Finite Group Databasebeta

← #13140997

Subgroups — #13140997

Order 1728.

726 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 726 conjugacy classes of subgroups — too many to draw. Only its 79 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×#88654C211yesmaximal, characteristic
2C2×#88549C211yesmaximal, characteristic, complemented
3#64155⋊C3C211yesmaximal, characteristic, complemented
4C2×#3755⋊C3C211yesmaximal, characteristic, complemented
5#11720324C211yesmaximal, characteristic, complemented
6#11720318C211yesmaximal, characteristic, complemented
7#11720317C211yesmaximal, characteristic, complemented
8#10591652—31nomaximal
9#10591651—31nomaximal
10C22×((C3⋊D3)⋊C3)⋊C2—21no—
11C2×Q8×H(3,𝔽3)C2211yescharacteristic, complemented
12C2×C4⋊#272C2211yescharacteristic
13C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
14C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
15C2×#6200—21no—
16#88654—22no—
17#88654—22no—
18#88549—22no—
19C2×#6180C2211yescharacteristic, complemented
20#3755⋊C3—22no—
21#3752⋊C3—21no—
22C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic, complemented
23#3750⋊C3—21no—
24C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic, complemented
25C2×#3802—31no—
26C2×D3×D12—31no—
27#64155D311yescharacteristic, complemented
28C2×#3755D311yescharacteristic, complemented
29C3×#946—31no—
30C2×C3×#215—31no—
31C2×C3×#215—31no—
32C2×C3×#215—31no—
33C2×C3×#215—61no—
34#63564—31no—
35#63563—31no—
36#63563—31no—
37#63562—31no—
38#63557—31no—
39(C2×D3)⋊(C3⋊C8)—31no—
40C22×#272—21no—
41C22×(C3⋊D3)⋊C3—21no—
42C22×(C3⋊D3)⋊C3D411yescharacteristic
43C2×((C3⋊D3)⋊C3)⋊C2—24no—
44C2×((C3⋊D3)⋊C3)⋊C2—42no—
45C2×((C3⋊D3)⋊C3)⋊C2—42no—
46Q8×H(3,𝔽3)—21no—
47Q8×H(3,𝔽3)—21no—
48C2×C4×H(3,𝔽3)—21no—
49C2×C4×H(3,𝔽3)C2311yescharacteristic
50C4⋊#272—21no—
51C4⋊#272D412yes—
52C2×(C3⋊Dic3)⋊C3—21no—
53C2×(C3⋊Dic3)⋊C3D411yescharacteristic
54C4⋊((C3⋊D3)⋊C3)—21no—
55C4⋊((C3⋊D3)⋊C3)D412yes—
56C4×(C3⋊D3)⋊C3—21no—
57C4×(C3⋊D3)⋊C3—21no—
58#6200—24no—
59#6180—21no—
60(C3⋊(C3⋊C8))⋊C3—21no—
61#5169—91no—
62#5169—91no—
63#5046—91no—
64C22×D32—61no—
65C22×D32—61no—
66C2×C32×Q8—21no—
67C2×C32×Q8C2×D311yescharacteristic, complemented
68C2×C32×Q8C2×D311yescharacteristic, complemented
69C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
70C2×C4×C3⋊D3C2×D311yescharacteristic
71C2×C3×D12—31no—
72C2×C3×D12—31no—
73C2×C3×D12—31no—
74C2×C3×D12—61no—
75C2×C3×C4×D3—31no—
76C2×#626—61no—
77C2×#625—61no—
78#3802—62no—
79#3802—62no—
80D3×D12—62no—
81D3×D12—62no—
82#3755—22no—
83#3752—21no—
84C4⋊(C3⋊Dic3)C2×D311yescharacteristic, complemented
85#3750—21no—
86C2×C3⋊(C3⋊C8)C2×D311yescharacteristic, complemented
87C3×#215—62no—
88C3×#215—62no—
89C3×#215—62no—
90C3×#215—64no—
91C3×(C2×D3)⋊C4—61no—
92C3×C4⋊Dic3—31no—
93C3×Dic3⋊C4—61no—
94C2×C3×C3⋊C8—31no—
95C2×C3×C3⋊C8—31no—
96C2×C3×C3⋊C8—31no—
97C2×C3×C3⋊C8—61no—
98C22×H(3,𝔽3)C2×D411yescharacteristic
99C2×#272—22no—
100C2×#272—41no—
101C2×#272—41no—
102C2×(C3⋊D3)⋊C3—21no—
103C2×(C3⋊D3)⋊C3—21no—
104C2×(C3⋊D3)⋊C3—22no—
105C2×(C3⋊D3)⋊C3—41no—
106C2×(C3⋊D3)⋊C3—41no—
107C2×(C3⋊D3)⋊C3SD1612yescomplemented
108((C3⋊D3)⋊C3)⋊C2—48no—
109C4×H(3,𝔽3)—41no—
110C4×H(3,𝔽3)C2×D411yescharacteristic
111C4×H(3,𝔽3)C2×D411yescharacteristic
112(C3⋊Dic3)⋊C3—21no—
113(C3⋊Dic3)⋊C3—41no—
114C2×D3×D4—91no—
115C2×D3×D4—91no—
116C22×D12—91no—
117C2×C3×SD16—91no—
118C2×C3×SD16—91no—
119C3×C22⋊Q8—91no—
120#946—31no—
121#946—91no—
122C2×#215—31no—
123C2×#215—31no—
124C2×#215—31no—
125C2×#215—61no—
126C2×#215—91no—
127#832—91no—
128#832—91no—
129#830—91no—
130#821—91no—
131#821—91no—
132(C2×D3)⋊C8—91no—
133(C2×D3)⋊C8—91no—
134#809—91no—
135#809—91no—
136C22×C3⋊D3—21no—
137C22×C3⋊D3#6711yescharacteristic
138C22×C3×D3—31no—
139C22×C3×D3—61no—
140C22×C3×D3—61no—
141C22×C3×D3—61no—
142C22×C3×D3—121no—
143C2×D32—64no—
144C2×D32—64no—
145C2×D32—122no—
146C2×D32—122no—
147C2×D32—122no—
148C2×D32—122no—
149C32×Q8—22no—
150C32×Q8—21no—
151C32×Q8—21no—
152C32×Q8—21no—
153C32×Q8—21no—
154C32×Q8—41no—
155C2×C32×C4—21no—
156C2×C32×C4—21no—
157C2×C32×C4—21no—
158C2×C32×C4—41no—
159C2×C32×C4C22×D311yescharacteristic
160C2×C32×C4C22×D311yescharacteristic
161C2×C3⋊Dic3—21no—
162C2×C3⋊Dic3#6711yescharacteristic
163C4⋊(C3⋊D3)—21no—
164C4⋊(C3⋊D3)#6712yes—
165C4×C3⋊D3—21no—
166C4×C3⋊D3—21no—
167C2×C3×Dic3—31no—
168C2×C3×Dic3—61no—
169C3×D12—32no—
170C3×D12—32no—
171C3×D12—32no—
172C3×D12—62no—
173C3×D12—61no—
174C3×D12—61no—
175C3×D12—61no—
176C3×D12—121no—
177C3×C4×D3—61no—
178C3×C4×D3—61no—
179#626—64no—
180#625—64no—
181C3⋊(C3⋊C8)—21no—
182C3×C3⋊C8—62no—
183C3×C3⋊C8—61no—
184C3×C3⋊C8—61no—
185C3×C3⋊C8—61no—
186#450—271noSylow
187C2×H(3,𝔽3)#12011yescharacteristic
188C2×H(3,𝔽3)C2×SD1611yescharacteristic
189C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
190#272—42no—
191(C3⋊D3)⋊C3—22no—
192(C3⋊D3)⋊C3—42no—
193C23×D3—181no—
194C23×D3—181no—
195C23×D3—181no—
196C2×C3×Q8—31no—
197C2×C3×Q8—31no—
198C2×C3×Q8—61no—
199C2×C3×Q8D3211yescharacteristic
200C2×C3×D4—91no—
201C2×C3×D4—91no—
202C22×C3×C4—91no—
203C2×#67—181no—
204C2×#67—181no—
205D3×D4—182no—
206D3×D4—182no—
207D3×D4—182no—
208D3×D4—182no—
209C2×D12—31no—
210C2×D12—31no—
211C2×D12—31no—
212C2×D12—61no—
213C2×D12—91no—
214C2×D12—182no—
215C2×D12—182no—
216C2×D12—181no—
217C2×C4×D3—31no—
218C2×C4×D3—91no—
219C3×SD16—182no—
220C3×SD16—182no—
221C2×C3×C8—91no—
222C2×C3×C8—91no—
223C3×C4⋊C4—91no—
224C3×C4⋊C4—181no—
225C3×C22⋊C4—181no—
226#215—62no—
227#215—62no—
228#215—62no—
229#215—64no—
230#215—182no—
231(C2×D3)⋊C4—61no—
232(C2×D3)⋊C4—181no—
233C4⋊Dic3—31no—
234C4⋊Dic3—91no—
235Dic3⋊C4—61no—
236Dic3⋊C4—181no—
237C2×C3⋊C8—31no—
238C2×C3⋊C8—31no—
239C2×C3⋊C8—31no—
240C2×C3⋊C8—61no—
241C2×C3⋊C8—91no—
242C22×C32—21no—
243C22×C32D3×D411yescharacteristic
244C22×C32C2×#6711yescharacteristic
245C2×C3⋊D3—21no—
246C2×C3⋊D3—21no—
247C2×C3⋊D3—22no—
248C2×C3⋊D3—41no—
249C2×C3⋊D3—41no—
250C2×C3⋊D3#21512yescomplemented
251C2×C3×D3—32no—
252C2×C3×D3—61no—
253C2×C3×D3—61no—
254C2×C3×D3—62no—
255C2×C3×D3—62no—
256C2×C3×D3—62no—
257C2×C3×D3—122no—
258C2×C3×D3—121no—
259C2×C3×D3—121no—
260C2×C3×D3—121no—
261C2×C3×D3—121no—
262C2×C3×D3—121no—
263C2×C3×D3—121no—
264C2×C3×D3—241no—
265C2×C3×D3—241no—
266D32—128no—
267D32—128no—
268C32×C4—21no—
269C32×C4—21no—
270C32×C4—42no—
271C32×C4—41no—
272C32×C4—41no—
273C32×C4D3×D411yescharacteristic
274C32×C4D3×D411yescharacteristic
275C32×C4D3×D411yescharacteristic
276C32×C4C2×#6711yescharacteristic
277C3⋊Dic3—21no—
278C3⋊Dic3—41no—
279C3×Dic3—61no—
280C3×Dic3—121no—
281C22×D4—271no—
282C2×SD16—91no—
283C2×SD16—91no—
284C22⋊Q8—91no—
285#102—271no—
286#102—271no—
287C22⋊C8—271no—
288H(3,𝔽3)#45011yescharacteristic, complemented, Sylow
289C23×C3—91no—
290C23×C3—181no—
291C23×C3—181no—
292C22×D3—31no—
293C22×D3—61no—
294C22×D3—61no—
295C22×D3—61no—
296C22×D3—91no—
297C22×D3—121no—
298C22×D3—181no—
299C22×D3—184no—
300C22×D3—184no—
301C22×D3—184no—
302C22×D3—362no—
303C22×D3—362no—
304C22×D3—362no—
305C22×D3—362no—
306C22×D3—362no—
307C22×D3—362no—
308C3×Q8—21no—
309C3×Q8—21no—
310C3×Q8—61no—
311C3×Q8—61no—
312C3×Q8—61no—
313C3×Q8—61no—
314C3×Q8—62no—
315C3×Q8—121no—
316C3×D4—92no—
317C3×D4—92no—
318C3×D4—181no—
319C3×D4—181no—
320C2×C3×C4—21no—
321C2×C3×C4—31no—
322C2×C3×C4—31no—
323C2×C3×C4—61no—
324C2×C3×C4—61no—
325C2×C3×C4—61no—
326C2×C3×C4—91no—
327C2×C3×C4—121no—
328C2×C3×C4—181no—
329C2×C3×C4—181no—
330C2×C3×C4—181no—
331C2×C3×C4C2×D3211yescharacteristic
332#67—184no—
333#67—184no—
334C2×Dic3—31no—
335C2×Dic3—61no—
336C2×Dic3—91no—
337C2×Dic3—181no—
338D12—32no—
339D12—32no—
340D12—32no—
341D12—62no—
342D12—61no—
343D12—61no—
344D12—61no—
345D12—92no—
346D12—121no—
347D12—181no—
348D12—184no—
349C4×D3—61no—
350C4×D3—61no—
351C4×D3—181no—
352C4×D3—181no—
353C3×C8—181no—
354C3×C8—181no—
355C3⋊C8—61no—
356C3⋊C8—61no—
357C3⋊C8—62no—
358C3⋊C8—61no—
359C3⋊C8—181no—
360C2×C32—21no—
361C2×C32—21no—
362C2×C32—21no—
363C2×C32D3×SD1611yescharacteristic
364C2×C32#91411yescharacteristic
365C2×C32#93711yescharacteristic
366C2×C32#93711yescharacteristic
367C2×C32C2×#21511yescharacteristic
368C2×C32#94911yescharacteristic
369C3⋊D3—22no—
370C3⋊D3—42no—
371C3×D3—62no—
372C3×D3—122no—
373C3×D3—122no—
374C3×D3—122no—
375C3×D3—242no—
376C24—541no—
377C2×Q8((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
378C2×D4—91no—
379C2×D4—91no—
380C2×D4—541no—
381C2×D4—542no—
382C2×D4—542no—
383C22×C4—91no—
384SD16—182no—
385SD16—182no—
386C2×C8—91no—
387C2×C8—91no—
388C4⋊C4—91no—
389C4⋊C4—181no—
390C22⋊C4—181no—
391C22×C3—31no—
392C22×C3—31no—
393C22×C3—61no—
394C22×C3—92no—
395C22×C3—181no—
396C22×C3—181no—
397C22×C3—182no—
398C22×C3—182no—
399C22×C3—361no—
400C22×C3—361no—
401C22×C3—361no—
402C22×C3—361no—
403C22×C3D3×#6711yescharacteristic
404C2×D3—32no—
405C2×D3—61no—
406C2×D3—61no—
407C2×D3—62no—
408C2×D3—62no—
409C2×D3—62no—
410C2×D3—92no—
411C2×D3—122no—
412C2×D3—121no—
413C2×D3—121no—
414C2×D3—121no—
415C2×D3—121no—
416C2×D3—121no—
417C2×D3—121no—
418C2×D3—181no—
419C2×D3—181no—
420C2×D3—182no—
421C2×D3—241no—
422C2×D3—241no—
423C2×D3—361no—
424C2×D3—361no—
425C2×D3—368no—
426C2×D3—368no—
427C2×D3—368no—
428C3×C4—31no—
429C3×C4—31no—
430C3×C4—31no—
431C3×C4—31no—
432C3×C4—41no—
433C3×C4—61no—
434C3×C4—61no—
435C3×C4—121no—
436C3×C4—121no—
437C3×C4—122no—
438C3×C4—181no—
439C3×C4—361no—
440C3×C4D3×#6711yescharacteristic
441C3×C4#381111yescharacteristic
442Dic3—61no—
443Dic3—121no—
444Dic3—181no—
445Dic3—361no—
446C32—21no—
447C32#504611yescharacteristic, complemented
448C32#516911yescharacteristic, complemented
449C23—91no—
450C23—181no—
451C23—181no—
452C23—544no—
453C23—1082no—
454C23—1082no—
455Q8—21no—
456Q8—21no—
457D4—92no—
458D4—92no—
459D4—181no—
460D4—181no—
461D4—544no—
462C2×C4—21no—
463C2×C4—91no—
464C2×C4—181no—
465C2×C4—181no—
466C2×C4—181no—
467C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
468C8—181no—
469C8—181no—
470C2×C3—31no—
471C2×C3—31no—
472C2×C3—31no—
473C2×C3—31no—
474C2×C3—31no—
475C2×C3—31no—
476C2×C3—61no—
477C2×C3—61no—
478C2×C3—61no—
479C2×C3—182no—
480C2×C3—362no—
481C2×C3—362no—
482C2×C3D3×#21511yescharacteristic
483C2×C3#6393911yescharacteristic
484C2×C3#6398311yescharacteristic
485D3—62no—
486D3—122no—
487D3—122no—
488D3—122no—
489D3—182no—
490D3—242no—
491D3—362no—
492C22—92no—
493C22—181no—
494C22—181no—
495C22—182no—
496C22—182no—
497C22—361no—
498C22—361no—
499C22—361no—
500C22—361no—
501C22—1088no—
502C22#8867611yescharacteristic
503C4—41no—
504C4—181no—
505C4—361no—
506C4#8867611yescharacteristic
507C4#8867711yescharacteristic
508C3—31no—
509C3—31no—
510C3—61no—
511C3#1059165111yescharacteristic
512C2—182no—
513C2—362no—
514C2—362no—
515C2#1172123211yescharacteristic
516C2#1172123311yescharacteristic
517C2#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×#88654C2
hypercenterC2×Q8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#450
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×#3755⋊C3C2semidirect
5. #11720324C2semidirect
6. #11720318C2semidirect
7. #11720317C2semidirect
11. C2×Q8×H(3,𝔽3)C22semidirect
19. C2×#6180C22semidirect
22. C4⋊((C3⋊Dic3)⋊C3)C22semidirect
24. C2×(C3⋊(C3⋊C8))⋊C3C22semidirect
27. #64155D3semidirect
28. C2×#3755D3semidirect
67. C2×C32×Q8C2×D3semidirect
68. C2×C32×Q8C2×D3semidirect
84. C4⋊(C3⋊Dic3)C2×D3semidirect
86. C2×C3⋊(C3⋊C8)C2×D3semidirect
107. C2×(C3⋊D3)⋊C3SD16semidirect
250. C2×C3⋊D3#215semidirect
288. H(3,𝔽3)#450semidirect
377. C2×Q8((C3⋊D3)⋊C3)⋊C2semidirect
447. C32#5046semidirect
448. C32#5169semidirect

Maximal subgroups

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

subgroupindex
C2×#886542
C2×#885492
#64155⋊C32
C2×#3755⋊C32
#117203242
#117203182
#117203172
#105916523
#105916513