Finite Group Databasebeta

← #13140985

Subgroups — #13140985

Order 1728.

592 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 592 conjugacy classes of subgroups — too many to draw. Only its 83 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×#88722C211yesmaximal, characteristic
2C2×#88550C211yesmaximal, characteristic, complemented
3#11721107C211yesmaximal, characteristic, complemented
4C2×#88438C211yesmaximal, characteristic, complemented
5#63662⋊C3C211yesmaximal, characteristic, complemented
6#11720318C211yesmaximal, characteristic, complemented
7#11720317C211yesmaximal, characteristic, complemented
8#10591602—31nomaximal
9#10591597—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C2×#6229C2211yescharacteristic
12#88722—22no—
13#88722—22no—
14C2×C4⋊((C3⋊D3)⋊C3)—21no—
15C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
16C2×C4×(C3⋊D3)⋊C3—21no—
17C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
18#88550—22no—
19C2×#6180C2211yescharacteristic
20C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
21C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
22#88450—21no—
23#88448—21no—
24#88438—22no—
25C2×#3832D311yescharacteristic
26C2×C3×#239—31no—
27C2×C3×#216—31no—
28C2×C3×#216—31no—
29C2×C3×#216—61no—
30#63871—31no—
31#63856—31no—
32C2×#3717—31no—
33C2×#3715—31no—
34#63662D311yescharacteristic, complemented
35C3×#835—31no—
36#63564—31no—
37#63563—31no—
38#63557—31no—
39(C2×D3)⋊(C3⋊C8)—31no—
40C22×(C3⋊D3)⋊C3—21no—
41C22×(C3⋊D3)⋊C3D411yescharacteristic
42Q8×H(3,𝔽3)—21no—
43Q8×H(3,𝔽3)D412yes—
44C2×C4×H(3,𝔽3)—21no—
45C2×C4×H(3,𝔽3)C2311yescharacteristic
46C2×#1084—21no—
47#6229—21no—
48#6229—21no—
49C2×(C3⋊Dic3)⋊C3—21no—
50C2×(C3⋊Dic3)⋊C3D411yescharacteristic
51C4⋊((C3⋊D3)⋊C3)—24no—
52C4⋊((C3⋊D3)⋊C3)—21no—
53C4⋊((C3⋊D3)⋊C3)D412yes—
54C4×(C3⋊D3)⋊C3—24no—
55C4×(C3⋊D3)⋊C3—21no—
56C4×(C3⋊D3)⋊C3—21no—
57#6180—21no—
58(C3⋊(C3⋊C8))⋊C3—21no—
59#5239—91no—
60#5187—91no—
61#4808—91no—
62C2×C32×Q8—21no—
63C2×C32×Q8C2×D311yescharacteristic
64C2×C32×Q8C2×D311yescharacteristic
65#3832—22no—
66#3832—22no—
67C2×C4⋊(C3⋊D3)—21no—
68C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
69C2×C4×C3⋊D3—21no—
70C2×C4×C3⋊D3C2×D311yescharacteristic
71C3×#239—62no—
72C3×#239—62no—
73C2×C3×D12—31no—
74C2×C3×D12—61no—
75C2×C3×C4×D3—31no—
76C2×C3×C4×D3—61no—
77C2×C3×Dic6—31no—
78C2×C3×Dic6—31no—
79C2×C3×Dic6—61no—
80C4⋊(C3⋊Dic3)C2×D311yescharacteristic
81C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
82C3×#216—64no—
83C3×#216—62no—
84C3×#216—62no—
85C3×C4⋊Dic3—31no—
86C2×C3×C3⋊C8—31no—
87C2×C3×C3⋊C8—31no—
88C2×C3×C3⋊C8—31no—
89C2×C3×C3⋊C8—61no—
90#3724—61no—
91Dic3⋊Dic3—61no—
92#3722—61no—
93(C2×D3)⋊Dic3—61no—
94#3717—62no—
95#3715—62no—
96C22×H(3,𝔽3)C2×D411yescharacteristic
97C2×(C3⋊D3)⋊C3—21no—
98C2×(C3⋊D3)⋊C3—22no—
99C2×(C3⋊D3)⋊C3—21no—
100C2×(C3⋊D3)⋊C3—21no—
101C2×(C3⋊D3)⋊C3—41no—
102C2×(C3⋊D3)⋊C3—41no—
103C4×H(3,𝔽3)—22no—
104C4×H(3,𝔽3)C2×D411yescharacteristic
105C4×H(3,𝔽3)C2×D411yescharacteristic
106#1084—41no—
107(C3⋊Dic3)⋊C3—21no—
108(C3⋊Dic3)⋊C3—41no—
109C2×C3×#41—91no—
110C2×#239—31no—
111C2×#239—91no—
112C2×C3×Q16—91no—
113#946—91no—
114C2×#216—31no—
115C2×#216—31no—
116C2×#216—61no—
117C2×#215—91no—
118C2×#214—91no—
119#924—91no—
120C2×#204—91no—
121#897—91no—
122C3×#103—91no—
123#835—31no—
124#835—91no—
125#832—91no—
126#830—91no—
127#821—91no—
128(C2×D3)⋊C8—91no—
129(C2×D3)⋊C8—91no—
130#809—91no—
131C22×C3⋊D3—21no—
132C22×C3⋊D3#6711yescharacteristic
133C22×C3×D3—31no—
134C22×C3×D3—61no—
135C32×Q8—21no—
136C32×Q8—21no—
137C32×Q8—22no—
138C32×Q8—41no—
139C32×Q8D1212yes—
140C32×Q8#6712yes—
141C2×C32×C4—21no—
142C2×C32×C4—21no—
143C2×C32×C4—21no—
144C2×C32×C4—41no—
145C2×C32×C4C22×D311yescharacteristic
146C2×C32×C4C22×D311yescharacteristic
147C2×C3⋊Dic3—21no—
148C2×C3⋊Dic3#6711yescharacteristic
149C4⋊(C3⋊D3)—24no—
150C4⋊(C3⋊D3)—21no—
151C4⋊(C3⋊D3)#6712yes—
152C4×C3⋊D3—24no—
153C4×C3⋊D3—21no—
154C4×C3⋊D3—21no—
155C2×C3×Dic3—31no—
156C2×C3×Dic3—61no—
157C2×C3×Dic3—61no—
158C2×C3×Dic3—61no—
159C2×C3×Dic3—121no—
160C3×D12—32no—
161C3×D12—64no—
162C3×D12—61no—
163C3×C4×D3—64no—
164C3×C4×D3—61no—
165C3×C4×D3—61no—
166C3×Dic6—62no—
167C3×Dic6—61no—
168C3×Dic6—61no—
169C3×Dic6—61no—
170C3×Dic6—61no—
171C3×Dic6—121no—
172C3⋊(C3⋊C8)—21no—
173C3×C3⋊C8—62no—
174C3×C3⋊C8—61no—
175C3×C3⋊C8—61no—
176C3×C3⋊C8—61no—
177#452—271noSylow
178C2×H(3,𝔽3)#12011yescharacteristic
179C2×H(3,𝔽3)#13511yescharacteristic
180C2×H(3,𝔽3)#13711yescharacteristic
181(C3⋊D3)⋊C3—41no—
182(C3⋊D3)⋊C3—42no—
183C3×#41—182no—
184C3×#41—182no—
185C2×C3×Q8—31no—
186C2×C3×Q8—31no—
187C2×C3×Q8—61no—
188C2×C3×Q8—91no—
189C2×C3×Q8D3211yescharacteristic
190C2×C3×D4—91no—
191C2×C3×D4—181no—
192C22×C3×C4—91no—
193C22×C3×C4—181no—
194#239—62no—
195#239—62no—
196#239—182no—
197#239—182no—
198C2×D12—31no—
199C2×D12—61no—
200C2×D12—91no—
201C2×D12—181no—
202C2×C4×D3—31no—
203C2×C4×D3—61no—
204C2×C4×D3—91no—
205C2×C4×D3—181no—
206C2×Dic6—31no—
207C2×Dic6—31no—
208C2×Dic6—61no—
209C3×Q16—182no—
210C2×C3×C8—91no—
211C2×C3×C8—91no—
212C3×C4⋊C4—91no—
213#217—181no—
214#216—62no—
215#216—62no—
216#216—64no—
217#215—182no—
218#214—182no—
219(C2×D3)⋊C4—181no—
220(C2×D3)⋊C4—181no—
221C4⋊Dic3—31no—
222C4⋊Dic3—91no—
223C4⋊Dic3—181no—
224Dic3⋊C4—181no—
225Dic3⋊C4—181no—
226C2×C3⋊C8—31no—
227C2×C3⋊C8—31no—
228C2×C3⋊C8—31no—
229C2×C3⋊C8—61no—
230C2×C3⋊C8—91no—
231#204—182no—
232C22×C32—21no—
233C22×C32D3×D411yescharacteristic
234C22×C32C2×#6711yescharacteristic
235C2×C3⋊D3—21no—
236C2×C3⋊D3—21no—
237C2×C3⋊D3—21no—
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—121no—
246C2×C3×D3—121no—
247C32×C4—22no—
248C32×C4—22no—
249C32×C4—21no—
250C32×C4—21no—
251C32×C4—42no—
252C32×C4C2×D1211yescharacteristic
253C32×C4D3×D411yescharacteristic
254C32×C4C2×#6711yescharacteristic
255C32×C4C2×#6711yescharacteristic
256C3⋊Dic3—21no—
257C3⋊Dic3—41no—
258C3×Dic3—61no—
259C3×Dic3—122no—
260C3×Dic3—121no—
261C3×Dic3—121no—
262C3×Dic3—121no—
263C2×#41—91no—
264C2×Q16—91no—
265C2×SD16—271no—
266C22⋊Q8—271no—
267#103—91no—
268#102—271no—
269C22⋊C8—271no—
270H(3,𝔽3)#45211yescharacteristic, complemented, Sylow
271C23×C3—91no—
272C23×C3—181no—
273C22×D3—31no—
274C22×D3—61no—
275C22×D3—91no—
276C22×D3—181no—
277C3×Q8—21no—
278C3×Q8—32no—
279C3×Q8—32no—
280C3×Q8—61no—
281C3×Q8—61no—
282C3×Q8—62no—
283C3×Q8—121no—
284C3×Q8—181no—
285C3×Q8—181no—
286C3×Q8#62612yes—
287C3×D4—92no—
288C3×D4—181no—
289C3×D4—184no—
290C2×C3×C4—21no—
291C2×C3×C4—31no—
292C2×C3×C4—31no—
293C2×C3×C4—61no—
294C2×C3×C4—61no—
295C2×C3×C4—61no—
296C2×C3×C4—91no—
297C2×C3×C4—121no—
298C2×C3×C4—181no—
299C2×C3×C4—181no—
300C2×C3×C4—184no—
301C2×C3×C4—181no—
302C2×C3×C4—181no—
303C2×C3×C4C2×D3211yescharacteristic
304C2×Dic3—31no—
305C2×Dic3—61no—
306C2×Dic3—61no—
307C2×Dic3—61no—
308C2×Dic3—91no—
309C2×Dic3—121no—
310C2×Dic3—181no—
311D12—32no—
312D12—64no—
313D12—61no—
314D12—92no—
315D12—181no—
316D12—184no—
317C4×D3—64no—
318C4×D3—61no—
319C4×D3—61no—
320C4×D3—181no—
321C4×D3—181no—
322C4×D3—184no—
323Dic6—61no—
324Dic6—61no—
325Dic6—61no—
326Dic6—61no—
327Dic6—62no—
328Dic6—121no—
329C3×C8—181no—
330C3×C8—181no—
331C3⋊C8—61no—
332C3⋊C8—61no—
333C3⋊C8—62no—
334C3⋊C8—61no—
335C3⋊C8—181no—
336C2×C32—21no—
337C2×C32—21no—
338C2×C32—21no—
339C2×C32#88211yescharacteristic
340C2×C32#91611yescharacteristic
341C2×C32#91811yescharacteristic
342C2×C32#95011yescharacteristic
343C2×C32#95111yescharacteristic
344C2×C32#95311yescharacteristic
345C3⋊D3—41no—
346C3⋊D3—42no—
347C3×D3—121no—
348C3×D3—122no—
349#41—182no—
350#41—182no—
351C2×Q8—91no—
352C2×Q8((C3⋊D3)⋊C3)⋊C211yescharacteristic
353C2×D4—91no—
354C2×D4—181no—
355C22×C4—91no—
356C22×C4—181no—
357Q16—182no—
358SD16—542no—
359C2×C8—91no—
360C2×C8—91no—
361C4⋊C4—91no—
362C4⋊C4—541no—
363C22⋊C4—541no—
364C22×C3—31no—
365C22×C3—31no—
366C22×C3—61no—
367C22×C3—182no—
368C22×C3—181no—
369C22×C3—181no—
370C22×C3—181no—
371C22×C3—361no—
372C22×C3—361no—
373C22×C3D3×#6711yescharacteristic
374C2×D3—61no—
375C2×D3—61no—
376C2×D3—62no—
377C2×D3—61no—
378C2×D3—121no—
379C2×D3—121no—
380C2×D3—182no—
381C2×D3—181no—
382C2×D3—181no—
383C2×D3—181no—
384C2×D3—361no—
385C2×D3—361no—
386C3×C4—22no—
387C3×C4—31no—
388C3×C4—31no—
389C3×C4—31no—
390C3×C4—31no—
391C3×C4—62no—
392C3×C4—62no—
393C3×C4—61no—
394C3×C4—61no—
395C3×C4—122no—
396C3×C4—181no—
397C3×C4—361no—
398C3×C4—361no—
399C3×C4C2×#62611yescharacteristic
400C3×C4D3×#6711yescharacteristic
401Dic3—61no—
402Dic3—121no—
403Dic3—121no—
404Dic3—122no—
405Dic3—121no—
406Dic3—181no—
407Dic3—361no—
408C32—21no—
409C32#480811yescharacteristic, complemented
410C32#523911yescharacteristic, complemented
411C23—91no—
412C23—181no—
413Q8—21no—
414Q8—181no—
415Q8—181no—
416Q8#620012yes—
417D4—92no—
418D4—184no—
419D4—181no—
420C2×C4—21no—
421C2×C4—91no—
422C2×C4—184no—
423C2×C4—181no—
424C2×C4—181no—
425C2×C4—181no—
426C2×C4—181no—
427C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
428C8—181no—
429C8—181no—
430C2×C3—31no—
431C2×C3—31no—
432C2×C3—31no—
433C2×C3—31no—
434C2×C3—31no—
435C2×C3—31no—
436C2×C3—61no—
437C2×C3—61no—
438C2×C3—61no—
439C2×C3—361no—
440C2×C3—362no—
441C2×C3#6394311yescharacteristic
442C2×C3#6394711yescharacteristic
443C2×C3#6397711yescharacteristic
444D3—121no—
445D3—122no—
446D3—361no—
447D3—362no—
448C22—181no—
449C22—181no—
450C22—182no—
451C22—181no—
452C22—361no—
453C22—361no—
454C22#8867611yescharacteristic
455C4—22no—
456C4—181no—
457C4—361no—
458C4—361no—
459C4C2×#620011yescharacteristic
460C4#8867611yescharacteristic
461C3—31no—
462C3—31no—
463C3—61no—
464C3#1059159911yescharacteristic
465C2—361no—
466C2—362no—
467C2#1172123711yescharacteristic
468C2#1172123911yescharacteristic
469C2#1172132111yescharacteristic
subgroupG/H
centerC22#88676
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×Q8×H(3,𝔽3)C22
socleC22×C3D3×#67
prime radicalC2×#88722C2
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. C2×#88550C2semidirect
3. #11721107C2semidirect
4. C2×#88438C2semidirect
5. #63662⋊C3C2semidirect
6. #11720318C2semidirect
7. #11720317C2semidirect
34. #63662D3semidirect
270. H(3,𝔽3)#452semidirect
409. C32#4808semidirect
410. C32#5239semidirect

Maximal subgroups

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

subgroupindex
C2×#887222
C2×#885502
#117211072
C2×#884382
#63662⋊C32
#117203182
#117203172
#105916023
#105915973