Finite Group Databasebeta

← #13156637

Subgroups — #13156637

Order 1728.

710 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 710 conjugacy classes of subgroups — too many to draw. Only its 108 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×#88811C211yesmaximal, characteristic
2#11723451C211yesmaximal, characteristic, complemented
3#11723436C211yesmaximal, characteristic
4C2×C3×#3719C211yesmaximal, characteristic, complemented
5C2×C3×#3718C211yesmaximal, characteristic, complemented
6C2×C3×#3717C211yesmaximal, characteristic
7#11722021C211yesmaximal, characteristic, complemented
8#10591629—31nomaximal
9#10591611—31nomaximal
10#10590870—31nomaximal
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C2×C32×Dic6C2211yescharacteristic
15C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18#88811—22no—
19#88806—22no—
20#88803—22no—
21C3×#3719—22no—
22C3×#3718—22no—
23C3×#3717—22no—
24#64156—31no—
25#64153—31no—
26C2×C3×#216—31no—
27C2×C3×#216—31no—
28C2×C3×#216—31no—
29C2×C3×#215—31no—
30C2×C3×Dic12—31no—
31C2×C3×#204—31no—
32#63873—31no—
33#63873—31no—
34#63873—61no—
35#63863—31no—
36#63863—31no—
37#63863—61no—
38C2×#3719D311yescharacteristic, complemented
39C2×#3718D311yescharacteristic, complemented
40C2×#3717—31no—
41C2×#3717D311yescharacteristic
42C2×#3716—31no—
43C2×#3715—31no—
44#63642—31no—
45#63555—31no—
46#63555—31no—
47#63555—61no—
48C22×C3×C3⋊D3—21no—
49C2×C33×C4C2311yescharacteristic
50C2×C3⋊(C3⋊Dic3)D412yes—
51C2×C3×C3⋊Dic3—21no—
52C3×C4⋊(C3⋊D3)—21no—
53C3×C4⋊(C3⋊D3)—21no—
54C3×#634—21no—
55C3×#634—21no—
56C2×C32×Dic3—21no—
57C2×C32×Dic3—21no—
58C32×Dic6—21no—
59C32×Dic6—21no—
60C32×Dic6—21no—
61C32×Dic6—21no—
62C3×C3⋊(C3⋊C8)D412yes—
63C32×C3⋊C8D412yes—
64#5190—91no—
65#5190—91no—
66#5167—91no—
67C2×C32×Q8—31no—
68C2×C32×Q8—31no—
69C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
70C2×#634C2×D311yescharacteristic
71C2×C3×D12—31no—
72C2×C3×D12—31no—
73C2×C3×D12—61no—
74C2×C3×Dic6—21no—
75C2×C3×Dic6—21no—
76C2×C3×Dic6—31no—
77C2×C3×Dic6—31no—
78C2×C3×Dic6—61no—
79C2×C3×Dic6C2×D311yescharacteristic
80C2×C3×Dic6C2×D311yescharacteristic
81C2×C3×Dic6C2×D311yescharacteristic
82C2×C3×Dic6C2×D311yescharacteristic
83C2×C32×C8—31no—
84#3752—62no—
85#3750—62no—
86C4×C3⋊Dic3—31no—
87C4×C3⋊Dic3—31no—
88C4×C3⋊Dic3—31no—
89C4×C3⋊Dic3—61no—
90C4×C3⋊Dic3—61no—
91C4×C3⋊Dic3—61no—
92C4×C3⋊Dic3—121no—
93C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
94C3×#216—62no—
95C3×#216—62no—
96C3×#216—62no—
97C3×#215—62no—
98C2×C3×C3⋊C8—21no—
99C2×C3×C3⋊C8—31no—
100C2×C3×C3⋊C8—31no—
101C2×C3×C3⋊C8—61no—
102C2×C3×C3⋊C8C2×D311yescharacteristic
103C2×C3×C3⋊C8C2×D311yescharacteristic
104C3×Dic12—62no—
105C3×#204—62no—
106#3724—62no—
107#3724—62no—
108#3724—64no—
109(C2×D3)⋊Dic3—62no—
110(C2×D3)⋊Dic3—62no—
111(C2×D3)⋊Dic3—64no—
112#3719—22no—
113#3718—22no—
114#3717—22no—
115#3717—62no—
116#3716—62no—
117#3715—62no—
118C22×C33C2×D411yescharacteristic
119C2×C3×C3⋊D3—41no—
120C2×C3×C3⋊D3—42no—
121C33×C4C2×D411yescharacteristic
122C33×C4C2×D411yescharacteristic
123C3⋊(C3⋊Dic3)—22no—
124C3×C3⋊Dic3—41no—
125C32×Dic3—41no—
126C32×Dic3—41no—
127C2×C3×Q16—91no—
128C2×C3×Q16—91no—
129C2×C3×SD16—91no—
130#947—91no—
131#947—91no—
132#947—181no—
133#944—91no—
134#944—91no—
135#944—91no—
136#944—181no—
137C2×#216—31no—
138C2×#216—31no—
139C2×#216—31no—
140C2×#215—31no—
141C2×#215—91no—
142#936—91no—
143C2×#214—91no—
144C2×Dic12—31no—
145C2×#204—31no—
146C2×#204—91no—
147#815—91no—
148#815—91no—
149#815—91no—
150#815—181no—
151C22×C3⋊D3—21no—
152C22×C3×D3—62no—
153C22×C3×D3—61no—
154C22×C3×D3—61no—
155C32×Q8—61no—
156C32×Q8—61no—
157C32×Q8—61no—
158C32×Q8—61no—
159C2×C32×C4—21no—
160C2×C32×C4—21no—
161C2×C32×C4—21no—
162C2×C32×C4—41no—
163C2×C32×C4—61no—
164C2×C32×C4—61no—
165C2×C32×C4C22×D311yescharacteristic
166C2×C32×C4C22×D311yescharacteristic
167C2×C32×C4C22×D311yescharacteristic
168C2×C3⋊Dic3—21no—
169C2×C3⋊Dic3—32no—
170C2×C3⋊Dic3—32no—
171C2×C3⋊Dic3—32no—
172C2×C3⋊Dic3—62no—
173C2×C3⋊Dic3—62no—
174C2×C3⋊Dic3—62no—
175C2×C3⋊Dic3—122no—
176C4⋊(C3⋊D3)—21no—
177C4⋊(C3⋊D3)—21no—
178#634—21no—
179#634—21no—
180C2×C3×Dic3—22no—
181C2×C3×Dic3—22no—
182C2×C3×Dic3—21no—
183C2×C3×Dic3—21no—
184C2×C3×Dic3—21no—
185C2×C3×Dic3—21no—
186C2×C3×Dic3—62no—
187C2×C3×Dic3—61no—
188C2×C3×Dic3—61no—
189C3×D12—61no—
190C3×D12—61no—
191C3×D12—61no—
192C3×D12—61no—
193C3×D12—121no—
194C3×D12—121no—
195C3×Dic6—21no—
196C3×Dic6—21no—
197C3×Dic6—21no—
198C3×Dic6—21no—
199C3×Dic6—21no—
200C3×Dic6—21no—
201C3×Dic6—21no—
202C3×Dic6—21no—
203C3×Dic6—41no—
204C3×Dic6—41no—
205C3×Dic6—41no—
206C3×Dic6—41no—
207C3×Dic6—61no—
208C3×Dic6—61no—
209C3×Dic6—61no—
210C3×Dic6—61no—
211C3×Dic6—121no—
212C3×Dic6—121no—
213C32×C8—32no—
214C3⋊(C3⋊C8)#6712yes—
215C3×C3⋊C8—22no—
216C3×C3⋊C8—32no—
217C3×C3⋊C8—32no—
218C3×C3⋊C8—62no—
219C3×C3⋊C8#6712yes—
220C3×C3⋊C8#6712yes—
221#497—271noSylow
222C2×C33#12711yescharacteristic
223C2×C33#13712yes—
224C3×C3⋊D3—81no—
225C2×C3×Q8—31no—
226C2×C3×Q8—31no—
227C2×C3×Q8—31no—
228C2×C3×Q8—31no—
229C2×C3×Q8—61no—
230C2×C3×Q8—61no—
231C2×C3×Q8—91no—
232C2×C3×D4—91no—
233C2×D12—31no—
234C2×D12—31no—
235C2×D12—61no—
236C2×Dic6—31no—
237C2×Dic6—31no—
238C2×Dic6—61no—
239C2×Dic6D3211yescharacteristic
240C2×Dic6D3211yescharacteristic
241C3×Q16—182no—
242C3×Q16—182no—
243C3×SD16—182no—
244C2×C3×C8—31no—
245C2×C3×C8—31no—
246C2×C3×C8—61no—
247C2×C3×C8—91no—
248#217—182no—
249#216—62no—
250#216—62no—
251#216—62no—
252#215—62no—
253#215—182no—
254#214—182no—
255(C2×D3)⋊C4—182no—
256(C2×D3)⋊C4—182no—
257(C2×D3)⋊C4—184no—
258Dic3⋊C4—182no—
259Dic3⋊C4—182no—
260Dic3⋊C4—182no—
261Dic3⋊C4—184no—
262C4×Dic3—91no—
263C4×Dic3—91no—
264C4×Dic3—91no—
265C4×Dic3—181no—
266C4×Dic3—181no—
267C4×Dic3—181no—
268C4×Dic3—361no—
269C2×C3⋊C8—31no—
270C2×C3⋊C8—31no—
271C2×C3⋊C8—61no—
272C2×C3⋊C8D3211yescharacteristic
273Dic12—62no—
274#204—62no—
275#204—182no—
276C22×C32—21no—
277C22×C32—21no—
278C22×C32—21no—
279C22×C32—41no—
280C22×C32D3×D411yescharacteristic
281C22×C32D3×D411yescharacteristic
282C22×C32D3×D411yescharacteristic
283C2×C3⋊D3—42no—
284C2×C3⋊D3—41no—
285C2×C3×D3—122no—
286C2×C3×D3—124no—
287C2×C3×D3—122no—
288C2×C3×D3—122no—
289C2×C3×D3—121no—
290C2×C3×D3—121no—
291C32×C4—21no—
292C32×C4—21no—
293C32×C4—21no—
294C32×C4—21no—
295C32×C4—21no—
296C32×C4—21no—
297C32×C4—41no—
298C32×C4—41no—
299C32×C4—121no—
300C32×C4—121no—
301C32×C4D3×D411yescharacteristic
302C32×C4D3×D411yescharacteristic
303C32×C4D3×D411yescharacteristic
304C32×C4C2×#6711yescharacteristic
305C32×C4C2×#6711yescharacteristic
306C32×C4C2×#6711yescharacteristic
307C3⋊Dic3—41no—
308C3⋊Dic3—62no—
309C3⋊Dic3—64no—
310C3⋊Dic3—62no—
311C3⋊Dic3—62no—
312C3⋊Dic3—122no—
313C3⋊Dic3—122no—
314C3⋊Dic3—124no—
315C3×Dic3—42no—
316C3×Dic3—42no—
317C3×Dic3—41no—
318C3×Dic3—41no—
319C3×Dic3—41no—
320C3×Dic3—41no—
321C3×Dic3—122no—
322C3×Dic3—121no—
323C3×Dic3—121no—
324C2×Q16—91no—
325C2×Q16—91no—
326C2×SD16—91no—
327C2×SD16—271no—
328C4⋊Q8—271no—
329#124—271no—
330#97—271no—
331C33#49711yescharacteristic, complemented, Sylow
332C23×C3—181no—
333C22×D3—62no—
334C22×D3—61no—
335C22×D3—61no—
336C3×Q8—61no—
337C3×Q8—61no—
338C3×Q8—61no—
339C3×Q8—61no—
340C3×Q8—61no—
341C3×Q8—61no—
342C3×Q8—61no—
343C3×Q8—61no—
344C3×Q8—121no—
345C3×Q8—121no—
346C3×Q8—121no—
347C3×Q8—121no—
348C3×Q8—181no—
349C3×Q8—181no—
350C3×D4—181no—
351C3×D4—181no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—41no—
356C2×C3×C4—62no—
357C2×C3×C4—62no—
358C2×C3×C4—61no—
359C2×C3×C4—61no—
360C2×C3×C4—61no—
361C2×C3×C4—61no—
362C2×C3×C4—181no—
363C2×C3×C4C2×D3211yescharacteristic
364C2×C3×C4C2×D3211yescharacteristic
365C2×C3×C4C2×D3211yescharacteristic
366C2×Dic3—21no—
367C2×Dic3—21no—
368C2×Dic3—62no—
369C2×Dic3—61no—
370C2×Dic3—61no—
371C2×Dic3—92no—
372C2×Dic3—92no—
373C2×Dic3—92no—
374C2×Dic3—182no—
375C2×Dic3—182no—
376C2×Dic3—182no—
377C2×Dic3—362no—
378D12—61no—
379D12—61no—
380D12—61no—
381D12—61no—
382D12—121no—
383D12—121no—
384Dic6—21no—
385Dic6—21no—
386Dic6—21no—
387Dic6—21no—
388Dic6—61no—
389Dic6—61no—
390Dic6—61no—
391Dic6—61no—
392Dic6—121no—
393Dic6—121no—
394C3×C8—32no—
395C3×C8—32no—
396C3×C8—62no—
397C3×C8—92no—
398C3⋊C8—32no—
399C3⋊C8—32no—
400C3⋊C8—62no—
401C3⋊C8#62512yes—
402C2×C32—22no—
403C2×C32—21no—
404C2×C32—22no—
405C2×C32—22no—
406C2×C32—21no—
407C2×C32—21no—
408C2×C32—41no—
409C2×C32—42no—
410C2×C32#91512yes—
411C2×C32#91812yes—
412C2×C32#91812yes—
413C2×C32#94011yescharacteristic
414C2×C32#94011yescharacteristic
415C2×C32#94011yescharacteristic
416C3⋊D3—81no—
417C3×D3—242no—
418C3×D3—241no—
419C3×D3—241no—
420C2×Q8—31no—
421C2×Q8—31no—
422C2×Q8—91no—
423C2×D4—91no—
424Q16—182no—
425Q16—182no—
426SD16—182no—
427SD16—542no—
428C2×C8—31no—
429C2×C8—91no—
430C4⋊C4—542no—
431C22⋊C4—542no—
432C42—271no—
433C22×C3—21no—
434C22×C3—21no—
435C22×C3—21no—
436C22×C3—41no—
437C22×C3—362no—
438C22×C3—361no—
439C22×C3#380211yescharacteristic
440C22×C3#381111yescharacteristic
441C22×C3#381111yescharacteristic
442C2×D3—122no—
443C2×D3—124no—
444C2×D3—122no—
445C2×D3—121no—
446C2×D3—122no—
447C2×D3—121no—
448C3×C4—21no—
449C3×C4—21no—
450C3×C4—21no—
451C3×C4—21no—
452C3×C4—21no—
453C3×C4—21no—
454C3×C4—41no—
455C3×C4—41no—
456C3×C4—122no—
457C3×C4—122no—
458C3×C4—121no—
459C3×C4—121no—
460C3×C4—121no—
461C3×C4—121no—
462C3×C4—361no—
463C3×C4#380211yescharacteristic
464C3×C4C2×#62511yescharacteristic
465C3×C4D3×#6711yescharacteristic
466C3×C4D3×#6711yescharacteristic
467C3×C4D3×#6711yescharacteristic
468C3×C4D3×#6711yescharacteristic
469Dic3—41no—
470Dic3—41no—
471Dic3—122no—
472Dic3—121no—
473Dic3—121no—
474Dic3—184no—
475Dic3—182no—
476Dic3—182no—
477Dic3—182no—
478Dic3—362no—
479Dic3—362no—
480Dic3—364no—
481C32—21no—
482C32—21no—
483C32—21no—
484C32—41no—
485C32#516711yescharacteristic, complemented
486C32#519011yescharacteristic, complemented
487C32#519011yescharacteristic, complemented
488C23—181no—
489Q8—61no—
490Q8—61no—
491Q8—61no—
492Q8—61no—
493Q8—181no—
494Q8—181no—
495D4—181no—
496D4—181no—
497C2×C4—61no—
498C2×C4—61no—
499C2×C4—181no—
500C2×C4—272no—
501C2×C4D3311yescharacteristic
502C8—32no—
503C8—92no—
504C2×C3—21no—
505C2×C3—21no—
506C2×C3—21no—
507C2×C3—22no—
508C2×C3—22no—
509C2×C3—22no—
510C2×C3—41no—
511C2×C3—42no—
512C2×C3—721no—
513C2×C3#6381112yes—
514C2×C3#6391411yescharacteristic
515C2×C3#6394512yes—
516C2×C3#6395112yes—
517C2×C3#6397911yescharacteristic
518C2×C3#6397911yescharacteristic
519D3—242no—
520D3—241no—
521D3—241no—
522C22—361no—
523C22—362no—
524C22#8895411yescharacteristic
525C4—121no—
526C4—121no—
527C4—361no—
528C4—542no—
529C4D3×#62511yescharacteristic
530C4#8895311yescharacteristic
531C3—21no—
532C3—21no—
533C3—21no—
534C3—41no—
535C3#1059087011yescharacteristic, complemented
536C3#1059161111yescharacteristic, complemented
537C3#1059162911yescharacteristic, complemented
538C2—721no—
539C2#1172289412yes—
540C2#1172302911yescharacteristic
subgroupG/H
centerC22#88954
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
hypocenterC33#497
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
2. #11723451C2semidirect
4. C2×C3×#3719C2semidirect
5. C2×C3×#3718C2semidirect
7. #11722021C2semidirect
38. C2×#3719D3semidirect
39. C2×#3718D3semidirect
331. C33#497semidirect
485. C32#5167semidirect
486. C32#5190semidirect
487. C32#5190semidirect
535. C3#10590870semidirect
536. C3#10591611semidirect
537. C3#10591629semidirect

Maximal subgroups

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

subgroupindex
C2×#888112
#117234512
#117234362
C2×C3×#37192
C2×C3×#37182
C2×C3×#37172
#117220212
#105916293
#105916113
#105908703