Finite Group Databasebeta

← #13139033

Subgroups — #13139033

Order 1728.

708 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 708 conjugacy classes of subgroups — too many to draw. Only its 77 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×#88655C211yesmaximal, characteristic
2C2×#3744⋊C3C211yesmaximal, characteristic, complemented
3#11721094C211yesmaximal, characteristic, complemented
4C2×#88431C211yesmaximal, characteristic, complemented
5#11720553C211yesmaximal, characteristic, complemented
6#63648⋊C3C211yesmaximal, characteristic, complemented
7#11720318C211yesmaximal, characteristic, complemented
8#10590946—31nomaximal
9#10590942—31nomaximal
10C22×((C3⋊D3)⋊C3)⋊C2—21no—
11C2×C4×#272C2211yescharacteristic
12C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
13C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
14C2×#634⋊C3C2211yescharacteristic, complemented
15C2×#6198—21no—
16#88655—22no—
17#88655—22no—
18C2×C8×H(3,𝔽3)C2211yescharacteristic, complemented
19C2×#6180C2211yescharacteristic, complemented
20C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic, complemented
21#3744⋊C3—22no—
22#88451—21no—
23#88448—21no—
24#88431—22no—
25C2×D3×D12—31no—
26C2×D3×D12—31no—
27C2×#3744D311yescharacteristic, complemented
28C2×C3×#204—31no—
29#63857—31no—
30#63856—31no—
31C2×#3717—31no—
32C2×#3716—31no—
33#63648D311yescharacteristic, complemented
34C3×#821—31no—
35C3×(C2×D3)⋊C8—31no—
36C3×(C2×D3)⋊C8—31no—
37C3×(C2×D3)⋊C8—61no—
38#63564—31no—
39#63563—31no—
40C22×#272D411yescharacteristic
41C22×(C3⋊D3)⋊C3—21no—
42C22×(C3⋊D3)⋊C3—21no—
43C2×((C3⋊D3)⋊C3)⋊C2—24no—
44C2×((C3⋊D3)⋊C3)⋊C2—42no—
45C2×((C3⋊D3)⋊C3)⋊C2—42no—
46C2×C4×H(3,𝔽3)C2311yescharacteristic
47C2×#1084D411yescharacteristic
48C4×#272—21no—
49C4×#272—21no—
50C2×(C3⋊Dic3)⋊C3—21no—
51C2×(C3⋊Dic3)⋊C3—21no—
52C4⋊((C3⋊D3)⋊C3)—21no—
53C4⋊((C3⋊D3)⋊C3)—21no—
54C4⋊((C3⋊D3)⋊C3)D412yes—
55C4⋊((C3⋊D3)⋊C3)D412yes—
56#634⋊C3—21no—
57#634⋊C3—21no—
58#6198—24no—
59C8×H(3,𝔽3)—21no—
60#6180—21no—
61#5169—91no—
62#4776—91no—
63#4731—91no—
64C22×D32—61no—
65C22×D32—61no—
66C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
67C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
68C2×#634C2×D311yescharacteristic, complemented
69C2×C3×D12—31no—
70C2×C3×D12—31no—
71C2×C3×C4×D3—31no—
72C2×C3×C4×D3—31no—
73C2×C3×C4×D3—61no—
74C2×C3×Dic6—31no—
75C2×#626—61no—
76C2×#626—61no—
77D3×D12—62no—
78D3×D12—62no—
79D3×D12—62no—
80D3×D12—62no—
81C2×C32×C8—21no—
82C2×C32×C8C2×D311yescharacteristic, complemented
83C2×C32×C8C2×D311yescharacteristic, complemented
84C4⋊(C3⋊Dic3)C2×D311yescharacteristic, complemented
85#3744—22no—
86C3×C4⋊Dic3—31no—
87C2×C3×C3⋊C8—31no—
88C2×C3×C3⋊C8—31no—
89C2×C3×C3⋊C8—61no—
90C3×#204—62no—
91#3724—61no—
92Dic3⋊Dic3—61no—
93(C2×D3)⋊Dic3—61no—
94(C2×D3)⋊Dic3—61no—
95#3717—62no—
96#3716—62no—
97C22×H(3,𝔽3)C2×D411yescharacteristic
98C2×#272—21no—
99C2×#272—21no—
100C2×#272SD1612yescomplemented
101C2×(C3⋊D3)⋊C3—22no—
102C2×(C3⋊D3)⋊C3—22no—
103C2×(C3⋊D3)⋊C3—41no—
104C2×(C3⋊D3)⋊C3—41no—
105C2×(C3⋊D3)⋊C3—41no—
106C2×(C3⋊D3)⋊C3—41no—
107((C3⋊D3)⋊C3)⋊C2—48no—
108C4×H(3,𝔽3)C2×D411yescharacteristic
109C4×H(3,𝔽3)C2×D411yescharacteristic
110#1084—21no—
111(C3⋊Dic3)⋊C3—41no—
112(C3⋊Dic3)⋊C3—41no—
113C2×D3×D4—91no—
114C2×D3×D4—91no—
115C22×D12—91no—
116C2×C3×SD16—91no—
117#946—91no—
118C2×#215—91no—
119C2×#214—91no—
120#924—91no—
121C2×#204—31no—
122C2×#204—91no—
123C2×#204—91no—
124#897—91no—
125C3×#102—91no—
126C3×C22⋊C8—91no—
127#832—91no—
128#821—31no—
129#821—91no—
130#821—91no—
131(C2×D3)⋊C8—31no—
132(C2×D3)⋊C8—31no—
133(C2×D3)⋊C8—61no—
134#809—91no—
135C22×C3⋊D3—21no—
136C22×C3⋊D3—21no—
137C22×C3×D3—31no—
138C22×C3×D3—31no—
139C22×C3×D3—61no—
140C22×C3×D3—61no—
141C22×C3×D3—61no—
142C2×D32—64no—
143C2×D32—64no—
144C2×D32—122no—
145C2×D32—122no—
146C2×D32—122no—
147C2×D32—122no—
148C2×C32×C4—21no—
149C2×C32×C4C22×D311yescharacteristic
150C2×C32×C4C22×D311yescharacteristic
151C2×C3⋊Dic3—21no—
152C2×C3⋊Dic3—21no—
153C4⋊(C3⋊D3)—21no—
154C4⋊(C3⋊D3)—21no—
155C4⋊(C3⋊D3)D1212yes—
156C4⋊(C3⋊D3)#6712yes—
157#634—21no—
158#634—21no—
159C2×C3×Dic3—31no—
160C2×C3×Dic3—31no—
161C2×C3×Dic3—61no—
162C2×C3×Dic3—61no—
163C2×C3×Dic3—61no—
164C3×D12—32no—
165C3×D12—32no—
166C3×D12—61no—
167C3×D12—61no—
168C3×C4×D3—61no—
169C3×C4×D3—61no—
170C3×C4×D3—61no—
171C3×C4×D3—61no—
172C3×C4×D3—121no—
173C3×C4×D3—121no—
174C3×Dic6—61no—
175C3×Dic6—61no—
176#626—64no—
177#626—64no—
178C32×C8—21no—
179C32×C8—21no—
180C32×C8—41no—
181C3×C3⋊C8—61no—
182C3×C3⋊C8—61no—
183C3×C3⋊C8—121no—
184#450—271noSylow
185C2×H(3,𝔽3)#12011yescharacteristic
186C2×H(3,𝔽3)C2×SD1611yescharacteristic
187C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
188#272—22no—
189(C3⋊D3)⋊C3—42no—
190(C3⋊D3)⋊C3—42no—
191C23×D3—181no—
192C23×D3—181no—
193C23×D3—181no—
194C2×C3×Q8—91no—
195C2×C3×D4—91no—
196C2×C3×D4—91no—
197C22×C3×C4—91no—
198C2×#67—181no—
199C2×#67—181no—
200D3×D4—182no—
201D3×D4—182no—
202D3×D4—182no—
203D3×D4—182no—
204C2×D12—31no—
205C2×D12—31no—
206C2×D12—91no—
207C2×D12—91no—
208C2×D12—182no—
209C2×D12—182no—
210C2×D12—181no—
211C2×C4×D3—31no—
212C2×C4×D3—31no—
213C2×C4×D3—61no—
214C2×Dic6—31no—
215C2×Dic6—91no—
216C3×SD16—182no—
217C2×C3×C8—31no—
218C2×C3×C8—31no—
219C2×C3×C8—61no—
220C2×C3×C8—91no—
221C2×C3×C8D3211yescharacteristic
222C3×C4⋊C4—91no—
223#217—181no—
224#215—182no—
225#214—182no—
226(C2×D3)⋊C4—181no—
227(C2×D3)⋊C4—181no—
228C4⋊Dic3—31no—
229C4⋊Dic3—91no—
230C4⋊Dic3—181no—
231Dic3⋊C4—181no—
232Dic3⋊C4—181no—
233C2×C3⋊C8—31no—
234C2×C3⋊C8—31no—
235C2×C3⋊C8—61no—
236#204—62no—
237#204—182no—
238#204—182no—
239C22×C32—21no—
240C22×C32D3×D411yescharacteristic
241C22×C32D3×D411yescharacteristic
242C2×C3⋊D3—22no—
243C2×C3⋊D3—22no—
244C2×C3⋊D3—41no—
245C2×C3⋊D3—41no—
246C2×C3⋊D3—41no—
247C2×C3⋊D3—41no—
248C2×C3×D3—32no—
249C2×C3×D3—32no—
250C2×C3×D3—62no—
251C2×C3×D3—61no—
252C2×C3×D3—61no—
253C2×C3×D3—61no—
254C2×C3×D3—61no—
255C2×C3×D3—62no—
256C2×C3×D3—62no—
257C2×C3×D3—121no—
258C2×C3×D3—121no—
259C2×C3×D3—121no—
260C2×C3×D3—121no—
261C2×C3×D3—121no—
262C2×C3×D3—121no—
263D32—128no—
264D32—128no—
265C32×C4—21no—
266C32×C4—21no—
267C32×C4C2×D1211yescharacteristic
268C32×C4D3×D411yescharacteristic
269C32×C4D3×D411yescharacteristic
270C32×C4C2×#6711yescharacteristic
271C3⋊Dic3—41no—
272C3⋊Dic3—41no—
273C3×Dic3—61no—
274C3×Dic3—61no—
275C3×Dic3—121no—
276C3×Dic3—121no—
277C3×Dic3—121no—
278C22×D4—271no—
279C2×SD16—91no—
280C2×SD16—271no—
281C22⋊Q8—271no—
282#102—91no—
283#102—271no—
284C22⋊C8—91no—
285H(3,𝔽3)#45011yescharacteristic, complemented, Sylow
286C23×C3—91no—
287C23×C3—181no—
288C23×C3—181no—
289C22×D3—31no—
290C22×D3—31no—
291C22×D3—61no—
292C22×D3—61no—
293C22×D3—61no—
294C22×D3—181no—
295C22×D3—181no—
296C22×D3—184no—
297C22×D3—184no—
298C22×D3—184no—
299C22×D3—362no—
300C22×D3—362no—
301C22×D3—362no—
302C22×D3—362no—
303C22×D3—362no—
304C22×D3—362no—
305C3×Q8—181no—
306C3×Q8—181no—
307C3×D4—92no—
308C3×D4—92no—
309C3×D4—181no—
310C3×D4—181no—
311C2×C3×C4—31no—
312C2×C3×C4—31no—
313C2×C3×C4—61no—
314C2×C3×C4—91no—
315C2×C3×C4—181no—
316C2×C3×C4—181no—
317C2×C3×C4—181no—
318C2×C3×C4—181no—
319C2×C3×C4C2×D3211yescharacteristic
320#67—184no—
321#67—184no—
322C2×Dic3—31no—
323C2×Dic3—31no—
324C2×Dic3—61no—
325C2×Dic3—61no—
326C2×Dic3—61no—
327C2×Dic3—181no—
328C2×Dic3—181no—
329D12—32no—
330D12—32no—
331D12—61no—
332D12—61no—
333D12—92no—
334D12—92no—
335D12—181no—
336D12—181no—
337D12—184no—
338C4×D3—61no—
339C4×D3—61no—
340C4×D3—61no—
341C4×D3—61no—
342C4×D3—121no—
343C4×D3—121no—
344Dic6—61no—
345Dic6—61no—
346Dic6—181no—
347Dic6—181no—
348C3×C8—21no—
349C3×C8—61no—
350C3×C8—61no—
351C3×C8—121no—
352C3×C8—181no—
353C3⋊C8—61no—
354C3⋊C8—61no—
355C3⋊C8—121no—
356C2×C32—21no—
357C2×C32—21no—
358C2×C32—21no—
359C2×C32#88211yescharacteristic
360C2×C32#91111yescharacteristic
361C2×C32D3×SD1611yescharacteristic
362C2×C32D3×SD1611yescharacteristic
363C2×C32#91411yescharacteristic
364C2×C32#93711yescharacteristic
365C3⋊D3—42no—
366C3⋊D3—42no—
367C3×D3—62no—
368C3×D3—62no—
369C3×D3—122no—
370C3×D3—122no—
371C3×D3—122no—
372C24—541no—
373C2×Q8—91no—
374C2×D4—91no—
375C2×D4—91no—
376C2×D4—541no—
377C2×D4—542no—
378C2×D4—542no—
379C22×C4—91no—
380SD16—182no—
381SD16—542no—
382C2×C8—91no—
383C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
384C4⋊C4—91no—
385C4⋊C4—541no—
386C22⋊C4—541no—
387C22×C3—31no—
388C22×C3—31no—
389C22×C3—61no—
390C22×C3—92no—
391C22×C3—182no—
392C22×C3—182no—
393C22×C3—181no—
394C22×C3—181no—
395C22×C3—361no—
396C22×C3—361no—
397C22×C3—361no—
398C22×C3—361no—
399C22×C3#380211yescharacteristic
400C2×D3—32no—
401C2×D3—32no—
402C2×D3—62no—
403C2×D3—61no—
404C2×D3—61no—
405C2×D3—61no—
406C2×D3—61no—
407C2×D3—62no—
408C2×D3—62no—
409C2×D3—121no—
410C2×D3—121no—
411C2×D3—121no—
412C2×D3—121no—
413C2×D3—121no—
414C2×D3—121no—
415C2×D3—182no—
416C2×D3—182no—
417C2×D3—361no—
418C2×D3—361no—
419C2×D3—361no—
420C2×D3—361no—
421C2×D3—368no—
422C2×D3—368no—
423C2×D3—368no—
424C3×C4—31no—
425C3×C4—31no—
426C3×C4—31no—
427C3×C4—31no—
428C3×C4—61no—
429C3×C4—61no—
430C3×C4—181no—
431C3×C4—361no—
432C3×C4—361no—
433C3×C4D3×D1211yescharacteristic
434C3×C4D3×#6711yescharacteristic
435Dic3—61no—
436Dic3—61no—
437Dic3—121no—
438Dic3—121no—
439Dic3—121no—
440Dic3—361no—
441Dic3—361no—
442C32—21no—
443C32#477611yescharacteristic, complemented
444C32#516911yescharacteristic, complemented
445C23—91no—
446C23—181no—
447C23—181no—
448C23—544no—
449C23—1082no—
450C23—1082no—
451Q8—181no—
452Q8—181no—
453D4—92no—
454D4—92no—
455D4—181no—
456D4—181no—
457D4—544no—
458C2×C4—91no—
459C2×C4—181no—
460C2×C4—181no—
461C2×C4—181no—
462C2×C4—181no—
463C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
464C8—21no—
465C8—181no—
466C2×C3—31no—
467C2×C3—31no—
468C2×C3—31no—
469C2×C3—31no—
470C2×C3—31no—
471C2×C3—31no—
472C2×C3—61no—
473C2×C3—61no—
474C2×C3—61no—
475C2×C3—182no—
476C2×C3—362no—
477C2×C3—362no—
478C2×C3#6379611yescharacteristic
479C2×C3#6379811yescharacteristic
480C2×C3#6392211yescharacteristic
481D3—62no—
482D3—62no—
483D3—122no—
484D3—122no—
485D3—122no—
486D3—362no—
487D3—362no—
488C22—92no—
489C22—181no—
490C22—181no—
491C22—182no—
492C22—182no—
493C22—361no—
494C22—361no—
495C22—361no—
496C22—361no—
497C22—1088no—
498C22#8865511yescharacteristic
499C4—181no—
500C4—361no—
501C4—361no—
502C4#8865411yescharacteristic
503C4#8867611yescharacteristic
504C3—31no—
505C3—31no—
506C3—61no—
507C3#1059094311yescharacteristic
508C2—182no—
509C2—362no—
510C2—362no—
511C2#1172083711yescharacteristic
512C2#1172083911yescharacteristic
513C2#1172116211yescharacteristic
subgroupG/H
centerC22#88655
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3#3802
prime radicalC2×#88655C2
hypercenterC2×C8((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×C8

Complements

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

NH ≅ G/Ntype
2. C2×#3744⋊C3C2semidirect
3. #11721094C2semidirect
4. C2×#88431C2semidirect
5. #11720553C2semidirect
6. #63648⋊C3C2semidirect
7. #11720318C2semidirect
14. C2×#634⋊C3C22semidirect
18. C2×C8×H(3,𝔽3)C22semidirect
19. C2×#6180C22semidirect
20. C4⋊((C3⋊Dic3)⋊C3)C22semidirect
27. C2×#3744D3semidirect
33. #63648D3semidirect
68. C2×#634C2×D3semidirect
82. C2×C32×C8C2×D3semidirect
83. C2×C32×C8C2×D3semidirect
84. C4⋊(C3⋊Dic3)C2×D3semidirect
100. C2×#272SD16semidirect
285. H(3,𝔽3)#450semidirect
383. C2×C8((C3⋊D3)⋊C3)⋊C2semidirect
443. C32#4776semidirect
444. C32#5169semidirect

Maximal subgroups

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

subgroupindex
C2×#886552
C2×#3744⋊C32
#117210942
C2×#884312
#117205532
#63648⋊C32
#117203182
#105909463
#105909423