Finite Group Databasebeta

← #13137963

Subgroups — #13137963

Order 1728.

508 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 73 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#11721765C211yesmaximal, characteristic, complemented
2#11721159C211yesmaximal, characteristic
3C2×#88438C211yesmaximal, characteristic, complemented
4C2×#88434C211yesmaximal, characteristic, complemented
5#63636⋊C3C211yesmaximal, characteristic, complemented
6#63635⋊C3C211yesmaximal, characteristic, complemented
7#11720326C211yesmaximal, characteristic, complemented
8#10590577—31nomaximal
9#10590501—31nomaximal
10C2×C4×#272C2211yescharacteristic
11C2×#6229C2211yescharacteristic
12C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
13C2×#6200—21no—
14C4⋊C4×H(3,𝔽3)C2211yescharacteristic
15#88542—21no—
16C4⋊#1084—21no—
17C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
18C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20#88450—21no—
21#88438—22no—
22#88434—22no—
23C3×#897—31no—
24C3×#897—31no—
25C3×#897—61no—
26#63909—31no—
27#63908—31no—
28C2×#3717—31no—
29C2×#3715—31no—
30C2×D3×C3⋊C8—31no—
31C2×D3×C3⋊C8—31no—
32#63636D311yescharacteristic, complemented
33#63635D311yescharacteristic, complemented
34C3×#809—31no—
35C3×#808—31no—
36#63566—31no—
37#63565—31no—
38C22×#272D411yescharacteristic
39C22×(C3⋊D3)⋊C3—21no—
40C2×C4×H(3,𝔽3)—21no—
41C2×C4×H(3,𝔽3)C2311yescharacteristic
42C2×#1084—21no—
43C2×#1084D411yescharacteristic
44C4×#272—21no—
45C4×#272—21no—
46#6229—21no—
47#6229—21no—
48C2×(C3⋊Dic3)⋊C3—21no—
49C4⋊((C3⋊D3)⋊C3)—21no—
50C4⋊((C3⋊D3)⋊C3)—21no—
51#6200—42no—
52(C3⋊(C3⋊C8))⋊C3—21no—
53(C3⋊(C3⋊C8))⋊C3D412yescomplemented
54#5048—91no—
55#4864—91no—
56#4778—91no—
57C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
58C2×C3×D12—31no—
59C2×C3×C4×D3—31no—
60C2×C3×C4×D3—31no—
61C2×C3×C4×D3—61no—
62C2×C3×Dic6—31no—
63C2×C3×Dic6—31no—
64C2×C3×Dic6—61no—
65C2×#626—61no—
66C2×#625—61no—
67C32×C4⋊C4—21no—
68C32×C4⋊C4C2×D311yescharacteristic
69C32×C4⋊C4C2×D311yescharacteristic
70C4⋊(C3⋊Dic3)C2×D311yescharacteristic
71C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
72C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
73C3×(C2×D3)⋊C4—61no—
74C3×(C2×D3)⋊C4—61no—
75C3×(C2×D3)⋊C4—62no—
76C3×C4⋊Dic3—31no—
77C3×C4⋊Dic3—61no—
78C3×C4⋊Dic3—61no—
79C3×C4⋊Dic3—62no—
80C2×C3×C3⋊C8—31no—
81C2×C3×C3⋊C8—31no—
82#3722—61no—
83(C2×D3)⋊Dic3—61no—
84#3717—62no—
85#3715—62no—
86D3×C3⋊C8—62no—
87D3×C3⋊C8—62no—
88C22×H(3,𝔽3)C2×D411yescharacteristic
89C2×#272—21no—
90C2×#272—21no—
91C2×#272SD1612yescomplemented
92C2×(C3⋊D3)⋊C3—41no—
93C2×(C3⋊D3)⋊C3—41no—
94C2×(C3⋊D3)⋊C3—41no—
95C4×H(3,𝔽3)—41no—
96C4×H(3,𝔽3)C2×D411yescharacteristic
97C4×H(3,𝔽3)#4111yescharacteristic
98#1084—21no—
99#1084—41no—
100(C3⋊Dic3)⋊C3—41no—
101C3×C22⋊Q8—91no—
102C2×#215—91no—
103#938—91no—
104C2×#214—91no—
105#930—91no—
106C22×C3⋊C8—91no—
107C2×#204—91no—
108C2×D3×C8—91no—
109C2×D3×C8—91no—
110#897—31no—
111#897—31no—
112#897—61no—
113#895—91no—
114C3×#106—91no—
115C3×#102—91no—
116#835—91no—
117#816—91no—
118#810—91no—
119#809—31no—
120#809—91no—
121#808—31no—
122#808—91no—
123C22×C3⋊D3—21no—
124C22×C3×D3—31no—
125C22×C3×D3—31no—
126C22×C3×D3—61no—
127C22×C3×D3—61no—
128C2×C32×C4—22no—
129C2×C32×C4—21no—
130C2×C32×C4—21no—
131C2×C32×C4—21no—
132C2×C32×C4C22×D311yescharacteristic
133C2×C32×C4C22×D311yescharacteristic
134C2×C3⋊Dic3—21no—
135C4⋊(C3⋊D3)—21no—
136C4⋊(C3⋊D3)—21no—
137C2×C3×Dic3—31no—
138C2×C3×Dic3—31no—
139C2×C3×Dic3—61no—
140C2×C3×Dic3—62no—
141C2×C3×Dic3—61no—
142C2×C3×Dic3—61no—
143C2×C3×Dic3—61no—
144C3×D12—61no—
145C3×D12—61no—
146C3×C4×D3—61no—
147C3×C4×D3—61no—
148C3×C4×D3—61no—
149C3×C4×D3—61no—
150C3×C4×D3—121no—
151C3×C4×D3—121no—
152C3×Dic6—61no—
153C3×Dic6—61no—
154C3×Dic6—61no—
155C3×Dic6—61no—
156C3×Dic6—121no—
157C3×Dic6—121no—
158#626—122no—
159#625—122no—
160C3⋊(C3⋊C8)—21no—
161C3⋊(C3⋊C8)D1212yescomplemented
162C3×C3⋊C8—32no—
163C3×C3⋊C8—61no—
164#465—271noSylow
165C2×H(3,𝔽3)#12111yescharacteristic
166C2×H(3,𝔽3)C2×SD1611yescharacteristic
167C2×H(3,𝔽3)#13511yescharacteristic
168#272—22no—
169(C3⋊D3)⋊C3—81no—
170C2×C3×Q8—91no—
171C2×C3×D4—91no—
172C22×C3×C4—91no—
173C2×#67—181no—
174C2×#67—181no—
175C2×D12—31no—
176C2×D12—91no—
177C2×D12—181no—
178C2×C4×D3—31no—
179C2×C4×D3—31no—
180C2×C4×D3—61no—
181C2×Dic6—31no—
182C2×Dic6—31no—
183C2×Dic6—61no—
184C2×C3×C8—91no—
185C2×C3×C8—91no—
186C3×C4⋊C4—31no—
187C3×C4⋊C4—31no—
188C3×C4⋊C4—61no—
189C3×C4⋊C4—91no—
190C3×C4⋊C4—181no—
191C3×C4⋊C4D3211yescharacteristic
192C3×C22⋊C4—181no—
193#217—181no—
194#215—182no—
195#214—182no—
196(C2×D3)⋊C4—61no—
197(C2×D3)⋊C4—61no—
198(C2×D3)⋊C4—62no—
199(C2×D3)⋊C4—181no—
200(C2×D3)⋊C4—181no—
201C4⋊Dic3—31no—
202C4⋊Dic3—61no—
203C4⋊Dic3—61no—
204C4⋊Dic3—62no—
205C4⋊Dic3—91no—
206C2×C3⋊C8—31no—
207C2×C3⋊C8—31no—
208C2×C3⋊C8—91no—
209C2×C3⋊C8—91no—
210C2×C3⋊C8—182no—
211#204—182no—
212D3×C8—182no—
213D3×C8—182no—
214C22×C32—21no—
215C22×C32D3×D411yescharacteristic
216C22×C32D3×D411yescharacteristic
217C2×C3⋊D3—41no—
218C2×C3⋊D3—41no—
219C2×C3⋊D3—41no—
220C2×C3×D3—32no—
221C2×C3×D3—32no—
222C2×C3×D3—62no—
223C2×C3×D3—61no—
224C2×C3×D3—61no—
225C2×C3×D3—61no—
226C2×C3×D3—61no—
227C2×C3×D3—121no—
228C2×C3×D3—121no—
229C2×C3×D3—121no—
230C2×C3×D3—121no—
231C2×C3×D3—121no—
232C32×C4—21no—
233C32×C4—21no—
234C32×C4—42no—
235C32×C4—41no—
236C32×C4—41no—
237C32×C4C2×D1211yescharacteristic
238C32×C4#23511yescharacteristic
239C32×C4D3×D411yescharacteristic
240C32×C4#23911yescharacteristic
241C3⋊Dic3—41no—
242C3×Dic3—61no—
243C3×Dic3—61no—
244C3×Dic3—121no—
245C3×Dic3—122no—
246C3×Dic3—121no—
247C3×Dic3—121no—
248C3×Dic3—121no—
249C2×SD16—271no—
250C22×C8—271no—
251C22⋊Q8—91no—
252#121—271no—
253#106—91no—
254#103—271no—
255#102—91no—
256H(3,𝔽3)#46511yescharacteristic, complemented, Sylow
257C23×C3—91no—
258C23×C3—181no—
259C22×D3—31no—
260C22×D3—31no—
261C22×D3—61no—
262C22×D3—61no—
263C22×D3—181no—
264C3×Q8—181no—
265C3×Q8—181no—
266C3×D4—181no—
267C3×D4—181no—
268C2×C3×C4—21no—
269C2×C3×C4—31no—
270C2×C3×C4—31no—
271C2×C3×C4—61no—
272C2×C3×C4—61no—
273C2×C3×C4—62no—
274C2×C3×C4—61no—
275C2×C3×C4—91no—
276C2×C3×C4—181no—
277C2×C3×C4—181no—
278C2×C3×C4—181no—
279C2×C3×C4—181no—
280C2×C3×C4C2×D3211yescharacteristic
281#67—362no—
282#67—362no—
283C2×Dic3—31no—
284C2×Dic3—31no—
285C2×Dic3—61no—
286C2×Dic3—61no—
287C2×Dic3—61no—
288C2×Dic3—62no—
289C2×Dic3—61no—
290C2×Dic3—181no—
291D12—61no—
292D12—61no—
293D12—181no—
294D12—181no—
295D12—362no—
296C4×D3—61no—
297C4×D3—61no—
298C4×D3—61no—
299C4×D3—61no—
300C4×D3—121no—
301C4×D3—121no—
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—21no—
315C2×C32—21no—
316C2×C32—21no—
317C2×C32#88411yescharacteristic
318C2×C32#89511yescharacteristic
319C2×C32#91211yescharacteristic
320C2×C32D3×SD1611yescharacteristic
321C2×C32D3×SD1611yescharacteristic
322C2×C32#91611yescharacteristic
323C3⋊D3—81no—
324C3×D3—62no—
325C3×D3—62no—
326C3×D3—122no—
327C3×D3—241no—
328C2×Q8—91no—
329C2×D4—91no—
330C2×D4—541no—
331C22×C4—91no—
332SD16—542no—
333C2×C8—91no—
334C2×C8—91no—
335C2×C8—542no—
336C4⋊C4—91no—
337C4⋊C4—181no—
338C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
339C22⋊C4—181no—
340C22⋊C4—541no—
341C22×C3—31no—
342C22×C3—31no—
343C22×C3—61no—
344C22×C3—92no—
345C22×C3—181no—
346C22×C3—181no—
347C22×C3—361no—
348C22×C3—361no—
349C22×C3—361no—
350C22×C3#381111yescharacteristic
351C2×D3—32no—
352C2×D3—32no—
353C2×D3—62no—
354C2×D3—61no—
355C2×D3—61no—
356C2×D3—61no—
357C2×D3—61no—
358C2×D3—121no—
359C2×D3—121no—
360C2×D3—121no—
361C2×D3—121no—
362C2×D3—121no—
363C2×D3—361no—
364C2×D3—361no—
365C2×D3—361no—
366C3×C4—31no—
367C3×C4—31no—
368C3×C4—31no—
369C3×C4—31no—
370C3×C4—41no—
371C3×C4—61no—
372C3×C4—61no—
373C3×C4—121no—
374C3×C4—121no—
375C3×C4—122no—
376C3×C4—181no—
377C3×C4—361no—
378C3×C4—361no—
379C3×C4#379911yescharacteristic
380C3×C4D3×D1211yescharacteristic
381Dic3—61no—
382Dic3—61no—
383Dic3—121no—
384Dic3—121no—
385Dic3—121no—
386Dic3—122no—
387Dic3—121no—
388Dic3—361no—
389C32—21no—
390C32#477811yescharacteristic, complemented
391C32#486411yescharacteristic, complemented
392C23—91no—
393C23—181no—
394Q8—181no—
395Q8—181no—
396D4—181no—
397D4—181no—
398D4—1082no—
399C2×C4—21no—
400C2×C4—91no—
401C2×C4—181no—
402C2×C4—181no—
403C2×C4—181no—
404C2×C4—181no—
405C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
406C8—92no—
407C8—181no—
408C2×C3—31no—
409C2×C3—31no—
410C2×C3—31no—
411C2×C3—31no—
412C2×C3—31no—
413C2×C3—31no—
414C2×C3—61no—
415C2×C3—61no—
416C2×C3—61no—
417C2×C3—182no—
418C2×C3—721no—
419C2×C3#6391011yescharacteristic
420C2×C3#6393611yescharacteristic
421C2×C3#6394011yescharacteristic
422D3—62no—
423D3—62no—
424D3—122no—
425D3—241no—
426D3—721no—
427C22—92no—
428C22—181no—
429C22—181no—
430C22—361no—
431C22—361no—
432C22—361no—
433C22#8867711yescharacteristic
434C4—41no—
435C4—181no—
436C4—361no—
437C4—361no—
438C4#8865211yescharacteristic
439C4#8865411yescharacteristic
440C3—31no—
441C3—31no—
442C3—61no—
443C3#1059050711yescharacteristic
444C2—182no—
445C2—721no—
446C2#1172115711yescharacteristic
447C2#1172119211yescharacteristic
448C2#1172122811yescharacteristic
subgroupG/H
centerC22#88677
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC4⋊C4×H(3,𝔽3)C22
socleC22×C3#3811
prime radical#11721159C2
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#465
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
1. #11721765C2semidirect
3. C2×#88438C2semidirect
4. C2×#88434C2semidirect
5. #63636⋊C3C2semidirect
6. #63635⋊C3C2semidirect
7. #11720326C2semidirect
32. #63636D3semidirect
33. #63635D3semidirect
53. (C3⋊(C3⋊C8))⋊C3D4semidirect
91. C2×#272SD16semidirect
161. C3⋊(C3⋊C8)D12semidirect
256. H(3,𝔽3)#465semidirect
390. C32#4778semidirect
391. C32#4864semidirect

Maximal subgroups

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

subgroupindex
#117217652
#117211592
C2×#884382
C2×#884342
#63636⋊C32
#63635⋊C32
#117203262
#105905773
#105905013