Finite Group Databasebeta

← #13140195

Subgroups — #13140195

Order 1728.

508 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 89 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
1#11721824C211yesmaximal, characteristic, complemented
2#64139⋊C3C211yesmaximal, characteristic
3C22×(C3⋊(C3⋊C8))⋊C3C211yesmaximal, characteristic, complemented
4C2×#88438C211yesmaximal, characteristic, complemented
5#11720326C211yesmaximal, characteristic, complemented
6#11720308C211yesmaximal, characteristic, complemented
7#11720307C211yesmaximal, characteristic, complemented
8#10591179—31nomaximal
9#10591177—31nomaximal
10C22×C4×H(3,𝔽3)C2211yescharacteristic
11C2×#6229C2211yescharacteristic
12C2×#638⋊C3—21no—
13C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
14#88552—21no—
15C4⋊#1084C2211yescharacteristic
16#88540—21no—
17#3752⋊C3—21no—
18C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
19C2×(C3⋊(C3⋊C8))⋊C3—22no—
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
22#88438—22no—
23#64139D311yescharacteristic
24C22×C3⋊(C3⋊C8)D311yescharacteristic, complemented
25C3×#930—31no—
26C3×#924—31no—
27C3×#924—31no—
28C3×#924—61no—
29C22×C3×C3⋊C8—31no—
30C2×#3717—31no—
31C2×#3715—31no—
32#63574—31no—
33#63573—31no—
34#63566—31no—
35#63565—31no—
36#63564—31no—
37#63562—31no—
38C23×H(3,𝔽3)D411yescharacteristic
39C22×(C3⋊D3)⋊C3—21no—
40C2×C4×H(3,𝔽3)—21no—
41C2×C4×H(3,𝔽3)—21no—
42C2×C4×H(3,𝔽3)D411yescharacteristic
43C2×C4×H(3,𝔽3)C2311yescharacteristic
44C2×#1084—21no—
45C2×#1084—21no—
46#6229—21no—
47#6229—21no—
48#638⋊C3—42no—
49C2×(C3⋊Dic3)⋊C3—21no—
50C4⋊((C3⋊D3)⋊C3)—21no—
51C4⋊((C3⋊D3)⋊C3)—21no—
52(C3⋊(C3⋊C8))⋊C3—21no—
53(C3⋊(C3⋊C8))⋊C3D412yescomplemented
54#5114—91no—
55#5048—91no—
56#5041—91no—
57C22×C32×C4—21no—
58C22×C32×C4C2×D311yescharacteristic
59C22×C32×C4C2×D311yescharacteristic
60C2×#638—21no—
61C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
62C2×C3×#67—61no—
63C2×C3×D12—31no—
64C2×C3×Dic6—31no—
65C2×C3×Dic6—31no—
66C2×C3×Dic6—61no—
67#3752—21no—
68C4⋊(C3⋊Dic3)C2×D311yescharacteristic
69C2×C3⋊(C3⋊C8)—22no—
70C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
71C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
72C3×#217—62no—
73C3×#217—61no—
74C3×#217—61no—
75C3×(C2×D3)⋊C4—61no—
76C3×C4⋊Dic3—31no—
77C3×C4⋊Dic3—31no—
78C3×C4⋊Dic3—31no—
79C3×C4⋊Dic3—61no—
80C3×Dic3⋊C4—62no—
81C3×Dic3⋊C4—61no—
82C3×Dic3⋊C4—61no—
83C2×C3×C3⋊C8—31no—
84C2×C3×C3⋊C8—31no—
85C2×C3×C3⋊C8—62no—
86#3717—62no—
87#3715—62no—
88C22×H(3,𝔽3)—21no—
89C22×H(3,𝔽3)—21no—
90C22×H(3,𝔽3)SD1612yescomplemented
91C22×H(3,𝔽3)C2×D411yescharacteristic
92C2×(C3⋊D3)⋊C3—41no—
93C2×(C3⋊D3)⋊C3—41no—
94C2×(C3⋊D3)⋊C3—41no—
95C4×H(3,𝔽3)—21no—
96C4×H(3,𝔽3)C2×D411yescharacteristic
97C4×H(3,𝔽3)#4111yescharacteristic
98#1084—41no—
99#1084—41no—
100(C3⋊Dic3)⋊C3—41no—
101C22×C3×C8—91no—
102C3×C22⋊Q8—91no—
103C3×#121—91no—
104C2×#215—91no—
105C2×#214—91no—
106#930—31no—
107#930—91no—
108#924—31no—
109#924—31no—
110#924—61no—
111C22×C3⋊C8—31no—
112C22×C3⋊C8—91no—
113C2×#204—91no—
114#835—91no—
115#832—91no—
116#821—91no—
117#817—91no—
118#816—91no—
119#810—91no—
120#809—91no—
121#808—91no—
122#808—91no—
123C23×C32—21no—
124C23×C32D1211yescharacteristic
125C23×C32#6711yescharacteristic
126C22×C3⋊D3—21no—
127C22×C3×D3—61no—
128C2×C32×C4—21no—
129C2×C32×C4—21no—
130C2×C32×C4—21no—
131C2×C32×C4—21no—
132C2×C32×C4—21no—
133C2×C32×C4—21no—
134C2×C32×C4—41no—
135C2×C32×C4—41no—
136C2×C32×C4D1211yescharacteristic
137C2×C32×C4#6711yescharacteristic
138C2×C32×C4C22×D311yescharacteristic
139C2×C32×C4C22×D311yescharacteristic
140#638—42no—
141C2×C3⋊Dic3—21no—
142C4⋊(C3⋊D3)—21no—
143C4⋊(C3⋊D3)—21no—
144C3×#67—122no—
145C2×C3×Dic3—62no—
146C2×C3×Dic3—62no—
147C2×C3×Dic3—61no—
148C2×C3×Dic3—61no—
149C2×C3×Dic3—61no—
150C2×C3×Dic3—61no—
151C2×C3×Dic3—61no—
152C3×D12—61no—
153C3×D12—61no—
154C3×Dic6—61no—
155C3×Dic6—61no—
156C3×Dic6—61no—
157C3×Dic6—61no—
158C3×Dic6—121no—
159C3×Dic6—121no—
160C3⋊(C3⋊C8)—21no—
161C3⋊(C3⋊C8)#6712yescomplemented
162C3×C3⋊C8—32no—
163C3×C3⋊C8—61no—
164#465—271noSylow
165C2×H(3,𝔽3)—22no—
166C2×H(3,𝔽3)#12111yescharacteristic
167C2×H(3,𝔽3)C2×SD1611yescharacteristic
168C2×H(3,𝔽3)#13511yescharacteristic
169(C3⋊D3)⋊C3—81no—
170C2×C3×Q8—91no—
171C2×C3×D4—91no—
172C2×C3×D4—181no—
173C22×C3×C4—31no—
174C22×C3×C4—31no—
175C22×C3×C4—61no—
176C22×C3×C4D3211yescharacteristic
177C2×#67—61no—
178C2×#67—181no—
179C2×D12—31no—
180C2×D12—91no—
181C2×Dic6—31no—
182C2×Dic6—31no—
183C2×Dic6—61no—
184C2×C3×C8—91no—
185C2×C3×C8—91no—
186C2×C3×C8—182no—
187C3×C4⋊C4—91no—
188C3×C4⋊C4—91no—
189C3×C4⋊C4—181no—
190C3×C22⋊C4—181no—
191C3×C22⋊C4—181no—
192#217—61no—
193#217—61no—
194#217—62no—
195#215—182no—
196#214—182no—
197(C2×D3)⋊C4—61no—
198(C2×D3)⋊C4—181no—
199C4⋊Dic3—31no—
200C4⋊Dic3—31no—
201C4⋊Dic3—31no—
202C4⋊Dic3—61no—
203C4⋊Dic3—91no—
204Dic3⋊C4—61no—
205Dic3⋊C4—61no—
206Dic3⋊C4—62no—
207C2×C3⋊C8—31no—
208C2×C3⋊C8—31no—
209C2×C3⋊C8—62no—
210C2×C3⋊C8—91no—
211C2×C3⋊C8—91no—
212C2×C3⋊C8—182no—
213#204—182no—
214C22×C32—21no—
215C22×C32—21no—
216C22×C32—21no—
217C22×C32—21no—
218C22×C32—22no—
219C22×C32—21no—
220C22×C32—41no—
221C22×C32—41no—
222C22×C32#20412yescomplemented
223C22×C32#21412yescomplemented
224C22×C32C2×D1211yescharacteristic
225C22×C32C2×#6711yescharacteristic
226C2×C3⋊D3—41no—
227C2×C3⋊D3—41no—
228C2×C3⋊D3—41no—
229C2×C3×D3—121no—
230C2×C3×D3—121no—
231C2×C3×D3—121no—
232C32×C4—21no—
233C32×C4—21no—
234C32×C4—21no—
235C32×C4—21no—
236C32×C4—41no—
237C32×C4#23511yescharacteristic
238C32×C4D3×D411yescharacteristic
239C32×C4#23711yescharacteristic
240C32×C4C2×#6711yescharacteristic
241C3⋊Dic3—41no—
242C3×Dic3—122no—
243C3×Dic3—122no—
244C3×Dic3—121no—
245C3×Dic3—121no—
246C3×Dic3—121no—
247C3×Dic3—121no—
248C3×Dic3—121no—
249C2×SD16—271no—
250C22×C8—91no—
251C22⋊Q8—91no—
252#121—91no—
253#106—271no—
254#103—271no—
255#102—271no—
256H(3,𝔽3)#46511yescharacteristic, complemented, Sylow
257C23×C3—31no—
258C23×C3—31no—
259C23×C3—61no—
260C23×C3—181no—
261C23×C3#62611yescharacteristic
262C22×D3—61no—
263C22×D3—181no—
264C3×Q8—181no—
265C3×Q8—181no—
266C3×D4—181no—
267C3×D4—181no—
268C3×D4—362no—
269C2×C3×C4—21no—
270C2×C3×C4—21no—
271C2×C3×C4—31no—
272C2×C3×C4—31no—
273C2×C3×C4—31no—
274C2×C3×C4—31no—
275C2×C3×C4—61no—
276C2×C3×C4—61no—
277C2×C3×C4—61no—
278C2×C3×C4—61no—
279C2×C3×C4—61no—
280C2×C3×C4—61no—
281C2×C3×C4—121no—
282C2×C3×C4—121no—
283C2×C3×C4—181no—
284C2×C3×C4—181no—
285C2×C3×C4—181no—
286C2×C3×C4#62611yescharacteristic
287C2×C3×C4C2×D3211yescharacteristic
288#67—122no—
289#67—362no—
290C2×Dic3—61no—
291C2×Dic3—61no—
292C2×Dic3—61no—
293C2×Dic3—61no—
294C2×Dic3—62no—
295C2×Dic3—62no—
296C2×Dic3—61no—
297C2×Dic3—181no—
298D12—61no—
299D12—61no—
300D12—181no—
301D12—181no—
302Dic6—61no—
303Dic6—61no—
304Dic6—61no—
305Dic6—61no—
306Dic6—121no—
307Dic6—121no—
308C3×C8—92no—
309C3×C8—181no—
310C3⋊C8—32no—
311C3⋊C8—61no—
312C3⋊C8—92no—
313C3⋊C8—181no—
314C2×C32—22no—
315C2×C32—22no—
316C2×C32—21no—
317C2×C32—21no—
318C2×C32—21no—
319C2×C32—42no—
320C2×C32C2×#20411yescharacteristic
321C2×C32#90411yescharacteristic
322C2×C32#93011yescharacteristic
323C2×C32C2×#21411yescharacteristic
324C2×C32#93911yescharacteristic
325C2×C32#95011yescharacteristic
326C3⋊D3—81no—
327C3×D3—241no—
328C2×Q8—91no—
329C2×D4—91no—
330C2×D4—181no—
331C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
332SD16—542no—
333C2×C8—91no—
334C2×C8—91no—
335C2×C8—182no—
336C4⋊C4—91no—
337C4⋊C4—91no—
338C4⋊C4—181no—
339C22⋊C4—181no—
340C22⋊C4—181no—
341C22×C3—21no—
342C22×C3—21no—
343C22×C3—31no—
344C22×C3—31no—
345C22×C3—32no—
346C22×C3—32no—
347C22×C3—61no—
348C22×C3—61no—
349C22×C3—61no—
350C22×C3—61no—
351C22×C3—61no—
352C22×C3—62no—
353C22×C3—121no—
354C22×C3—121no—
355C22×C3—361no—
356C22×C3—361no—
357C22×C3—361no—
358C22×C3#371612yes—
359C22×C3C2×#62611yescharacteristic
360C2×D3—121no—
361C2×D3—121no—
362C2×D3—121no—
363C2×D3—361no—
364C2×D3—361no—
365C2×D3—361no—
366C3×C4—21no—
367C3×C4—31no—
368C3×C4—31no—
369C3×C4—31no—
370C3×C4—31no—
371C3×C4—61no—
372C3×C4—61no—
373C3×C4—61no—
374C3×C4—61no—
375C3×C4—121no—
376C3×C4—361no—
377C3×C4—361no—
378C3×C4—361no—
379C3×C4#380411yescharacteristic
380C3×C4D3×#6711yescharacteristic
381Dic3—121no—
382Dic3—121no—
383Dic3—121no—
384Dic3—121no—
385Dic3—122no—
386Dic3—122no—
387Dic3—121no—
388Dic3—361no—
389C32—21no—
390C32#504111yescharacteristic, complemented
391C32#511411yescharacteristic, complemented
392C23—181no—
393C23#620011yescharacteristic
394Q8—181no—
395Q8—181no—
396D4—181no—
397D4—181no—
398D4—362no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—181no—
402C2×C4—181no—
403C2×C4—181no—
404C2×C4#620011yescharacteristic
405C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
406C8—92no—
407C8—181no—
408C2×C3—22no—
409C2×C3—31no—
410C2×C3—31no—
411C2×C3—31no—
412C2×C3—31no—
413C2×C3—31no—
414C2×C3—31no—
415C2×C3—62no—
416C2×C3—62no—
417C2×C3—61no—
418C2×C3—61no—
419C2×C3—61no—
420C2×C3—122no—
421C2×C3—721no—
422C2×C3C2×#371611yescharacteristic
423C2×C3#6382911yescharacteristic
424C2×C3#6397811yescharacteristic
425D3—241no—
426D3—721no—
427C22—21no—
428C22—21no—
429C22—361no—
430C22—361no—
431C22—361no—
432C22#8843812yescomplemented
433C22C2×#620011yescharacteristic
434C4—21no—
435C4—361no—
436C4—361no—
437C4—361no—
438C4#8867111yescharacteristic
439C4#8867611yescharacteristic
440C3—31no—
441C3—31no—
442C3—61no—
443C3#1059117511yescharacteristic
444C2—22no—
445C2—721no—
446C2C2×#8843811yescharacteristic
447C2#1172107311yescharacteristic
448C2#1172132211yescharacteristic
subgroupG/H
centerC22C2×#6200
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC22×C4×H(3,𝔽3)C22
socleC22×C3C2×#626
prime radical#64139⋊C3C2
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#465
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. #11721824C2semidirect
3. C22×(C3⋊(C3⋊C8))⋊C3C2semidirect
4. C2×#88438C2semidirect
5. #11720326C2semidirect
6. #11720308C2semidirect
7. #11720307C2semidirect
24. C22×C3⋊(C3⋊C8)D3semidirect
53. (C3⋊(C3⋊C8))⋊C3D4semidirect
90. C22×H(3,𝔽3)SD16semidirect
161. C3⋊(C3⋊C8)#67semidirect
222. C22×C32#204semidirect
223. C22×C32#214semidirect
256. H(3,𝔽3)#465semidirect
390. C32#5041semidirect
391. C32#5114semidirect
432. C22#88438semidirect

Maximal subgroups

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

subgroupindex
#117218242
#64139⋊C32
C22×(C3⋊(C3⋊C8))⋊C32
C2×#884382
#117203262
#117203082
#117203072
#105911793
#105911773