Finite Group Databasebeta

← #13140967

Subgroups — #13140967

Order 1728.

565 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 565 conjugacy classes of subgroups — too many to draw. Only its 83 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×#88549C211yesmaximal, characteristic, complemented
2#64153⋊C3C211yesmaximal, characteristic, complemented
3C2×#3755⋊C3C211yesmaximal, characteristic, complemented
4#11721158C211yesmaximal, characteristic
5C2×#88437C211yesmaximal, characteristic, complemented
6C2×#88431C211yesmaximal, characteristic, complemented
7#11720310C211yesmaximal, characteristic, complemented
8#10591620—31nomaximal
9#10591618—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C2×C4⋊#272C2211yescharacteristic
12C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
13C2×#634⋊C3C2211yescharacteristic
14C2×#6200—22no—
15#88549—22no—
16C2×#6180C2211yescharacteristic
17#3755⋊C3—22no—
18#3750⋊C3—22no—
19C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21#88437—22no—
22#88431—22no—
23#64153D311yescharacteristic, complemented
24C2×#3755D311yescharacteristic, complemented
25C3×#944—31no—
26C2×C3×#215—31no—
27C2×C3×#215—31no—
28C2×C3×#215—31no—
29C2×C3×#215—61no—
30#63914—31no—
31#63911—31no—
32C2×#3717—31no—
33C2×#3716—31no—
34C2×#3716—31no—
35C2×#3715—31no—
36#63553—31no—
37Dic3×C3⋊C8—31no—
38C22×#272—21no—
39C22×(C3⋊D3)⋊C3—21no—
40Q8×H(3,𝔽3)—21no—
41Q8×H(3,𝔽3)—21no—
42C2×C4×H(3,𝔽3)—21no—
43C2×C4×H(3,𝔽3)C2311yescharacteristic
44C4⋊#272—21no—
45C4⋊#272—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—
53#6180D412yescomplemented
54(C3⋊(C3⋊C8))⋊C3D412yescomplemented
55#5175—91no—
56#5175—91no—
57#5085—91no—
58C2×C32×Q8—21no—
59C2×C32×Q8C2×D311yescharacteristic
60C2×C32×Q8C2×D311yescharacteristic
61C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
62C2×#634C2×D311yescharacteristic
63C2×C3×D12—31no—
64C2×C3×D12—31no—
65C2×C3×D12—31no—
66C2×C3×D12—61no—
67C2×C3×Dic6—31no—
68C2×#626—62no—
69C2×#625—62no—
70#3755—22no—
71#3750—22no—
72C4×C3⋊Dic3C2×D311yescharacteristic
73C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
74C3×#215—62no—
75C3×#215—62no—
76C3×#215—62no—
77C3×#215—64no—
78C3×Dic3⋊C4—62no—
79C3×C4×Dic3—31no—
80C2×C3×C3⋊C8—31no—
81C2×C3×C3⋊C8—31no—
82C2×C3×C3⋊C8—31no—
83C2×C3×C3⋊C8—61no—
84#3717—62no—
85#3716—62no—
86#3716—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)—41no—
94C4×H(3,𝔽3)C2×D411yescharacteristic
95C4×H(3,𝔽3)C2×D411yescharacteristic
96(C3⋊Dic3)⋊C3—41no—
97(C3⋊Dic3)⋊C3SD1614yescomplemented
98C2×C3×SD16—91no—
99C2×C3×SD16—91no—
100C3×C4⋊Q8—91no—
101#944—31no—
102#944—91no—
103C2×#215—31no—
104C2×#215—31no—
105C2×#215—31no—
106C2×#215—61no—
107C2×#215—91no—
108C2×#215—91no—
109C2×#215—91no—
110#940—91no—
111#940—91no—
112C2×#214—91no—
113C2×#214—91no—
114C2×#204—91no—
115C2×#204—91no—
116#874—91no—
117C8×Dic3—91no—
118C8×Dic3—91no—
119C4×C3⋊C8—91no—
120C22×C3⋊D3—21no—
121C22×C3×D3—61no—
122C22×C3×D3—61no—
123C22×C3×D3—61no—
124C22×C3×D3—121no—
125C32×Q8—22no—
126C32×Q8—21no—
127C32×Q8—21no—
128C32×Q8—21no—
129C32×Q8—21no—
130C32×Q8—41no—
131C2×C32×C4—21no—
132C2×C32×C4—21no—
133C2×C32×C4—21no—
134C2×C32×C4—41no—
135C2×C32×C4C22×D311yescharacteristic
136C2×C32×C4C22×D311yescharacteristic
137C2×C3⋊Dic3—21no—
138C2×C3⋊Dic3#6712yes—
139C4⋊(C3⋊D3)—21no—
140C4⋊(C3⋊D3)—21no—
141#634—21no—
142#634—21no—
143C2×C3×Dic3—32no—
144C2×C3×Dic3—61no—
145C3×D12—62no—
146C3×D12—61no—
147C3×D12—61no—
148C3×D12—61no—
149C3×D12—61no—
150C3×D12—61no—
151C3×D12—61no—
152C3×D12—121no—
153C3×Dic6—61no—
154C3×Dic6—61no—
155#626—124no—
156#625—124no—
157C3⋊(C3⋊C8)#6712yescomplemented
158C3×C3⋊C8—32no—
159C3×C3⋊C8—32no—
160C3×C3⋊C8—32no—
161C3×C3⋊C8—62no—
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—31no—
168C2×C3×Q8—31no—
169C2×C3×Q8—61no—
170C2×C3×Q8—91no—
171C2×C3×Q8D3211yescharacteristic
172C2×C3×D4—91no—
173C2×C3×D4—91no—
174C2×#67—182no—
175C2×#67—182no—
176C2×D12—31no—
177C2×D12—31no—
178C2×D12—31no—
179C2×D12—61no—
180C2×D12—91no—
181C2×D12—182no—
182C2×Dic6—31no—
183C2×Dic6—91no—
184C3×SD16—182no—
185C3×SD16—182no—
186C2×C3×C8—91no—
187C2×C3×C8—91no—
188C3×C4⋊C4—182no—
189C3×C42—91no—
190#215—62no—
191#215—62no—
192#215—62no—
193#215—64no—
194#215—182no—
195#215—182no—
196#215—182no—
197#214—182no—
198#214—182no—
199Dic3⋊C4—62no—
200Dic3⋊C4—182no—
201C4×Dic3—31no—
202C4×Dic3—91no—
203C2×C3⋊C8—31no—
204C2×C3⋊C8—31no—
205C2×C3⋊C8—31no—
206C2×C3⋊C8—61no—
207C2×C3⋊C8—91no—
208#204—182no—
209#204—182no—
210C22×C32—21no—
211C22×C32D3×D411yescharacteristic
212C22×C32C2×#6711yescharacteristic
213C2×C3⋊D3—42no—
214C2×C3⋊D3—41no—
215C2×C3×D3—122no—
216C2×C3×D3—122no—
217C2×C3×D3—121no—
218C2×C3×D3—121no—
219C2×C3×D3—121no—
220C2×C3×D3—122no—
221C2×C3×D3—122no—
222C2×C3×D3—242no—
223C32×C4—21no—
224C32×C4—21no—
225C32×C4—42no—
226C32×C4—41no—
227C32×C4—41no—
228C32×C4D3×D411yescharacteristic
229C32×C4D3×D411yescharacteristic
230C32×C4D3×D411yescharacteristic
231C32×C4C2×#6711yescharacteristic
232C3⋊Dic3—41no—
233C3⋊Dic3#21514yescomplemented
234C3×Dic3—34no—
235C3×Dic3—121no—
236C2×SD16—91no—
237C2×SD16—91no—
238C2×SD16—271no—
239C2×SD16—271no—
240C4⋊Q8—91no—
241#127—271no—
242C4×C8—271no—
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—21no—
252C3×Q8—21no—
253C3×Q8—61no—
254C3×Q8—61no—
255C3×Q8—61no—
256C3×Q8—61no—
257C3×Q8—62no—
258C3×Q8—121no—
259C3×Q8—181no—
260C3×Q8—181no—
261C3×D4—181no—
262C3×D4—181no—
263C3×D4—181no—
264C3×D4—181no—
265C2×C3×C4—21no—
266C2×C3×C4—31no—
267C2×C3×C4—31no—
268C2×C3×C4—61no—
269C2×C3×C4—61no—
270C2×C3×C4—61no—
271C2×C3×C4—92no—
272C2×C3×C4—121no—
273C2×C3×C4—181no—
274C2×C3×C4C2×D3211yescharacteristic
275#67—364no—
276#67—364no—
277C2×Dic3—32no—
278C2×Dic3—61no—
279C2×Dic3—92no—
280C2×Dic3—181no—
281D12—61no—
282D12—61no—
283D12—61no—
284D12—61no—
285D12—62no—
286D12—61no—
287D12—61no—
288D12—121no—
289D12—181no—
290D12—181no—
291D12—364no—
292Dic6—61no—
293Dic6—61no—
294Dic6—181no—
295Dic6—181no—
296C3×C8—92no—
297C3×C8—92no—
298C3⋊C8—32no—
299C3⋊C8—32no—
300C3⋊C8—32no—
301C3⋊C8—62no—
302C3⋊C8—92no—
303C2×C32—22no—
304C2×C32—21no—
305C2×C32D3×SD1612yes—
306C2×C32#94011yescharacteristic
307C2×C32#94011yescharacteristic
308C2×C32C2×#21512yes—
309C3⋊D3—81no—
310C3×D3—242no—
311C3×D3—241no—
312C3×D3—241no—
313C3×D3—241no—
314C2×Q8—91no—
315C2×Q8((C3⋊D3)⋊C3)⋊C211yescharacteristic
316C2×D4—91no—
317C2×D4—91no—
318C2×D4—542no—
319SD16—182no—
320SD16—182no—
321SD16—542no—
322SD16—542no—
323C2×C8—91no—
324C2×C8—91no—
325C4⋊C4—182no—
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×#6711yescharacteristic
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—41no—
350C3×C4—61no—
351C3×C4—61no—
352C3×C4—94no—
353C3×C4—121no—
354C3×C4—121no—
355C3×C4—122no—
356C3×C4—361no—
357C3×C4D3×#6711yescharacteristic
358C3×C4#381111yescharacteristic
359Dic3—34no—
360Dic3—94no—
361Dic3—121no—
362Dic3—361no—
363C32—21no—
364C32#508511yescharacteristic, complemented
365C32#517511yescharacteristic, complemented
366C23—181no—
367C23—181no—
368Q8—21no—
369Q8—21no—
370Q8—181no—
371Q8—181no—
372D4—181no—
373D4—181no—
374D4—181no—
375D4—181no—
376D4—1084no—
377C2×C4—21no—
378C2×C4—92no—
379C2×C4—181no—
380C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
381C8—92no—
382C8—92no—
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×#21512yes—
392C2×C3#6397911yescharacteristic
393D3—241no—
394D3—241no—
395D3—242no—
396D3—241no—
397D3—721no—
398C22—362no—
399C22—362no—
400C22—361no—
401C22—361no—
402C22#8867611yescharacteristic
403C4—41no—
404C4—94no—
405C4—361no—
406C4#8867611yescharacteristic
407C4#8867711yescharacteristic
408C3—31no—
409C3—31no—
410C3—61no—
411C3#1059161811yescharacteristic
412C2—721no—
413C2—721no—
414C2#1172123212yes—
415C2#1172131711yescharacteristic
subgroupG/H
centerC22#88676
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×Q8×H(3,𝔽3)C22
socleC22×C3D3×#67
prime radical#11721158C2
hypercenterC2×Q8((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×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#88549C2semidirect
2. #64153⋊C3C2semidirect
3. C2×#3755⋊C3C2semidirect
5. C2×#88437C2semidirect
6. C2×#88431C2semidirect
7. #11720310C2semidirect
23. #64153D3semidirect
24. C2×#3755D3semidirect
53. #6180D4semidirect
54. (C3⋊(C3⋊C8))⋊C3D4semidirect
97. (C3⋊Dic3)⋊C3SD16semidirect
157. C3⋊(C3⋊C8)#67semidirect
233. C3⋊Dic3#215semidirect
243. H(3,𝔽3)#492semidirect
364. C32#5085semidirect
365. C32#5175semidirect

Maximal subgroups

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

subgroupindex
C2×#885492
#64153⋊C32
C2×#3755⋊C32
#117211582
C2×#884372
C2×#884312
#117203102
#105916203
#105916183