Finite Group Databasebeta

← #13167219

Subgroups — #13167219

Order 1728.

1100 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 1100 conjugacy classes of subgroups — too many to draw. Only its 79 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
1C22×D32⋊D3C212yesmaximal, complemented
2C2×C4⋊#1113C211yesmaximal, characteristic, complemented
3C2×#88932C212yesmaximal, complemented
4C3×#63921C211yesmaximal, characteristic, complemented
5#11723106C211yesmaximal, characteristic, complemented
6#10593455—31nomaximal
7C22×#1113C2212yes—
8C22×C3×D32C2211yescharacteristic
9C22×C3⋊#157C2212yes—
10C2×D32⋊D3—24no—
11C2×D32⋊D3—24no—
12C2×D32⋊D3—24no—
13C2×C3×C4⋊(C3⋊D3)—21no—
14C2×C3×C4×C3⋊D3C2211yescharacteristic
15C2×#6295—21no—
16C4⋊#1113—24no—
17C2×C3×#624C2211yescharacteristic
18#88932—24no—
19C32×(C2×D3)⋊C4—21no—
20C3×(C2×D3)⋊Dic3—21no—
21C22×GO+(4,𝔽2)—32no—
22C2×#3802—31no—
23C2×D3×D12—61no—
24C2×D32⋊C4—32no—
25C3×D3×C22⋊C4—61no—
26#63921D311yescharacteristic, complemented
27#63781—31no—
28C22×C3×C3⋊D3—22no—
29C22×C3×C3⋊D3C2311yescharacteristic
30C22×C32×D3—21no—
31C2×#1113—22no—
32C2×#1113—22no—
33C2×#1113—22no—
34C2×#1113—22no—
35C2×#1113—22no—
36C2×#1113D414yescomplemented
37C2×C3×D32—22no—
38C2×C3×D32—22no—
39C2×C3×D32—21no—
40C2×C3×D32D412yescomplemented
41C2×C3⋊#157—22no—
42C2×C3⋊#157—22no—
43C2×C3⋊#157D414yescomplemented
44D32⋊D3—216no—
45C2×C33×C4D411yescharacteristic, complemented
46C2×C3×C3⋊Dic3D411yescharacteristic, complemented
47C3×C4⋊(C3⋊D3)—24no—
48C3×C4×C3⋊D3—22no—
49C2×C32×Dic3—21no—
50#6295—24no—
51C3×#624—22no—
52#4960—91nomaximal
53C23×C3×D3—61no—
54C22×D32—32no—
55C22×D32—62no—
56C22×D32C2×D311yescharacteristic
57C22×#157—32no—
58C2×GO+(4,𝔽2)—64no—
59C2×GO+(4,𝔽2)—64no—
60C2×GO+(4,𝔽2)—64no—
61C2×C4⋊(C3⋊D3)—21no—
62C2×C4×C3⋊D3C2×D311yescharacteristic
63C2×C3×D12—61no—
64C2×C3×D12—61no—
65C2×C3×D12—121no—
66C2×C3×C4×D3—61no—
67C2×C3×C4×D3—61no—
68C2×C3×C4×D3—61no—
69C2×#626—62no—
70C2×#625—61no—
71C2×#624C2×D311yescharacteristic
72#3802—64no—
73D3×D12—68no—
74D32⋊C4—64no—
75C32×C22⋊C4—61no—
76C3×#217—61no—
77C3×(C2×D3)⋊C4—21no—
78C3×(C2×D3)⋊C4—21no—
79C3×(C2×D3)⋊C4—41no—
80C3×(C2×D3)⋊C4—61no—
81(C2×D3)⋊Dic3—21no—
82C22×C33C2×D411yescharacteristic
83C2×C3×C3⋊D3—24no—
84C2×C3×C3⋊D3—42no—
85C2×C3×C3⋊D3—42no—
86C2×C3×C3⋊D3C2×D411yescharacteristic
87C2×C3×C3⋊D3C2×D412yes—
88C2×C3×C3⋊D3C2×D412yes—
89C2×C3×C3⋊D3#4111yescharacteristic
90C2×C32×D3—22no—
91C2×C32×D3—41no—
92C2×C32×D3—41no—
93#1113—24no—
94#1113—24no—
95#1113—42no—
96#1113—42no—
97C3×D32—24no—
98C3×D32—42no—
99C3⋊#157—24no—
100C33×C4—21no—
101C3×C3⋊Dic3—21no—
102C32×Dic3—41no—
103C22×#67—92no—
104C2×D3×D4—181no—
105C22×D12—91no—
106C2×C3×C22⋊C4—91no—
107C2×(C2×D3)⋊C4—92no—
108D3×C22⋊C4—61no—
109C3⋊#95—91no—
110C23×C32—61no—
111C22×C3⋊D3—22no—
112C22×C3⋊D3C22×D311yescharacteristic
113C22×C3×D3—21no—
114C22×C3×D3—21no—
115C22×C3×D3—41no—
116C22×C3×D3—62no—
117C22×C3×D3—61no—
118C22×C3×D3—61no—
119C22×C3×D3—62no—
120C22×C3×D3—62no—
121C22×C3×D3—62no—
122C22×C3×D3—122no—
123C22×C3×D3—122no—
124C22×C3×D3—122no—
125C2×D32—21no—
126C2×D32—22no—
127C2×D32—22no—
128C2×D32—34no—
129C2×D32—64no—
130C2×D32—64no—
131C2×D32—62no—
132C2×D32—62no—
133C2×D32—62no—
134C2×D32—62no—
135C2×D32—62no—
136C2×D32—122no—
137C2×D32—122no—
138C2×D32—122no—
139C2×D32—122no—
140C2×D32D1212yescomplemented
141C2×#157—34no—
142C2×#157—62no—
143C2×#157—62no—
144GO+(4,𝔽2)—616no—
145C2×C32×C4—21no—
146C2×C32×C4—21no—
147C2×C32×C4—41no—
148C2×C32×C4—41no—
149C2×C32×C4—61no—
150C2×C32×C4#6711yescharacteristic, complemented
151C2×C3⋊Dic3#6711yescharacteristic, complemented
152C4⋊(C3⋊D3)—24no—
153C4×C3⋊D3—22no—
154C2×C3×Dic3—21no—
155C2×C3×Dic3—21no—
156C2×C3×Dic3—41no—
157C2×C3×Dic3—61no—
158C2×C3×Dic3—61no—
159C3×D12—64no—
160C3×D12—64no—
161C3×D12—124no—
162C3×C4×D3—64no—
163C3×C4×D3—122no—
164C3×C4×D3—122no—
165#626—68no—
166#625—64no—
167#624—22no—
168#392—271noSylow
169C2×C33#12011yescharacteristic
170C2×C33#12111yescharacteristic
171C2×C33#12711yescharacteristic
172C3×C3⋊D3—44no—
173C3×C3⋊D3#12012yes—
174C3×C3⋊D3#12112yes—
175C32×D3—42no—
176C24×C3—91no—
177C23×D3—61no—
178C23×D3—92no—
179C23×D3—182no—
180C2×C3×D4—181no—
181C22×C3×C4—91no—
182C22×C3×C4—91no—
183C2×#67—182no—
184C2×#67—184no—
185C2×#67—184no—
186C2×#67—184no—
187C22×Dic3—92no—
188D3×D4—188no—
189C2×D12—61no—
190C2×D12—61no—
191C2×D12—121no—
192C2×D12—184no—
193C2×D12—181no—
194C2×C4×D3—61no—
195C2×C4×D3—61no—
196C2×C4×D3—61no—
197C3×C22⋊C4—61no—
198C3×C22⋊C4—61no—
199C3×C22⋊C4—121no—
200C3×C22⋊C4—181no—
201#217—61no—
202(C2×D3)⋊C4—21no—
203(C2×D3)⋊C4—61no—
204(C2×D3)⋊C4—184no—
205C22×C32—21no—
206C22×C32—21no—
207C22×C32—41no—
208C22×C32—41no—
209C22×C32—62no—
210C22×C32—121no—
211C22×C32—121no—
212C22×C32C2×#6711yescharacteristic
213C2×C3⋊D3—24no—
214C2×C3⋊D3—42no—
215C2×C3⋊D3—42no—
216C2×C3⋊D3C2×D1211yescharacteristic
217C2×C3⋊D3#23511yescharacteristic
218C2×C3⋊D3D3×D412yes—
219C2×C3⋊D3D3×D412yes—
220C2×C3×D3—22no—
221C2×C3×D3—22no—
222C2×C3×D3—42no—
223C2×C3×D3—41no—
224C2×C3×D3—41no—
225C2×C3×D3—41no—
226C2×C3×D3—41no—
227C2×C3×D3—62no—
228C2×C3×D3—62no—
229C2×C3×D3—62no—
230C2×C3×D3—61no—
231C2×C3×D3—62no—
232C2×C3×D3—61no—
233C2×C3×D3—61no—
234C2×C3×D3—61no—
235C2×C3×D3—64no—
236C2×C3×D3—64no—
237C2×C3×D3—81no—
238C2×C3×D3—81no—
239C2×C3×D3—124no—
240C2×C3×D3—124no—
241C2×C3×D3—124no—
242C2×C3×D3—122no—
243C2×C3×D3—122no—
244C2×C3×D3—122no—
245C2×C3×D3—122no—
246C2×C3×D3—242no—
247C2×C3×D3—242no—
248D32—24no—
249D32—42no—
250D32—64no—
251D32—64no—
252D32—124no—
253D32—124no—
254D32—124no—
255D32—124no—
256D32—122no—
257D32—122no—
258#157—64no—
259C32×C4—22no—
260C32×C4—21no—
261C32×C4—41no—
262C32×C4—42no—
263C32×C4—81no—
264C32×C4—121no—
265C3⋊Dic3—21no—
266C3×Dic3—41no—
267C3×Dic3—41no—
268C3×Dic3—62no—
269C3×Dic3—81no—
270C3×Dic3—121no—
271C22×D4—271no—
272C22×D4—272no—
273C2×C22⋊C4—91no—
274C2×C22⋊C4—272no—
275#95—271no—
276C33#39211yescharacteristic, complemented, Sylow
277C23×C3—61no—
278C23×C3—61no—
279C23×C3—92no—
280C23×C3—91no—
281C23×C3—121no—
282C23×C3—182no—
283C23×C3—181no—
284C23×C3—182no—
285C23×C3—182no—
286C22×D3—21no—
287C22×D3—62no—
288C22×D3—61no—
289C22×D3—61no—
290C22×D3—62no—
291C22×D3—62no—
292C22×D3—62no—
293C22×D3—94no—
294C22×D3—122no—
295C22×D3—122no—
296C22×D3—122no—
297C22×D3—184no—
298C22×D3—182no—
299C22×D3—184no—
300C22×D3—182no—
301C22×D3—182no—
302C22×D3—182no—
303C22×D3—182no—
304C22×D3—362no—
305C22×D3—362no—
306C22×D3—362no—
307C22×D3—362no—
308C3×D4—184no—
309C2×C3×C4—21no—
310C2×C3×C4—21no—
311C2×C3×C4—41no—
312C2×C3×C4—41no—
313C2×C3×C4—61no—
314C2×C3×C4—61no—
315C2×C3×C4—91no—
316C2×C3×C4—121no—
317C2×C3×C4—182no—
318C2×C3×C4—182no—
319C2×C3×C4GO+(4,𝔽2)11yescharacteristic, complemented
320#67—188no—
321#67—1816no—
322C2×Dic3—21no—
323C2×Dic3—61no—
324C2×Dic3—61no—
325C2×Dic3—94no—
326C2×Dic3—182no—
327C2×Dic3—182no—
328D12—64no—
329D12—64no—
330D12—124no—
331D12—184no—
332C4×D3—64no—
333C4×D3—122no—
334C4×D3—122no—
335C2×C32—21no—
336C2×C32—21no—
337C2×C32—21no—
338C2×C32—21no—
339C2×C32—21no—
340C2×C32—21no—
341C2×C32—41no—
342C2×C32—41no—
343C2×C32—41no—
344C2×C32—41no—
345C2×C32—41no—
346C2×C32—41no—
347C2×C32—122no—
348C2×C32#93011yescharacteristic
349C2×C32#93711yescharacteristic
350C2×C32#94011yescharacteristic
351C3⋊D3—44no—
352C3⋊D3#88212yes—
353C3⋊D3#88412yes—
354C3×D3—42no—
355C3×D3—42no—
356C3×D3—62no—
357C3×D3—62no—
358C3×D3—62no—
359C3×D3—62no—
360C3×D3—82no—
361C3×D3—124no—
362C3×D3—124no—
363C3×D3—244no—
364C24—91no—
365C24—272no—
366C2×D4—181no—
367C2×D4—541no—
368C2×D4—544no—
369C2×D4—544no—
370C2×D4—544no—
371C2×D4—544no—
372C22×C4—91no—
373C22×C4—91no—
374C22×C4—272no—
375C22⋊C4—61no—
376C22⋊C4—181no—
377C22⋊C4—544no—
378C22×C3—21no—
379C22×C3—21no—
380C22×C3—41no—
381C22×C3—41no—
382C22×C3—62no—
383C22×C3—62no—
384C22×C3—91no—
385C22×C3—91no—
386C22×C3—92no—
387C22×C3—92no—
388C22×C3—122no—
389C22×C3—121no—
390C22×C3—121no—
391C22×C3—121no—
392C22×C3—121no—
393C22×C3—184no—
394C22×C3—184no—
395C22×C3—241no—
396C22×C3—241no—
397C22×C3—362no—
398C22×C3—362no—
399C22×C3—362no—
400C22×C3C2×GO+(4,𝔽2)11yescharacteristic
401C2×D3—22no—
402C2×D3—41no—
403C2×D3—41no—
404C2×D3—62no—
405C2×D3—62no—
406C2×D3—62no—
407C2×D3—61no—
408C2×D3—61no—
409C2×D3—61no—
410C2×D3—61no—
411C2×D3—62no—
412C2×D3—64no—
413C2×D3—64no—
414C2×D3—124no—
415C2×D3—124no—
416C2×D3—124no—
417C2×D3—122no—
418C2×D3—122no—
419C2×D3—122no—
420C2×D3—122no—
421C2×D3—184no—
422C2×D3—184no—
423C2×D3—242no—
424C2×D3—242no—
425C2×D3—364no—
426C2×D3—364no—
427C2×D3—364no—
428C2×D3—364no—
429C2×D3—362no—
430C2×D3—362no—
431C3×C4—22no—
432C3×C4—21no—
433C3×C4—41no—
434C3×C4—42no—
435C3×C4—81no—
436C3×C4—121no—
437C3×C4—121no—
438C3×C4—181no—
439C3×C4—241no—
440Dic3—41no—
441Dic3—62no—
442Dic3—121no—
443Dic3—184no—
444C32—21no—
445C32—21no—
446C32—41no—
447C32—41no—
448C32#496011yescharacteristic, complemented
449C23—61no—
450C23—92no—
451C23—91no—
452C23—181no—
453C23—182no—
454C23—182no—
455C23—182no—
456C23—274no—
457C23—542no—
458C23—542no—
459C23—542no—
460C23—542no—
461C23—542no—
462D4—184no—
463D4—544no—
464D4—5416no—
465C2×C4—61no—
466C2×C4—91no—
467C2×C4—182no—
468C2×C4—182no—
469C2×C4—274no—
470C2×C4—542no—
471C2×C4—542no—
472C2×C4D32⋊D311yescharacteristic, complemented
473C2×C3—21no—
474C2×C3—21no—
475C2×C3—21no—
476C2×C3—21no—
477C2×C3—21no—
478C2×C3—21no—
479C2×C3—41no—
480C2×C3—41no—
481C2×C3—41no—
482C2×C3—41no—
483C2×C3—41no—
484C2×C3—41no—
485C2×C3—92no—
486C2×C3—92no—
487C2×C3—122no—
488C2×C3—122no—
489C2×C3—242no—
490C2×C3—364no—
491C2×C3#6423011yescharacteristic
492C2×C3#6423111yescharacteristic
493C2×C3#6424111yescharacteristic
494D3—42no—
495D3—62no—
496D3—62no—
497D3—62no—
498D3—62no—
499D3—124no—
500D3—124no—
501D3—244no—
502C22—62no—
503C22—91no—
504C22—91no—
505C22—92no—
506C22—92no—
507C22—121no—
508C22—121no—
509C22—184no—
510C22—184no—
511C22—362no—
512C22—362no—
513C22—362no—
514C22—544no—
515C22—544no—
516C22—1082no—
517C22—1082no—
518C22C2×D32⋊D311yescharacteristic
519C4—21no—
520C4—121no—
521C4—181no—
522C4—544no—
523C3—21no—
524C3—21no—
525C3—41no—
526C3—41no—
527C3#1059345511yescharacteristic, complemented
528C2—92no—
529C2—92no—
530C2—122no—
531C2—364no—
532C2#1172431411yescharacteristic
533C2#1172431511yescharacteristic
534C2#1172433011yescharacteristic
subgroupG/H
center, FrattiniC22C2×D32⋊D3
commutator, cosocleC22×C3×C3⋊D3C23
FittingC2×C33×C4D4
socleC22×C33C2×D4
hypercenterC2×C4D32⋊D3
hypocenterC33#392
LCSG ▷ C22×C3×C3⋊D3 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C22×D32⋊D3C2semidirect
2. C2×C4⋊#1113C2semidirect
3. C2×#88932C2semidirect
4. C3×#63921C2semidirect
5. #11723106C2semidirect
26. #63921D3semidirect
36. C2×#1113D4semidirect
40. C2×C3×D32D4semidirect
43. C2×C3⋊#157D4semidirect
45. C2×C33×C4D4semidirect
46. C2×C3×C3⋊Dic3D4semidirect
140. C2×D32D12semidirect
150. C2×C32×C4#67semidirect
151. C2×C3⋊Dic3#67semidirect
276. C33#392semidirect
319. C2×C3×C4GO+(4,𝔽2)semidirect
448. C32#4960semidirect
472. C2×C4D32⋊D3semidirect
527. C3#10593455semidirect

Maximal subgroups

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

subgroupindex
C22×D32⋊D32
C2×C4⋊#11132
C2×#889322
C3×#639212
#117231062
#105934553
#49609