Finite Group Databasebeta

← #13138056

Subgroups — #13138056

Order 1728.

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

normal subgroups only

This group has 708 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
2#64106⋊C3C211yesmaximal, characteristic, complemented
3C2×#88437C211yesmaximal, characteristic, complemented
4C2×#88431C211yesmaximal, characteristic, complemented
5#11720531C211yesmaximal, characteristic, complemented
6#63636⋊C3C211yesmaximal, characteristic, complemented
7#11720317C211yesmaximal, characteristic, complemented
8#10590597—31nomaximal
9#10590596—31nomaximal
10C22×((C3⋊D3)⋊C3)⋊C2—21no—
11C2×C4⋊#272C2211yescharacteristic
12C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
13C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
14C2×#634⋊C3C2211yescharacteristic, complemented
15C2×#6200—21no—
16#88654—22no—
17#88654—22no—
18C4⋊C4×H(3,𝔽3)C2211yescharacteristic, complemented
19C2×#6180C2211yescharacteristic, complemented
20#3752⋊C3—21no—
21C4⋊((C3⋊Dic3)⋊C3)—21no—
22C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic, complemented
23#88437—22no—
24#88431—22no—
25C2×#3802—31no—
26C2×D3×D12—31no—
27#64106D311yescharacteristic, complemented
28C3×#897—31no—
29C2×#3717—31no—
30C2×#3716—31no—
31C2×#3716—31no—
32C2×#3715—31no—
33#63636D311yescharacteristic, complemented
34C3×#809—31no—
35C3×#809—31no—
36C3×#809—31no—
37C3×#809—61no—
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—
46C2×C4×H(3,𝔽3)—21no—
47C2×C4×H(3,𝔽3)C2311yescharacteristic
48C4⋊#272—21no—
49C4⋊#272D412yes—
50C2×(C3⋊Dic3)⋊C3—21no—
51C2×(C3⋊Dic3)⋊C3D411yescharacteristic
52C4⋊((C3⋊D3)⋊C3)—21no—
53C4⋊((C3⋊D3)⋊C3)D412yes—
54C4×(C3⋊D3)⋊C3—21no—
55C4×(C3⋊D3)⋊C3—21no—
56#634⋊C3—21no—
57#634⋊C3—21no—
58#6200—24no—
59#6180—21no—
60(C3⋊(C3⋊C8))⋊C3—21no—
61#5046—91no—
62#4776—91no—
63#4776—91no—
64C22×D32—61no—
65C22×D32—61no—
66C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
67C2×C4×C3⋊D3C2×D311yescharacteristic
68C2×#634C2×D311yescharacteristic, complemented
69C2×C3×D12—31no—
70C2×C3×D12—31no—
71C2×C3×D12—31no—
72C2×C3×D12—61no—
73C2×C3×C4×D3—31no—
74C2×C3×Dic6—31no—
75C2×#626—61no—
76C2×#625—61no—
77#3802—62no—
78#3802—62no—
79D3×D12—62no—
80D3×D12—62no—
81C32×C4⋊C4—21no—
82C32×C4⋊C4C2×D311yescharacteristic, complemented
83C32×C4⋊C4C2×D311yescharacteristic, complemented
84#3752—21no—
85C4⋊(C3⋊Dic3)—21no—
86C2×C3⋊(C3⋊C8)C2×D311yescharacteristic, complemented
87C3×(C2×D3)⋊C4—61no—
88C3×C4⋊Dic3—61no—
89C2×C3×C3⋊C8—31no—
90C2×C3×C3⋊C8—31no—
91C2×C3×C3⋊C8—31no—
92C2×C3×C3⋊C8—61no—
93#3717—62no—
94#3716—62no—
95#3716—62no—
96#3715—62no—
97C22×H(3,𝔽3)C2×D411yescharacteristic
98C2×#272—22no—
99C2×#272—41no—
100C2×#272—41no—
101C2×(C3⋊D3)⋊C3—21no—
102C2×(C3⋊D3)⋊C3—21no—
103C2×(C3⋊D3)⋊C3—22no—
104C2×(C3⋊D3)⋊C3—41no—
105C2×(C3⋊D3)⋊C3—41no—
106C2×(C3⋊D3)⋊C3SD1612yescomplemented
107((C3⋊D3)⋊C3)⋊C2—48no—
108C4×H(3,𝔽3)—41no—
109C4×H(3,𝔽3)C2×D411yescharacteristic
110C4×H(3,𝔽3)C2×D411yescharacteristic
111(C3⋊Dic3)⋊C3—21no—
112(C3⋊Dic3)⋊C3—41no—
113C2×D3×D4—91no—
114C2×D3×D4—91no—
115C22×D12—91no—
116C3×C22⋊Q8—91no—
117C2×#215—91no—
118C2×#215—91no—
119C2×#214—91no—
120C2×#214—91no—
121C2×#204—91no—
122C2×#204—91no—
123#897—31no—
124#897—91no—
125C3×#102—91no—
126C3×#102—91no—
127#830—91no—
128(C2×D3)⋊C8—91no—
129(C2×D3)⋊C8—91no—
130#809—31no—
131#809—31no—
132#809—31no—
133#809—61no—
134#809—91no—
135C22×C3⋊D3—21no—
136C22×C3⋊D3#6711yescharacteristic
137C22×C3×D3—31no—
138C22×C3×D3—61no—
139C22×C3×D3—61no—
140C22×C3×D3—61no—
141C22×C3×D3—121no—
142C2×D32—64no—
143C2×D32—64no—
144C2×D32—122no—
145C2×D32—122no—
146C2×D32—122no—
147C2×D32—122no—
148C2×C32×C4—21no—
149C2×C32×C4—21no—
150C2×C32×C4—21no—
151C2×C32×C4—41no—
152C2×C32×C4C22×D311yescharacteristic
153C2×C32×C4C22×D311yescharacteristic
154C2×C3⋊Dic3—21no—
155C2×C3⋊Dic3#6711yescharacteristic
156C4⋊(C3⋊D3)—21no—
157C4⋊(C3⋊D3)D1212yes—
158C4×C3⋊D3—21no—
159C4×C3⋊D3—21no—
160#634—21no—
161#634—21no—
162C2×C3×Dic3—31no—
163C2×C3×Dic3—61no—
164C3×D12—32no—
165C3×D12—32no—
166C3×D12—32no—
167C3×D12—62no—
168C3×D12—61no—
169C3×D12—61no—
170C3×D12—61no—
171C3×D12—121no—
172C3×C4×D3—61no—
173C3×C4×D3—61no—
174C3×Dic6—61no—
175C3×Dic6—61no—
176#626—64no—
177#625—64no—
178C3⋊(C3⋊C8)—21no—
179C3×C3⋊C8—61no—
180C3×C3⋊C8—61no—
181C3×C3⋊C8—61no—
182C3×C3⋊C8—121no—
183#450—271noSylow
184C2×H(3,𝔽3)#12011yescharacteristic
185C2×H(3,𝔽3)C2×SD1611yescharacteristic
186C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
187#272—42no—
188(C3⋊D3)⋊C3—22no—
189(C3⋊D3)⋊C3—42no—
190C23×D3—181no—
191C23×D3—181no—
192C23×D3—181no—
193C2×C3×Q8—91no—
194C2×C3×D4—91no—
195C2×C3×D4—91no—
196C22×C3×C4—91no—
197C2×#67—181no—
198C2×#67—181no—
199D3×D4—182no—
200D3×D4—182no—
201D3×D4—182no—
202D3×D4—182no—
203C2×D12—31no—
204C2×D12—31no—
205C2×D12—31no—
206C2×D12—61no—
207C2×D12—91no—
208C2×D12—182no—
209C2×D12—182no—
210C2×D12—181no—
211C2×C4×D3—31no—
212C2×C4×D3—91no—
213C2×Dic6—31no—
214C2×Dic6—91no—
215C2×C3×C8—91no—
216C2×C3×C8—91no—
217C3×C4⋊C4—31no—
218C3×C4⋊C4—31no—
219C3×C4⋊C4—61no—
220C3×C4⋊C4—181no—
221C3×C4⋊C4D3211yescharacteristic
222C3×C22⋊C4—181no—
223#215—182no—
224#215—182no—
225#214—182no—
226#214—182no—
227(C2×D3)⋊C4—61no—
228(C2×D3)⋊C4—181no—
229C4⋊Dic3—61no—
230C4⋊Dic3—181no—
231C2×C3⋊C8—31no—
232C2×C3⋊C8—31no—
233C2×C3⋊C8—31no—
234C2×C3⋊C8—61no—
235C2×C3⋊C8—91no—
236#204—182no—
237#204—182no—
238C22×C32—21no—
239C22×C32D3×D411yescharacteristic
240C22×C32C2×#6711yescharacteristic
241C2×C3⋊D3—21no—
242C2×C3⋊D3—21no—
243C2×C3⋊D3—22no—
244C2×C3⋊D3—41no—
245C2×C3⋊D3—41no—
246C2×C3⋊D3#21512yescomplemented
247C2×C3×D3—32no—
248C2×C3×D3—61no—
249C2×C3×D3—61no—
250C2×C3×D3—62no—
251C2×C3×D3—62no—
252C2×C3×D3—62no—
253C2×C3×D3—122no—
254C2×C3×D3—121no—
255C2×C3×D3—121no—
256C2×C3×D3—121no—
257C2×C3×D3—121no—
258C2×C3×D3—121no—
259C2×C3×D3—121no—
260C2×C3×D3—241no—
261C2×C3×D3—241no—
262D32—128no—
263D32—128no—
264C32×C4—21no—
265C32×C4—21no—
266C32×C4—41no—
267C32×C4—41no—
268C32×C4—81no—
269C32×C4C2×D1211yescharacteristic
270C32×C4D3×D411yescharacteristic
271C32×C4D3×D411yescharacteristic
272C32×C4D3×D411yescharacteristic
273C3⋊Dic3—21no—
274C3⋊Dic3—41no—
275C3×Dic3—61no—
276C3×Dic3—121no—
277C22×D4—271no—
278C2×SD16—271no—
279C2×SD16—271no—
280C22⋊Q8—91no—
281#102—91no—
282#102—91no—
283C22⋊C8—271no—
284H(3,𝔽3)#45011yescharacteristic, complemented, Sylow
285C23×C3—91no—
286C23×C3—181no—
287C23×C3—181no—
288C22×D3—31no—
289C22×D3—61no—
290C22×D3—61no—
291C22×D3—61no—
292C22×D3—91no—
293C22×D3—121no—
294C22×D3—181no—
295C22×D3—184no—
296C22×D3—184no—
297C22×D3—184no—
298C22×D3—362no—
299C22×D3—362no—
300C22×D3—362no—
301C22×D3—362no—
302C22×D3—362no—
303C22×D3—362no—
304C3×Q8—181no—
305C3×Q8—181no—
306C3×D4—92no—
307C3×D4—92no—
308C3×D4—181no—
309C3×D4—181no—
310C2×C3×C4—21no—
311C2×C3×C4—31no—
312C2×C3×C4—31no—
313C2×C3×C4—61no—
314C2×C3×C4—61no—
315C2×C3×C4—61no—
316C2×C3×C4—91no—
317C2×C3×C4—121no—
318C2×C3×C4—181no—
319C2×C3×C4—181no—
320C2×C3×C4—181no—
321C2×C3×C4C2×D3211yescharacteristic
322#67—184no—
323#67—184no—
324C2×Dic3—31no—
325C2×Dic3—61no—
326C2×Dic3—91no—
327C2×Dic3—181no—
328D12—32no—
329D12—32no—
330D12—32no—
331D12—62no—
332D12—61no—
333D12—61no—
334D12—61no—
335D12—92no—
336D12—121no—
337D12—181no—
338D12—184no—
339C4×D3—61no—
340C4×D3—61no—
341C4×D3—181no—
342C4×D3—181no—
343Dic6—61no—
344Dic6—61no—
345Dic6—181no—
346Dic6—181no—
347C3×C8—181no—
348C3×C8—181no—
349C3⋊C8—61no—
350C3⋊C8—61no—
351C3⋊C8—61no—
352C3⋊C8—121no—
353C3⋊C8—181no—
354C2×C32—21no—
355C2×C32—21no—
356C2×C32—21no—
357C2×C32#88211yescharacteristic
358C2×C32#91111yescharacteristic
359C2×C32D3×SD1611yescharacteristic
360C2×C32#93711yescharacteristic
361C2×C32C2×#21511yescharacteristic
362C2×C32#94911yescharacteristic
363C3⋊D3—22no—
364C3⋊D3—42no—
365C3×D3—62no—
366C3×D3—122no—
367C3×D3—122no—
368C3×D3—122no—
369C3×D3—242no—
370C24—541no—
371C2×Q8—91no—
372C2×D4—91no—
373C2×D4—91no—
374C2×D4—541no—
375C2×D4—542no—
376C2×D4—542no—
377C22×C4—91no—
378SD16—542no—
379SD16—542no—
380C2×C8—91no—
381C2×C8—91no—
382C4⋊C4—181no—
383C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
384C22⋊C4—181no—
385C22×C3—31no—
386C22×C3—31no—
387C22×C3—61no—
388C22×C3—92no—
389C22×C3—181no—
390C22×C3—181no—
391C22×C3—182no—
392C22×C3—182no—
393C22×C3—361no—
394C22×C3—361no—
395C22×C3—361no—
396C22×C3—361no—
397C22×C3D3×#6711yescharacteristic
398C2×D3—32no—
399C2×D3—61no—
400C2×D3—61no—
401C2×D3—62no—
402C2×D3—62no—
403C2×D3—62no—
404C2×D3—92no—
405C2×D3—122no—
406C2×D3—121no—
407C2×D3—121no—
408C2×D3—121no—
409C2×D3—121no—
410C2×D3—121no—
411C2×D3—121no—
412C2×D3—181no—
413C2×D3—181no—
414C2×D3—182no—
415C2×D3—241no—
416C2×D3—241no—
417C2×D3—361no—
418C2×D3—361no—
419C2×D3—368no—
420C2×D3—368no—
421C2×D3—368no—
422C3×C4—31no—
423C3×C4—31no—
424C3×C4—31no—
425C3×C4—31no—
426C3×C4—41no—
427C3×C4—61no—
428C3×C4—61no—
429C3×C4—121no—
430C3×C4—121no—
431C3×C4—181no—
432C3×C4—241no—
433C3×C4—361no—
434C3×C4D3×D1211yescharacteristic
435C3×C4#380211yescharacteristic
436Dic3—61no—
437Dic3—121no—
438Dic3—181no—
439Dic3—361no—
440C32—21no—
441C32#477611yescharacteristic, complemented
442C32#504611yescharacteristic, complemented
443C23—91no—
444C23—181no—
445C23—181no—
446C23—544no—
447C23—1082no—
448C23—1082no—
449Q8—181no—
450Q8—181no—
451D4—92no—
452D4—92no—
453D4—181no—
454D4—181no—
455D4—544no—
456C2×C4—21no—
457C2×C4—91no—
458C2×C4—181no—
459C2×C4—181no—
460C2×C4—181no—
461C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
462C8—181no—
463C8—181no—
464C2×C3—31no—
465C2×C3—31no—
466C2×C3—31no—
467C2×C3—31no—
468C2×C3—31no—
469C2×C3—31no—
470C2×C3—61no—
471C2×C3—61no—
472C2×C3—61no—
473C2×C3—182no—
474C2×C3—362no—
475C2×C3—362no—
476C2×C3#6392211yescharacteristic
477C2×C3#6392511yescharacteristic
478C2×C3D3×#21511yescharacteristic
479D3—62no—
480D3—122no—
481D3—122no—
482D3—122no—
483D3—182no—
484D3—242no—
485D3—362no—
486C22—92no—
487C22—181no—
488C22—181no—
489C22—182no—
490C22—182no—
491C22—361no—
492C22—361no—
493C22—361no—
494C22—361no—
495C22—1088no—
496C22#8867611yescharacteristic
497C4—41no—
498C4—181no—
499C4—361no—
500C4#8865411yescharacteristic
501C4#8865511yescharacteristic
502C3—31no—
503C3—31no—
504C3—61no—
505C3#1059059611yescharacteristic
506C2—182no—
507C2—362no—
508C2—362no—
509C2#1172116211yescharacteristic
510C2#1172119511yescharacteristic
511C2#1172123211yescharacteristic
subgroupG/H
centerC22#88676
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC4⋊C4×H(3,𝔽3)C22
socleC22×C3D3×#67
prime radicalC2×#88654C2
hypercenterC4⋊C4((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 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #64106⋊C3C2semidirect
3. C2×#88437C2semidirect
4. C2×#88431C2semidirect
5. #11720531C2semidirect
6. #63636⋊C3C2semidirect
7. #11720317C2semidirect
14. C2×#634⋊C3C22semidirect
18. C4⋊C4×H(3,𝔽3)C22semidirect
19. C2×#6180C22semidirect
22. C2×(C3⋊(C3⋊C8))⋊C3C22semidirect
27. #64106D3semidirect
33. #63636D3semidirect
68. C2×#634C2×D3semidirect
82. C32×C4⋊C4C2×D3semidirect
83. C32×C4⋊C4C2×D3semidirect
86. C2×C3⋊(C3⋊C8)C2×D3semidirect
106. C2×(C3⋊D3)⋊C3SD16semidirect
246. C2×C3⋊D3#215semidirect
284. H(3,𝔽3)#450semidirect
383. C4⋊C4((C3⋊D3)⋊C3)⋊C2semidirect
441. C32#4776semidirect
442. C32#5046semidirect

Maximal subgroups

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

subgroupindex
C2×#886542
#64106⋊C32
C2×#884372
C2×#884312
#117205312
#63636⋊C32
#117203172
#105905973
#105905963