Finite Group Databasebeta

← #13140265

Subgroups — #13140265

Order 1728.

1259 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 1259 conjugacy classes of subgroups — too many to draw. Only its 125 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×#6200C212yesmaximal, complemented
2C22×#6198C211yesmaximal, characteristic, complemented
3C2×#88542C211yesmaximal, characteristic, complemented
4C2×#3752⋊C3C212yesmaximal, complemented
5#11720334C211yesmaximal, characteristic, complemented
6#10591393—32nomaximal
7C23×#272C2211yescharacteristic
8C23×(C3⋊D3)⋊C3C2212yes—
9C22×C4×H(3,𝔽3)C2211yescharacteristic
10C22×#1084C2211yescharacteristic
11C22×(C3⋊Dic3)⋊C3C2212yes—
12C2×#6200—24no—
13C2×#6200—24no—
14C2×#6200—24no—
15C2×#6198—24no—
16C2×#6198—22no—
17#88542—22no—
18#3752⋊C3—24no—
19C22×#626—32no—
20C22×#626—32no—
21C22×#625—32no—
22C2×#3752D312yescomplemented
23C2×C3×(C2×D3)⋊C4—32no—
24C2×C3×(C2×D3)⋊C4—32no—
25C2×C3×(C2×D3)⋊C4—61no—
26#63579—32no—
27C23×H(3,𝔽3)C2311yescharacteristic
28C22×#272—21no—
29C22×#272—22no—
30C22×#272—22no—
31C22×#272—21no—
32C22×#272D412yescomplemented
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)⋊C3D414yescomplemented
40C2×C4×H(3,𝔽3)—21no—
41C2×C4×H(3,𝔽3)—22no—
42C2×#1084—22no—
43C2×#1084D412yescomplemented
44C2×(C3⋊Dic3)⋊C3—22no—
45C2×(C3⋊Dic3)⋊C3—22no—
46C2×(C3⋊Dic3)⋊C3D414yescomplemented
47#6200—216no—
48#6198—28no—
49#4960—91no—
50#4960—92no—
51C23×C3⋊D3C2×D312yes—
52C23×C3×D3—32no—
53C23×C3×D3—32no—
54C23×C3×D3—61no—
55C22×C32×C4—21no—
56C22×C32×C4C2×D312yes—
57C22×C3⋊Dic3C2×D312yes—
58C22×C3×Dic3—32no—
59C22×C3×Dic3—32no—
60C22×C3×Dic3—61no—
61C2×#626—64no—
62C2×#626—64no—
63C2×#626—64no—
64C2×#626—64no—
65C2×#626—64no—
66C2×#626—64no—
67C2×#625—64no—
68C2×#625—64no—
69C2×#625—64no—
70#3752—24no—
71C3×(C2×D3)⋊C4—64no—
72C3×(C2×D3)⋊C4—64no—
73C3×(C2×D3)⋊C4—64no—
74C22×H(3,𝔽3)C2×D412yes—
75C22×H(3,𝔽3)C2×D411yescharacteristic
76C22×H(3,𝔽3)C2×D411yescharacteristic
77C22×H(3,𝔽3)C2×D412yes—
78C22×H(3,𝔽3)#4111yescharacteristic
79C2×#272—24no—
80C2×#272—22no—
81C2×#272—41no—
82C2×#272—42no—
83C2×#272—41no—
84C2×(C3⋊D3)⋊C3—24no—
85C2×(C3⋊D3)⋊C3—24no—
86C2×(C3⋊D3)⋊C3—24no—
87C2×(C3⋊D3)⋊C3—42no—
88C2×(C3⋊D3)⋊C3—42no—
89C2×(C3⋊D3)⋊C3—42no—
90C2×(C3⋊D3)⋊C3—42no—
91C4×H(3,𝔽3)—41no—
92#1084—22no—
93(C3⋊Dic3)⋊C3—24no—
94C22×#67—92no—
95C22×#67—92no—
96C22×#67—92no—
97C22×D12—91no—
98C22×D12—92no—
99C2×C3×C22⋊C4—92no—
100C2×C3×C22⋊C4—91no—
101C2×(C2×D3)⋊C4—32no—
102C2×(C2×D3)⋊C4—32no—
103C2×(C2×D3)⋊C4—61no—
104C2×(C2×D3)⋊C4—92no—
105C3⋊#95—91no—
106C3⋊#95—92no—
107C23×C32—21no—
108C23×C32C22×D312yes—
109C22×C3⋊D3—22no—
110C22×C3⋊D3—22no—
111C22×C3⋊D3—22no—
112C22×C3⋊D3—22no—
113C22×C3⋊D3—22no—
114C22×C3⋊D3—22no—
115C22×C3⋊D3D1214yescomplemented
116C22×C3×D3—34no—
117C22×C3×D3—34no—
118C22×C3×D3—62no—
119C22×C3×D3—62no—
120C22×C3×D3—62no—
121C22×C3×D3—62no—
122C22×C3×D3—62no—
123C22×C3×D3—62no—
124C22×C3×D3—62no—
125C22×C3×D3—62no—
126C22×C3×D3—62no—
127C22×C3×D3—62no—
128C22×C3×D3—62no—
129C22×C3×D3—62no—
130C22×C3×D3—62no—
131C22×C3×D3—62no—
132C22×C3×D3—62no—
133C22×C3×D3—122no—
134C22×C3×D3—122no—
135C2×C32×C4—22no—
136C2×C32×C4—22no—
137C2×C32×C4—22no—
138C2×C32×C4—22no—
139C2×C32×C4—42no—
140C2×C3⋊Dic3—22no—
141C2×C3⋊Dic3—22no—
142C2×C3⋊Dic3#6714yescomplemented
143C2×C3×Dic3—34no—
144C2×C3×Dic3—34no—
145C2×C3×Dic3—62no—
146C2×C3×Dic3—62no—
147C2×C3×Dic3—62no—
148C2×C3×Dic3—62no—
149C2×C3×Dic3—62no—
150C2×C3×Dic3—122no—
151#626—616no—
152#626—616no—
153#625—616no—
154#392—271noSylow
155C2×H(3,𝔽3)#12011yescharacteristic
156C2×H(3,𝔽3)#12012yes—
157C2×H(3,𝔽3)#12111yescharacteristic
158C2×H(3,𝔽3)#12112yes—
159C2×H(3,𝔽3)#12711yescharacteristic
160#272—42no—
161(C3⋊D3)⋊C3—44no—
162C24×C3—92no—
163C24×C3—91no—
164C23×D3—32no—
165C23×D3—32no—
166C23×D3—61no—
167C23×D3—92no—
168C22×C3×C4—32no—
169C22×C3×C4—61no—
170C22×C3×C4—92no—
171C22×C3×C4—91no—
172C22×C3×C4D3211yescharacteristic
173C2×#67—184no—
174C2×#67—184no—
175C2×#67—184no—
176C2×#67—184no—
177C2×#67—184no—
178C2×#67—184no—
179C2×#67—184no—
180C2×#67—184no—
181C2×#67—184no—
182C22×Dic3—32no—
183C22×Dic3—32no—
184C22×Dic3—61no—
185C22×Dic3—92no—
186C2×D12—182no—
187C2×D12—184no—
188C2×D12—184no—
189C2×D12—184no—
190C2×D12—184no—
191C3×C22⋊C4—182no—
192C3×C22⋊C4—184no—
193(C2×D3)⋊C4—64no—
194(C2×D3)⋊C4—64no—
195(C2×D3)⋊C4—64no—
196(C2×D3)⋊C4—184no—
197C22×C32—21no—
198C22×C32—22no—
199C22×C32—21no—
200C22×C32—22no—
201C22×C32—21no—
202C22×C32C2×D1212yes—
203C22×C32#23512yes—
204C22×C32D3×D412yes—
205C22×C32D3×D412yes—
206C22×C32D3×D412yes—
207C22×C32D3×D412yes—
208C22×C32C2×#6712yes—
209C2×C3⋊D3—24no—
210C2×C3⋊D3—24no—
211C2×C3⋊D3—24no—
212C2×C3⋊D3—42no—
213C2×C3⋊D3—42no—
214C2×C3⋊D3—42no—
215C2×C3⋊D3—42no—
216C2×C3×D3—64no—
217C2×C3×D3—64no—
218C2×C3×D3—64no—
219C2×C3×D3—64no—
220C2×C3×D3—64no—
221C2×C3×D3—64no—
222C2×C3×D3—64no—
223C2×C3×D3—124no—
224C2×C3×D3—124no—
225C2×C3×D3—122no—
226C2×C3×D3—122no—
227C2×C3×D3—122no—
228C2×C3×D3—122no—
229C2×C3×D3—122no—
230C2×C3×D3—122no—
231C2×C3×D3—122no—
232C2×C3×D3—122no—
233C2×C3×D3—122no—
234C2×C3×D3—122no—
235C32×C4—42no—
236C32×C4—42no—
237C3⋊Dic3—24no—
238C3×Dic3—64no—
239C3×Dic3—64no—
240C3×Dic3—122no—
241C22×D4—271no—
242C22×D4—272no—
243C2×C22⋊C4—91no—
244C2×C22⋊C4—92no—
245#95—271no—
246H(3,𝔽3)#39211yescharacteristic, complemented, Sylow
247C23×C3—32no—
248C23×C3—61no—
249C23×C3—94no—
250C23×C3—92no—
251C23×C3—182no—
252C23×C3—182no—
253C23×C3—182no—
254C23×C3—182no—
255C23×C3—182no—
256C23×C3—182no—
257C23×C3—181no—
258C23×C3—182no—
259C23×C3—182no—
260C23×C3—181no—
261C23×C3C2×D3211yescharacteristic
262C22×D3—34no—
263C22×D3—34no—
264C22×D3—62no—
265C22×D3—62no—
266C22×D3—62no—
267C22×D3—62no—
268C22×D3—62no—
269C22×D3—62no—
270C22×D3—62no—
271C22×D3—62no—
272C22×D3—62no—
273C22×D3—62no—
274C22×D3—62no—
275C22×D3—62no—
276C22×D3—62no—
277C22×D3—62no—
278C22×D3—62no—
279C22×D3—94no—
280C22×D3—122no—
281C22×D3—122no—
282C22×D3—182no—
283C22×D3—182no—
284C22×D3—182no—
285C22×D3—182no—
286C22×D3—182no—
287C22×D3—182no—
288C2×C3×C4—22no—
289C2×C3×C4—21no—
290C2×C3×C4—62no—
291C2×C3×C4—62no—
292C2×C3×C4—62no—
293C2×C3×C4—62no—
294C2×C3×C4—94no—
295C2×C3×C4—92no—
296C2×C3×C4—122no—
297C2×C3×C4—182no—
298C2×C3×C4—182no—
299C2×C3×C4—182no—
300#67—1816no—
301#67—1816no—
302#67—1816no—
303C2×Dic3—34no—
304C2×Dic3—34no—
305C2×Dic3—62no—
306C2×Dic3—62no—
307C2×Dic3—62no—
308C2×Dic3—62no—
309C2×Dic3—62no—
310C2×Dic3—94no—
311C2×Dic3—122no—
312C2×Dic3—182no—
313C2×Dic3—182no—
314D12—188no—
315D12—1816no—
316C2×C32—22no—
317C2×C32—21no—
318C2×C32—22no—
319C2×C32—21no—
320C2×C32—21no—
321C2×C32#88212yes—
322C2×C32#88212yes—
323C2×C32#88412yes—
324C2×C32#88412yes—
325C2×C32#93012yes—
326C2×C32#93712yes—
327C2×C32#94012yes—
328C3⋊D3—44no—
329C3×D3—124no—
330C3×D3—124no—
331C3×D3—124no—
332C24—91no—
333C24—92no—
334C2×D4—544no—
335C2×D4—542no—
336C2×D4—544no—
337C2×D4—544no—
338C2×D4—544no—
339C22×C4—91no—
340C22×C4—92no—
341C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
342C22⋊C4—182no—
343C22⋊C4—184no—
344C22×C3—32no—
345C22×C3—32no—
346C22×C3—32no—
347C22×C3—32no—
348C22×C3—32no—
349C22×C3—32no—
350C22×C3—32no—
351C22×C3—61no—
352C22×C3—62no—
353C22×C3—61no—
354C22×C3—62no—
355C22×C3—61no—
356C22×C3—184no—
357C22×C3—184no—
358C22×C3—184no—
359C22×C3—182no—
360C22×C3—184no—
361C22×C3—362no—
362C22×C3—362no—
363C22×C3—362no—
364C22×C3—362no—
365C22×C3—362no—
366C22×C3—361no—
367C22×C3—361no—
368C22×C3#379811yescharacteristic
369C22×C3D3×D1212yes—
370C22×C3#380211yescharacteristic
371C22×C3D3×#6712yes—
372C22×C3#381111yescharacteristic
373C2×D3—64no—
374C2×D3—64no—
375C2×D3—64no—
376C2×D3—64no—
377C2×D3—64no—
378C2×D3—64no—
379C2×D3—64no—
380C2×D3—124no—
381C2×D3—122no—
382C2×D3—122no—
383C2×D3—122no—
384C2×D3—122no—
385C2×D3—122no—
386C2×D3—124no—
387C2×D3—122no—
388C2×D3—122no—
389C2×D3—122no—
390C2×D3—122no—
391C2×D3—122no—
392C2×D3—184no—
393C2×D3—184no—
394C2×D3—184no—
395C2×D3—362no—
396C2×D3—362no—
397C2×D3—362no—
398C2×D3—362no—
399C3×C4—41no—
400C3×C4—122no—
401C3×C4—122no—
402C3×C4—184no—
403C3×C4—182no—
404Dic3—64no—
405Dic3—64no—
406Dic3—122no—
407Dic3—184no—
408C32—21no—
409C32#496012yescomplemented
410C23—94no—
411C23—92no—
412C23—181no—
413C23—182no—
414C23—182no—
415C23—182no—
416C23—182no—
417C23—182no—
418C23—182no—
419C23—182no—
420C23—181no—
421C23—182no—
422C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
423D4—5416no—
424D4—548no—
425C2×C4—22no—
426C2×C4—21no—
427C2×C4—94no—
428C2×C4—92no—
429C2×C4—182no—
430C2×C4—182no—
431C2×C4—182no—
432C2×C3—32no—
433C2×C3—32no—
434C2×C3—32no—
435C2×C3—32no—
436C2×C3—32no—
437C2×C3—32no—
438C2×C3—32no—
439C2×C3—61no—
440C2×C3—62no—
441C2×C3—61no—
442C2×C3—62no—
443C2×C3—61no—
444C2×C3—364no—
445C2×C3—362no—
446C2×C3#6390612yes—
447C2×C3#6391211yescharacteristic
448C2×C3#6392212yes—
449C2×C3#6392311yescharacteristic
450C2×C3#6397911yescharacteristic
451D3—124no—
452D3—124no—
453D3—124no—
454D3—364no—
455C22—184no—
456C22—182no—
457C22—184no—
458C22—184no—
459C22—184no—
460C22—361no—
461C22—362no—
462C22—362no—
463C22—362no—
464C22—362no—
465C22—362no—
466C22—361no—
467C22#8865011yescharacteristic
468C22#8865412yes—
469C22#8865511yescharacteristic
470C22#8867612yes—
471C22#8867711yescharacteristic
472C4—41no—
473C4—184no—
474C4—182no—
475C3—32no—
476C3—61no—
477C3#1059139311yescharacteristic
478C2—362no—
479C2—364no—
480C2#1172114012yes—
481C2#1172114211yescharacteristic
482C2#1172116212yes—
483C2#1172116311yescharacteristic
484C2#1172131711yescharacteristic
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×#6200C2semidirect
2. C22×#6198C2semidirect
3. C2×#88542C2semidirect
4. C2×#3752⋊C3C2semidirect
5. #11720334C2semidirect
22. C2×#3752D3semidirect
32. C22×#272D4semidirect
39. C22×(C3⋊D3)⋊C3D4semidirect
43. C2×#1084D4semidirect
46. C2×(C3⋊Dic3)⋊C3D4semidirect
115. C22×C3⋊D3D12semidirect
142. C2×C3⋊Dic3#67semidirect
246. H(3,𝔽3)#392semidirect
409. C32#4960semidirect

Maximal subgroups

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

subgroupindex
C22×#62002
C22×#61982
C2×#885422
C2×#3752⋊C32
#117203342
#105913933