Finite Group Databasebeta

← #13138979

Subgroups — #13138979

Order 1728.

565 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 565 conjugacy classes of subgroups — too many to draw. Only its 91 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×#88528C211yesmaximal, characteristic, complemented
2C2×#3744⋊C3C211yesmaximal, characteristic, complemented
3#11721158C211yesmaximal, characteristic
4#11721090C211yesmaximal, characteristic, complemented
5C2×#88438C211yesmaximal, characteristic, complemented
6C2×#88437C211yesmaximal, characteristic, complemented
7C8×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
8#10590884—31nomaximal
9#10590877—31nomaximal
10C2×C4⋊#272C2211yescharacteristic
11C2×#6229C2211yescharacteristic
12C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
13C2×#634⋊C3C2211yescharacteristic
14C2×#6200—22no—
15C2×C8×H(3,𝔽3)C2211yescharacteristic
16#88528—22no—
17C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
18C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
19#3744⋊C3—22no—
20#88448—22no—
21#88438—22no—
22#88437—22no—
23C2×#3744D311yescharacteristic, complemented
24C2×C3×#204—31no—
25C2×C3×#204—31no—
26C2×C3×#204—31no—
27C2×C3×#204—61no—
28#63914—31no—
29#63911—31no—
30#63873—31no—
31#63866—31no—
32C2×#3717—31no—
33C2×#3716—31no—
34C2×#3715—31no—
35C2×#3715—31no—
36C8×C3⋊Dic3D311yescharacteristic, complemented
37C3×C8×Dic3—31no—
38C22×#272—21no—
39C22×(C3⋊D3)⋊C3—21no—
40C2×C4×H(3,𝔽3)C2311yescharacteristic
41C2×#1084—21no—
42C4⋊#272—21no—
43C4⋊#272—21no—
44#6229—21no—
45#6229—21no—
46C2×(C3⋊Dic3)⋊C3—21no—
47C2×(C3⋊Dic3)⋊C3D412yes—
48C4⋊((C3⋊D3)⋊C3)—21no—
49C4⋊((C3⋊D3)⋊C3)—21no—
50#634⋊C3—21no—
51#634⋊C3—21no—
52#6200—44no—
53C8×H(3,𝔽3)D412yescomplemented
54(C3⋊(C3⋊C8))⋊C3D412yescomplemented
55#5175—91no—
56#5175—91no—
57#4693—91no—
58C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
59C2×#634C2×D311yescharacteristic
60C2×C3×D12—31no—
61C2×C3×D12—31no—
62C2×C3×D12—31no—
63C2×C3×D12—61no—
64C2×C3×Dic6—31no—
65C2×C3×Dic6—31no—
66C2×C3×Dic6—31no—
67C2×C3×Dic6—61no—
68C2×#626—62no—
69C2×#625—62no—
70C2×C32×C8—21no—
71C2×C32×C8C2×D311yescharacteristic
72C2×C32×C8C2×D311yescharacteristic
73C4×C3⋊Dic3C2×D311yescharacteristic
74C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
75#3744—22no—
76C3×C4×Dic3—31no—
77C2×C3×C3⋊C8—31no—
78C3×#204—64no—
79C3×#204—62no—
80C3×#204—62no—
81C3×#204—62no—
82#3724—62no—
83Dic3⋊Dic3—62no—
84#3717—62no—
85#3716—62no—
86#3715—62no—
87#3715—62no—
88C22×H(3,𝔽3)C2×D411yescharacteristic
89C2×#272—41no—
90C2×#272—42no—
91C2×(C3⋊D3)⋊C3—41no—
92C2×(C3⋊D3)⋊C3—42no—
93C4×H(3,𝔽3)C2×D411yescharacteristic
94C4×H(3,𝔽3)C2×D411yescharacteristic
95#1084—41no—
96(C3⋊Dic3)⋊C3—41no—
97(C3⋊Dic3)⋊C3SD1614yescomplemented
98C2×C3×SD16—91no—
99C2×C3×SD16—91no—
100#944—91no—
101#944—91no—
102C2×#215—91no—
103C2×#215—91no—
104#940—91no—
105#940—91no—
106C2×#214—91no—
107C2×#214—91no—
108C2×#204—31no—
109C2×#204—31no—
110C2×#204—31no—
111C2×#204—61no—
112C2×#204—91no—
113C2×#204—91no—
114C2×#204—91no—
115#874—91no—
116#869—91no—
117C3×C4×C8—91no—
118C8×Dic3—31no—
119C8×Dic3—91no—
120C22×C3⋊D3—21no—
121C22×C3×D3—61no—
122C22×C3×D3—61no—
123C22×C3×D3—61no—
124C22×C3×D3—121no—
125C2×C32×C4—21no—
126C2×C32×C4C22×D311yescharacteristic
127C2×C32×C4C22×D311yescharacteristic
128C2×C3⋊Dic3—21no—
129C2×C3⋊Dic3D1212yes—
130C4⋊(C3⋊D3)—21no—
131C4⋊(C3⋊D3)—21no—
132#634—21no—
133#634—21no—
134C2×C3×Dic3—32no—
135C2×C3×Dic3—61no—
136C2×C3×Dic3—61no—
137C2×C3×Dic3—61no—
138C2×C3×Dic3—121no—
139C3×D12—62no—
140C3×D12—61no—
141C3×D12—61no—
142C3×D12—61no—
143C3×D12—61no—
144C3×D12—61no—
145C3×D12—61no—
146C3×D12—121no—
147C3×Dic6—62no—
148C3×Dic6—61no—
149C3×Dic6—61no—
150C3×Dic6—61no—
151C3×Dic6—61no—
152C3×Dic6—61no—
153C3×Dic6—61no—
154C3×Dic6—121no—
155#626—124no—
156#625—124no—
157C32×C8—22no—
158C32×C8D1212yescomplemented
159C32×C8#6712yescomplemented
160C3⋊(C3⋊C8)D1212yescomplemented
161C3×C3⋊C8—32no—
162#492—271noSylow
163C2×H(3,𝔽3)#12711yescharacteristic
164C2×H(3,𝔽3)C2×SD1612yes—
165#272—81no—
166(C3⋊D3)⋊C3—81no—
167C2×C3×Q8—91no—
168C2×C3×Q8—91no—
169C2×C3×D4—91no—
170C2×C3×D4—91no—
171C2×#67—182no—
172C2×#67—182no—
173C2×D12—31no—
174C2×D12—31no—
175C2×D12—31no—
176C2×D12—61no—
177C2×D12—91no—
178C2×D12—182no—
179C2×Dic6—31no—
180C2×Dic6—31no—
181C2×Dic6—31no—
182C2×Dic6—61no—
183C2×Dic6—91no—
184C3×SD16—182no—
185C3×SD16—182no—
186C2×C3×C8—31no—
187C2×C3×C8—31no—
188C2×C3×C8—61no—
189C2×C3×C8—91no—
190C2×C3×C8D3211yescharacteristic
191C3×C42—91no—
192#215—182no—
193#215—182no—
194#214—182no—
195#214—182no—
196C4⋊Dic3—182no—
197Dic3⋊C4—182no—
198Dic3⋊C4—182no—
199C4×Dic3—31no—
200C4×Dic3—91no—
201C2×C3⋊C8—31no—
202C2×C3⋊C8—91no—
203#204—62no—
204#204—62no—
205#204—64no—
206#204—62no—
207#204—182no—
208#204—182no—
209#204—182no—
210C22×C32—21no—
211C22×C32C2×D1211yescharacteristic
212C22×C32D3×D411yescharacteristic
213C2×C3⋊D3—42no—
214C2×C3⋊D3—41no—
215C2×C3×D3—122no—
216C2×C3×D3—121no—
217C2×C3×D3—121no—
218C2×C3×D3—122no—
219C2×C3×D3—122no—
220C2×C3×D3—122no—
221C2×C3×D3—121no—
222C2×C3×D3—242no—
223C32×C4—21no—
224C32×C4—21no—
225C32×C4C2×D1211yescharacteristic
226C32×C4C2×D1211yescharacteristic
227C32×C4D3×D411yescharacteristic
228C32×C4C2×#6711yescharacteristic
229C3⋊Dic3—41no—
230C3⋊Dic3#20414yescomplemented
231C3×Dic3—34no—
232C3×Dic3—122no—
233C3×Dic3—121no—
234C3×Dic3—121no—
235C3×Dic3—121no—
236C2×SD16—91no—
237C2×SD16—91no—
238C2×SD16—271no—
239C2×SD16—271no—
240C4⋊Q8—271no—
241#127—271no—
242C4×C8—91no—
243H(3,𝔽3)#49211yescharacteristic, complemented, Sylow
244C23×C3—181no—
245C23×C3—181no—
246C22×D3—61no—
247C22×D3—61no—
248C22×D3—61no—
249C22×D3—121no—
250C22×D3—181no—
251C3×Q8—181no—
252C3×Q8—181no—
253C3×Q8—181no—
254C3×Q8—181no—
255C3×D4—181no—
256C3×D4—181no—
257C3×D4—181no—
258C3×D4—181no—
259C2×C3×C4—31no—
260C2×C3×C4—31no—
261C2×C3×C4—61no—
262C2×C3×C4—92no—
263C2×C3×C4—181no—
264C2×C3×C4—181no—
265C2×C3×C4C2×D3211yescharacteristic
266#67—364no—
267#67—364no—
268C2×Dic3—32no—
269C2×Dic3—61no—
270C2×Dic3—61no—
271C2×Dic3—61no—
272C2×Dic3—92no—
273C2×Dic3—121no—
274C2×Dic3—181no—
275D12—61no—
276D12—61no—
277D12—61no—
278D12—61no—
279D12—62no—
280D12—61no—
281D12—61no—
282D12—121no—
283D12—181no—
284D12—181no—
285D12—364no—
286Dic6—61no—
287Dic6—61no—
288Dic6—61no—
289Dic6—61no—
290Dic6—62no—
291Dic6—61no—
292Dic6—61no—
293Dic6—121no—
294Dic6—181no—
295Dic6—181no—
296C3×C8—32no—
297C3×C8—32no—
298C3×C8—62no—
299C3×C8—92no—
300C3×C8#62612yes—
301C3⋊C8—32no—
302C3⋊C8—92no—
303C2×C32—22no—
304C2×C32—21no—
305C2×C32#87411yescharacteristic
306C2×C32C2×#20412yes—
307C2×C32D3×SD1612yes—
308C2×C32#94011yescharacteristic
309C3⋊D3—81no—
310C3×D3—242no—
311C3×D3—241no—
312C3×D3—241no—
313C3×D3—241no—
314C2×Q8—91no—
315C2×Q8—91no—
316C2×D4—91no—
317C2×D4—91no—
318C2×D4—542no—
319SD16—182no—
320SD16—182no—
321SD16—542no—
322SD16—542no—
323C2×C8—91no—
324C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
325C4⋊C4—542no—
326C42—91no—
327C22×C3—31no—
328C22×C3—31no—
329C22×C3—61no—
330C22×C3—362no—
331C22×C3—361no—
332C22×C3—362no—
333C22×C3—361no—
334C22×C3D3×D1211yescharacteristic
335C2×D3—121no—
336C2×D3—121no—
337C2×D3—122no—
338C2×D3—122no—
339C2×D3—122no—
340C2×D3—122no—
341C2×D3—121no—
342C2×D3—242no—
343C2×D3—361no—
344C2×D3—362no—
345C3×C4—31no—
346C3×C4—31no—
347C3×C4—31no—
348C3×C4—31no—
349C3×C4—61no—
350C3×C4—61no—
351C3×C4—94no—
352C3×C4—361no—
353C3×C4—361no—
354C3×C4D3×D1211yescharacteristic
355C3×C4C2×#62611yescharacteristic
356Dic3—34no—
357Dic3—94no—
358Dic3—121no—
359Dic3—121no—
360Dic3—122no—
361Dic3—121no—
362Dic3—361no—
363C32—21no—
364C32#469311yescharacteristic, complemented
365C32#517511yescharacteristic, complemented
366C23—181no—
367C23—181no—
368Q8—181no—
369Q8—181no—
370Q8—181no—
371Q8—181no—
372D4—181no—
373D4—181no—
374D4—181no—
375D4—181no—
376D4—1084no—
377C2×C4—92no—
378C2×C4—181no—
379C2×C4—181no—
380C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
381C8—92no—
382C8#620012yescomplemented
383C2×C3—32no—
384C2×C3—32no—
385C2×C3—31no—
386C2×C3—31no—
387C2×C3—62no—
388C2×C3—61no—
389C2×C3—721no—
390C2×C3—721no—
391C2×C3D3×#20412yes—
392C2×C3#6391111yescharacteristic
393D3—241no—
394D3—241no—
395D3—242no—
396D3—241no—
397D3—721no—
398C22—362no—
399C22—362no—
400C22—361no—
401C22—361no—
402C22#8865411yescharacteristic
403C4—94no—
404C4—361no—
405C4—361no—
406C4#8865411yescharacteristic
407C4C2×#620011yescharacteristic
408C3—31no—
409C3—31no—
410C3—61no—
411C3#1059087711yescharacteristic
412C2—721no—
413C2—721no—
414C2#1172084612yes—
415C2#1172115811yescharacteristic
subgroupG/H
centerC22#88654
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3D3×D12
prime radical#11721158C2
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#492
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
1. C2×#88528C2semidirect
2. C2×#3744⋊C3C2semidirect
4. #11721090C2semidirect
5. C2×#88438C2semidirect
6. C2×#88437C2semidirect
7. C8×(C3⋊Dic3)⋊C3C2semidirect
23. C2×#3744D3semidirect
36. C8×C3⋊Dic3D3semidirect
53. C8×H(3,𝔽3)D4semidirect
54. (C3⋊(C3⋊C8))⋊C3D4semidirect
97. (C3⋊Dic3)⋊C3SD16semidirect
158. C32×C8D12semidirect
159. C32×C8#67semidirect
160. C3⋊(C3⋊C8)D12semidirect
230. C3⋊Dic3#204semidirect
243. H(3,𝔽3)#492semidirect
364. C32#4693semidirect
365. C32#5175semidirect
382. C8#6200semidirect

Maximal subgroups

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

subgroupindex
C2×#885282
C2×#3744⋊C32
#117211582
#117210902
C2×#884382
C2×#884372
C8×(C3⋊Dic3)⋊C32
#105908843
#105908773