Finite Group Databasebeta

← #13155749

Subgroups — #13155749

Order 1728.

602 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723354C211yesmaximal, characteristic, complemented
2#11723350C211yesmaximal, characteristic, complemented
3C3×#63829C211yesmaximal, characteristic
4#11722077C211yesmaximal, characteristic, complemented
5#11722020C211yesmaximal, characteristic, complemented
6C3×#63577C211yesmaximal, characteristic, complemented
7C3×#63568C211yesmaximal, characteristic, complemented
8#10591676—31nomaximal
9#10590978—31nomaximal
10#10590920—31nomaximal
11C3×#3828C2211yescharacteristic
12C32×#235C2211yescharacteristic
13C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
14C3×C4×C3⋊Dic3C2211yescharacteristic
15C3×#3748C2211yescharacteristic
16C32×#208C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18#88787—21no—
19#88786—21no—
20C3⋊D3×C3⋊C8—21no—
21D3×C3⋊(C3⋊C8)—21no—
22C3×#3719C411yescharacteristic, complemented
23C3×#3717C411yescharacteristic, complemented
24C3×#3716C411yescharacteristic, complemented
25C3×#3714C411yescharacteristic, complemented
26#64157—31no—
27#64116—31no—
28C3×#950—31no—
29C3×#904—31no—
30#63829D311yescharacteristic
31#63815—31no—
32#63815—31no—
33#63815—31no—
34#63815—31no—
35#63815—61no—
36#63815—61no—
37#63642—31no—
38C3×#837—31no—
39C3×#825—31no—
40C3×#822—31no—
41C3×#819—31no—
42#63577D311yescharacteristic, complemented
43#63570—31no—
44#63569—31no—
45#63568—31no—
46#63568D311yescharacteristic, complemented
47#63554—31no—
48#63554—31no—
49#63554—61no—
50C2×C33×C4C2311yescharacteristic
51C3×#638—21no—
52C2×C3×C3⋊Dic3—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))D411yescharacteristic
61C3⋊(C3⋊(C3⋊C8))D411yescharacteristic
62C3×C3⋊(C3⋊C8)—21no—
63C32×C3⋊C8—21no—
64C32×C3⋊C8C2×C411yescharacteristic
65C32×C3⋊C8C2×C411yescharacteristic
66#5197—91no—
67#4894—91no—
68#4823—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⋊Dic3C2×D311yescharacteristic
80#3748—61no—
81#3748C2×D311yescharacteristic
82C2×C3⋊(C3⋊C8)—31no—
83C2×C3⋊(C3⋊C8)—31no—
84C2×C3⋊(C3⋊C8)—31no—
85C2×C3⋊(C3⋊C8)—61no—
86C2×C3⋊(C3⋊C8)—61no—
87C2×C3⋊(C3⋊C8)—61no—
88C2×C3⋊(C3⋊C8)—61no—
89C2×C3⋊(C3⋊C8)—121no—
90#3743—61no—
91C8×C3⋊D3—61no—
92C3×#216—31no—
93C3×#215—31no—
94C3×#214—31no—
95C3×#213—31no—
96C3×C4×Dic3—31no—
97C3×C4×Dic3—31no—
98C3×C4×Dic3—61no—
99C3×#208—21no—
100C3×#208—31no—
101C3×#208—31no—
102C3×#208—61no—
103C3×#208C2×D311yescharacteristic
104C3×#208C2×D311yescharacteristic
105C2×C3×C3⋊C8—21no—
106C2×C3×C3⋊C8C2×D311yescharacteristic
107C2×C3×C3⋊C8C2×D311yescharacteristic
108C3×Dic12—31no—
109C3×D24—31no—
110C3×#204—31no—
111C3×#204—31no—
112#3719Dic311yescharacteristic, complemented
113#3717Dic311yescharacteristic, complemented
114#3716Dic311yescharacteristic, complemented
115#3714Dic311yescharacteristic, complemented
116#3711—61no—
117#3711—61no—
118#3711—62no—
119#3711—61no—
120#3711—61no—
121#3711—62no—
122D3×C3⋊C8—61no—
123D3×C3⋊C8—61no—
124D3×C3⋊C8—62no—
125D3×C3⋊C8—61no—
126D3×C3⋊C8—61no—
127D3×C3⋊C8—62no—
128C22×C33#4111yescharacteristic
129C2×C3×C3⋊D3—21no—
130C2×C32×D3—21no—
131C33×C4C22×C411yescharacteristic
132C33×C4C2×D411yescharacteristic
133C3×C3⋊Dic3—21no—
134C3×C3⋊Dic3—41no—
135C32×Dic3—21no—
136C3×#135—91no—
137#950—31no—
138#948—91no—
139#948—91no—
140#948—91no—
141#948—181no—
142#907—91no—
143#907—91no—
144#907—91no—
145#907—181no—
146#904—31no—
147C3×#108—91no—
148C3×#104—91no—
149#837—31no—
150#837—91no—
151#825—31no—
152#825—91no—
153#822—31no—
154#819—31no—
155#815—91no—
156#815—91no—
157#815—181no—
158#805—91no—
159#803—91no—
160C32×Q8—31no—
161C32×D4—31no—
162C32×D4—61no—
163C2×C32×C4—21no—
164C2×C32×C4—21no—
165C2×C32×C4—21no—
166C2×C32×C4—41no—
167C2×C32×C4—61no—
168C2×C32×C4C22×D311yescharacteristic
169C2×C32×C4C22×D311yescharacteristic
170C2×C32×C4C22×D311yescharacteristic
171#638—21no—
172C2×C3⋊Dic3—21no—
173C4⋊(C3⋊D3)C2×Dic311yescharacteristic
174C4×C3⋊D3—21no—
175#634C2×Dic311yescharacteristic
176C3×#67—22no—
177C3×#67—21no—
178C3×#67—21no—
179C3×#67—62no—
180C3×#67—61no—
181C3×#67—61no—
182C2×C3×Dic3—62no—
183C2×C3×Dic3—61no—
184C2×C3×Dic3—61no—
185C3×D12—21no—
186C3×D12—31no—
187C3×D12—31no—
188C3×D12—61no—
189C3×D12C4×D311yescharacteristic
190C3×D12C2×Dic311yescharacteristic
191C3×C4×D3—22no—
192C3×C4×D3—21no—
193C3×C4×D3—21no—
194C3×C4×D3—62no—
195C3×C4×D3—61no—
196C3×C4×D3—61no—
197C3×Dic6—21no—
198C3×Dic6—31no—
199C3×Dic6—31no—
200C3×Dic6—61no—
201C3×Dic6C4×D311yescharacteristic
202C3×Dic6C2×Dic311yescharacteristic
203C32×C8—31no—
204C32×C8—31no—
205C32×C8—61no—
206C3⋊(C3⋊C8)—21no—
207C3⋊(C3⋊C8)—31no—
208C3⋊(C3⋊C8)—31no—
209C3⋊(C3⋊C8)—31no—
210C3⋊(C3⋊C8)—31no—
211C3⋊(C3⋊C8)—31no—
212C3⋊(C3⋊C8)—31no—
213C3⋊(C3⋊C8)—61no—
214C3⋊(C3⋊C8)—61no—
215C3⋊(C3⋊C8)—61no—
216C3⋊(C3⋊C8)—61no—
217C3⋊(C3⋊C8)—61no—
218C3⋊(C3⋊C8)—61no—
219C3⋊(C3⋊C8)—121no—
220C3⋊(C3⋊C8)—121no—
221C3×C3⋊C8—22no—
222C3×C3⋊C8—21no—
223C3×C3⋊C8—21no—
224C3×C3⋊C8—21no—
225C3×C3⋊C8—21no—
226C3×C3⋊C8—62no—
227C3×C3⋊C8—61no—
228C3×C3⋊C8—61no—
229C3×C3⋊C8C4×D311yescharacteristic
230C3×C3⋊C8C4×D311yescharacteristic
231C3×C3⋊C8C2×Dic311yescharacteristic
232C3×C3⋊C8C2×Dic311yescharacteristic
233#444—271noSylow
234C2×C33—21no—
235C2×C33C4×D411yescharacteristic
236C3×C3⋊D3—41no—
237C32×D3—41no—
238C3×#41—31no—
239C3×#41—31no—
240C3×#41—61no—
241C3×#41—91no—
242#235—31no—
243#235—31no—
244#235—61no—
245#235D3211yescharacteristic
246C3×Q16—91no—
247C3×SD16—91no—
248C3×SD16—91no—
249C3×D8—91no—
250C3×M16—31no—
251C3×M16—31no—
252C3×M16—61no—
253C3×M16—91no—
254C2×C3×C8—31no—
255C2×C3×C8—31no—
256C2×C3×C8—61no—
257C3×C42—91no—
258#216—31no—
259#215—31no—
260#214—31no—
261#213—31no—
262C4×Dic3—31no—
263C4×Dic3—31no—
264C4×Dic3—61no—
265#208—31no—
266#208—31no—
267#208—61no—
268#208—181no—
269#208—181no—
270#208—181no—
271#208—182no—
272#208D3211yescharacteristic
273C2×C3⋊C8—91no—
274C2×C3⋊C8—91no—
275C2×C3⋊C8—91no—
276C2×C3⋊C8—181no—
277C2×C3⋊C8—181no—
278C2×C3⋊C8—181no—
279C2×C3⋊C8—181no—
280C2×C3⋊C8—181no—
281C2×C3⋊C8—181no—
282C2×C3⋊C8—182no—
283C2×C3⋊C8—361no—
284C2×C3⋊C8D3211yescharacteristic
285Dic12—31no—
286D24—31no—
287#204—31no—
288#204—31no—
289ΓL(1,𝔽25)—181no—
290ΓL(1,𝔽25)—181no—
291ΓL(1,𝔽25)—181no—
292ΓL(1,𝔽25)—182no—
293D3×C8—181no—
294D3×C8—181no—
295D3×C8—181no—
296D3×C8—182no—
297C22×C32—21no—
298C22×C32—21no—
299C22×C32—21no—
300C22×C32—41no—
301C22×C32—61no—
302C22×C32#23711yescharacteristic
303C22×C32#23711yescharacteristic
304C22×C32#23911yescharacteristic
305C2×C3⋊D3—21no—
306C2×C3×D3—22no—
307C2×C3×D3—21no—
308C2×C3×D3—21no—
309C2×C3×D3—62no—
310C2×C3×D3—61no—
311C2×C3×D3—61no—
312C32×C4—21no—
313C32×C4—21no—
314C32×C4—21no—
315C32×C4—21no—
316C32×C4—21no—
317C32×C4—21no—
318C32×C4—41no—
319C32×C4—41no—
320C32×C4—61no—
321C32×C4C2×C4×D311yescharacteristic
322C32×C4C2×C4×D311yescharacteristic
323C32×C4D3×D411yescharacteristic
324C32×C4D3×D411yescharacteristic
325C32×C4D3×D411yescharacteristic
326C32×C4C22×Dic311yescharacteristic
327C3⋊Dic3—21no—
328C3⋊Dic3—41no—
329C3×Dic3—22no—
330C3×Dic3—21no—
331C3×Dic3—21no—
332C3×Dic3—62no—
333C3×Dic3—61no—
334C3×Dic3—61no—
335C3×Dic3—122no—
336C3×Dic3—121no—
337C3×Dic3—121no—
338#135—91no—
339#131—271no—
340#131—271no—
341#108—91no—
342#104—91no—
343#104—271no—
344#97—271no—
345C33#44411yescharacteristic, complemented, Sylow
346C3×Q8—31no—
347C3×Q8—31no—
348C3×Q8—61no—
349C3×Q8—91no—
350C3×D4—31no—
351C3×D4—31no—
352C3×D4—61no—
353C3×D4—62no—
354C3×D4—61no—
355C3×D4—61no—
356C3×D4—91no—
357C3×D4—181no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—41no—
362C2×C3×C4—62no—
363C2×C3×C4—61no—
364C2×C3×C4—61no—
365C2×C3×C4—181no—
366C2×C3×C4—181no—
367C2×C3×C4C2×D3211yescharacteristic
368C2×C3×C4C2×D3211yescharacteristic
369C2×C3×C4C2×D3211yescharacteristic
370#67—21no—
371#67—62no—
372#67—61no—
373#67—61no—
374C2×Dic3—62no—
375C2×Dic3—61no—
376C2×Dic3—61no—
377D12—31no—
378D12—31no—
379D12—61no—
380D12D3×Dic311yescharacteristic
381C4×D3—21no—
382C4×D3—62no—
383C4×D3—61no—
384C4×D3—61no—
385Dic6—31no—
386Dic6—31no—
387Dic6—61no—
388Dic6D3×Dic311yescharacteristic
389C3×C8—31no—
390C3×C8—31no—
391C3×C8—31no—
392C3×C8—31no—
393C3×C8—61no—
394C3×C8—61no—
395C3×C8—62no—
396C3×C8—61no—
397C3×C8—61no—
398C3×C8—181no—
399C3⋊C8—21no—
400C3⋊C8—62no—
401C3⋊C8—61no—
402C3⋊C8—61no—
403C3⋊C8—91no—
404C3⋊C8—91no—
405C3⋊C8—91no—
406C3⋊C8—91no—
407C3⋊C8—91no—
408C3⋊C8—91no—
409C3⋊C8—181no—
410C3⋊C8—181no—
411C3⋊C8—181no—
412C3⋊C8—181no—
413C3⋊C8—181no—
414C3⋊C8—181no—
415C3⋊C8—361no—
416C3⋊C8—361no—
417C3⋊C8D3×Dic311yescharacteristic
418C3⋊C8D3×Dic311yescharacteristic
419C2×C32—21no—
420C2×C32—22no—
421C2×C32—21no—
422C2×C32—21no—
423C2×C32—21no—
424C2×C32—21no—
425C2×C32—21no—
426C2×C32—41no—
427C2×C32—41no—
428C2×C32—41no—
429C2×C32—42no—
430C2×C32—121no—
431C2×C32#67⋊C411yescharacteristic
432C2×C32#89311yescharacteristic
433C2×C32D4×Dic311yescharacteristic
434C3⋊D3—41no—
435C3×D3—42no—
436C3×D3—41no—
437C3×D3—41no—
438C3×D3—122no—
439C3×D3—121no—
440C3×D3—121no—
441#41—31no—
442#41—91no—
443Q16—91no—
444SD16—91no—
445SD16—91no—
446D8—91no—
447M16—31no—
448M16—91no—
449M16—541no—
450M16—541no—
451C2×C8—31no—
452C2×C8—271no—
453C2×C8—541no—
454C2×C8—541no—
455C42—91no—
456C22×C3—21no—
457C22×C3—21no—
458C22×C3—21no—
459C22×C3—41no—
460C22×C3—62no—
461C22×C3—61no—
462C22×C3—61no—
463C22×C3—181no—
464C22×C3#379611yescharacteristic
465C22×C3#379611yescharacteristic
466C22×C3#380511yescharacteristic
467C2×D3—21no—
468C2×D3—62no—
469C2×D3—61no—
470C2×D3—61no—
471C3×C4—21no—
472C3×C4—21no—
473C3×C4—21no—
474C3×C4—21no—
475C3×C4—21no—
476C3×C4—21no—
477C3×C4—41no—
478C3×C4—41no—
479C3×C4—62no—
480C3×C4—61no—
481C3×C4—61no—
482C3×C4—181no—
483C3×C4—361no—
484C3×C4C4×D3211yescharacteristic
485C3×C4#380211yescharacteristic
486C3×C4C2×D3×Dic311yescharacteristic
487C3×C4C2×D3×Dic311yescharacteristic
488C3×C4#381111yescharacteristic
489C3×C4#381111yescharacteristic
490Dic3—21no—
491Dic3—62no—
492Dic3—61no—
493Dic3—61no—
494Dic3—122no—
495Dic3—121no—
496Dic3—121no—
497C32—21no—
498C32—21no—
499C32—21no—
500C32—41no—
501C32#482311yescharacteristic, complemented
502C32#489411yescharacteristic, complemented
503C32#519711yescharacteristic, complemented
504Q8—31no—
505Q8—91no—
506D4—31no—
507D4—61no—
508D4—91no—
509D4—181no—
510C2×C4—61no—
511C2×C4—181no—
512C2×C4—181no—
513C2×C4D3311yescharacteristic
514C8—31no—
515C8—31no—
516C8—61no—
517C8—181no—
518C8—271no—
519C8—271no—
520C2×C3—22no—
521C2×C3—21no—
522C2×C3—21no—
523C2×C3—21no—
524C2×C3—21no—
525C2×C3—21no—
526C2×C3—21no—
527C2×C3—41no—
528C2×C3—41no—
529C2×C3—42no—
530C2×C3—41no—
531C2×C3—122no—
532C2×C3—121no—
533C2×C3—121no—
534C2×C3—361no—
535C2×C3#6388111yescharacteristic
536C2×C3#6389811yescharacteristic
537C2×C3#67⋊Dic311yescharacteristic
538D3—41no—
539D3—122no—
540D3—121no—
541D3—121no—
542C22—61no—
543C22—181no—
544C22#8896411yescharacteristic
545C4—61no—
546C4—181no—
547C4—361no—
548C4D32×Dic311yescharacteristic
549C4#8895411yescharacteristic
550C3—21no—
551C3—21no—
552C3—21no—
553C3—41no—
554C3#1059092011yescharacteristic, complemented
555C3#1059097811yescharacteristic, complemented
556C3#1059167611yescharacteristic, complemented
557C2—21no—
558C2—121no—
559C2—361no—
560C2#1172292911yescharacteristic
subgroupG/H
centerC4#88954
commutatorC33×C4C22×C4
Frattini, hypercenterC2×C4D33
Fitting, cosocleC2×C33×C4C23
socleC2×C33C4×D4
prime radicalC3×#63829C2
hypocenterC33#444
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. #11723354C2semidirect
2. #11723350C2semidirect
4. #11722077C2semidirect
5. #11722020C2semidirect
6. C3×#63577C2semidirect
7. C3×#63568C2semidirect
22. C3×#3719C4semidirect
23. C3×#3717C4semidirect
24. C3×#3716C4semidirect
25. C3×#3714C4semidirect
42. #63577D3semidirect
46. #63568D3semidirect
112. #3719Dic3semidirect
113. #3717Dic3semidirect
114. #3716Dic3semidirect
115. #3714Dic3semidirect
345. C33#444semidirect
501. C32#4823semidirect
502. C32#4894semidirect
503. C32#5197semidirect
554. C3#10590920semidirect
555. C3#10590978semidirect
556. C3#10591676semidirect

Maximal subgroups

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

subgroupindex
#117233542
#117233502
C3×#638292
#117220772
#117220202
C3×#635772
C3×#635682
#105916763
#105909783
#105909203