Finite Group Databasebeta

← #13139038

Subgroups — #13139038

Order 1728.

635 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 635 conjugacy classes of subgroups — too many to draw. Only its 87 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×#88649C211yesmaximal, characteristic
2C2×#88527C211yesmaximal, characteristic, complemented
3#11721152C211yesmaximal, characteristic, complemented
4#63648⋊C3C211yesmaximal, characteristic, complemented
5#63643⋊C3C211yesmaximal, characteristic, complemented
6#11720318C211yesmaximal, characteristic, complemented
7#11720313C211yesmaximal, characteristic, complemented
8#10590671—31nomaximal
9#10590670—31nomaximal
10C2×C4×#272C2211yescharacteristic
11C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
12C2×#634⋊C3C2211yescharacteristic
13C2×#6200—21no—
14C2×#6199—21no—
15C2×((C3⋊D3)⋊C3)⋊C4—21no—
16#88649—22no—
17#88649C414yescomplemented
18C2×C8×H(3,𝔽3)C2211yescharacteristic
19C2×#6180C2211yescharacteristic
20#88527—22no—
21C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
22C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
23#88449—21no—
24C2×#3799—31no—
25C2×#3795—31no—
26C2×C3×ΓL(1,𝔽25)—31no—
27C2×C3×ΓL(1,𝔽25)—31no—
28C2×C3×ΓL(1,𝔽25)—61no—
29D3×C4⋊Dic3—31no—
30D3×C4⋊Dic3—31no—
31#63648D311yescharacteristic, complemented
32#63643D311yescharacteristic, complemented
33C3×#821—31no—
34C3×#816—31no—
35#63567—31no—
36#63566—31no—
37#63564—31no—
38#63563—31no—
39C22×#272D411yescharacteristic
40C22×(C3⋊D3)⋊C3—21no—
41C2×C4×H(3,𝔽3)C2311yescharacteristic
42C2×#1084D411yescharacteristic
43C4×#272C2×C412yes—
44C4×#272D412yes—
45C2×(C3⋊Dic3)⋊C3—21no—
46C2×(C3⋊Dic3)⋊C3—21no—
47C2×(C3⋊Dic3)⋊C3—21no—
48C4⋊((C3⋊D3)⋊C3)—21no—
49C4⋊((C3⋊D3)⋊C3)C2×C412yescomplemented
50#634⋊C3—21no—
51#634⋊C3C2×C412yescomplemented
52#6200—24no—
53#6199—42no—
54((C3⋊D3)⋊C3)⋊C4—24no—
55C8×H(3,𝔽3)—21no—
56#6180—21no—
57#5133—91no—
58#4803—91no—
59#4771—91no—
60C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
61C2×#634C2×D311yescharacteristic
62C2×C3×D12—31no—
63C2×C3×C4×D3—31no—
64C2×C3×C4×D3—31no—
65C2×C3×C4×D3—61no—
66C2×C3×Dic6—31no—
67C2×#626—61no—
68C2×#625—61no—
69C2×#624—61no—
70C2×D3×Dic3—61no—
71C2×D3×Dic3—61no—
72C2×D3×Dic3—61no—
73#3799—34no—
74#3799—62no—
75#3795—34no—
76#3795—62no—
77C2×C32×C8—21no—
78C2×C32×C8C2×D311yescharacteristic
79C2×C32×C8C2×D311yescharacteristic
80C4⋊(C3⋊Dic3)C2×D311yescharacteristic
81C4⋊(C3⋊Dic3)C2×D311yescharacteristic
82C3×C4⋊Dic3—31no—
83C3×C4⋊Dic3—31no—
84C2×C3×C3⋊C8—31no—
85C2×C3×C3⋊C8—31no—
86C2×C3×C3⋊C8—61no—
87C3×ΓL(1,𝔽25)—62no—
88C3×ΓL(1,𝔽25)—62no—
89C3×ΓL(1,𝔽25)—64no—
90Dic3⋊Dic3—61no—
91Dic3⋊Dic3—61no—
92C22×H(3,𝔽3)C2×D411yescharacteristic
93C2×#272—21no—
94C2×#272—21no—
95C2×#272C22⋊C412yes—
96C2×(C3⋊D3)⋊C3—22no—
97C2×(C3⋊D3)⋊C3—41no—
98C2×(C3⋊D3)⋊C3—41no—
99C4×H(3,𝔽3)C22×C411yescharacteristic
100C4×H(3,𝔽3)C2×D411yescharacteristic
101#1084C22⋊C412yes—
102(C3⋊Dic3)⋊C3—22no—
103(C3⋊Dic3)⋊C3—41no—
104(C3⋊Dic3)⋊C3—41no—
105C2×#239—91no—
106C2×#237—91no—
107C2×#235—91no—
108C2×C3×M16—91no—
109C2×C4⋊Dic3—91no—
110C2×ΓL(1,𝔽25)—31no—
111C2×ΓL(1,𝔽25)—31no—
112C2×ΓL(1,𝔽25)—61no—
113D3×C4⋊C4—91no—
114D3×C4⋊C4—91no—
115C3×#103—91no—
116C3×#102—91no—
117#835—91no—
118#832—91no—
119#821—31no—
120#821—91no—
121#821—91no—
122#816—31no—
123#816—91no—
124#816—91no—
125#810—91no—
126#809—91no—
127C22×C3⋊D3—21no—
128C22×C3×D3—31no—
129C22×C3×D3—31no—
130C22×C3×D3—61no—
131C22×C3×D3—61no—
132C2×C32×C4—21no—
133C2×C32×C4C22×D311yescharacteristic
134C2×C32×C4C22×D311yescharacteristic
135C2×C3⋊Dic3—21no—
136C2×C3⋊Dic3—21no—
137C2×C3⋊Dic3—21no—
138C4⋊(C3⋊D3)—21no—
139C4⋊(C3⋊D3)C4×D312yescomplemented
140#634—21no—
141#634C4×D312yescomplemented
142C2×C3×Dic3—31no—
143C2×C3×Dic3—31no—
144C2×C3×Dic3—61no—
145C2×C3×Dic3—61no—
146C2×C3×Dic3—61no—
147C2×C3×Dic3—61no—
148C3×D12—32no—
149C3×D12—61no—
150C3×C4×D3—32no—
151C3×C4×D3—32no—
152C3×C4×D3—32no—
153C3×C4×D3—32no—
154C3×C4×D3—62no—
155C3×C4×D3—62no—
156C3×Dic6—32no—
157C3×Dic6—61no—
158#626—64no—
159#625—64no—
160#624—64no—
161D3×Dic3—64no—
162D3×Dic3—122no—
163D3×Dic3—122no—
164C32×C8—22no—
165C32×C8—21no—
166C32×C8—21no—
167C3×C3⋊C8—62no—
168C3×C3⋊C8—61no—
169C3×C3⋊C8—61no—
170#417—271noSylow
171C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
172C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
173C2×H(3,𝔽3)#13711yescharacteristic
174#272—22no—
175(C3⋊D3)⋊C3—42no—
176C2×C3×Q8—91no—
177C2×C3×D4—91no—
178C22×C3×C4—91no—
179C2×#67—181no—
180C2×#67—181no—
181C22×Dic3—181no—
182C22×Dic3—181no—
183#239—94no—
184#239—182no—
185#237—94no—
186#237—182no—
187#235—94no—
188#235—182no—
189C2×D12—31no—
190C2×D12—91no—
191C2×D12—181no—
192C2×C4×D3—31no—
193C2×C4×D3—31no—
194C2×C4×D3—61no—
195C2×C4×D3—181no—
196C2×C4×D3—181no—
197C2×C4×D3—181no—
198C2×C4×D3—181no—
199C2×Dic6—31no—
200C2×Dic6—91no—
201C3×M16—182no—
202C2×C3×C8—31no—
203C2×C3×C8—31no—
204C2×C3×C8—61no—
205C2×C3×C8—91no—
206C2×C3×C8D3211yescharacteristic
207C3×C4⋊C4—91no—
208C3×C4⋊C4—91no—
209C4⋊Dic3—31no—
210C4⋊Dic3—31no—
211C4⋊Dic3—91no—
212C4⋊Dic3—91no—
213C4⋊Dic3—181no—
214Dic3⋊C4—181no—
215Dic3⋊C4—181no—
216C2×C3⋊C8—31no—
217C2×C3⋊C8—31no—
218C2×C3⋊C8—61no—
219ΓL(1,𝔽25)—62no—
220ΓL(1,𝔽25)—62no—
221ΓL(1,𝔽25)—64no—
222C22×C32—21no—
223C22×C32D3×D411yescharacteristic
224C22×C32D3×D411yescharacteristic
225C2×C3⋊D3—22no—
226C2×C3⋊D3—41no—
227C2×C3⋊D3—41no—
228C2×C3×D3—32no—
229C2×C3×D3—32no—
230C2×C3×D3—62no—
231C2×C3×D3—61no—
232C2×C3×D3—61no—
233C2×C3×D3—61no—
234C2×C3×D3—61no—
235C2×C3×D3—62no—
236C2×C3×D3—121no—
237C2×C3×D3—121no—
238C2×C3×D3—121no—
239C2×C3×D3—121no—
240C32×C4—21no—
241C32×C4—21no—
242C32×C4C2×C4×D311yescharacteristic
243C32×C4C2×C4×D311yescharacteristic
244C32×C4D3×D411yescharacteristic
245C32×C4D3×D411yescharacteristic
246C3⋊Dic3—22no—
247C3⋊Dic3—41no—
248C3⋊Dic3—41no—
249C3×Dic3—32no—
250C3×Dic3—32no—
251C3×Dic3—62no—
252C3×Dic3—62no—
253C3×Dic3—121no—
254C3×Dic3—121no—
255C2×#41—271no—
256C2×M16—91no—
257C2×C4⋊C4—271no—
258#103—91no—
259#103—271no—
260#102—91no—
261#102—271no—
262H(3,𝔽3)#41711yescharacteristic, complemented, Sylow
263C23×C3—91no—
264C23×C3—181no—
265C22×D3—31no—
266C22×D3—31no—
267C22×D3—61no—
268C22×D3—61no—
269C22×D3—181no—
270C3×Q8—92no—
271C3×Q8—181no—
272C3×D4—92no—
273C3×D4—181no—
274C2×C3×C4—31no—
275C2×C3×C4—31no—
276C2×C3×C4—61no—
277C2×C3×C4—91no—
278C2×C3×C4—92no—
279C2×C3×C4—92no—
280C2×C3×C4—181no—
281C2×C3×C4—181no—
282C2×C3×C4—181no—
283C2×C3×C4C2×D3211yescharacteristic
284#67—184no—
285#67—184no—
286C2×Dic3—31no—
287C2×Dic3—31no—
288C2×Dic3—61no—
289C2×Dic3—61no—
290C2×Dic3—61no—
291C2×Dic3—61no—
292C2×Dic3—181no—
293C2×Dic3—181no—
294C2×Dic3—181no—
295C2×Dic3—184no—
296C2×Dic3—362no—
297D12—32no—
298D12—61no—
299D12—92no—
300D12—181no—
301D12—184no—
302C4×D3—32no—
303C4×D3—32no—
304C4×D3—32no—
305C4×D3—32no—
306C4×D3—62no—
307C4×D3—62no—
308C4×D3—184no—
309C4×D3—184no—
310C4×D3—362no—
311C4×D3—362no—
312Dic6—32no—
313Dic6—61no—
314Dic6—92no—
315Dic6—181no—
316C3×C8—21no—
317C3×C8—61no—
318C3×C8—61no—
319C3×C8—62no—
320C3×C8—181no—
321C3⋊C8—61no—
322C3⋊C8—61no—
323C3⋊C8—62no—
324C2×C32—21no—
325C2×C32—21no—
326C2×C32—21no—
327C2×C32D3×C22⋊C411yescharacteristic
328C2×C32D3×C22⋊C411yescharacteristic
329C2×C32#91111yescharacteristic
330C2×C32#91411yescharacteristic
331C2×C32#91511yescharacteristic
332C2×C32#91811yescharacteristic
333C3⋊D3—42no—
334C3×D3—62no—
335C3×D3—62no—
336C3×D3—122no—
337C3×D3—122no—
338#41—274no—
339#41—542no—
340C2×Q8—91no—
341C2×D4—91no—
342C2×D4—541no—
343C22×C4—91no—
344C22×C4—541no—
345C22×C4—541no—
346M16—182no—
347C2×C8—91no—
348C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
349C4⋊C4—91no—
350C4⋊C4—91no—
351C4⋊C4—541no—
352C22×C3—31no—
353C22×C3—31no—
354C22×C3—61no—
355C22×C3—92no—
356C22×C3—182no—
357C22×C3—181no—
358C22×C3—181no—
359C22×C3—361no—
360C22×C3—361no—
361C22×C3#380211yescharacteristic
362C2×D3—32no—
363C2×D3—32no—
364C2×D3—62no—
365C2×D3—61no—
366C2×D3—61no—
367C2×D3—61no—
368C2×D3—61no—
369C2×D3—62no—
370C2×D3—121no—
371C2×D3—121no—
372C2×D3—121no—
373C2×D3—121no—
374C2×D3—182no—
375C2×D3—361no—
376C2×D3—361no—
377C3×C4—31no—
378C3×C4—31no—
379C3×C4—31no—
380C3×C4—31no—
381C3×C4—61no—
382C3×C4—61no—
383C3×C4—92no—
384C3×C4—182no—
385C3×C4—361no—
386C3×C4—361no—
387C3×C4C4×D3211yescharacteristic
388C3×C4#381111yescharacteristic
389Dic3—32no—
390Dic3—32no—
391Dic3—62no—
392Dic3—62no—
393Dic3—121no—
394Dic3—121no—
395Dic3—182no—
396Dic3—361no—
397Dic3—361no—
398C32—21no—
399C32#477111yescharacteristic, complemented
400C32#480311yescharacteristic, complemented
401C23—91no—
402C23—181no—
403Q8—92no—
404Q8—181no—
405D4—92no—
406D4—181no—
407D4—544no—
408C2×C4—91no—
409C2×C4—92no—
410C2×C4—92no—
411C2×C4—181no—
412C2×C4—181no—
413C2×C4—181no—
414C2×C4—544no—
415C2×C4—1082no—
416C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
417C8—21no—
418C8—181no—
419C2×C3—31no—
420C2×C3—31no—
421C2×C3—31no—
422C2×C3—31no—
423C2×C3—31no—
424C2×C3—31no—
425C2×C3—61no—
426C2×C3—61no—
427C2×C3—61no—
428C2×C3—182no—
429C2×C3—362no—
430C2×C3#6379811yescharacteristic
431C2×C3#6381111yescharacteristic
432C2×C3#6392111yescharacteristic
433D3—62no—
434D3—62no—
435D3—122no—
436D3—122no—
437D3—362no—
438C22—92no—
439C22—181no—
440C22—181no—
441C22—182no—
442C22—361no—
443C22—361no—
444C22#8865511yescharacteristic
445C4—92no—
446C4—182no—
447C4—361no—
448C4—361no—
449C4C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
450C4#8867711yescharacteristic
451C3—31no—
452C3—31no—
453C3—61no—
454C3#1059095711yescharacteristic
455C2—182no—
456C2—362no—
457C2#1172083911yescharacteristic
458C2#1172084511yescharacteristic
459C2#1172116111yescharacteristic
subgroupG/H
centerC22#88655
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3#3802
prime radicalC2×#88649C2
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#417
LCSG ▷ 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×#88527C2semidirect
3. #11721152C2semidirect
4. #63648⋊C3C2semidirect
5. #63643⋊C3C2semidirect
6. #11720318C2semidirect
7. #11720313C2semidirect
17. #88649C4semidirect
31. #63648D3semidirect
32. #63643D3semidirect
49. C4⋊((C3⋊D3)⋊C3)C2×C4semidirect
51. #634⋊C3C2×C4semidirect
139. C4⋊(C3⋊D3)C4×D3semidirect
141. #634C4×D3semidirect
262. H(3,𝔽3)#417semidirect
399. C32#4771semidirect
400. C32#4803semidirect

Maximal subgroups

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

subgroupindex
C2×#886492
C2×#885272
#117211522
#63648⋊C32
#63643⋊C32
#117203182
#117203132
#105906713
#105906703