Finite Group Databasebeta

← #13155731

Subgroups — #13155731

Order 1728.

586 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723657C211yesmaximal, characteristic, complemented
2C3×#63829C211yesmaximal, characteristic
3C3×#63815C211yesmaximal, characteristic, complemented
4#11722045C211yesmaximal, characteristic, complemented
5#11722043C211yesmaximal, characteristic, complemented
6#11722018C211yesmaximal, characteristic, complemented
7C3×#63577C211yesmaximal, characteristic, complemented
8#10591673—31nomaximal
9#10590995—31nomaximal
10#10590919—31nomaximal
11C3×#3828C2211yescharacteristic
12C32×#235C2211yescharacteristic
13C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
14C3×#3748C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×#208C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18#88809—21no—
19#88808—21no—
20C3×#3719C411yescharacteristic, complemented
21C3×#3717C411yescharacteristic, complemented
22C3×#3716C411yescharacteristic, complemented
23C3×#3714C411yescharacteristic, complemented
24C3×#3711—21no—
25C3×D3×C3⋊C8—21no—
26C3×#950—31no—
27C3×#948—31no—
28C3×#907—31no—
29C3×#904—31no—
30#63829D311yescharacteristic
31#63817—31no—
32#63815—31no—
33#63815D311yescharacteristic, complemented
34#63814—31no—
35#63664—31no—
36#63652—31no—
37C8×C3⋊Dic3—31no—
38C3×#822—31no—
39C3×#819—31no—
40#63577D311yescharacteristic, complemented
41#63568—31no—
42#63568—31no—
43#63568—31no—
44#63568—31no—
45#63568—61no—
46#63568—61no—
47#63553—31no—
48#63553—31no—
49#63553—61no—
50C2×C33×C4C2311yescharacteristic
51C2×C3⋊(C3⋊Dic3)—21no—
52C3×#638—21no—
53C3×C4⋊(C3⋊D3)C2×C411yescharacteristic
54C3×C4×C3⋊D3—21no—
55C3×#634C2×C411yescharacteristic
56C32×#67—21no—
57C32×D12C2×C411yescharacteristic
58C32×C4×D3—21no—
59C32×Dic6C2×C411yescharacteristic
60C3×C3⋊(C3⋊C8)—21no—
61C3×C3⋊(C3⋊C8)D411yescharacteristic
62C3×C3⋊(C3⋊C8)D411yescharacteristic
63C32×C3⋊C8—21no—
64C32×C3⋊C8C2×C411yescharacteristic
65C32×C3⋊C8C2×C411yescharacteristic
66#5196—91no—
67#5144—91no—
68#4895—91no—
69C32×#41—31no—
70#3828C2×D311yescharacteristic
71C3×#235—21no—
72C3×#235—31no—
73C3×#235—31no—
74C3×#235—61no—
75C3×#235C2×D311yescharacteristic
76C3×#235C2×D311yescharacteristic
77C32×M16—31no—
78C2×C32×C8—31no—
79C4×C3⋊Dic3—31no—
80C4×C3⋊Dic3—31no—
81C4×C3⋊Dic3—31no—
82C4×C3⋊Dic3—61no—
83C4×C3⋊Dic3—61no—
84C4×C3⋊Dic3—61no—
85C4×C3⋊Dic3—121no—
86#3748C2×D311yescharacteristic
87C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
88C3×#216—31no—
89C3×#215—31no—
90C3×#214—31no—
91C3×#213—31no—
92C3×#208—21no—
93C3×#208—31no—
94C3×#208—31no—
95C3×#208—61no—
96C3×#208—61no—
97C3×#208C2×D311yescharacteristic
98C3×#208C2×D311yescharacteristic
99C2×C3×C3⋊C8—21no—
100C2×C3×C3⋊C8—31no—
101C2×C3×C3⋊C8—31no—
102C2×C3×C3⋊C8—61no—
103C2×C3×C3⋊C8—61no—
104C2×C3×C3⋊C8C2×D311yescharacteristic
105C2×C3×C3⋊C8C2×D311yescharacteristic
106C3×Dic12—31no—
107C3×D24—31no—
108C3×#204—31no—
109C3×#204—31no—
110C3×ΓL(1,𝔽25)—61no—
111C3×D3×C8—61no—
112#3719Dic311yescharacteristic, complemented
113#3717Dic311yescharacteristic, complemented
114#3716Dic311yescharacteristic, complemented
115#3714Dic311yescharacteristic, complemented
116#3712—61no—
117#3711—21no—
118#3711—61no—
119#3711—61no—
120#3710—61no—
121D3×C3⋊C8—21no—
122D3×C3⋊C8—61no—
123D3×C3⋊C8—61no—
124C22×C33#4111yescharacteristic
125C2×C3×C3⋊D3—21no—
126C2×C32×D3—21no—
127C33×C4C22×C411yescharacteristic
128C33×C4C2×D411yescharacteristic
129C3⋊(C3⋊Dic3)—22no—
130C3×C3⋊Dic3—21no—
131C32×Dic3—21no—
132C3×#135—91no—
133C3×#131—91no—
134#950—31no—
135#948—31no—
136#948—91no—
137#907—31no—
138#907—91no—
139#904—31no—
140#901—91no—
141C3×#108—91no—
142#837—91no—
143#837—91no—
144#837—91no—
145#837—181no—
146#825—91no—
147#825—91no—
148#825—91no—
149#825—181no—
150#822—31no—
151#819—31no—
152C8×Dic3—91no—
153C8×Dic3—91no—
154C8×Dic3—91no—
155C8×Dic3—181no—
156C32×Q8—31no—
157C32×D4—31no—
158C32×D4—61no—
159C2×C32×C4—21no—
160C2×C32×C4—21no—
161C2×C32×C4—21no—
162C2×C32×C4—41no—
163C2×C32×C4—61no—
164C2×C32×C4C22×D311yescharacteristic
165C2×C32×C4C22×D311yescharacteristic
166C2×C32×C4C22×D311yescharacteristic
167#638—21no—
168C2×C3⋊Dic3—61no—
169C2×C3⋊Dic3—62no—
170C2×C3⋊Dic3—61no—
171C2×C3⋊Dic3—61no—
172C2×C3⋊Dic3—121no—
173C2×C3⋊Dic3—121no—
174C2×C3⋊Dic3—122no—
175C4⋊(C3⋊D3)C2×Dic311yescharacteristic
176C4×C3⋊D3—21no—
177#634C2×Dic311yescharacteristic
178C3×#67—21no—
179C3×#67—21no—
180C3×#67—41no—
181C3×#67—61no—
182C3×#67—61no—
183C3×#67—121no—
184C3×D12—21no—
185C3×D12—31no—
186C3×D12—31no—
187C3×D12—61no—
188C3×D12C4×D311yescharacteristic
189C3×D12C2×Dic311yescharacteristic
190C3×C4×D3—21no—
191C3×C4×D3—21no—
192C3×C4×D3—41no—
193C3×C4×D3—61no—
194C3×C4×D3—61no—
195C3×C4×D3—121no—
196C3×Dic6—21no—
197C3×Dic6—31no—
198C3×Dic6—31no—
199C3×Dic6—61no—
200C3×Dic6C4×D311yescharacteristic
201C3×Dic6C2×Dic311yescharacteristic
202C32×C8—31no—
203C32×C8—31no—
204C32×C8—61no—
205C3⋊(C3⋊C8)—21no—
206C3⋊(C3⋊C8)#6711yescharacteristic
207C3⋊(C3⋊C8)#6711yescharacteristic
208C3×C3⋊C8—21no—
209C3×C3⋊C8—21no—
210C3×C3⋊C8—21no—
211C3×C3⋊C8—21no—
212C3×C3⋊C8—31no—
213C3×C3⋊C8—31no—
214C3×C3⋊C8—31no—
215C3×C3⋊C8—31no—
216C3×C3⋊C8—41no—
217C3×C3⋊C8—61no—
218C3×C3⋊C8—61no—
219C3×C3⋊C8—61no—
220C3×C3⋊C8—61no—
221C3×C3⋊C8—121no—
222C3×C3⋊C8C4×D311yescharacteristic
223C3×C3⋊C8C4×D311yescharacteristic
224C3×C3⋊C8C2×Dic311yescharacteristic
225C3×C3⋊C8C2×Dic311yescharacteristic
226#443—271noSylow
227C2×C33—21no—
228C2×C33C4×D411yescharacteristic
229C3×C3⋊D3—41no—
230C32×D3—41no—
231C3×#41—31no—
232C3×#41—31no—
233C3×#41—61no—
234C3×#41—91no—
235#235—31no—
236#235—31no—
237#235—61no—
238#235D3211yescharacteristic
239C3×Q16—91no—
240C3×SD16—91no—
241C3×SD16—91no—
242C3×D8—91no—
243C3×M16—31no—
244C3×M16—31no—
245C3×M16—61no—
246C3×M16—91no—
247C3×M16—181no—
248C2×C3×C8—31no—
249C2×C3×C8—31no—
250C2×C3×C8—61no—
251C2×C3×C8—91no—
252C2×C3×C8—181no—
253#216—31no—
254#215—31no—
255#214—31no—
256#213—31no—
257C4×Dic3—91no—
258C4×Dic3—91no—
259C4×Dic3—91no—
260C4×Dic3—181no—
261C4×Dic3—181no—
262C4×Dic3—181no—
263C4×Dic3—361no—
264#208—31no—
265#208—31no—
266#208—61no—
267#208—61no—
268#208—181no—
269#208D3211yescharacteristic
270C2×C3⋊C8—31no—
271C2×C3⋊C8—31no—
272C2×C3⋊C8—61no—
273C2×C3⋊C8—61no—
274C2×C3⋊C8—181no—
275C2×C3⋊C8D3211yescharacteristic
276Dic12—31no—
277D24—31no—
278#204—31no—
279#204—31no—
280ΓL(1,𝔽25)—61no—
281ΓL(1,𝔽25)—181no—
282ΓL(1,𝔽25)—181no—
283D3×C8—61no—
284D3×C8—181no—
285D3×C8—181no—
286C22×C32—21no—
287C22×C32—21no—
288C22×C32—21no—
289C22×C32—41no—
290C22×C32—61no—
291C22×C32#23511yescharacteristic
292C22×C32#23711yescharacteristic
293C22×C32#23911yescharacteristic
294C2×C3⋊D3—21no—
295C2×C3×D3—21no—
296C2×C3×D3—21no—
297C2×C3×D3—41no—
298C2×C3×D3—61no—
299C2×C3×D3—61no—
300C2×C3×D3—121no—
301C32×C4—21no—
302C32×C4—21no—
303C32×C4—21no—
304C32×C4—21no—
305C32×C4—21no—
306C32×C4—21no—
307C32×C4—41no—
308C32×C4—41no—
309C32×C4—61no—
310C32×C4C2×C4×D311yescharacteristic
311C32×C4C2×C4×D311yescharacteristic
312C32×C4D3×D411yescharacteristic
313C32×C4D3×D411yescharacteristic
314C32×C4C22×Dic311yescharacteristic
315C32×C4C2×#6711yescharacteristic
316C3⋊Dic3—21no—
317C3⋊Dic3—62no—
318C3⋊Dic3—64no—
319C3⋊Dic3—62no—
320C3⋊Dic3—62no—
321C3⋊Dic3—122no—
322C3⋊Dic3—122no—
323C3⋊Dic3—124no—
324C3×Dic3—21no—
325C3×Dic3—21no—
326C3×Dic3—41no—
327C3×Dic3—61no—
328C3×Dic3—61no—
329C3×Dic3—121no—
330#135—91no—
331#131—91no—
332#131—271no—
333#108—91no—
334#104—271no—
335#104—271no—
336C4×C8—271no—
337C33#44311yescharacteristic, complemented, Sylow
338C3×Q8—31no—
339C3×Q8—31no—
340C3×Q8—61no—
341C3×Q8—91no—
342C3×D4—31no—
343C3×D4—31no—
344C3×D4—61no—
345C3×D4—61no—
346C3×D4—61no—
347C3×D4—91no—
348C3×D4—121no—
349C3×D4—181no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—41no—
354C2×C3×C4—61no—
355C2×C3×C4—61no—
356C2×C3×C4—121no—
357C2×C3×C4—181no—
358C2×C3×C4C2×D3211yescharacteristic
359C2×C3×C4C2×D3211yescharacteristic
360C2×C3×C4C2×D3211yescharacteristic
361#67—21no—
362#67—61no—
363#67—61no—
364#67—121no—
365C2×Dic3—182no—
366C2×Dic3—181no—
367C2×Dic3—181no—
368C2×Dic3—181no—
369C2×Dic3—361no—
370C2×Dic3—361no—
371C2×Dic3—362no—
372D12—31no—
373D12—31no—
374D12—61no—
375D12D3×Dic311yescharacteristic
376C4×D3—21no—
377C4×D3—61no—
378C4×D3—61no—
379C4×D3—121no—
380Dic6—31no—
381Dic6—31no—
382Dic6—61no—
383Dic6D3×Dic311yescharacteristic
384C3×C8—31no—
385C3×C8—31no—
386C3×C8—31no—
387C3×C8—31no—
388C3×C8—61no—
389C3×C8—61no—
390C3×C8—61no—
391C3×C8—61no—
392C3×C8—91no—
393C3×C8—91no—
394C3×C8—121no—
395C3×C8—181no—
396C3⋊C8—21no—
397C3⋊C8—31no—
398C3⋊C8—31no—
399C3⋊C8—31no—
400C3⋊C8—31no—
401C3⋊C8—61no—
402C3⋊C8—61no—
403C3⋊C8—61no—
404C3⋊C8—61no—
405C3⋊C8—121no—
406C3⋊C8D3×Dic311yescharacteristic
407C3⋊C8D3×Dic311yescharacteristic
408C2×C32—21no—
409C2×C32—22no—
410C2×C32—21no—
411C2×C32—21no—
412C2×C32—21no—
413C2×C32—21no—
414C2×C32—21no—
415C2×C32—41no—
416C2×C32—41no—
417C2×C32—41no—
418C2×C32—42no—
419C2×C32—121no—
420C2×C32#89311yescharacteristic
421C2×C32C4×#6711yescharacteristic
422C2×C32D4×Dic311yescharacteristic
423C3⋊D3—41no—
424C3×D3—41no—
425C3×D3—41no—
426C3×D3—81no—
427C3×D3—121no—
428C3×D3—121no—
429C3×D3—241no—
430#41—31no—
431#41—91no—
432Q16—91no—
433SD16—91no—
434SD16—91no—
435D8—91no—
436M16—31no—
437M16—91no—
438M16—181no—
439M16—541no—
440C2×C8—31no—
441C2×C8—91no—
442C2×C8—181no—
443C2×C8—541no—
444C42—271no—
445C22×C3—21no—
446C22×C3—21no—
447C22×C3—21no—
448C22×C3—41no—
449C22×C3—61no—
450C22×C3—61no—
451C22×C3—121no—
452C22×C3—181no—
453C22×C3#379611yescharacteristic
454C22×C3#379911yescharacteristic
455C22×C3#380411yescharacteristic
456C2×D3—21no—
457C2×D3—61no—
458C2×D3—61no—
459C2×D3—121no—
460C3×C4—21no—
461C3×C4—21no—
462C3×C4—21no—
463C3×C4—21no—
464C3×C4—21no—
465C3×C4—21no—
466C3×C4—41no—
467C3×C4—41no—
468C3×C4—61no—
469C3×C4—61no—
470C3×C4—121no—
471C3×C4—181no—
472C3×C4C4×D3211yescharacteristic
473C3×C4#380211yescharacteristic
474C3×C4C2×D3×Dic311yescharacteristic
475C3×C4C2×D3×Dic311yescharacteristic
476C3×C4D3×#6711yescharacteristic
477C3×C4D3×#6711yescharacteristic
478Dic3—21no—
479Dic3—61no—
480Dic3—61no—
481Dic3—121no—
482Dic3—182no—
483Dic3—184no—
484Dic3—182no—
485Dic3—182no—
486Dic3—362no—
487Dic3—362no—
488Dic3—364no—
489C32—21no—
490C32—21no—
491C32—21no—
492C32—41no—
493C32#489511yescharacteristic, complemented
494C32#514411yescharacteristic, complemented
495C32#519611yescharacteristic, complemented
496Q8—31no—
497Q8—91no—
498D4—31no—
499D4—61no—
500D4—91no—
501D4—181no—
502C2×C4—61no—
503C2×C4—181no—
504C2×C4—541no—
505C2×C4D3311yescharacteristic
506C8—31no—
507C8—31no—
508C8—61no—
509C8—91no—
510C8—91no—
511C8—181no—
512C2×C3—22no—
513C2×C3—21no—
514C2×C3—21no—
515C2×C3—21no—
516C2×C3—21no—
517C2×C3—21no—
518C2×C3—21no—
519C2×C3—41no—
520C2×C3—41no—
521C2×C3—42no—
522C2×C3—41no—
523C2×C3—121no—
524C2×C3—121no—
525C2×C3—241no—
526C2×C3—361no—
527C2×C3#6389811yescharacteristic
528C2×C3#6390411yescharacteristic
529C2×C3Dic3×#6711yescharacteristic
530D3—41no—
531D3—121no—
532D3—121no—
533D3—241no—
534C22—61no—
535C22—181no—
536C22#8896211yescharacteristic
537C4—61no—
538C4—181no—
539C4—542no—
540C4D32×Dic311yescharacteristic
541C4#8895311yescharacteristic
542C3—21no—
543C3—21no—
544C3—21no—
545C3—41no—
546C3#1059091911yescharacteristic, complemented
547C3#1059099511yescharacteristic, complemented
548C3#1059167311yescharacteristic, complemented
549C2—21no—
550C2—121no—
551C2—361no—
552C2#1172292711yescharacteristic
subgroupG/H
centerC4#88953
commutatorC33×C4C22×C4
Frattini, hypercenterC2×C4D33
Fitting, cosocleC2×C33×C4C23
socleC2×C33C4×D4
prime radicalC3×#63829C2
hypocenterC33#443
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. #11723657C2semidirect
3. C3×#63815C2semidirect
4. #11722045C2semidirect
5. #11722043C2semidirect
6. #11722018C2semidirect
7. C3×#63577C2semidirect
20. C3×#3719C4semidirect
21. C3×#3717C4semidirect
22. C3×#3716C4semidirect
23. C3×#3714C4semidirect
33. #63815D3semidirect
40. #63577D3semidirect
112. #3719Dic3semidirect
113. #3717Dic3semidirect
114. #3716Dic3semidirect
115. #3714Dic3semidirect
337. C33#443semidirect
493. C32#4895semidirect
494. C32#5144semidirect
495. C32#5196semidirect
546. C3#10590919semidirect
547. C3#10590995semidirect
548. C3#10591673semidirect

Maximal subgroups

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

subgroupindex
#117236572
C3×#638292
C3×#638152
#117220452
#117220432
#117220182
C3×#635772
#105916733
#105909953
#105909193