Finite Group Databasebeta

← #13140998

Subgroups — #13140998

Order 1728.

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

normal subgroups only

This group has 600 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×#88652C211yesmaximal, characteristic
2C2×#88549C211yesmaximal, characteristic, complemented
3#64155⋊C3C211yesmaximal, characteristic, complemented
4C2×#3756⋊C3C211yesmaximal, characteristic, complemented
5#11720324C211yesmaximal, characteristic, complemented
6#11720317C211yesmaximal, characteristic, complemented
7#11720313C211yesmaximal, characteristic, complemented
8#10591656—31nomaximal
9#10591655—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C2×C4⋊#272C2211yescharacteristic
12C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
13C2×#634⋊C3C2211yescharacteristic
14C2×#6200—21no—
15C2×#6199—21no—
16#88652—22no—
17#88652—22no—
18#88549—22no—
19C2×#6180C2211yescharacteristic
20#3756⋊C3—22no—
21#3752⋊C3—21no—
22C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
23#3750⋊C3—21no—
24C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
25C2×#3796—31no—
26C2×#3795—31no—
27#64155D311yescharacteristic, complemented
28C2×#3756D311yescharacteristic, complemented
29C3×#946—31no—
30C2×C3×#216—31no—
31C2×C3×#215—31no—
32C2×C3×#215—31no—
33C2×C3×#215—61no—
34#63567—31no—
35#63566—31no—
36#63563—31no—
37#63562—31no—
38#63557—31no—
39(C2×D3)⋊(C3⋊C8)—31no—
40C22×#272—21no—
41C22×(C3⋊D3)⋊C3D411yescharacteristic
42Q8×H(3,𝔽3)—21no—
43Q8×H(3,𝔽3)—21no—
44C2×C4×H(3,𝔽3)—21no—
45C2×C4×H(3,𝔽3)C2311yescharacteristic
46C4⋊#272—21no—
47C4⋊#272D412yes—
48C2×(C3⋊Dic3)⋊C3—21no—
49C2×(C3⋊Dic3)⋊C3—21no—
50C2×(C3⋊Dic3)⋊C3D411yescharacteristic
51C4×(C3⋊D3)⋊C3—21no—
52C4×(C3⋊D3)⋊C3—21no—
53#634⋊C3—21no—
54#634⋊C3D412yes—
55#6200—24no—
56#6199—24no—
57#6180—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⋊D3C2×D311yescharacteristic
66C2×#634C2×D311yescharacteristic
67C2×C3×D12—31no—
68C2×C3×D12—31no—
69C2×C3×D12—61no—
70C2×C3×C4×D3—31no—
71C2×C3×Dic6—31no—
72C2×#626—61no—
73C2×#625—61no—
74C2×D3×Dic3—61no—
75C2×D3×Dic3—61no—
76#3796—62no—
77#3796—62no—
78#3795—62no—
79#3795—62no—
80#3756—22no—
81#3752—21no—
82C4⋊(C3⋊Dic3)C2×D311yescharacteristic
83#3750—21no—
84C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
85C3×#216—62no—
86C3×#215—62no—
87C3×#215—62no—
88C3×#215—64no—
89C3×(C2×D3)⋊C4—61no—
90C3×C4⋊Dic3—31no—
91C3×Dic3⋊C4—61no—
92C2×C3×C3⋊C8—31no—
93C2×C3×C3⋊C8—31no—
94C2×C3×C3⋊C8—31no—
95C2×C3×C3⋊C8—61no—
96C22×H(3,𝔽3)C2×D411yescharacteristic
97C2×#272—22no—
98C2×#272—41no—
99C2×#272—41no—
100C2×(C3⋊D3)⋊C3—21no—
101C2×(C3⋊D3)⋊C3—21no—
102C2×(C3⋊D3)⋊C3—21no—
103C4×H(3,𝔽3)—41no—
104C4×H(3,𝔽3)C2×D411yescharacteristic
105C4×H(3,𝔽3)C2×D411yescharacteristic
106(C3⋊Dic3)⋊C3—21no—
107(C3⋊Dic3)⋊C3—22no—
108(C3⋊Dic3)⋊C3—41no—
109C2×#239—91no—
110C2×#237—91no—
111C2×#235—91no—
112C2×C3×Q16—91no—
113C2×C3×SD16—91no—
114C3×C22⋊Q8—91no—
115#946—31no—
116#946—91no—
117C2×#216—31no—
118C2×#216—91no—
119C2×#215—31no—
120C2×#215—31no—
121C2×#215—61no—
122#835—91no—
123#832—91no—
124#830—91no—
125#821—91no—
126(C2×D3)⋊C8—91no—
127(C2×D3)⋊C8—91no—
128#816—91no—
129#810—91no—
130#809—91no—
131C22×C3⋊D3#6711yescharacteristic
132C22×C3×D3—31no—
133C22×C3×D3—61no—
134C22×C3×D3—61no—
135C22×C3×D3—121no—
136C32×Q8—22no—
137C32×Q8—21no—
138C32×Q8—21no—
139C32×Q8—21no—
140C32×Q8—21no—
141C32×Q8—41no—
142C2×C32×C4—21no—
143C2×C32×C4—21no—
144C2×C32×C4—21no—
145C2×C32×C4—41no—
146C2×C32×C4C22×D311yescharacteristic
147C2×C32×C4C22×D311yescharacteristic
148C2×C3⋊Dic3—21no—
149C2×C3⋊Dic3—21no—
150C2×C3⋊Dic3#6711yescharacteristic
151C4×C3⋊D3—21no—
152C4×C3⋊D3—21no—
153#634—21no—
154#634#6712yes—
155C2×C3×Dic3—31no—
156C2×C3×Dic3—61no—
157C2×C3×Dic3—61no—
158C3×D12—32no—
159C3×D12—32no—
160C3×D12—62no—
161C3×D12—61no—
162C3×D12—61no—
163C3×D12—121no—
164C3×C4×D3—61no—
165C3×C4×D3—61no—
166C3×Dic6—32no—
167C3×Dic6—61no—
168#626—64no—
169#625—64no—
170D3×Dic3—64no—
171D3×Dic3—64no—
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#272—42no—
182(C3⋊D3)⋊C3—41no—
183C2×C3×Q8—31no—
184C2×C3×Q8—31no—
185C2×C3×Q8—61no—
186C2×C3×Q8—91no—
187C2×C3×Q8D3211yescharacteristic
188C2×C3×D4—91no—
189C22×C3×C4—91no—
190C2×#67—181no—
191C2×#67—181no—
192C22×Dic3—181no—
193#239—182no—
194#239—182no—
195#237—182no—
196#237—182no—
197#235—182no—
198#235—182no—
199C2×D12—31no—
200C2×D12—31no—
201C2×D12—61no—
202C2×D12—181no—
203C2×C4×D3—31no—
204C2×C4×D3—91no—
205C2×C4×D3—181no—
206C2×C4×D3—181no—
207C2×Dic6—31no—
208C2×Dic6—91no—
209C3×Q16—182no—
210C3×SD16—182no—
211C2×C3×C8—91no—
212C2×C3×C8—91no—
213C3×C4⋊C4—91no—
214C3×C4⋊C4—181no—
215C3×C22⋊C4—181no—
216#216—62no—
217#216—182no—
218#215—62no—
219#215—62no—
220#215—64no—
221(C2×D3)⋊C4—61no—
222(C2×D3)⋊C4—181no—
223C4⋊Dic3—31no—
224C4⋊Dic3—91no—
225Dic3⋊C4—61no—
226Dic3⋊C4—181no—
227C2×C3⋊C8—31no—
228C2×C3⋊C8—31no—
229C2×C3⋊C8—31no—
230C2×C3⋊C8—61no—
231C2×C3⋊C8—91no—
232C22×C32—21no—
233C22×C32D3×D411yescharacteristic
234C22×C32C2×#6711yescharacteristic
235C2×C3⋊D3—21no—
236C2×C3⋊D3—21no—
237C2×C3⋊D3—21no—
238C2×C3×D3—61no—
239C2×C3×D3—61no—
240C2×C3×D3—61no—
241C2×C3×D3—62no—
242C2×C3×D3—62no—
243C2×C3×D3—122no—
244C2×C3×D3—121no—
245C2×C3×D3—121no—
246C2×C3×D3—121no—
247C2×C3×D3—121no—
248C2×C3×D3—241no—
249C2×C3×D3—241no—
250C32×C4—21no—
251C32×C4—21no—
252C32×C4—42no—
253C32×C4—41no—
254C32×C4—41no—
255C32×C4D3×D411yescharacteristic
256C32×C4D3×D411yescharacteristic
257C32×C4D3×D411yescharacteristic
258C32×C4C2×#6711yescharacteristic
259C3⋊Dic3—21no—
260C3⋊Dic3—22no—
261C3⋊Dic3—41no—
262C3×Dic3—61no—
263C3×Dic3—62no—
264C3×Dic3—121no—
265C2×#41—271no—
266C2×Q16—91no—
267C2×SD16—91no—
268C22⋊Q8—91no—
269#103—271no—
270#102—271no—
271C22⋊C8—271no—
272H(3,𝔽3)#45211yescharacteristic, complemented, Sylow
273C23×C3—91no—
274C23×C3—181no—
275C22×D3—31no—
276C22×D3—61no—
277C22×D3—61no—
278C22×D3—91no—
279C22×D3—121no—
280C3×Q8—21no—
281C3×Q8—21no—
282C3×Q8—61no—
283C3×Q8—61no—
284C3×Q8—61no—
285C3×Q8—61no—
286C3×Q8—62no—
287C3×Q8—92no—
288C3×Q8—121no—
289C3×Q8—181no—
290C3×D4—92no—
291C3×D4—181no—
292C2×C3×C4—21no—
293C2×C3×C4—31no—
294C2×C3×C4—31no—
295C2×C3×C4—61no—
296C2×C3×C4—61no—
297C2×C3×C4—61no—
298C2×C3×C4—91no—
299C2×C3×C4—121no—
300C2×C3×C4—181no—
301C2×C3×C4—181no—
302C2×C3×C4—181no—
303C2×C3×C4—181no—
304C2×C3×C4C2×D3211yescharacteristic
305#67—184no—
306#67—184no—
307C2×Dic3—31no—
308C2×Dic3—61no—
309C2×Dic3—61no—
310C2×Dic3—91no—
311C2×Dic3—181no—
312C2×Dic3—181no—
313C2×Dic3—184no—
314D12—32no—
315D12—32no—
316D12—62no—
317D12—61no—
318D12—61no—
319D12—121no—
320D12—184no—
321C4×D3—61no—
322C4×D3—61no—
323C4×D3—181no—
324C4×D3—181no—
325C4×D3—184no—
326C4×D3—184no—
327Dic6—32no—
328Dic6—61no—
329Dic6—92no—
330Dic6—181no—
331C3×C8—181no—
332C3×C8—181no—
333C3⋊C8—61no—
334C3⋊C8—61no—
335C3⋊C8—62no—
336C3⋊C8—61no—
337C3⋊C8—181no—
338C2×C32—21no—
339C2×C32—21no—
340C2×C32—21no—
341C2×C32#91811yescharacteristic
342C2×C32#91911yescharacteristic
343C2×C32#93711yescharacteristic
344C2×C32#93711yescharacteristic
345C2×C32#94211yescharacteristic
346C2×C32#95011yescharacteristic
347C3⋊D3—41no—
348C3×D3—121no—
349C3×D3—122no—
350C3×D3—122no—
351C3×D3—242no—
352#41—542no—
353#41—542no—
354C2×Q8—91no—
355C2×Q8((C3⋊D3)⋊C3)⋊C211yescharacteristic
356C2×D4—91no—
357C2×D4—541no—
358C22×C4—91no—
359C22×C4—541no—
360Q16—182no—
361SD16—182no—
362C2×C8—91no—
363C2×C8—91no—
364C4⋊C4—91no—
365C4⋊C4—181no—
366C22⋊C4—181no—
367C22×C3—31no—
368C22×C3—31no—
369C22×C3—61no—
370C22×C3—181no—
371C22×C3—181no—
372C22×C3—181no—
373C22×C3—182no—
374C22×C3—361no—
375C22×C3—361no—
376C22×C3D3×#6711yescharacteristic
377C2×D3—62no—
378C2×D3—62no—
379C2×D3—61no—
380C2×D3—61no—
381C2×D3—61no—
382C2×D3—122no—
383C2×D3—121no—
384C2×D3—121no—
385C2×D3—121no—
386C2×D3—121no—
387C2×D3—181no—
388C2×D3—181no—
389C2×D3—181no—
390C2×D3—241no—
391C2×D3—241no—
392C3×C4—31no—
393C3×C4—31no—
394C3×C4—31no—
395C3×C4—31no—
396C3×C4—41no—
397C3×C4—61no—
398C3×C4—61no—
399C3×C4—121no—
400C3×C4—121no—
401C3×C4—122no—
402C3×C4—181no—
403C3×C4—182no—
404C3×C4—361no—
405C3×C4D3×#6711yescharacteristic
406C3×C4#381111yescharacteristic
407Dic3—62no—
408Dic3—61no—
409Dic3—121no—
410Dic3—182no—
411Dic3—181no—
412Dic3—361no—
413C32—21no—
414C32#504711yescharacteristic, complemented
415C32#518711yescharacteristic, complemented
416C23—91no—
417C23—181no—
418Q8—21no—
419Q8—21no—
420Q8—92no—
421Q8—181no—
422D4—92no—
423D4—181no—
424D4—544no—
425C2×C4—21no—
426C2×C4—91no—
427C2×C4—181no—
428C2×C4—181no—
429C2×C4—181no—
430C2×C4—181no—
431C2×C4—544no—
432C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
433C8—181no—
434C8—181no—
435C2×C3—31no—
436C2×C3—31no—
437C2×C3—31no—
438C2×C3—31no—
439C2×C3—31no—
440C2×C3—31no—
441C2×C3—61no—
442C2×C3—61no—
443C2×C3—61no—
444C2×C3—361no—
445C2×C3—362no—
446C2×C3#6394811yescharacteristic
447C2×C3#6394911yescharacteristic
448C2×C3#6398311yescharacteristic
449D3—122no—
450D3—122no—
451D3—121no—
452D3—242no—
453D3—361no—
454C22—181no—
455C22—181no—
456C22—181no—
457C22—182no—
458C22—361no—
459C22—361no—
460C22#8867611yescharacteristic
461C4—41no—
462C4—181no—
463C4—182no—
464C4—361no—
465C4#8867611yescharacteristic
466C4#8867711yescharacteristic
467C3—31no—
468C3—31no—
469C3—61no—
470C3#1059165711yescharacteristic
471C2—361no—
472C2—362no—
473C2#1172123411yescharacteristic
474C2#1172123511yescharacteristic
475C2#1172132411yescharacteristic
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×#88652C2
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×#88549C2semidirect
3. #64155⋊C3C2semidirect
4. C2×#3756⋊C3C2semidirect
5. #11720324C2semidirect
6. #11720317C2semidirect
7. #11720313C2semidirect
27. #64155D3semidirect
28. C2×#3756D3semidirect
272. H(3,𝔽3)#452semidirect
414. C32#5047semidirect
415. C32#5187semidirect

Maximal subgroups

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

subgroupindex
C2×#886522
C2×#885492
#64155⋊C32
C2×#3756⋊C32
#117203242
#117203172
#117203132
#105916563
#105916553