Finite Group Databasebeta

← #13140968

Subgroups — #13140968

Order 1728.

572 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 75 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#64156⋊C3C211yesmaximal, characteristic, complemented
3C2×#3755⋊C3C211yesmaximal, characteristic, complemented
4#11721158C211yesmaximal, characteristic, complemented
5C2×#88436C211yesmaximal, characteristic, complemented
6C2×#88432C211yesmaximal, characteristic, complemented
7#11720312C211yesmaximal, characteristic, complemented
8#10591621—31nomaximal
9#10591619—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C2×C4⋊#272C2211yescharacteristic
12C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
13C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
14C2×#6200—22no—
15#88549—22no—
16C2×#6180C2211yescharacteristic
17#3755⋊C3—22no—
18#3752⋊C3—22no—
19C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21#88436—22no—
22#88432—22no—
23#64156D311yescharacteristic, complemented
24C2×#3755D311yescharacteristic, complemented
25C3×#947—31no—
26C2×C3×#215—31no—
27C2×C3×#215—31no—
28C2×C3×#215—31no—
29C2×C3×#215—61no—
30#63914—31no—
31#63911—31no—
32C2×#3714—31no—
33C2×#3714—31no—
34C2×#3714—31no—
35C2×#3713—31no—
36#63555—31no—
37#63554—31no—
38C22×#272—21no—
39C22×(C3⋊D3)⋊C3—21no—
40C22×(C3⋊D3)⋊C3—21no—
41Q8×H(3,𝔽3)—21no—
42Q8×H(3,𝔽3)—21no—
43C2×C4×H(3,𝔽3)—21no—
44C2×C4×H(3,𝔽3)C2311yescharacteristic
45C4⋊#272—21no—
46C4⋊#272—21no—
47C2×(C3⋊Dic3)⋊C3D412yes—
48C4⋊((C3⋊D3)⋊C3)—21no—
49C4⋊((C3⋊D3)⋊C3)—21no—
50C4⋊((C3⋊D3)⋊C3)—21no—
51C4⋊((C3⋊D3)⋊C3)—21no—
52#6200—44no—
53#6180D412yescomplemented
54(C3⋊(C3⋊C8))⋊C3D412yescomplemented
55#5176—91no—
56#5176—91no—
57#5058—91no—
58C2×C32×Q8—21no—
59C2×C32×Q8C2×D311yescharacteristic
60C2×C32×Q8C2×D311yescharacteristic
61C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
62C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
63C2×C3×D12—31no—
64C2×C3×D12—31no—
65C2×C3×D12—31no—
66C2×C3×D12—31no—
67C2×C3×D12—61no—
68C2×#626—62no—
69C2×#625—62no—
70#3755—22no—
71#3752—22no—
72C4×C3⋊Dic3C2×D311yescharacteristic
73C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
74C3×#215—62no—
75C3×#215—62no—
76C3×#215—62no—
77C3×#215—64no—
78C3×(C2×D3)⋊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#3714—62no—
85#3714—62no—
86#3714—62no—
87#3713—62no—
88C22×H(3,𝔽3)C2×D411yescharacteristic
89C2×#272—41no—
90C2×#272—42no—
91C2×(C3⋊D3)⋊C3—41no—
92C2×(C3⋊D3)⋊C3—41no—
93C2×(C3⋊D3)⋊C3—42no—
94C2×(C3⋊D3)⋊C3—42no—
95C4×H(3,𝔽3)—41no—
96C4×H(3,𝔽3)C2×D411yescharacteristic
97C4×H(3,𝔽3)C2×D411yescharacteristic
98(C3⋊Dic3)⋊C3—22no—
99C2×C3×SD16—91no—
100C2×C3×SD16—91no—
101C3×#124—91no—
102#947—31no—
103#947—91no—
104C2×#215—31no—
105C2×#215—31no—
106C2×#215—31no—
107C2×#215—61no—
108C2×#215—91no—
109#940—91no—
110#940—91no—
111C2×#213—91no—
112C2×#213—91no—
113C2×#213—91no—
114C2×#213—91no—
115C2×D24—91no—
116C2×D24—91no—
117#874—91no—
118#815—91no—
119#815—91no—
120#803—91no—
121C22×C3⋊D3—21no—
122C22×C3⋊D3—21no—
123C22×C3×D3—61no—
124C22×C3×D3—61no—
125C22×C3×D3—61no—
126C22×C3×D3—61no—
127C22×C3×D3—121no—
128C32×Q8—22no—
129C32×Q8—21no—
130C32×Q8—21no—
131C32×Q8—21no—
132C32×Q8—21no—
133C32×Q8—41no—
134C2×C32×C4—21no—
135C2×C32×C4—21no—
136C2×C32×C4—21no—
137C2×C32×C4—41no—
138C2×C32×C4C22×D311yescharacteristic
139C2×C32×C4C22×D311yescharacteristic
140C2×C3⋊Dic3#6712yes—
141C4⋊(C3⋊D3)—21no—
142C4⋊(C3⋊D3)—21no—
143C4⋊(C3⋊D3)—21no—
144C4⋊(C3⋊D3)—21no—
145C2×C3×Dic3—32no—
146C3×D12—62no—
147C3×D12—61no—
148C3×D12—61no—
149C3×D12—61no—
150C3×D12—61no—
151C3×D12—61no—
152C3×D12—61no—
153C3×D12—61no—
154C3×D12—61no—
155C3×D12—121no—
156#626—124no—
157#625—124no—
158C3⋊(C3⋊C8)#6712yescomplemented
159C3×C3⋊C8—32no—
160C3×C3⋊C8—32no—
161C3×C3⋊C8—32no—
162C3×C3⋊C8—62no—
163#496—271noSylow
164C2×H(3,𝔽3)#12711yescharacteristic
165C2×H(3,𝔽3)AGL(1,ℤ/8)12yes—
166#272—81no—
167(C3⋊D3)⋊C3—81no—
168(C3⋊D3)⋊C3—81no—
169C2×C3×Q8—31no—
170C2×C3×Q8—31no—
171C2×C3×Q8—61no—
172C2×C3×Q8D3211yescharacteristic
173C2×C3×D4—91no—
174C2×C3×D4—91no—
175C2×C3×D4—91no—
176C2×#67—182no—
177C2×#67—182no—
178C2×D12—31no—
179C2×D12—31no—
180C2×D12—31no—
181C2×D12—31no—
182C2×D12—61no—
183C2×D12—91no—
184C2×D12—91no—
185C2×D12—182no—
186C3×SD16—182no—
187C3×SD16—182no—
188C2×C3×C8—91no—
189C2×C3×C8—91no—
190C3×C22⋊C4—182no—
191C3×C42—91no—
192#215—62no—
193#215—62no—
194#215—62no—
195#215—64no—
196#215—182no—
197#213—182no—
198#213—182no—
199#213—182no—
200#213—182no—
201(C2×D3)⋊C4—62no—
202(C2×D3)⋊C4—182no—
203C4×Dic3—31no—
204C4×Dic3—91no—
205C2×C3⋊C8—31no—
206C2×C3⋊C8—31no—
207C2×C3⋊C8—31no—
208C2×C3⋊C8—61no—
209C2×C3⋊C8—91no—
210D24—182no—
211D24—182no—
212C22×C32—21no—
213C22×C32D3×D411yescharacteristic
214C22×C32C2×#6711yescharacteristic
215C2×C3⋊D3—42no—
216C2×C3⋊D3—41no—
217C2×C3⋊D3—41no—
218C2×C3⋊D3—42no—
219C2×C3×D3—122no—
220C2×C3×D3—122no—
221C2×C3×D3—122no—
222C2×C3×D3—121no—
223C2×C3×D3—121no—
224C2×C3×D3—121no—
225C2×C3×D3—121no—
226C2×C3×D3—122no—
227C2×C3×D3—122no—
228C2×C3×D3—242no—
229C32×C4—21no—
230C32×C4—21no—
231C32×C4—42no—
232C32×C4—41no—
233C32×C4—41no—
234C32×C4D3×D411yescharacteristic
235C32×C4D3×D411yescharacteristic
236C32×C4D3×D411yescharacteristic
237C32×C4C2×#6711yescharacteristic
238C3⋊Dic3—22no—
239C3×Dic3—62no—
240C2×SD16—91no—
241C2×SD16—91no—
242C2×D8—271no—
243C2×D8—271no—
244#127—271no—
245#124—91no—
246#97—271no—
247H(3,𝔽3)#49611yescharacteristic, complemented, Sylow
248C23×C3—181no—
249C23×C3—181no—
250C23×C3—181no—
251C22×D3—61no—
252C22×D3—61no—
253C22×D3—61no—
254C22×D3—61no—
255C22×D3—121no—
256C22×D3—181no—
257C22×D3—181no—
258C3×Q8—21no—
259C3×Q8—21no—
260C3×Q8—61no—
261C3×Q8—61no—
262C3×Q8—61no—
263C3×Q8—61no—
264C3×Q8—62no—
265C3×Q8—121no—
266C3×D4—181no—
267C3×D4—181no—
268C3×D4—181no—
269C3×D4—181no—
270C3×D4—181no—
271C3×D4—181no—
272C2×C3×C4—21no—
273C2×C3×C4—31no—
274C2×C3×C4—31no—
275C2×C3×C4—61no—
276C2×C3×C4—61no—
277C2×C3×C4—61no—
278C2×C3×C4—92no—
279C2×C3×C4—121no—
280C2×C3×C4C2×D3211yescharacteristic
281#67—364no—
282#67—364no—
283C2×Dic3—32no—
284C2×Dic3—92no—
285D12—61no—
286D12—61no—
287D12—61no—
288D12—61no—
289D12—62no—
290D12—61no—
291D12—61no—
292D12—61no—
293D12—61no—
294D12—121no—
295D12—181no—
296D12—181no—
297D12—181no—
298D12—181no—
299D12—364no—
300C3×C8—92no—
301C3×C8—92no—
302C3⋊C8—32no—
303C3⋊C8—32no—
304C3⋊C8—32no—
305C3⋊C8—62no—
306C3⋊C8—92no—
307C2×C32—22no—
308C2×C32—21no—
309C2×C32#91412yes—
310C2×C32#94011yescharacteristic
311C2×C32#94011yescharacteristic
312C2×C32#94912yes—
313C3⋊D3—81no—
314C3⋊D3—81no—
315C3×D3—242no—
316C3×D3—241no—
317C3×D3—241no—
318C3×D3—241no—
319C3×D3—241no—
320C2×Q8((C3⋊D3)⋊C3)⋊C211yescharacteristic
321C2×D4—91no—
322C2×D4—91no—
323C2×D4—91no—
324C2×D4—542no—
325SD16—182no—
326SD16—182no—
327D8—542no—
328D8—542no—
329C2×C8—91no—
330C2×C8—91no—
331C22⋊C4—182no—
332C42—91no—
333C22×C3—31no—
334C22×C3—31no—
335C22×C3—61no—
336C22×C3—362no—
337C22×C3—361no—
338C22×C3—361no—
339C22×C3—362no—
340C22×C3—362no—
341C22×C3—361no—
342C22×C3D3×#6711yescharacteristic
343C2×D3—121no—
344C2×D3—121no—
345C2×D3—122no—
346C2×D3—122no—
347C2×D3—122no—
348C2×D3—122no—
349C2×D3—122no—
350C2×D3—121no—
351C2×D3—121no—
352C2×D3—242no—
353C2×D3—361no—
354C2×D3—361no—
355C2×D3—362no—
356C2×D3—362no—
357C3×C4—31no—
358C3×C4—31no—
359C3×C4—31no—
360C3×C4—31no—
361C3×C4—41no—
362C3×C4—61no—
363C3×C4—61no—
364C3×C4—121no—
365C3×C4—121no—
366C3×C4—122no—
367C3×C4—182no—
368C3×C4D3×#6711yescharacteristic
369C3×C4#381111yescharacteristic
370Dic3—62no—
371Dic3—182no—
372C32—21no—
373C32#505811yescharacteristic, complemented
374C32#517611yescharacteristic, complemented
375C23—181no—
376C23—181no—
377C23—181no—
378Q8—21no—
379Q8—21no—
380D4—181no—
381D4—181no—
382D4—181no—
383D4—181no—
384D4—181no—
385D4—181no—
386D4—1084no—
387C2×C4—21no—
388C2×C4—92no—
389C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
390C8—92no—
391C8—92no—
392C2×C3—32no—
393C2×C3—32no—
394C2×C3—31no—
395C2×C3—31no—
396C2×C3—62no—
397C2×C3—61no—
398C2×C3—721no—
399C2×C3—721no—
400C2×C3—721no—
401C2×C3#6393912yes—
402C2×C3#6397911yescharacteristic
403D3—241no—
404D3—241no—
405D3—242no—
406D3—241no—
407D3—241no—
408D3—721no—
409D3—721no—
410C22—362no—
411C22—362no—
412C22—362no—
413C22—361no—
414C22—361no—
415C22—361no—
416C22#8867611yescharacteristic
417C4—41no—
418C4—182no—
419C4#8867611yescharacteristic
420C4#8867711yescharacteristic
421C3—31no—
422C3—31no—
423C3—61no—
424C3#1059161911yescharacteristic
425C2—721no—
426C2—721no—
427C2—721no—
428C2#1172123312yes—
429C2#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
hypercenterC2×Q8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#496
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. #64156⋊C3C2semidirect
3. C2×#3755⋊C3C2semidirect
4. #11721158C2semidirect
5. C2×#88436C2semidirect
6. C2×#88432C2semidirect
7. #11720312C2semidirect
23. #64156D3semidirect
24. C2×#3755D3semidirect
53. #6180D4semidirect
54. (C3⋊(C3⋊C8))⋊C3D4semidirect
158. C3⋊(C3⋊C8)#67semidirect
247. H(3,𝔽3)#496semidirect
373. C32#5058semidirect
374. C32#5176semidirect

Maximal subgroups

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

subgroupindex
C2×#885492
#64156⋊C32
C2×#3755⋊C32
#117211582
C2×#884362
C2×#884322
#117203122
#105916213
#105916193