Finite Group Databasebeta

← #13140247

Subgroups — #13140247

Order 1728.

1203 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 1203 conjugacy classes of subgroups — too many to draw. Only its 131 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×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic, complemented
2C22×#6198C212yesmaximal, complemented
3C2×#3752⋊C3C211yesmaximal, characteristic, complemented
4C2×#88450C212yesmaximal, complemented
5#11720574C211yesmaximal, characteristic, complemented
6#10591395—31nomaximal
7#10591391—31nomaximal
8C23×(C3⋊D3)⋊C3C2211yescharacteristic
9C23×(C3⋊D3)⋊C3C2212yes—
10C22×C4×H(3,𝔽3)C2211yescharacteristic
11C22×#1084C2212yes—
12C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
13C2×C4⋊((C3⋊D3)⋊C3)—24no—
14C2×C4⋊((C3⋊D3)⋊C3)—22no—
15C2×#6198—24no—
16C2×#6198—24no—
17C2×#6198—24no—
18#3752⋊C3—22no—
19#88450—24no—
20C22×C4⋊(C3⋊D3)D311yescharacteristic, complemented
21C22×C3×D12—31no—
22C22×#626—32no—
23C22×#626—32no—
24C2×#3752D311yescharacteristic, complemented
25C2×C3×(C2×D3)⋊C4—31no—
26C2×#3722—32no—
27C2×(C2×D3)⋊Dic3—32no—
28C3×C3⋊#95—31no—
29C3×C3⋊#95—31no—
30C3×C3⋊#95—61no—
31C23×H(3,𝔽3)C2311yescharacteristic
32C22×(C3⋊D3)⋊C3—22no—
33C22×(C3⋊D3)⋊C3—22no—
34C22×(C3⋊D3)⋊C3—22no—
35C22×(C3⋊D3)⋊C3—22no—
36C22×(C3⋊D3)⋊C3—22no—
37C22×(C3⋊D3)⋊C3—22no—
38C22×(C3⋊D3)⋊C3—22no—
39C22×(C3⋊D3)⋊C3—21no—
40C22×(C3⋊D3)⋊C3—22no—
41C22×(C3⋊D3)⋊C3—21no—
42C22×(C3⋊D3)⋊C3D412yescomplemented
43C22×(C3⋊D3)⋊C3D414yescomplemented
44C2×C4×H(3,𝔽3)—22no—
45C2×C4×H(3,𝔽3)D412yescomplemented
46C2×#1084—22no—
47C2×#1084—22no—
48C2×#1084D414yescomplemented
49C2×(C3⋊Dic3)⋊C3—22no—
50C2×(C3⋊Dic3)⋊C3—21no—
51C4⋊((C3⋊D3)⋊C3)—28no—
52#6198—216no—
53#5223—91no—
54#4960—91no—
55#4671—91no—
56C23×C3⋊D3C2×D311yescharacteristic
57C23×C3⋊D3C2×D312yes—
58C23×C3×D3—31no—
59C23×C3×D3—32no—
60C22×C32×C4—21no—
61C22×C32×C4C2×D311yescharacteristic
62C22×C32×C4C2×D311yescharacteristic
63C22×C3⋊Dic3C2×D311yescharacteristic
64C2×C4⋊(C3⋊D3)—24no—
65C2×C4⋊(C3⋊D3)—22no—
66C22×C3×Dic3—32no—
67C22×C3×Dic3—32no—
68C22×C3×Dic3—31no—
69C22×C3×Dic3—62no—
70C2×C3×D12—64no—
71C2×C3×D12—62no—
72C2×#626—64no—
73C2×#626—64no—
74C2×#626—64no—
75C2×#626—64no—
76C2×#626—64no—
77C2×#626—64no—
78#3752—22no—
79C3×(C2×D3)⋊C4—62no—
80#3722—64no—
81(C2×D3)⋊Dic3—64no—
82C22×H(3,𝔽3)C2×D411yescharacteristic
83C22×H(3,𝔽3)C2×D412yes—
84C22×H(3,𝔽3)C2×D412yes—
85C22×H(3,𝔽3)C2×D411yescharacteristic
86C22×H(3,𝔽3)#4111yescharacteristic
87C2×(C3⋊D3)⋊C3—24no—
88C2×(C3⋊D3)⋊C3—24no—
89C2×(C3⋊D3)⋊C3—24no—
90C2×(C3⋊D3)⋊C3—24no—
91C2×(C3⋊D3)⋊C3—22no—
92C2×(C3⋊D3)⋊C3—42no—
93C2×(C3⋊D3)⋊C3—41no—
94C2×(C3⋊D3)⋊C3—42no—
95C2×(C3⋊D3)⋊C3—42no—
96C2×(C3⋊D3)⋊C3—42no—
97C2×(C3⋊D3)⋊C3—41no—
98C2×(C3⋊D3)⋊C3—42no—
99C4×H(3,𝔽3)—22no—
100#1084—24no—
101(C3⋊Dic3)⋊C3—41no—
102C22×C3×D4—91no—
103C22×#67—92no—
104C22×#67—92no—
105C22×D12—31no—
106C22×D12—91no—
107C22×D12—92no—
108C2×C3×C22⋊C4—91no—
109C2×#217—92no—
110C2×(C2×D3)⋊C4—31no—
111C2×(C2×D3)⋊C4—91no—
112C2×(C2×D3)⋊C4—92no—
113C2×(C2×D3)⋊C4—92no—
114C3×#95—91no—
115C3⋊#95—31no—
116C3⋊#95—31no—
117C3⋊#95—61no—
118C23×C32—21no—
119C23×C32C22×D311yescharacteristic
120C23×C32C22×D311yescharacteristic
121C22×C3⋊D3—21no—
122C22×C3⋊D3—22no—
123C22×C3⋊D3—22no—
124C22×C3⋊D3—22no—
125C22×C3⋊D3—22no—
126C22×C3⋊D3—22no—
127C22×C3⋊D3—22no—
128C22×C3⋊D3—21no—
129C22×C3⋊D3—22no—
130C22×C3⋊D3—22no—
131C22×C3⋊D3D1212yescomplemented
132C22×C3⋊D3#6714yescomplemented
133C22×C3×D3—32no—
134C22×C3×D3—34no—
135C22×C3×D3—62no—
136C22×C3×D3—62no—
137C22×C3×D3—62no—
138C22×C3×D3—62no—
139C22×C3×D3—62no—
140C22×C3×D3—62no—
141C22×C3×D3—61no—
142C22×C3×D3—61no—
143C22×C3×D3—62no—
144C22×C3×D3—62no—
145C2×C32×C4—22no—
146C2×C32×C4—22no—
147C2×C32×C4—22no—
148C2×C32×C4—42no—
149C2×C32×C4#6712yescomplemented
150C2×C32×C4#6712yescomplemented
151C2×C3⋊Dic3—22no—
152C2×C3⋊Dic3—21no—
153C4⋊(C3⋊D3)—28no—
154C2×C3×Dic3—34no—
155C2×C3×Dic3—34no—
156C2×C3×Dic3—64no—
157C2×C3×Dic3—62no—
158C2×C3×Dic3—62no—
159C2×C3×Dic3—62no—
160C2×C3×Dic3—62no—
161C2×C3×Dic3—62no—
162C2×C3×Dic3—61no—
163C2×C3×Dic3—122no—
164C2×C3×Dic3—122no—
165C3×D12—68no—
166#626—616no—
167#626—616no—
168#392—271noSylow
169C2×H(3,𝔽3)#12012yes—
170C2×H(3,𝔽3)#12011yescharacteristic
171C2×H(3,𝔽3)#12112yes—
172C2×H(3,𝔽3)#12111yescharacteristic
173C2×H(3,𝔽3)#12711yescharacteristic
174(C3⋊D3)⋊C3—44no—
175(C3⋊D3)⋊C3—42no—
176C24×C3—92no—
177C24×C3—91no—
178C23×D3—31no—
179C23×D3—32no—
180C23×D3—92no—
181C23×D3—91no—
182C2×C3×D4—182no—
183C2×C3×D4—184no—
184C22×C3×C4—31no—
185C22×C3×C4—31no—
186C22×C3×C4—61no—
187C22×C3×C4—92no—
188C22×C3×C4—91no—
189C22×C3×C4D3211yescharacteristic
190C2×#67—184no—
191C2×#67—184no—
192C2×#67—184no—
193C2×#67—184no—
194C2×#67—184no—
195C2×#67—184no—
196C22×Dic3—32no—
197C22×Dic3—32no—
198C22×Dic3—31no—
199C22×Dic3—62no—
200C22×Dic3—91no—
201C2×D12—64no—
202C2×D12—62no—
203C2×D12—182no—
204C2×D12—184no—
205C2×D12—184no—
206C2×D12—184no—
207C2×D12—184no—
208C3×C22⋊C4—182no—
209#217—184no—
210(C2×D3)⋊C4—62no—
211(C2×D3)⋊C4—182no—
212(C2×D3)⋊C4—184no—
213(C2×D3)⋊C4—184no—
214C22×C32—21no—
215C22×C32—21no—
216C22×C32—22no—
217C22×C32—21no—
218C22×C32—22no—
219C22×C32C2×D1211yescharacteristic
220C22×C32#23511yescharacteristic
221C22×C32D3×D411yescharacteristic
222C22×C32D3×D412yes—
223C22×C32D3×D412yes—
224C22×C32D3×D412yes—
225C22×C32#23711yescharacteristic
226C22×C32C2×#6711yescharacteristic
227C22×C32C2×#6711yescharacteristic
228C22×C32C2×#6712yes—
229C2×C3⋊D3—24no—
230C2×C3⋊D3—22no—
231C2×C3⋊D3—24no—
232C2×C3⋊D3—24no—
233C2×C3⋊D3—24no—
234C2×C3⋊D3—42no—
235C2×C3⋊D3—42no—
236C2×C3⋊D3—41no—
237C2×C3⋊D3—42no—
238C2×C3⋊D3—42no—
239C2×C3⋊D3—41no—
240C2×C3⋊D3—42no—
241C2×C3×D3—64no—
242C2×C3×D3—64no—
243C2×C3×D3—64no—
244C2×C3×D3—64no—
245C2×C3×D3—62no—
246C2×C3×D3—122no—
247C2×C3×D3—122no—
248C2×C3×D3—122no—
249C2×C3×D3—122no—
250C2×C3×D3—122no—
251C2×C3×D3—121no—
252C2×C3×D3—121no—
253C32×C4—22no—
254C32×C4—22no—
255C32×C4—42no—
256C3⋊Dic3—41no—
257C3×Dic3—64no—
258C3×Dic3—64no—
259C3×Dic3—124no—
260C3×Dic3—121no—
261C22×D4—91no—
262C22×D4—272no—
263C2×C22⋊C4—91no—
264C2×C22⋊C4—272no—
265#95—91no—
266H(3,𝔽3)#39211yescharacteristic, complemented, Sylow
267C23×C3—31no—
268C23×C3—31no—
269C23×C3—61no—
270C23×C3—92no—
271C23×C3—94no—
272C23×C3—182no—
273C23×C3—181no—
274C23×C3—182no—
275C23×C3—182no—
276C23×C3—182no—
277C23×C3—182no—
278C23×C3—182no—
279C23×C3—182no—
280C23×C3—181no—
281C23×C3—182no—
282C23×C3C2×D3211yescharacteristic
283C22×D3—32no—
284C22×D3—34no—
285C22×D3—62no—
286C22×D3—61no—
287C22×D3—62no—
288C22×D3—62no—
289C22×D3—62no—
290C22×D3—62no—
291C22×D3—61no—
292C22×D3—62no—
293C22×D3—62no—
294C22×D3—62no—
295C22×D3—92no—
296C22×D3—94no—
297C22×D3—182no—
298C22×D3—181no—
299C22×D3—182no—
300C22×D3—182no—
301C22×D3—182no—
302C22×D3—182no—
303C22×D3—182no—
304C22×D3—182no—
305C22×D3—182no—
306C22×D3—181no—
307C3×D4—188no—
308C2×C3×C4—22no—
309C2×C3×C4—32no—
310C2×C3×C4—32no—
311C2×C3×C4—62no—
312C2×C3×C4—62no—
313C2×C3×C4—62no—
314C2×C3×C4—94no—
315C2×C3×C4—122no—
316C2×C3×C4—182no—
317C2×C3×C4—182no—
318C2×C3×C4—182no—
319C2×C3×C4—181no—
320C2×C3×C4#62512yes—
321#67—1816no—
322#67—1816no—
323C2×Dic3—34no—
324C2×Dic3—34no—
325C2×Dic3—64no—
326C2×Dic3—62no—
327C2×Dic3—62no—
328C2×Dic3—62no—
329C2×Dic3—62no—
330C2×Dic3—62no—
331C2×Dic3—61no—
332C2×Dic3—122no—
333C2×Dic3—122no—
334C2×Dic3—181no—
335C2×Dic3—182no—
336D12—68no—
337D12—188no—
338D12—1816no—
339C2×C32—22no—
340C2×C32—21no—
341C2×C32—21no—
342C2×C32—22no—
343C2×C32—21no—
344C2×C32#88212yes—
345C2×C32#88412yes—
346C2×C32#93011yescharacteristic
347C2×C32#93712yes—
348C2×C32#93711yescharacteristic
349C2×C32#93811yescharacteristic
350C2×C32#93912yes—
351C2×C32#94011yescharacteristic
352C2×C32#94011yescharacteristic
353C2×C32#95311yescharacteristic
354C3⋊D3—42no—
355C3⋊D3—44no—
356C3×D3—124no—
357C3×D3—122no—
358C24—91no—
359C24—92no—
360C2×D4—184no—
361C2×D4—182no—
362C2×D4—544no—
363C2×D4—544no—
364C2×D4—544no—
365C22×C4—92no—
366C22×C4—91no—
367C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
368C22⋊C4—182no—
369C22⋊C4—544no—
370C22×C3—31no—
371C22×C3—31no—
372C22×C3—31no—
373C22×C3—31no—
374C22×C3—32no—
375C22×C3—32no—
376C22×C3—31no—
377C22×C3—31no—
378C22×C3—32no—
379C22×C3—32no—
380C22×C3—61no—
381C22×C3—61no—
382C22×C3—62no—
383C22×C3—61no—
384C22×C3—62no—
385C22×C3—184no—
386C22×C3—184no—
387C22×C3—184no—
388C22×C3—184no—
389C22×C3—182no—
390C22×C3—361no—
391C22×C3—362no—
392C22×C3—362no—
393C22×C3—362no—
394C22×C3—362no—
395C22×C3—361no—
396C22×C3—362no—
397C22×C3#379511yescharacteristic
398C22×C3D3×D1211yescharacteristic
399C22×C3#380212yes—
400C22×C3C2×#62511yescharacteristic
401C22×C3D3×#6712yes—
402C2×D3—64no—
403C2×D3—62no—
404C2×D3—64no—
405C2×D3—64no—
406C2×D3—64no—
407C2×D3—122no—
408C2×D3—122no—
409C2×D3—122no—
410C2×D3—122no—
411C2×D3—121no—
412C2×D3—122no—
413C2×D3—121no—
414C2×D3—184no—
415C2×D3—184no—
416C2×D3—184no—
417C2×D3—184no—
418C2×D3—182no—
419C2×D3—361no—
420C2×D3—362no—
421C2×D3—362no—
422C2×D3—362no—
423C2×D3—361no—
424C2×D3—362no—
425C2×D3—362no—
426C3×C4—22no—
427C3×C4—62no—
428C3×C4—62no—
429C3×C4—122no—
430C3×C4—184no—
431C3×C4—361no—
432Dic3—64no—
433Dic3—64no—
434Dic3—124no—
435Dic3—121no—
436Dic3—361no—
437C32—21no—
438C32#496011yescharacteristic, complemented
439C32#522311yescharacteristic, complemented
440C23—92no—
441C23—94no—
442C23—182no—
443C23—181no—
444C23—182no—
445C23—182no—
446C23—182no—
447C23—182no—
448C23—181no—
449C23—182no—
450C23—182no—
451C23—182no—
452C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
453D4—188no—
454D4—5416no—
455C2×C4—22no—
456C2×C4—94no—
457C2×C4—182no—
458C2×C4—182no—
459C2×C4—182no—
460C2×C4—181no—
461C2×C4#619812yescomplemented
462C2×C3—31no—
463C2×C3—31no—
464C2×C3—32no—
465C2×C3—32no—
466C2×C3—31no—
467C2×C3—31no—
468C2×C3—31no—
469C2×C3—31no—
470C2×C3—32no—
471C2×C3—32no—
472C2×C3—61no—
473C2×C3—62no—
474C2×C3—61no—
475C2×C3—61no—
476C2×C3—62no—
477C2×C3—362no—
478C2×C3—364no—
479C2×C3#6388512yes—
480C2×C3#6390811yescharacteristic
481C2×C3#6391411yescharacteristic
482C2×C3#6392212yes—
483C2×C3#6397611yescharacteristic
484D3—124no—
485D3—122no—
486D3—364no—
487D3—362no—
488C22—184no—
489C22—184no—
490C22—182no—
491C22—184no—
492C22—184no—
493C22—361no—
494C22—362no—
495C22—362no—
496C22—362no—
497C22—362no—
498C22—361no—
499C22—362no—
500C22#8864811yescharacteristic
501C22#8865411yescharacteristic
502C22#8865512yes—
503C22C2×#619811yescharacteristic
504C22#8867612yes—
505C4—22no—
506C4—184no—
507C4—361no—
508C3—31no—
509C3—31no—
510C3—61no—
511C3#1059138911yescharacteristic
512C2—364no—
513C2—362no—
514C2#1172112612yes—
515C2#1172114111yescharacteristic
516C2#1172114411yescharacteristic
517C2#1172116212yes—
518C2#1172131611yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC23×H(3,𝔽3)C23
Frattini, socleC23×C3C2×D32
FittingC22×C4×H(3,𝔽3)C22
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#392
LCSG ▷ C23×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×C4⋊((C3⋊D3)⋊C3)C2semidirect
2. C22×#6198C2semidirect
3. C2×#3752⋊C3C2semidirect
4. C2×#88450C2semidirect
5. #11720574C2semidirect
20. C22×C4⋊(C3⋊D3)D3semidirect
24. C2×#3752D3semidirect
42. C22×(C3⋊D3)⋊C3D4semidirect
43. C22×(C3⋊D3)⋊C3D4semidirect
45. C2×C4×H(3,𝔽3)D4semidirect
48. C2×#1084D4semidirect
131. C22×C3⋊D3D12semidirect
132. C22×C3⋊D3#67semidirect
149. C2×C32×C4#67semidirect
150. C2×C32×C4#67semidirect
266. H(3,𝔽3)#392semidirect
438. C32#4960semidirect
439. C32#5223semidirect
461. C2×C4#6198semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊((C3⋊D3)⋊C3)2
C22×#61982
C2×#3752⋊C32
C2×#884502
#117205742
#105913953
#105913913