69no—
317C33—63no—
318C33—66no—
319C33—63no—
320C33—66no—
321C33—618no—
322C33—636no—
323C33—618no—
324C33—66no—
325C33—612no—
326C33—62no—
327C33—61no—
328C33—61no—
329C33—61no—
330C33—62no—
331C33—181no—
332C33—182no—
333C33—181no—
334C33—182no—
335C33C2×(C3⋊D3)⋊C311yescharacteristic
336C33C2×(C3⋊D3)⋊C312yes—
337C33C2×C32×D311yescharacteristic
338C33C2×C32×D313yescomplemented
339C33C2×C3×C3⋊D311yescharacteristic
3403−1+2—23no—
3413−1+2—26no—
3423−1+2—612no—
3433−1+2—66no—
3443−1+2—62no—
3453−1+2—64no—
346H(3,𝔽3)—22no—
347H(3,𝔽3)—21no—
348H(3,𝔽3)—32no—
349H(3,𝔽3)—64no—
350H(3,𝔽3)—62no—
351H(3,𝔽3)—62no—
352H(3,𝔽3)—61no—
353H(3,𝔽3)—62no—
354H(3,𝔽3)—62no—
355H(3,𝔽3)C2×C3×C3⋊D312yescomplemented
356H(3,𝔽3)C2×C3×C3⋊D311yescharacteristic, complemented
357C3×C9—66no—
358C3×C9—61no—
359C3×C9—62no—
360C3×C9—63no—
361C2×C32—21no—
362C2×C32—22no—
363C2×C32—21no—
364C2×C32—22no—
365C2×C32—21no—
366C2×C32—318no—
367C2×C32—31no—
368C2×C32—32no—
369C2×C32—33no—
370C2×C32—36no—
371C2×C32—33no—
372C2×C32—39no—
373C2×C32—32no—
374C2×C32—31no—
375C2×C32—39no—
376C2×C32—64no—
377C2×C32—62no—
378C2×C32—62no—
379C2×C32—61no—
380C2×C32—618no—
381C2×C32—69no—
382C2×C32—69no—
383C2×C32—62no—
384C2×C32—61no—
385C2×C32—636no—
386C2×C32—618no—
387C2×C32—618no—
388C2×C32—618no—
389C2×C32—69no—
390C2×C32—69no—
391C2×C32—61no—
392C2×C32—62no—
393C2×C32—63no—
394C2×C32—66no—
395C2×C32—63no—
396C2×C32—62no—
397C2×C32—61no—
398C2×C32—62no—
399C2×C32—61no—
400C2×C32—91no—
401C2×C32—181no—
402C2×C32—182no—
403C2×C32—181no—
404C2×C32—184no—
405C2×C32—182no—
406C2×C32—182no—
407C2×C32—181no—
408C2×C32—181no—
409C2×C32—182no—
410C2×C32—182no—
411C2×C32—184no—
412C2×C32—812no—
413C2×C32#417511yescharacteristic, complemented
414C2×C32#417512yescomplemented
415C2×C32C3×(C3⋊D3)⋊C312yes—
416C2×C32C3×(C3⋊D3)⋊C311yescharacteristic
417C2×C32C3⋊((C3⋊D3)⋊C3)11yescharacteristic
418C2×C32C32×C3⋊D311yescharacteristic
419C3⋊D3—92no—
420C3⋊D3—94no—
421C3⋊D3—92no—
422C3⋊D3—276no—
423C3⋊D3—272no—
424C3⋊D3—274no—
425C3⋊D3—276no—
426C3⋊D3—2712no—
427C3⋊D3—2736no—
428C3⋊D3—2718no—
429C3×D3—274no—
430C3×D3—272no—
431C3×D3—272no—
432C3×D3—274no—
433C3×D3—274no—
434C3×D3—278no—
435C3×D3—272no—
436C3×D3—274no—
437C3×D3—272no—
438C3×D3—814no—
439C3×D3—812no—
440C3×D3—8118no—
441C2×C9—63no—
442C2×C9—66no—
443C22×C3—811no—
444C22×C3—812no—
445C22×C3—811no—
446C2×D3—271no—
447C2×D3—272no—
448C2×D3—271no—
449C2×D3—812no—
450C2×D3—811no—
451C2×D3—819no—
452C32—21no—
453C32—21no—
454C32—22no—
455C32—22no—
456C32—21no—
457C32—39no—
458C32—318no—
459C32—33no—
460C32—33no—
461C32—36no—
462C32—31no—
463C32—32no—
464C32—39no—
465C32—32no—
466C32—31no—
467C32—62no—
468C32—61no—
469C32—64no—
470C32—62no—
471C32—62no—
472C32—61no—
473C32—618no—
474C32—69no—
475C32—69no—
476C32—69no—
477C32—69no—
478C32—618no—
479C32—636no—
480C32—618no—
481C32—618no—
482C32—63no—
483C32—61no—
484C32—62no—
485C32—63no—
486C32—66no—
487C32—62no—
488C32—61no—
489C32—62no—
490C32—61no—
491C32—91no—
492C32—181no—
493C32—182no—
494C32—181no—
495C32—184no—
496C32—182no—
497C32—182no—
498C32—181no—
499C32—181no—
500C32—182no—
501C32—182no—
502C32—184no—
503C32C2×#417511yescharacteristic, complemented
504C32C2×#417512yescomplemented
505C32C2×C3×(C3⋊D3)⋊C312yes—
506C32C2×C3×(C3⋊D3)⋊C311yescharacteristic
507C32C2×C3⋊((C3⋊D3)⋊C3)11yescharacteristic
508C32C2×C32×C3⋊D311yescharacteristic
509C9—63no—
510C9—66no—
511C2×C3—22no—
512C2×C3—21no—
513C2×C3—21no—
514C2×C3—39no—
515C2×C3—32no—
516C2×C3—31no—
517C2×C3—62no—
518C2×C3—61no—
519C2×C3—69no—
520C2×C3—181no—
521C2×C3—182no—
522C2×C3—181no—
523C2×C3—182no—
524C2×C3—812no—
525C2×C3—814no—
526C2×C3—812no—
527C2×C3C3×#417511yescharacteristic, complemented
528C2×C3C3×#417512yescomplemented
529C2×C3C3⋊#417511yescharacteristic, direct product factor
530C2×C3C3×C3⋊((C3⋊D3)⋊C3)11yescharacteristic
531D3—272no—
532D3—272no—
533D3—274no—
534D3—814no—
535D3—812no—
536D3—8118no—
537C22—811noSylow
538C3—22no—
539C3—21no—
540C3—21no—
541C3—39no—
542C3—32no—
543C3—31no—
544C3—62no—
545C3—61no—
546C3—69no—
547C3—181no—
548C3—182no—
549C3—181no—
550C3—182no—
551C3C2×C3×#417511yescharacteristic, complemented
552C3C2×C3×#417512yescomplemented
553C3C2×C3⋊#417511yescharacteristic, direct product factor
554C3C2×C3×C3⋊((C3⋊D3)⋊C3)11yescharacteristic
555C2—812no—
556C2C3×C3⋊#417511yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2×C3C3⋊#4175
commutator, cosocle, hypocenterC34C22×C32
FrattiniC32C2×C32×C3⋊D3
FittingC2×C32×AΣL(1,𝔽27)C2
socleC2×C33(C3⋊D3)⋊C3
LCSG ▷ C34
UCSC1 ◁ C2×C3

Complements

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

NH ≅ G/Ntype
1. C3×C3⋊#4175C2direct (N × H)
2. C2×C32×AΣL(1,𝔽27)C2semidirect
3. C2×C3×C3⋊(C3⋊(C3⋊D3))C3semidirect
5. C2×C3⋊#4175C3direct (N × H)
6. C2×C3⋊#4175C3direct (N × H)
8. C32×AΣL(1,𝔽27)C22semidirect
9. C2×C35C2×C3semidirect
10. C3×C3⋊(C3⋊(C3⋊D3))C2×C3semidirect
12. C2×C32×H(3,𝔽3)D3semidirect
19. C2×C3×AΣL(1,𝔽27)D3semidirect
20. C2×C3×AΣL(1,𝔽27)C2×C3semidirect
21. C2×C3×AΣL(1,𝔽27)C2×C3semidirect
22. C3⋊#4175C2×C3direct (N × H)
23. C3⋊#4175C2×C3direct (N × H)
25. C2×C3⋊(C3⋊(C3⋊D3))C32semidirect
39. C35C22×C3semidirect
41. C32×H(3,𝔽3)C2×D3semidirect
47. C3×AΣL(1,𝔽27)C2×D3semidirect
48. C3×AΣL(1,𝔽27)C22×C3semidirect
49. C3×AΣL(1,𝔽27)C22×C3semidirect
62. C2×C34C3×D3semidirect
63. C2×C34C2×C32semidirect
64. C3⋊(C3⋊(C3⋊D3))C2×C32semidirect
85. C2×C3×H(3,𝔽3)C3×D3semidirect
86. C2×C3×H(3,𝔽3)C3×D3semidirect
87. C2×C3×H(3,𝔽3)C3⋊D3semidirect
99. C2×AΣL(1,𝔽27)C3×D3semidirect
100. C2×AΣL(1,𝔽27)C3×D3semidirect
141. C34C2×C3×D3semidirect
142. C34C22×C32semidirect
156. C3×H(3,𝔽3)C2×C3×D3semidirect
157. C3×H(3,𝔽3)C2×C3×D3semidirect
158. C3×H(3,𝔽3)C2×C3⋊D3semidirect
166. AΣL(1,𝔽27)C2×C3×D3semidirect
167. AΣL(1,𝔽27)C2×C3×D3semidirect
217. C2×C33C32×D3semidirect
255. C2×H(3,𝔽3)C3×C3⋊D3semidirect
256. C2×H(3,𝔽3)C3×C3⋊D3semidirect
338. C33C2×C32×D3semidirect
355. H(3,𝔽3)C2×C3×C3⋊D3semidirect
356. H(3,𝔽3)C2×C3×C3⋊D3semidirect
413. C2×C32#4175semidirect
414. C2×C32#4175semidirect
503. C32C2×#4175semidirect
504. C32C2×#4175semidirect
527. C2×C3C3×#4175semidirect
528. C2×C3C3×#4175semidirect
529. C2×C3C3⋊#4175direct (N × H)
551. C3C2×C3×#4175semidirect
552. C3C2×C3×#4175semidirect
553. C3C2×C3⋊#4175direct (N × H)
556. C2C3×C3⋊#4175direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C3⋊#41752
C2×C32×AΣL(1,𝔽27)2
C2×C3×C3⋊(C3⋊(C3⋊D3))3
C2×C3×C3⋊((C3⋊D3)⋊C3)3
C2×C3⋊#41753
C2×C3×#41753