Finite Group Databasebeta

← #13156524

Subgroups — #13156524

Order 1728.

602 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 602 conjugacy classes of subgroups — too many to draw. Only its 100 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
1#11723536C211yesmaximal, characteristic
2#11723451C211yesmaximal, characteristic
3#11722076C211yesmaximal, characteristic
4#11722018C211yesmaximal, characteristic
5C3×#63567C211yesmaximal, characteristic
6C3×#63566C211yesmaximal, characteristic
7C3×#63565C211yesmaximal, characteristic
8#10591538—31nomaximal
9#10590901—31nomaximal
10#10590567—31nomaximal
11C2×C3×#634C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C2×#6293—21no—
14C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88806—21no—
20#88806—21no—
21#88806—21no—
22#64153—31no—
23#64097—31no—
24#63919—31no—
25#63919—31no—
26#63919—61no—
27#63873—31no—
28#63873—31no—
29#63873—61no—
30C8×C3⋊Dic3—31no—
31C3×#835—31no—
32C3×#835—31no—
33C3×#816—31no—
34C3×#816—31no—
35C3×#810—31no—
36C3×#810—31no—
37#63567—31no—
38#63567D311yescharacteristic
39#63566—31no—
40#63566D311yescharacteristic
41#63565—31no—
42#63565D311yescharacteristic
43#63553—31no—
44#63553—31no—
45#63553—61no—
46C2×C33×C4C2311yescharacteristic
47C2×C3⋊(C3⋊Dic3)D411yescharacteristic
48C2×C3⋊(C3⋊Dic3)D411yescharacteristic
49C2×C3×C3⋊Dic3—21no—
50C2×C3×C3⋊Dic3—21no—
51C3×#634—21no—
52C3×#634—21no—
53C2×C32×Dic3—21no—
54C2×C32×Dic3—21no—
55C32×Dic6—21no—
56C32×Dic6—21no—
57#6293—42no—
58C3×C3⋊(C3⋊C8)—21no—
59C32×C3⋊C8—21no—
60#4792—91no—
61#4792—91no—
62#4792—91no—
63C2×C32×Q8—31no—
64C2×#634—61no—
65C2×#634C2×D311yescharacteristic
66C2×C3×Dic6—21no—
67C2×C3×Dic6—31no—
68C2×C3×Dic6—31no—
69C2×C3×Dic6—61no—
70C2×C3×Dic6C2×D311yescharacteristic
71C2×C3×Dic6C2×D311yescharacteristic
72C2×#627—61no—
73C2×#627—61no—
74C2×#627—62no—
75C2×C32×C8—31no—
76C32×C4⋊C4—31no—
77C4⋊(C3⋊Dic3)C2×D311yescharacteristic
78#3750—61no—
79#3750—61no—
80#3750—61no—
81C4×C3⋊Dic3—31no—
82C4×C3⋊Dic3—31no—
83C4×C3⋊Dic3—31no—
84C4×C3⋊Dic3—61no—
85C4×C3⋊Dic3—61no—
86C4×C3⋊Dic3—61no—
87C4×C3⋊Dic3—121no—
88C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
89C3×C4⋊Dic3—21no—
90C3×C4⋊Dic3—31no—
91C3×C4⋊Dic3—31no—
92C3×C4⋊Dic3—61no—
93C3×C4⋊Dic3C2×D311yescharacteristic
94C3×C4⋊Dic3C2×D311yescharacteristic
95C2×C3×C3⋊C8—21no—
96C2×C3×C3⋊C8—31no—
97C2×C3×C3⋊C8—31no—
98C2×C3×C3⋊C8—61no—
99C2×C3×C3⋊C8C2×D311yescharacteristic
100C2×C3×C3⋊C8C2×D311yescharacteristic
101#3724—61no—
102#3724—61no—
103#3724—61no—
104#3724—61no—
105#3724—61no—
106#3724—61no—
107#3724—62no—
108#3724—62no—
109#3724—62no—
110C22×C33C2×D411yescharacteristic
111C33×C4#4111yescharacteristic
112C33×C4#4111yescharacteristic
113C3⋊(C3⋊Dic3)SD1612yes—
114C3⋊(C3⋊Dic3)Q1612yes—
115C3×C3⋊Dic3—41no—
116C3×C3⋊Dic3—41no—
117C32×Dic3—41no—
118C32×Dic3—41no—
119#944—91no—
120#944—91no—
121#944—91no—
122#944—181no—
123#888—91no—
124#888—91no—
125#888—91no—
126#888—181no—
127C3×#103—91no—
128C3×#103—91no—
129C3×#103—91no—
130#835—31no—
131#835—31no—
132#835—91no—
133#816—31no—
134#816—31no—
135#816—91no—
136C8×Dic3—91no—
137C8×Dic3—91no—
138C8×Dic3—91no—
139C8×Dic3—181no—
140#810—31no—
141#810—31no—
142#810—91no—
143C32×Q8—61no—
144C32×Q8—61no—
145C2×C32×C4—21no—
146C2×C32×C4—21no—
147C2×C32×C4—21no—
148C2×C32×C4—41no—
149C2×C32×C4—61no—
150C2×C32×C4—61no—
151C2×C32×C4C22×D311yescharacteristic
152C2×C32×C4C22×D311yescharacteristic
153C2×C32×C4C22×D311yescharacteristic
154C2×C3⋊Dic3—21no—
155C2×C3⋊Dic3—21no—
156C2×C3⋊Dic3—31no—
157C2×C3⋊Dic3—31no—
158C2×C3⋊Dic3—31no—
159C2×C3⋊Dic3—31no—
160C2×C3⋊Dic3—31no—
161C2×C3⋊Dic3—31no—
162C2×C3⋊Dic3—61no—
163C2×C3⋊Dic3—61no—
164C2×C3⋊Dic3—61no—
165C2×C3⋊Dic3—61no—
166C2×C3⋊Dic3—61no—
167C2×C3⋊Dic3—61no—
168C2×C3⋊Dic3—121no—
169C2×C3⋊Dic3—121no—
170#634—21no—
171#634—21no—
172#634—122no—
173C2×C3×Dic3—22no—
174C2×C3×Dic3—22no—
175C2×C3×Dic3—21no—
176C2×C3×Dic3—21no—
177C2×C3×Dic3—21no—
178C2×C3×Dic3—21no—
179C2×C3×Dic3—62no—
180C2×C3×Dic3—62no—
181C2×C3×Dic3—61no—
182C2×C3×Dic3—61no—
183C2×C3×Dic3—61no—
184C2×C3×Dic3—61no—
185C3×Dic6—21no—
186C3×Dic6—21no—
187C3×Dic6—21no—
188C3×Dic6—21no—
189C3×Dic6—41no—
190C3×Dic6—41no—
191C3×Dic6—61no—
192C3×Dic6—61no—
193C3×Dic6—61no—
194C3×Dic6—61no—
195C3×Dic6—121no—
196C3×Dic6—121no—
197#627—122no—
198#627—122no—
199#627—124no—
200C32×C8—61no—
201C3⋊(C3⋊C8)—21no—
202C3×C3⋊C8—22no—
203C3×C3⋊C8—21no—
204C3×C3⋊C8—21no—
205C3×C3⋊C8—62no—
206C3×C3⋊C8—61no—
207C3×C3⋊C8—61no—
208#487—271noSylow
209C2×C33#12411yescharacteristic
210C2×C33C2×SD1611yescharacteristic
211C2×C33C2×Q1611yescharacteristic
212C2×C3×Q8—31no—
213C2×C3×Q8—31no—
214C2×C3×Q8—61no—
215C2×C3×Q8—91no—
216C2×Dic6—31no—
217C2×Dic6—31no—
218C2×Dic6—61no—
219C2×Dic6—181no—
220C2×Dic6—181no—
221C2×Dic6—181no—
222C2×Dic6—182no—
223C2×Dic6D3211yescharacteristic
224C2×C3×C8—31no—
225C2×C3×C8—31no—
226C2×C3×C8—61no—
227C2×C3×C8—91no—
228C3×C4⋊C4—31no—
229C3×C4⋊C4—31no—
230C3×C4⋊C4—61no—
231C3×C4⋊C4—91no—
232C4⋊Dic3—31no—
233C4⋊Dic3—31no—
234C4⋊Dic3—61no—
235C4⋊Dic3D3211yescharacteristic
236Dic3⋊C4—181no—
237Dic3⋊C4—181no—
238Dic3⋊C4—181no—
239Dic3⋊C4—181no—
240Dic3⋊C4—181no—
241Dic3⋊C4—181no—
242Dic3⋊C4—181no—
243Dic3⋊C4—181no—
244Dic3⋊C4—181no—
245Dic3⋊C4—182no—
246Dic3⋊C4—182no—
247Dic3⋊C4—182no—
248C4×Dic3—91no—
249C4×Dic3—91no—
250C4×Dic3—91no—
251C4×Dic3—181no—
252C4×Dic3—181no—
253C4×Dic3—181no—
254C4×Dic3—361no—
255C2×C3⋊C8—31no—
256C2×C3⋊C8—31no—
257C2×C3⋊C8—61no—
258C2×C3⋊C8D3211yescharacteristic
259C22×C32—21no—
260C22×C32—21no—
261C22×C32—21no—
262C22×C32—41no—
263C22×C32D3×D411yescharacteristic
264C22×C32D3×D411yescharacteristic
265C22×C32D3×D411yescharacteristic
266C32×C4—21no—
267C32×C4—21no—
268C32×C4—21no—
269C32×C4—21no—
270C32×C4—21no—
271C32×C4—21no—
272C32×C4—41no—
273C32×C4—41no—
274C32×C4—121no—
275C32×C4—121no—
276C32×C4#23511yescharacteristic
277C32×C4#23511yescharacteristic
278C32×C4#23511yescharacteristic
279C32×C4#23711yescharacteristic
280C32×C4#23711yescharacteristic
281C32×C4#23711yescharacteristic
282C3⋊Dic3—32no—
283C3⋊Dic3—32no—
284C3⋊Dic3—32no—
285C3⋊Dic3—32no—
286C3⋊Dic3—32no—
287C3⋊Dic3—32no—
288C3⋊Dic3—41no—
289C3⋊Dic3—41no—
290C3⋊Dic3—62no—
291C3⋊Dic3—62no—
292C3⋊Dic3—62no—
293C3⋊Dic3—62no—
294C3⋊Dic3—62no—
295C3⋊Dic3—62no—
296C3⋊Dic3—122no—
297C3⋊Dic3—122no—
298C3×Dic3—42no—
299C3×Dic3—42no—
300C3×Dic3—41no—
301C3×Dic3—41no—
302C3×Dic3—41no—
303C3×Dic3—41no—
304C3×Dic3—122no—
305C3×Dic3—122no—
306C3×Dic3—121no—
307C3×Dic3—121no—
308C3×Dic3—121no—
309C3×Dic3—121no—
310C4⋊Q8—271no—
311C4⋊Q8—271no—
312#103—91no—
313#103—91no—
314#103—91no—
315#103—271no—
316C4×C8—271no—
317C33#48711yescharacteristic, complemented, Sylow
318C3×Q8—61no—
319C3×Q8—61no—
320C3×Q8—61no—
321C3×Q8—61no—
322C3×Q8—121no—
323C3×Q8—121no—
324C3×Q8—181no—
325C3×Q8—181no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—41no—
330C2×C3×C4—62no—
331C2×C3×C4—62no—
332C2×C3×C4—61no—
333C2×C3×C4—61no—
334C2×C3×C4—61no—
335C2×C3×C4—61no—
336C2×C3×C4—181no—
337C2×C3×C4—181no—
338C2×C3×C4C2×D3211yescharacteristic
339C2×C3×C4C2×D3211yescharacteristic
340C2×C3×C4C2×D3211yescharacteristic
341C2×Dic3—21no—
342C2×Dic3—21no—
343C2×Dic3—62no—
344C2×Dic3—62no—
345C2×Dic3—61no—
346C2×Dic3—61no—
347C2×Dic3—61no—
348C2×Dic3—61no—
349C2×Dic3—91no—
350C2×Dic3—91no—
351C2×Dic3—91no—
352C2×Dic3—91no—
353C2×Dic3—91no—
354C2×Dic3—91no—
355C2×Dic3—181no—
356C2×Dic3—181no—
357C2×Dic3—181no—
358C2×Dic3—181no—
359C2×Dic3—181no—
360C2×Dic3—181no—
361C2×Dic3—361no—
362C2×Dic3—361no—
363Dic6—21no—
364Dic6—21no—
365Dic6—61no—
366Dic6—61no—
367Dic6—61no—
368Dic6—61no—
369Dic6—121no—
370Dic6—121no—
371Dic6—362no—
372Dic6—362no—
373Dic6—362no—
374Dic6—364no—
375C3×C8—62no—
376C3×C8—61no—
377C3×C8—61no—
378C3×C8—181no—
379C3⋊C8—21no—
380C3⋊C8—62no—
381C3⋊C8—61no—
382C3⋊C8—61no—
383C2×C32—21no—
384C2×C32—21no—
385C2×C32—21no—
386C2×C32—21no—
387C2×C32—21no—
388C2×C32—21no—
389C2×C32—21no—
390C2×C32—21no—
391C2×C32—21no—
392C2×C32—41no—
393C2×C32—41no—
394C2×C32—41no—
395C2×C32#88511yescharacteristic
396C2×C32#88511yescharacteristic
397C2×C32#88511yescharacteristic
398C2×C32D3×SD1611yescharacteristic
399C2×C32D3×SD1611yescharacteristic
400C2×C32D3×SD1611yescharacteristic
401C2×C32D3×Q1611yescharacteristic
402C2×C32D3×Q1611yescharacteristic
403C2×C32D3×Q1611yescharacteristic
404C2×Q8—31no—
405C2×Q8—91no—
406C2×Q8—541no—
407C2×C8—31no—
408C2×C8—91no—
409C4⋊C4—31no—
410C4⋊C4—91no—
411C4⋊C4—541no—
412C4⋊C4—541no—
413C4⋊C4—541no—
414C42—271no—
415C22×C3—21no—
416C22×C3—21no—
417C22×C3—21no—
418C22×C3—41no—
419C22×C3#380211yescharacteristic
420C22×C3#381111yescharacteristic
421C22×C3#381111yescharacteristic
422C3×C4—21no—
423C3×C4—21no—
424C3×C4—21no—
425C3×C4—21no—
426C3×C4—21no—
427C3×C4—21no—
428C3×C4—41no—
429C3×C4—41no—
430C3×C4—122no—
431C3×C4—122no—
432C3×C4—121no—
433C3×C4—121no—
434C3×C4—121no—
435C3×C4—121no—
436C3×C4—361no—
437C3×C4—361no—
438C3×C4#379511yescharacteristic
439C3×C4#379511yescharacteristic
440C3×C4#379811yescharacteristic
441C3×C4#380411yescharacteristic
442C3×C4#380411yescharacteristic
443C3×C4#380511yescharacteristic
444Dic3—41no—
445Dic3—41no—
446Dic3—92no—
447Dic3—92no—
448Dic3—92no—
449Dic3—92no—
450Dic3—92no—
451Dic3—92no—
452Dic3—122no—
453Dic3—122no—
454Dic3—121no—
455Dic3—121no—
456Dic3—121no—
457Dic3—121no—
458Dic3—182no—
459Dic3—182no—
460Dic3—182no—
461Dic3—182no—
462Dic3—182no—
463Dic3—182no—
464Dic3—362no—
465Dic3—362no—
466C32—21no—
467C32—21no—
468C32—21no—
469C32—41no—
470C32#479211yescharacteristic, complemented
471C32#479211yescharacteristic, complemented
472C32#479211yescharacteristic, complemented
473Q8—61no—
474Q8—61no—
475Q8—181no—
476Q8—181no—
477Q8—1082no—
478C2×C4—61no—
479C2×C4—61no—
480C2×C4—181no—
481C2×C4—181no—
482C2×C4—271no—
483C2×C4—271no—
484C2×C4D3311yescharacteristic
485C8—61no—
486C8—181no—
487C2×C3—21no—
488C2×C3—21no—
489C2×C3—21no—
490C2×C3—21no—
491C2×C3—21no—
492C2×C3—21no—
493C2×C3—21no—
494C2×C3—21no—
495C2×C3—21no—
496C2×C3—41no—
497C2×C3—41no—
498C2×C3—41no—
499C2×C3#6379611yescharacteristic
500C2×C3#6380211yescharacteristic
501C2×C3#6391311yescharacteristic
502C2×C3#6391511yescharacteristic
503C2×C3#6393011yescharacteristic
504C2×C3#6394011yescharacteristic
505C2×C3#6394411yescharacteristic
506C2×C3#6394411yescharacteristic
507C2×C3#6395311yescharacteristic
508C22#8895411yescharacteristic
509C4—121no—
510C4—121no—
511C4—272no—
512C4—272no—
513C4—361no—
514C4—361no—
515C4#8896011yescharacteristic
516C4#8896111yescharacteristic
517C3—21no—
518C3—21no—
519C3—21no—
520C3—41no—
521C3#1059056711yescharacteristic, complemented
522C3#1059090111yescharacteristic, complemented
523C3#1059153811yescharacteristic, complemented
524C2#1172284111yescharacteristic
525C2#1172284811yescharacteristic
526C2#1172300911yescharacteristic
subgroupG/H
centerC22#88954
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socle, prime radicalC22×C33C2×D4
hypocenterC33#487
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
317. C33#487semidirect
470. C32#4792semidirect
471. C32#4792semidirect
472. C32#4792semidirect
521. C3#10590567semidirect
522. C3#10590901semidirect
523. C3#10591538semidirect

Maximal subgroups

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

subgroupindex
#117235362
#117234512
#117220762
#117220182
C3×#635672
C3×#635662
C3×#635652
#105915383
#105909013
#105905673