Finite Group Databasebeta

← C2×#72221890 #72254017

Subgroups — C2×#72221890 #72254017

Order 8748.

3744 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 3744 conjugacy classes of subgroups — too many to draw. Only its 160 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×C3⋊#92285—32nomaximal
2C2×C3⋊#92283—32nomaximal
3C2×C3×#92404C311yesmaximal, characteristic, complemented
4C2×#13091148C313yesmaximal, complemented
5C2×#14554343C211yesmaximal, characteristic, complemented
6#72221890C212yesmaximal, direct product factor
7#14554343C2211yescharacteristic, complemented, Sylow
8C3×#92404C2×C312yescomplemented
9C3⋊#92285—34no—
10C3⋊#92283—34no—
11#13091148C2×C316yescomplemented
12C2×C3×H(5,𝔽3)C2×C311yescharacteristic
13C2×C3×#6911—22no—
14C2×C3×#6911D312yescomplemented
15C2×C3×#6910D312yescomplemented
16C2×C3×#6909—22no—
17C2×#10624760—26no—
18C2×#10624760—26no—
19C2×#10624760C2×C313yescomplemented
20C2×#92404C3211yescharacteristic, complemented
21C2×C3×C3⋊#272—32no—
22C2×C3×C3⋊#272—32no—
23C2×C3×C3⋊#272—96no—
24C2×C3×C3⋊#272—96no—
25C2×C3⋊D3×3−1+2—91no—
26C2×C3⋊#4178—36no—
27C2×C3⋊#4176—36no—
28C2×#92285—96no—
29C2×#92283—96no—
30C3×H(5,𝔽3)C22×C311yescharacteristic
31C3×#6911—22no—
32C3×#6911C2×D312yescomplemented
33C3×#6910C2×D312yescomplemented
34C3×#6909—22no—
35#10624760—26no—
36#10624760—26no—
37#10624760C22×C313yescomplemented
38C2×H(5,𝔽3)—22no—
39C2×H(5,𝔽3)—22no—
40C2×H(5,𝔽3)—66no—
41C2×H(5,𝔽3)—66no—
42C2×H(5,𝔽3)C2×C3211yescharacteristic
43C2×C32×3−1+2—22no—
44C2×C32×3−1+2C3⋊D311yescharacteristic, complemented
45C2×C32×H(3,𝔽3)—22no—
46C2×C32×H(3,𝔽3)—36no—
47C2×C32×H(3,𝔽3)—36no—
48C2×C32×H(3,𝔽3)C3×D312yes—
49C2×C32×H(3,𝔽3)C3×D312yes—
50#92404C2×C3212yescomplemented
51C3×C3⋊#272—34no—
52C3×C3⋊#272—34no—
53C3×C3⋊#272—912no—
54C3×C3⋊#272—912no—
55C3⋊D3×3−1+2—92no—
56C2×#6911—212no—
57C2×#6911—26no—
58C2×#6911—36no—
59C2×#6911—66no—
60C2×#6911—66no—
61C2×#6911—66no—
62C2×#6910—36no—
63C2×#6910—66no—
64C2×#6909—212no—
65C2×#6909—26no—
66C2×#6909—66no—
67C2×#6909—66no—
68C2×C3×#732—26no—
69C2×C3×#732—66no—
70C2×C3×#732C3×D316yescomplemented
71C2×C3×#731—26no—
72C2×C3×#731C3×D316yescomplemented
73C2×C3×AΣL(1,𝔽27)—66no—
74C3⋊#4178—312no—
75C3⋊#4176—312no—
76#92285—912no—
77#92283—912no—
78C2×C32×C3⋊D3—91no—
79C2×C32×C3⋊D3—272no—
80C2×C32×C3⋊D3—272no—
81C2×C32×C3⋊D3—279no—
82C2×C3⋊#272—32no—
83C2×C3⋊#272—32no—
84C2×C3⋊#272—96no—
85C2×C3⋊#272—96no—
86C2×C3×#272—96no—
87C2×C3×#272—96no—
88C2×C3×#272—274no—
89C2×C3×#272—274no—
90C2×C3×#272—2718no—
91C2×C9×C3⋊D3—93no—
92C2×D3×3−1+2—272no—
93C2×D3×3−1+2—272no—
94C2×#4178—918no—
95C2×#4176—918no—
96H(5,𝔽3)—22no—
97H(5,𝔽3)—22no—
98H(5,𝔽3)—66no—
99H(5,𝔽3)—66no—
100H(5,𝔽3)C22×C3211yescharacteristic
101C32×3−1+2—22no—
102C32×3−1+2C2×C3⋊D311yescharacteristic, complemented
103C32×H(3,𝔽3)—22no—
104C32×H(3,𝔽3)—36no—
105C32×H(3,𝔽3)—36no—
106C32×H(3,𝔽3)C2×C3×D312yes—
107C32×H(3,𝔽3)C2×C3×D312yes—
108#6911—212no—
109#6911—26no—
110#6911—36no—
111#6911—66no—
112#6911—66no—
113#6911—66no—
114#6910—36no—
115#6910—66no—
116#6909—212no—
117#6909—26no—
118#6909—66no—
119#6909—66no—
120C3×#732—26no—
121C3×#732—66no—
122C3×#732C2×C3×D316yescomplemented
123C3×#731—26no—
124C3×#731C2×C3×D316yescomplemented
125C3×AΣL(1,𝔽27)—66no—
126C2×C34—39no—
127C2×C34—32no—
128C2×C34—32no—
129C2×C34—62no—
130C2×C34C3×C3⋊D311yescharacteristic
131C32×C3⋊D3—92no—
132C32×C3⋊D3—274no—
133C32×C3⋊D3—274no—
134C32×C3⋊D3—2718no—
135C2×C3×3−1+2—26no—
136C2×C3×3−1+2—32no—
137C2×C3×3−1+2—32no—
138C2×C3×3−1+2—62no—
139C2×C3×3−1+2—62no—
140C2×C3×3−1+2—62no—
141C2×C3×3−1+2—62no—
142C2×C3×3−1+2—64no—
143C2×C3×3−1+2—64no—
144C2×C3×3−1+2—64no—
145C2×C3×3−1+2—64no—
146C2×C3×3−1+2—62no—
147C2×C3×H(3,𝔽3)—22no—
148C2×C3×H(3,𝔽3)—22no—
149C2×C3×H(3,𝔽3)—26no—
150C2×C3×H(3,𝔽3)—26no—
151C2×C3×H(3,𝔽3)—26no—
152C2×C3×H(3,𝔽3)—36no—
153C2×C3×H(3,𝔽3)—36no—
154C2×C3×H(3,𝔽3)—318no—
155C2×C3×H(3,𝔽3)—34no—
156C2×C3×H(3,𝔽3)—34no—
157C2×C3×H(3,𝔽3)—612no—
158C2×C3×H(3,𝔽3)—612no—
159C2×C3×H(3,𝔽3)—66no—
160C2×C3×H(3,𝔽3)—66no—
161C2×C3×H(3,𝔽3)—636no—
162C2×C3×H(3,𝔽3)—636no—
163C2×C3×H(3,𝔽3)—64no—
164C2×C3×H(3,𝔽3)—64no—
165C2×C3×H(3,𝔽3)—62no—
166C2×C3×H(3,𝔽3)—62no—
167C2×C3×H(3,𝔽3)—618no—
168C2×C3×H(3,𝔽3)—618no—
169C2×C3×H(3,𝔽3)—66no—
170C2×C3×H(3,𝔽3)—66no—
171C2×C3×H(3,𝔽3)—64no—
172C2×C3×H(3,𝔽3)—64no—
173C2×C3×H(3,𝔽3)—64no—
174C2×C3×H(3,𝔽3)—62no—
175C2×C3×H(3,𝔽3)—64no—
176C2×C3×H(3,𝔽3)—62no—
177C2×C3×H(3,𝔽3)—62no—
178C2×C3×H(3,𝔽3)(C3⋊D3)⋊C316yes—
179C2×C3×H(3,𝔽3)(C3⋊D3)⋊C316yes—
180C2×C3×H(3,𝔽3)C32×D312yes—
181C2×C3×H(3,𝔽3)C32×D312yes—
182C2×C32×C9—62no—
183C2×C32×C9C3×C3⋊D313yescomplemented
184C3⋊#272—34no—
185C3⋊#272—34no—
186C3⋊#272—912no—
187C3⋊#272—912no—
188C3×#272—912no—
189C3×#272—912no—
190C3×#272—278no—
191C3×#272—278no—
192C3×#272—2736no—
193C9×C3⋊D3—96no—
194D3×3−1+2—274no—
195D3×3−1+2—274no—
196C2×#732—318no—
197C2×#732—618no—
198C2×#732—636no—
199C2×#732—618no—
200C2×#731—318no—
201C2×#731—618no—
202C2×AΣL(1,𝔽27)—636no—
203C2×AΣL(1,𝔽27)—618no—
204#4178—936no—
205#4176—936no—
206C2×C3×C3⋊D3—91no—
207C2×C3×C3⋊D3—272no—
208C2×C3×C3⋊D3—272no—
209C2×C3×C3⋊D3—279no—
210C2×C3×C3⋊D3—271no—
211C2×C3×C3⋊D3—276no—
212C2×C3×C3⋊D3—276no—
213C2×C3×C3⋊D3—2727no—
214C2×C32×D3—272no—
215C2×C32×D3—272no—
216C2×C32×D3—816no—
217C2×C32×D3—816no—
218C22×3−1+2—811no—
219C2×#272—96no—
220C2×#272—96no—
221C2×#272—274no—
222C2×#272—274no—
223C2×#272—2718no—
224C2×D3×C9—276no—
225C2×D3×C9—276no—
226C34—39no—
227C34—32no—
228C34—32no—
229C34—62no—
230C34C2×C3×C3⋊D311yescharacteristic
231C3×3−1+2—26no—
232C3×3−1+2—32no—
233C3×3−1+2—32no—
234C3×3−1+2—62no—
235C3×3−1+2—62no—
236C3×3−1+2—62no—
237C3×3−1+2—62no—
238C3×3−1+2—64no—
239C3×3−1+2—64no—
240C3×3−1+2—64no—
241C3×3−1+2—64no—
242C3×3−1+2—62no—
243C3×H(3,𝔽3)—22no—
244C3×H(3,𝔽3)—22no—
245C3×H(3,𝔽3)—26no—
246C3×H(3,𝔽3)—26no—
247C3×H(3,𝔽3)—26no—
248C3×H(3,𝔽3)—36no—
249C3×H(3,𝔽3)—36no—
250C3×H(3,𝔽3)—318no—
251C3×H(3,𝔽3)—34no—
252C3×H(3,𝔽3)—34no—
253C3×H(3,𝔽3)—612no—
254C3×H(3,𝔽3)—612no—
255C3×H(3,𝔽3)—66no—
256C3×H(3,𝔽3)—66no—
257C3×H(3,𝔽3)—636no—
258C3×H(3,𝔽3)—636no—
259C3×H(3,𝔽3)—64no—
260C3×H(3,𝔽3)—64no—
261C3×H(3,𝔽3)—62no—
262C3×H(3,𝔽3)—62no—
263C3×H(3,𝔽3)—618no—
264C3×H(3,𝔽3)—618no—
265C3×H(3,𝔽3)—66no—
266C3×H(3,𝔽3)—66no—
267C3×H(3,𝔽3)—64no—
268C3×H(3,𝔽3)—64no—
269C3×H(3,𝔽3)—64no—
270C3×H(3,𝔽3)—62no—
271C3×H(3,𝔽3)—64no—
272C3×H(3,𝔽3)—62no—
273C3×H(3,𝔽3)—62no—
274C3×H(3,𝔽3)C2×(C3⋊D3)⋊C316yes—
275C3×H(3,𝔽3)C2×(C3⋊D3)⋊C316yes—
276C3×H(3,𝔽3)C2×C32×D312yes—
277C3×H(3,𝔽3)C2×C32×D312yes—
278C32×C9—62no—
279C32×C9C2×C3×C3⋊D313yescomplemented
280#732—318no—
281#732—618no—
282#732—636no—
283#732—618no—
284#731—318no—
285#731—618no—
286AΣL(1,𝔽27)—636no—
287AΣL(1,𝔽27)—618no—
288C2×C33—22no—
289C2×C33—22no—
290C2×C33—39no—
291C2×C33—32no—
292C2×C33—32no—
293C2×C33—36no—
294C2×C33—36no—
295C2×C33—66no—
296C2×C33—66no—
297C2×C33—618no—
298C2×C33—618no—
299C2×C33—62no—
300C2×C33—62no—
301C2×C33—66no—
302C2×C33—182no—
303C2×C33—184no—
304C2×C33—271no—
305C2×C33—276no—
306C2×C33—276no—
307C2×C33—2727no—
308C2×C33—546no—
309C2×C33C3⋊((C3⋊D3)⋊C3)12yes—
310C2×C33C3⋊((C3⋊D3)⋊C3)12yes—
311C2×C33C32×C3⋊D311yescharacteristic
312C3×C3⋊D3—92no—
313C3×C3⋊D3—274no—
314C3×C3⋊D3—274no—
315C3×C3⋊D3—272no—
316C3×C3⋊D3—2718no—
317C3×C3⋊D3—2712no—
318C3×C3⋊D3—2712no—
319C3×C3⋊D3—2754no—
320C32×D3—274no—
321C32×D3—274no—
322C32×D3—8112no—
323C32×D3—8112no—
324C2×3−1+2—612no—
325C2×3−1+2—66no—
326C2×3−1+2—91no—
327C2×3−1+2—182no—
328C2×3−1+2—182no—
329C2×3−1+2—184no—
330C2×3−1+2—182no—
331C2×3−1+2—184no—
332C2×3−1+2—182no—
333C2×3−1+2—812no—
334C2×H(3,𝔽3)—26no—
335C2×H(3,𝔽3)—26no—
336C2×H(3,𝔽3)—318no—
337C2×H(3,𝔽3)—34no—
338C2×H(3,𝔽3)—34no—
339C2×H(3,𝔽3)—612no—
340C2×H(3,𝔽3)—612no—
341C2×H(3,𝔽3)—66no—
342C2×H(3,𝔽3)—66no—
343C2×H(3,𝔽3)—636no—
344C2×H(3,𝔽3)—636no—
345C2×H(3,𝔽3)—64no—
346C2×H(3,𝔽3)—64no—
347C2×H(3,𝔽3)—66no—
348C2×H(3,𝔽3)—612no—
349C2×H(3,𝔽3)—182no—
350C2×H(3,𝔽3)—184no—
351C2×H(3,𝔽3)—184no—
352C2×H(3,𝔽3)—182no—
353C2×H(3,𝔽3)C3×(C3⋊D3)⋊C316yes—
354C2×H(3,𝔽3)C3×(C3⋊D3)⋊C316yes—
355C2×C3×C9—36no—
356C2×C3×C9—36no—
357C2×C3×C9—66no—
358C2×C3×C9—184no—
359C2×C3×C9—182no—
360#272—912no—
361#272—912no—
362#272—278no—
363#272—278no—
364#272—2736no—
365D3×C9—2712no—
366D3×C9—2712no—
367C22×C32—811no—
368C2×C3⋊D3—91no—
369C2×C3⋊D3—272no—
370C2×C3⋊D3—272no—
371C2×C3⋊D3—279no—
372C2×C3×D3—272no—
373C2×C3×D3—272no—
374C2×C3×D3—812no—
375C2×C3×D3—812no—
376C2×C3×D3—816no—
377C2×C3×D3—816no—
378C2×C3×D3—8118no—
379C2×C3×D3—8118no—
380C22×C9—813no—
381C33—22no—
382C33—22no—
383C33—39no—
384C33—32no—
385C33—32no—
386C33—36no—
387C33—36no—
388C33—66no—
389C33—66no—
390C33—618no—
391C33—618no—
392C33—62no—
393C33—62no—
394C33—66no—
395C33—182no—
396C33—184no—
397C33—271no—
398C33—276no—
399C33—276no—
400C33—2727no—
401C33—546no—
402C33C2×C3⋊((C3⋊D3)⋊C3)12yes—
403C33C2×C3⋊((C3⋊D3)⋊C3)12yes—
404C33C2×C32×C3⋊D311yescharacteristic
4053−1+2—612no—
4063−1+2—66no—
4073−1+2—91no—
4083−1+2—182no—
4093−1+2—182no—
4103−1+2—184no—
4113−1+2—182no—
4123−1+2—184no—
4133−1+2—182no—
414H(3,𝔽3)—26no—
415H(3,𝔽3)—26no—
416H(3,𝔽3)—318no—
417H(3,𝔽3)—34no—
418H(3,𝔽3)—34no—
419H(3,𝔽3)—66no—
420H(3,𝔽3)—66no—
421H(3,𝔽3)—612no—
422H(3,𝔽3)—612no—
423H(3,𝔽3)—636no—
424H(3,𝔽3)—636no—
425H(3,𝔽3)—64no—
426H(3,𝔽3)—64no—
427H(3,𝔽3)—66no—
428H(3,𝔽3)—612no—
429H(3,𝔽3)—182no—
430H(3,𝔽3)—184no—
431H(3,𝔽3)—184no—
432H(3,𝔽3)—182no—
433H(3,𝔽3)C2×C3×(C3⋊D3)⋊C316yes—
434H(3,𝔽3)C2×C3×(C3⋊D3)⋊C316yes—
435C3×C9—36no—
436C3×C9—36no—
437C3×C9—66no—
438C3×C9—184no—
439C3×C9—182no—
440C2×C32—22no—
441C2×C32—22no—
442C2×C32—36no—
443C2×C32—36no—
444C2×C32—66no—
445C2×C32—66no—
446C2×C32—91no—
447C2×C32—92no—
448C2×C32—92no—
449C2×C32—918no—
450C2×C32—918no—
451C2×C32—182no—
452C2×C32—182no—
453C2×C32—186no—
454C2×C32—272no—
455C2×C32—272no—
456C2×C32—279no—
457C2×C32—542no—
458C2×C32—542no—
459C2×C32—546no—
460C2×C32—546no—
461C2×C32—546no—
462C2×C32—546no—
463C2×C32—5412no—
464C2×C32—5418no—
465C2×C32—5418no—
466C2×C32—812no—
467C2×C32#9230111yescharacteristic
468C2×C32C3×C3⋊((C3⋊D3)⋊C3)12yes—
469C2×C32C3×C3⋊((C3⋊D3)⋊C3)12yes—
470C3⋊D3—92no—
471C3⋊D3—274no—
472C3⋊D3—274no—
473C3⋊D3—2718no—
474C3×D3—274no—
475C3×D3—274no—
476C3×D3—814no—
477C3×D3—814no—
478C3×D3—8112no—
479C3×D3—8112no—
480C3×D3—8136no—
481C3×D3—8136no—
482C2×C9—93no—
483C2×C9—186no—
484C2×C9—816no—
485C22×C3—811no—
486C22×C3—2431no—
487C2×D3—272no—
488C2×D3—272no—
489C2×D3—816no—
490C2×D3—816no—
491C32—22no—
492C32—22no—
493C32—36no—
494C32—36no—
495C32—66no—
496C32—66no—
497C32—91no—
498C32—92no—
499C32—92no—
500C32—918no—
501C32—918no—
502C32—182no—
503C32—182no—
504C32—186no—
505C32—272no—
506C32—272no—
507C32—279no—
508C32—542no—
509C32—542no—
510C32—546no—
511C32—546no—
512C32—546no—
513C32—546no—
514C32—5412no—
515C32—5418no—
516C32—5418no—
517C32C2×#9230111yescharacteristic
518C32C2×C3×C3⋊((C3⋊D3)⋊C3)12yes—
519C32C2×C3×C3⋊((C3⋊D3)⋊C3)12yes—
520C9—93no—
521C9—186no—
522C2×C3—32no—
523C2×C3—32no—
524C2×C3—31no—
525C2×C3—62no—
526C2×C3—62no—
527C2×C3—96no—
528C2×C3—96no—
529C2×C3—186no—
530C2×C3—186no—
531C2×C3—546no—
532C2×C3—812no—
533C2×C3—2432no—
534C2×C3C3×#9230111yescharacteristic
535D3—274no—
536D3—274no—
537D3—8112no—
538D3—8112no—
539C22—811noSylow
540C3—32no—
541C3—32no—
542C3—31no—
543C3—62no—
544C3—62no—
545C3—96no—
546C3—96no—
547C3—186no—
548C3—186no—
549C3—546no—
550C3C2×C3×#9230111yescharacteristic
551C2—812no—
552C2#7222189011yescharacteristic, direct product factor
subgroupG/H
center, socleC2×C3C3×#92301
commutator, cosocle, hypocenterH(5,𝔽3)C22×C32
FrattiniC33C2×C32×C3⋊D3
FittingC2×#14554343C2
hypercenterC2×C32#92301
LCSG ▷ H(5,𝔽3)
UCSC1 ◁ C2×C3 ◁ C2×C32

Complements

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

NH ≅ G/Ntype
3. C2×C3×#92404C3semidirect
4. C2×#13091148C3semidirect
5. C2×#14554343C2semidirect
6. #72221890C2direct (N × H)
7. #14554343C22semidirect
8. C3×#92404C2×C3semidirect
11. #13091148C2×C3semidirect
14. C2×C3×#6911D3semidirect
15. C2×C3×#6910D3semidirect
19. C2×#10624760C2×C3semidirect
20. C2×#92404C32semidirect
32. C3×#6911C2×D3semidirect
33. C3×#6910C2×D3semidirect
37. #10624760C22×C3semidirect
44. C2×C32×3−1+2C3⋊D3semidirect
50. #92404C2×C32semidirect
70. C2×C3×#732C3×D3semidirect
72. C2×C3×#731C3×D3semidirect
102. C32×3−1+2C2×C3⋊D3semidirect
122. C3×#732C2×C3×D3semidirect
124. C3×#731C2×C3×D3semidirect
183. C2×C32×C9C3×C3⋊D3semidirect
279. C32×C9C2×C3×C3⋊D3semidirect
552. C2#72221890direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3⋊#922853
C2×C3⋊#922833
C2×C3×#924043
C2×#130911483
C2×#145543432
#722218902