Finite Group Databasebeta

← #13156633

Subgroups — #13156633

Order 1728.

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

normal subgroups only

This group has 686 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
2C2×#88798C211yesmaximal, characteristic, complemented
3C2×#88793C211yesmaximal, characteristic
4C3×#63866C211yesmaximal, characteristic, complemented
5C3×#63864C211yesmaximal, characteristic
6C2×C3×#3718C211yesmaximal, characteristic, complemented
7#11722019C211yesmaximal, characteristic, complemented
8#10591627—31nomaximal
9#10591608—31nomaximal
10#10590376—31nomaximal
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C2×C32×Dic6C2211yescharacteristic
15C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×C4×Dic3C2211yescharacteristic
18#88811—22no—
19#88798—22no—
20#88793—22no—
21C3×Dic3⋊Dic3—22no—
22C3×#3722—22no—
23C3×#3718—22no—
24C2×#3756—31no—
25C2×#3755—31no—
26C3×#947—31no—
27C3×#944—31no—
28C2×C3×#216—31no—
29C2×C3×#216—31no—
30C3×#875—31no—
31C3×#869—31no—
32#63866D311yescharacteristic, complemented
33#63864D311yescharacteristic
34C2×#3718—31no—
35C2×#3718—31no—
36C2×#3718—61no—
37C2×#3718D311yescharacteristic, complemented
38C2×#3717—31no—
39C2×#3716—31no—
40C2×#3715—31no—
41C2×#3715—31no—
42C2×#3715—31no—
43C2×#3715—61no—
44#63630—31no—
45#63554—31no—
46#63554—31no—
47#63554—61no—
48C22×C3×C3⋊D3—21no—
49C2×C33×C4C2311yescharacteristic
50C2×C3×C3⋊Dic3—21no—
51C3×C4⋊(C3⋊D3)—21no—
52C3×C4⋊(C3⋊D3)—21no—
53C3×#634—21no—
54C3×#634—21no—
55C2×C32×Dic3—21no—
56C2×C32×Dic3—21no—
57C2×C32×Dic3D412yes—
58C32×Dic6—21no—
59C32×Dic6—21no—
60C32×Dic6—21no—
61C32×Dic6—21no—
62C3⋊(C3⋊(C3⋊C8))D412yes—
63C3×C3⋊(C3⋊C8)D412yes—
64#5190—91no—
65#5086—91no—
66#5060—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
83C32×C42—31no—
84#3756—62no—
85#3755—62no—
86C2×C3⋊(C3⋊C8)—31no—
87C2×C3⋊(C3⋊C8)—31no—
88C2×C3⋊(C3⋊C8)—31no—
89C2×C3⋊(C3⋊C8)—61no—
90C2×C3⋊(C3⋊C8)—61no—
91C2×C3⋊(C3⋊C8)—61no—
92C2×C3⋊(C3⋊C8)—121no—
93C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
94C3×#216—62no—
95C3×#216—62no—
96C3×(C2×D3)⋊C4—62no—
97C3×(C2×D3)⋊C4—62no—
98C3×C4⋊Dic3—62no—
99C3×Dic3⋊C4—62no—
100C3×C4×Dic3—21no—
101C3×C4×Dic3C2×D311yescharacteristic
102C3×C4×Dic3C2×D311yescharacteristic
103C2×C3×C3⋊C8—31no—
104C2×C3×C3⋊C8—31no—
105C2×C3×C3⋊C8—61no—
106Dic3⋊Dic3—22no—
107#3722—22no—
108#3718—22no—
109#3718—62no—
110#3718—62no—
111#3718—64no—
112#3717—62no—
113#3716—62no—
114#3715—62no—
115#3715—62no—
116#3715—64no—
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—41no—
124C32×Dic3—22no—
125C32×Dic3—41no—
126C32×Dic3—41no—
127C2×C3×Q16—91no—
128C3×C4⋊Q8—91no—
129C3×#124—91no—
130#947—31no—
131#944—31no—
132C2×#216—31no—
133C2×#216—31no—
134C2×#216—91no—
135C2×#216—91no—
136C2×#216—91no—
137C2×#216—181no—
138C2×#215—91no—
139C2×#215—91no—
140C2×#215—91no—
141C2×#215—181no—
142C2×#214—91no—
143C2×#214—91no—
144C2×#204—91no—
145#875—31no—
146#869—31no—
147#815—91no—
148#803—91no—
149#803—91no—
150#803—181no—
151C22×C3⋊D3—21no—
152C22×C3×D3—61no—
153C22×C3×D3—61no—
154C22×C3×D3—121no—
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—32no—
163C2×C32×C4—41no—
164C2×C32×C4—61no—
165C2×C32×C4—61no—
166C2×C32×C4C22×D311yescharacteristic
167C2×C32×C4C22×D311yescharacteristic
168C2×C32×C4C22×D311yescharacteristic
169C2×C3⋊Dic3—21no—
170C4⋊(C3⋊D3)—21no—
171C4⋊(C3⋊D3)—21no—
172#634—21no—
173#634—21no—
174C2×C3×Dic3—21no—
175C2×C3×Dic3—21no—
176C2×C3×Dic3—21no—
177C2×C3×Dic3—21no—
178C2×C3×Dic3—22no—
179C2×C3×Dic3—41no—
180C2×C3×Dic3—41no—
181C2×C3×Dic3—61no—
182C2×C3×Dic3—61no—
183C2×C3×Dic3—121no—
184C2×C3×Dic3#6712yes—
185C2×C3×Dic3#6712yes—
186C3×D12—62no—
187C3×D12—61no—
188C3×D12—61no—
189C3×D12—61no—
190C3×D12—61no—
191C3×D12—121no—
192C3×Dic6—22no—
193C3×Dic6—22no—
194C3×Dic6—21no—
195C3×Dic6—21no—
196C3×Dic6—21no—
197C3×Dic6—21no—
198C3×Dic6—21no—
199C3×Dic6—21no—
200C3×Dic6—21no—
201C3×Dic6—21no—
202C3×Dic6—41no—
203C3×Dic6—41no—
204C3×Dic6—62no—
205C3×Dic6—61no—
206C3×Dic6—61no—
207C3×Dic6—61no—
208C3×Dic6—61no—
209C3×Dic6—121no—
210C3⋊(C3⋊C8)—32no—
211C3⋊(C3⋊C8)—32no—
212C3⋊(C3⋊C8)—32no—
213C3⋊(C3⋊C8)—62no—
214C3⋊(C3⋊C8)—62no—
215C3⋊(C3⋊C8)—62no—
216C3⋊(C3⋊C8)—122no—
217C3⋊(C3⋊C8)#6712yes—
218C3×C3⋊C8—32no—
219C3×C3⋊C8—32no—
220C3×C3⋊C8—62no—
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—
242C2×C3×C8—91no—
243C3×C4⋊C4—182no—
244C3×C22⋊C4—182no—
245C3×C42—31no—
246C3×C42—31no—
247C3×C42—61no—
248#216—62no—
249#216—62no—
250#216—182no—
251#216—182no—
252#216—182no—
253#216—184no—
254#215—182no—
255#215—182no—
256#215—182no—
257#215—184no—
258#214—182no—
259#214—182no—
260(C2×D3)⋊C4—62no—
261(C2×D3)⋊C4—62no—
262C4⋊Dic3—62no—
263Dic3⋊C4—62no—
264C4×Dic3D3211yescharacteristic
265C2×C3⋊C8—31no—
266C2×C3⋊C8—31no—
267C2×C3⋊C8—61no—
268C2×C3⋊C8—91no—
269C2×C3⋊C8—91no—
270C2×C3⋊C8—91no—
271C2×C3⋊C8—181no—
272C2×C3⋊C8—181no—
273C2×C3⋊C8—181no—
274C2×C3⋊C8—361no—
275#204—182no—
276C22×C32—21no—
277C22×C32—21no—
278C22×C32—21no—
279C22×C32—41no—
280C22×C32D3×D411yescharacteristic
281C22×C32C2×#6711yescharacteristic
282C22×C32C2×#6711yescharacteristic
283C2×C3⋊D3—41no—
284C2×C3⋊D3—42no—
285C2×C3×D3—122no—
286C2×C3×D3—122no—
287C2×C3×D3—121no—
288C2×C3×D3—121no—
289C2×C3×D3—122no—
290C2×C3×D3—242no—
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—62no—
300C32×C4—121no—
301C32×C4—121no—
302C32×C4D3×D411yescharacteristic
303C32×C4D3×D411yescharacteristic
304C32×C4D3×D411yescharacteristic
305C32×C4D3×D411yescharacteristic
306C32×C4D3×D411yescharacteristic
307C32×C4C2×#6711yescharacteristic
308C3⋊Dic3—41no—
309C3×Dic3—22no—
310C3×Dic3—22no—
311C3×Dic3—42no—
312C3×Dic3—42no—
313C3×Dic3—42no—
314C3×Dic3—41no—
315C3×Dic3—41no—
316C3×Dic3—41no—
317C3×Dic3—41no—
318C3×Dic3—122no—
319C3×Dic3—121no—
320C3×Dic3—121no—
321C2×Q16—91no—
322C2×Q16—271no—
323C2×SD16—271no—
324C2×SD16—271no—
325C4⋊Q8—91no—
326#124—91no—
327#97—271no—
328C33#49711yescharacteristic, complemented, Sylow
329C23×C3—181no—
330C22×D3—61no—
331C22×D3—61no—
332C22×D3—121no—
333C3×Q8—62no—
334C3×Q8—62no—
335C3×Q8—61no—
336C3×Q8—61no—
337C3×Q8—61no—
338C3×Q8—61no—
339C3×Q8—61no—
340C3×Q8—61no—
341C3×Q8—61no—
342C3×Q8—61no—
343C3×Q8—121no—
344C3×Q8—121no—
345C3×Q8—181no—
346C3×Q8—181no—
347C3×D4—181no—
348C3×D4—181no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—32no—
353C2×C3×C4—32no—
354C2×C3×C4—41no—
355C2×C3×C4—62no—
356C2×C3×C4—61no—
357C2×C3×C4—61no—
358C2×C3×C4—61no—
359C2×C3×C4—61no—
360C2×C3×C4—121no—
361C2×C3×C4—121no—
362C2×C3×C4—181no—
363C2×C3×C4C2×D3211yescharacteristic
364C2×C3×C4C2×D3211yescharacteristic
365C2×C3×C4C2×D3211yescharacteristic
366C2×Dic3—21no—
367C2×Dic3—21no—
368C2×Dic3—61no—
369C2×Dic3—61no—
370C2×Dic3—121no—
371C2×Dic3#62512yes—
372D12—62no—
373D12—61no—
374D12—61no—
375D12—61no—
376D12—61no—
377D12—121no—
378Dic6—21no—
379Dic6—21no—
380Dic6—21no—
381Dic6—21no—
382Dic6—62no—
383Dic6—61no—
384Dic6—61no—
385Dic6—61no—
386Dic6—61no—
387Dic6—121no—
388C3×C8—92no—
389C3⋊C8—32no—
390C3⋊C8—32no—
391C3⋊C8—62no—
392C3⋊C8—92no—
393C3⋊C8—92no—
394C3⋊C8—92no—
395C3⋊C8—182no—
396C3⋊C8—182no—
397C3⋊C8—182no—
398C3⋊C8—362no—
399C2×C32—22no—
400C2×C32—21no—
401C2×C32—22no—
402C2×C32—22no—
403C2×C32—21no—
404C2×C32—21no—
405C2×C32—41no—
406C2×C32—42no—
407C2×C32#91812yes—
408C2×C32#94011yescharacteristic
409C2×C32#94011yescharacteristic
410C2×C32#94011yescharacteristic
411C2×C32#94212yes—
412C2×C32#95112yes—
413C3⋊D3—81no—
414C3×D3—242no—
415C3×D3—241no—
416C3×D3—241no—
417C2×Q8—31no—
418C2×Q8—31no—
419C2×Q8—91no—
420C2×D4—91no—
421Q16—182no—
422Q16—542no—
423SD16—542no—
424SD16—542no—
425C2×C8—91no—
426C2×C8—271no—
427C4⋊C4—182no—
428C22⋊C4—182no—
429C42—31no—
430C22×C3—21no—
431C22×C3—21no—
432C22×C3—21no—
433C22×C3—41no—
434C22×C3—362no—
435C22×C3—361no—
436C22×C3C2×#62511yescharacteristic
437C22×C3D3×#6711yescharacteristic
438C22×C3D3×#6711yescharacteristic
439C2×D3—122no—
440C2×D3—122no—
441C2×D3—121no—
442C2×D3—122no—
443C2×D3—121no—
444C2×D3—242no—
445C3×C4—21no—
446C3×C4—21no—
447C3×C4—21no—
448C3×C4—21no—
449C3×C4—21no—
450C3×C4—21no—
451C3×C4—41no—
452C3×C4—41no—
453C3×C4—62no—
454C3×C4—62no—
455C3×C4—122no—
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×C4#380211yescharacteristic
465C3×C4D3×#6711yescharacteristic
466C3×C4D3×#6711yescharacteristic
467C3×C4#381111yescharacteristic
468C3×C4#381111yescharacteristic
469Dic3—22no—
470Dic3—41no—
471Dic3—41no—
472Dic3—122no—
473Dic3—121no—
474Dic3—121no—
475C32—21no—
476C32—21no—
477C32—21no—
478C32—41no—
479C32#506011yescharacteristic, complemented
480C32#508611yescharacteristic, complemented
481C32#519011yescharacteristic, complemented
482C23—181no—
483Q8—61no—
484Q8—61no—
485Q8—61no—
486Q8—61no—
487Q8—181no—
488Q8—181no—
489D4—181no—
490D4—181no—
491C2×C4—32no—
492C2×C4—61no—
493C2×C4—61no—
494C2×C4—181no—
495C2×C4D3311yescharacteristic
496C8—92no—
497C8—272no—
498C2×C3—21no—
499C2×C3—21no—
500C2×C3—21no—
501C2×C3—22no—
502C2×C3—22no—
503C2×C3—22no—
504C2×C3—41no—
505C2×C3—42no—
506C2×C3—721no—
507C2×C3#6382712yes—
508C2×C3#6391411yescharacteristic
509C2×C3#6394312yes—
510C2×C3#6394812yes—
511C2×C3#6397911yescharacteristic
512C2×C3#6397911yescharacteristic
513D3—242no—
514D3—241no—
515D3—241no—
516C22—361no—
517C22—362no—
518C22D3×#62511yescharacteristic
519C4—62no—
520C4—121no—
521C4—121no—
522C4—361no—
523C4#8895311yescharacteristic
524C4#8895411yescharacteristic
525C3—21no—
526C3—21no—
527C3—21no—
528C3—41no—
529C3#1059037611yescharacteristic, complemented
530C3#1059160811yescharacteristic, complemented
531C3#1059162711yescharacteristic, complemented
532C2—721no—
533C2#1172288312yes—
534C2#1172302911yescharacteristic
subgroupG/H
centerC22D3×#625
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. C2×#88798C2semidirect
4. C3×#63866C2semidirect
6. C2×C3×#3718C2semidirect
7. #11722019C2semidirect
32. #63866D3semidirect
37. C2×#3718D3semidirect
328. C33#497semidirect
479. C32#5060semidirect
480. C32#5086semidirect
481. C32#5190semidirect
529. C3#10590376semidirect
530. C3#10591608semidirect
531. C3#10591627semidirect

Maximal subgroups

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

subgroupindex
C2×#888112
C2×#887982
C2×#887932
C3×#638662
C3×#638642
C2×C3×#37182
#117220192
#105916273
#105916083
#105903763