Finite Group Databasebeta

← #13140996

Subgroups — #13140996

Order 1728.

606 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 606 conjugacy classes of subgroups — too many to draw. Only its 75 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×#88649C211yesmaximal, characteristic
2#11721863C211yesmaximal, characteristic, complemented
3C2×#3755⋊C3C211yesmaximal, characteristic, complemented
4C2×#3756⋊C3C211yesmaximal, characteristic, complemented
5#11720323C211yesmaximal, characteristic, complemented
6#11720309C211yesmaximal, characteristic, complemented
7#11720308C211yesmaximal, characteristic, complemented
8#10591657—31nomaximal
9#10591655—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C2×C4×#272C2211yescharacteristic
12C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
13C2×#634⋊C3C2211yescharacteristic
14C2×#6200—21no—
15C2×((C3⋊D3)⋊C3)⋊C4—21no—
16#88649—22no—
17#88649—22no—
18#88542—21no—
19C4⋊#1084C2211yescharacteristic
20#88540—21no—
21#3755⋊C3—22no—
22#3756⋊C3—22no—
23C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
24C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
25C2×#3799—31no—
26C2×#3795—31no—
27C2×#3756D311yescharacteristic, complemented
28C2×#3755D311yescharacteristic, complemented
29C3×#946—31no—
30C3×#946—31no—
31C3×#946—61no—
32C2×C3×#216—31no—
33C2×C3×#215—31no—
34#63567—31no—
35#63565—31no—
36#63564—31no—
37#63562—31no—
38(C2×D3)⋊(C3⋊C8)—31no—
39(C2×D3)⋊(C3⋊C8)—31no—
40C22×#272D411yescharacteristic
41C22×(C3⋊D3)⋊C3—21no—
42Q8×H(3,𝔽3)—21no—
43Q8×H(3,𝔽3)—21no—
44C2×C4×H(3,𝔽3)—21no—
45C2×C4×H(3,𝔽3)C2311yescharacteristic
46C2×#1084—21no—
47C2×#1084D411yescharacteristic
48C4×#272—21no—
49C4×#272—21no—
50C2×(C3⋊Dic3)⋊C3—21no—
51C4⋊((C3⋊D3)⋊C3)—21no—
52C4⋊((C3⋊D3)⋊C3)D412yes—
53#634⋊C3—21no—
54#634⋊C3D412yes—
55#6200—24no—
56((C3⋊D3)⋊C3)⋊C4—24no—
57(C3⋊(C3⋊C8))⋊C3—21no—
58(C3⋊(C3⋊C8))⋊C3—21no—
59#5187—91no—
60#5173—91no—
61#5047—91no—
62C2×C32×Q8—21no—
63C2×C32×Q8C2×D311yescharacteristic
64C2×C32×Q8C2×D311yescharacteristic
65C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
66C2×#634C2×D311yescharacteristic
67C2×C3×D12—31no—
68C2×C3×C4×D3—31no—
69C2×C3×C4×D3—31no—
70C2×C3×C4×D3—61no—
71C2×C3×Dic6—31no—
72C2×#626—61no—
73C2×#625—61no—
74C2×#624—61no—
75C2×D3×Dic3—61no—
76#3799—62no—
77#3799—62no—
78#3795—62no—
79#3795—62no—
80#3756—22no—
81#3755—22no—
82C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
83C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
84C3×#216—62no—
85C3×#215—62no—
86C3×(C2×D3)⋊C4—61no—
87C3×(C2×D3)⋊C4—61no—
88C3×(C2×D3)⋊C4—62no—
89C3×C4⋊Dic3—31no—
90C3×C4⋊Dic3—31no—
91C3×C4⋊Dic3—61no—
92C3×Dic3⋊C4—61no—
93C3×Dic3⋊C4—61no—
94C3×Dic3⋊C4—62no—
95C2×C3×C3⋊C8—31no—
96C2×C3×C3⋊C8—31no—
97C22×H(3,𝔽3)C2×D411yescharacteristic
98C2×#272—21no—
99C2×#272—21no—
100C2×#272—21no—
101C2×(C3⋊D3)⋊C3—22no—
102C2×(C3⋊D3)⋊C3—41no—
103C2×(C3⋊D3)⋊C3—41no—
104C4×H(3,𝔽3)—41no—
105C4×H(3,𝔽3)C2×D411yescharacteristic
106C4×H(3,𝔽3)C2×D411yescharacteristic
107#1084—21no—
108#1084—41no—
109(C3⋊Dic3)⋊C3—22no—
110C2×#239—91no—
111C2×#237—91no—
112C2×#235—91no—
113C2×C3×Q16—91no—
114C2×C3×SD16—91no—
115C3×C22⋊Q8—91no—
116#946—31no—
117#946—31no—
118#946—61no—
119C2×#216—31no—
120C2×#216—91no—
121C2×#215—31no—
122C2×#215—91no—
123#835—91no—
124#832—91no—
125#830—91no—
126#821—91no—
127(C2×D3)⋊C8—91no—
128(C2×D3)⋊C8—91no—
129#816—91no—
130#810—91no—
131#809—91no—
132C22×C3⋊D3—21no—
133C22×C3×D3—31no—
134C22×C3×D3—31no—
135C22×C3×D3—61no—
136C22×C3×D3—61no—
137C32×Q8—21no—
138C32×Q8—21no—
139C32×Q8—21no—
140C32×Q8—21no—
141C32×Q8—41no—
142C32×Q8—41no—
143C2×C32×C4—22no—
144C2×C32×C4—21no—
145C2×C32×C4—21no—
146C2×C32×C4—21no—
147C2×C32×C4C22×D311yescharacteristic
148C2×C32×C4C22×D311yescharacteristic
149C2×C3⋊Dic3—21no—
150C4⋊(C3⋊D3)—21no—
151C4⋊(C3⋊D3)#6712yes—
152#634—21no—
153#634#6712yes—
154C2×C3×Dic3—31no—
155C2×C3×Dic3—31no—
156C2×C3×Dic3—61no—
157C2×C3×Dic3—61no—
158C2×C3×Dic3—62no—
159C2×C3×Dic3—61no—
160C2×C3×Dic3—61no—
161C3×D12—32no—
162C3×D12—61no—
163C3×C4×D3—61no—
164C3×C4×D3—61no—
165C3×C4×D3—61no—
166C3×C4×D3—61no—
167C3×C4×D3—121no—
168C3×C4×D3—121no—
169C3×Dic6—32no—
170C3×Dic6—61no—
171#626—64no—
172#625—64no—
173#624—64no—
174D3×Dic3—64no—
175C3⋊(C3⋊C8)—21no—
176C3⋊(C3⋊C8)—21no—
177C3×C3⋊C8—61no—
178C3×C3⋊C8—61no—
179#452—271noSylow
180C2×H(3,𝔽3)#12011yescharacteristic
181C2×H(3,𝔽3)#13511yescharacteristic
182C2×H(3,𝔽3)#13711yescharacteristic
183#272—41no—
184(C3⋊D3)⋊C3—42no—
185C2×C3×Q8—31no—
186C2×C3×Q8—31no—
187C2×C3×Q8—61no—
188C2×C3×Q8—91no—
189C2×C3×Q8D3211yescharacteristic
190C2×C3×D4—91no—
191C22×C3×C4—91no—
192C2×#67—181no—
193C2×#67—181no—
194C22×Dic3—181no—
195#239—182no—
196#239—182no—
197#237—182no—
198#237—182no—
199#235—182no—
200#235—182no—
201C2×D12—31no—
202C2×D12—91no—
203C2×D12—181no—
204C2×C4×D3—31no—
205C2×C4×D3—31no—
206C2×C4×D3—61no—
207C2×C4×D3—181no—
208C2×C4×D3—181no—
209C2×Dic6—31no—
210C2×Dic6—91no—
211C3×Q16—182no—
212C3×SD16—182no—
213C2×C3×C8—91no—
214C2×C3×C8—91no—
215C3×C4⋊C4—91no—
216C3×C4⋊C4—181no—
217C3×C22⋊C4—181no—
218#216—62no—
219#216—182no—
220#215—62no—
221#215—182no—
222(C2×D3)⋊C4—61no—
223(C2×D3)⋊C4—61no—
224(C2×D3)⋊C4—62no—
225C4⋊Dic3—31no—
226C4⋊Dic3—31no—
227C4⋊Dic3—61no—
228Dic3⋊C4—61no—
229Dic3⋊C4—61no—
230Dic3⋊C4—62no—
231C2×C3⋊C8—31no—
232C2×C3⋊C8—31no—
233C2×C3⋊C8—91no—
234C2×C3⋊C8—91no—
235C22×C32—21no—
236C22×C32D3×D411yescharacteristic
237C22×C32D3×D411yescharacteristic
238C2×C3⋊D3—22no—
239C2×C3⋊D3—41no—
240C2×C3⋊D3—41no—
241C2×C3×D3—62no—
242C2×C3×D3—61no—
243C2×C3×D3—61no—
244C2×C3×D3—61no—
245C2×C3×D3—61no—
246C2×C3×D3—62no—
247C2×C3×D3—61no—
248C2×C3×D3—61no—
249C2×C3×D3—121no—
250C2×C3×D3—121no—
251C2×C3×D3—121no—
252C2×C3×D3—121no—
253C32×C4—21no—
254C32×C4—21no—
255C32×C4—42no—
256C32×C4—41no—
257C32×C4—41no—
258C32×C4D3×D411yescharacteristic
259C32×C4D3×D411yescharacteristic
260C32×C4C2×#6711yescharacteristic
261C32×C4C2×#6711yescharacteristic
262C3⋊Dic3—22no—
263C3×Dic3—62no—
264C3×Dic3—61no—
265C3×Dic3—61no—
266C3×Dic3—121no—
267C3×Dic3—122no—
268C3×Dic3—121no—
269C3×Dic3—121no—
270C2×#41—271no—
271C2×Q16—91no—
272C2×SD16—91no—
273C22⋊Q8—91no—
274#103—271no—
275#102—271no—
276C22⋊C8—271no—
277H(3,𝔽3)#45211yescharacteristic, complemented, Sylow
278C23×C3—91no—
279C23×C3—181no—
280C22×D3—31no—
281C22×D3—31no—
282C22×D3—61no—
283C22×D3—61no—
284C22×D3—181no—
285C3×Q8—21no—
286C3×Q8—21no—
287C3×Q8—61no—
288C3×Q8—61no—
289C3×Q8—61no—
290C3×Q8—61no—
291C3×Q8—92no—
292C3×Q8—121no—
293C3×Q8—121no—
294C3×Q8—181no—
295C3×D4—92no—
296C3×D4—181no—
297C2×C3×C4—21no—
298C2×C3×C4—31no—
299C2×C3×C4—31no—
300C2×C3×C4—61no—
301C2×C3×C4—61no—
302C2×C3×C4—62no—
303C2×C3×C4—61no—
304C2×C3×C4—91no—
305C2×C3×C4—181no—
306C2×C3×C4—181no—
307C2×C3×C4—181no—
308C2×C3×C4—181no—
309C2×C3×C4C2×D3211yescharacteristic
310#67—184no—
311#67—184no—
312C2×Dic3—31no—
313C2×Dic3—31no—
314C2×Dic3—61no—
315C2×Dic3—61no—
316C2×Dic3—61no—
317C2×Dic3—61no—
318C2×Dic3—62no—
319C2×Dic3—181no—
320C2×Dic3—184no—
321D12—32no—
322D12—61no—
323D12—92no—
324D12—181no—
325D12—184no—
326C4×D3—61no—
327C4×D3—61no—
328C4×D3—61no—
329C4×D3—61no—
330C4×D3—121no—
331C4×D3—121no—
332C4×D3—184no—
333C4×D3—184no—
334Dic6—32no—
335Dic6—61no—
336Dic6—92no—
337Dic6—181no—
338C3×C8—181no—
339C3×C8—181no—
340C3⋊C8—61no—
341C3⋊C8—61no—
342C3⋊C8—181no—
343C3⋊C8—181no—
344C2×C32—21no—
345C2×C32—21no—
346C2×C32—21no—
347C2×C32#91511yescharacteristic
348C2×C32#91611yescharacteristic
349C2×C32#91811yescharacteristic
350C2×C32#91911yescharacteristic
351C2×C32#93711yescharacteristic
352C2×C32#93711yescharacteristic
353C3⋊D3—42no—
354C3×D3—122no—
355C3×D3—121no—
356C3×D3—122no—
357C3×D3—121no—
358#41—542no—
359#41—542no—
360C2×Q8—91no—
361C2×Q8((C3⋊D3)⋊C3)⋊C211yescharacteristic
362C2×D4—91no—
363C2×D4—541no—
364C22×C4—91no—
365C22×C4—541no—
366Q16—182no—
367SD16—182no—
368C2×C8—91no—
369C2×C8—91no—
370C4⋊C4—91no—
371C4⋊C4—181no—
372C22⋊C4—181no—
373C22×C3—31no—
374C22×C3—31no—
375C22×C3—61no—
376C22×C3—182no—
377C22×C3—181no—
378C22×C3—181no—
379C22×C3—181no—
380C22×C3—361no—
381C22×C3—361no—
382C22×C3#381111yescharacteristic
383C2×D3—61no—
384C2×D3—61no—
385C2×D3—61no—
386C2×D3—61no—
387C2×D3—61no—
388C2×D3—61no—
389C2×D3—62no—
390C2×D3—62no—
391C2×D3—121no—
392C2×D3—121no—
393C2×D3—121no—
394C2×D3—121no—
395C2×D3—182no—
396C2×D3—361no—
397C2×D3—361no—
398C3×C4—31no—
399C3×C4—31no—
400C3×C4—31no—
401C3×C4—31no—
402C3×C4—41no—
403C3×C4—61no—
404C3×C4—61no—
405C3×C4—121no—
406C3×C4—121no—
407C3×C4—122no—
408C3×C4—182no—
409C3×C4—181no—
410C3×C4—361no—
411C3×C4D3×#6711yescharacteristic
412C3×C4D3×#6711yescharacteristic
413Dic3—61no—
414Dic3—61no—
415Dic3—62no—
416Dic3—121no—
417Dic3—121no—
418Dic3—121no—
419Dic3—122no—
420Dic3—182no—
421C32—21no—
422C32#517311yescharacteristic, complemented
423C32#518711yescharacteristic, complemented
424C23—91no—
425C23—181no—
426Q8—21no—
427Q8—21no—
428Q8—92no—
429Q8—181no—
430D4—92no—
431D4—181no—
432D4—544no—
433C2×C4—21no—
434C2×C4—91no—
435C2×C4—181no—
436C2×C4—181no—
437C2×C4—181no—
438C2×C4—181no—
439C2×C4—544no—
440C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
441C8—181no—
442C8—181no—
443C2×C3—31no—
444C2×C3—31no—
445C2×C3—31no—
446C2×C3—31no—
447C2×C3—31no—
448C2×C3—31no—
449C2×C3—61no—
450C2×C3—61no—
451C2×C3—61no—
452C2×C3—362no—
453C2×C3—361no—
454C2×C3#6395011yescharacteristic
455C2×C3#6395111yescharacteristic
456C2×C3#6398311yescharacteristic
457D3—121no—
458D3—121no—
459D3—122no—
460D3—122no—
461D3—362no—
462C22—181no—
463C22—181no—
464C22—181no—
465C22—182no—
466C22—361no—
467C22—361no—
468C22#8867711yescharacteristic
469C4—41no—
470C4—181no—
471C4—182no—
472C4—361no—
473C4#8867611yescharacteristic
474C4#8867611yescharacteristic
475C3—31no—
476C3—31no—
477C3—61no—
478C3#1059165611yescharacteristic
479C2—361no—
480C2—362no—
481C2#1172123011yescharacteristic
482C2#1172123111yescharacteristic
483C2#1172132411yescharacteristic
subgroupG/H
centerC22#88677
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×Q8×H(3,𝔽3)C22
socleC22×C3#3811
prime radicalC2×#88649C2
hypercenterC2×Q8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#452
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. #11721863C2semidirect
3. C2×#3755⋊C3C2semidirect
4. C2×#3756⋊C3C2semidirect
5. #11720323C2semidirect
6. #11720309C2semidirect
7. #11720308C2semidirect
27. C2×#3756D3semidirect
28. C2×#3755D3semidirect
277. H(3,𝔽3)#452semidirect
422. C32#5173semidirect
423. C32#5187semidirect

Maximal subgroups

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

subgroupindex
C2×#886492
#117218632
C2×#3755⋊C32
C2×#3756⋊C32
#117203232
#117203092
#117203082
#105916573
#105916553