Finite Group Databasebeta

← #13140933

Subgroups — #13140933

Order 1728.

494 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 71 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#11721863C211yesmaximal, characteristic, complemented
2C2×#3755⋊C3C211yesmaximal, characteristic, complemented
3#11721159C211yesmaximal, characteristic
4C2×#88435C211yesmaximal, characteristic, complemented
5#63662⋊C3C211yesmaximal, characteristic, complemented
6#11720308C211yesmaximal, characteristic, complemented
7#11720306C211yesmaximal, characteristic, complemented
8#10591589—31nomaximal
9#10591587—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C2×C4×#272C2211yescharacteristic
12C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
13C2×#6200—21no—
14#88542—21no—
15C4⋊#1084C2211yescharacteristic
16#88540—21no—
17#3755⋊C3—22no—
18C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21#88450—21no—
22#88435—22no—
23C2×#3755D311yescharacteristic, complemented
24C3×#946—31no—
25C3×#946—31no—
26C3×#946—61no—
27C2×C3×#215—31no—
28#63909—31no—
29#63908—31no—
30C2×#3711—31no—
31C2×#3711—31no—
32#63662D311yescharacteristic, complemented
33C3×#835—31no—
34#63576—31no—
35#63575—31no—
36#63564—31no—
37#63562—31no—
38C22×#272D411yescharacteristic
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
44C2×#1084—21no—
45C2×#1084D411yescharacteristic
46C4×#272—21no—
47C4×#272—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#5174—91no—
55#5049—91no—
56#4812—91no—
57C2×C32×Q8—21no—
58C2×C32×Q8C2×D311yescharacteristic
59C2×C32×Q8C2×D311yescharacteristic
60C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
61C2×C3×D12—31no—
62C2×C3×C4×D3—31no—
63C2×C3×C4×D3—31no—
64C2×C3×C4×D3—61no—
65C2×#626—61no—
66C2×#625—61no—
67#3755—22no—
68C4⋊(C3⋊Dic3)C2×D311yescharacteristic
69C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
70C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
71C3×#215—62no—
72C3×(C2×D3)⋊C4—61no—
73C3×(C2×D3)⋊C4—61no—
74C3×(C2×D3)⋊C4—62no—
75C3×C4⋊Dic3—31no—
76C3×C4⋊Dic3—31no—
77C3×C4⋊Dic3—31no—
78C3×C4⋊Dic3—61no—
79C3×Dic3⋊C4—61no—
80C3×Dic3⋊C4—61no—
81C3×Dic3⋊C4—62no—
82C2×C3×C3⋊C8—31no—
83C2×C3×C3⋊C8—31no—
84#3722—61no—
85(C2×D3)⋊Dic3—61no—
86#3711—62no—
87#3711—62no—
88C22×H(3,𝔽3)C2×D411yescharacteristic
89C2×#272—21no—
90C2×#272—21no—
91C2×#272—21no—
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—
101C2×C3×SD16—91no—
102C3×C22⋊Q8—91no—
103#946—31no—
104#946—31no—
105#946—61no—
106C2×#215—31no—
107C2×#215—91no—
108#938—91no—
109#930—91no—
110C2×#208—91no—
111C2×ΓL(1,𝔽25)—91no—
112C2×ΓL(1,𝔽25)—91no—
113#895—91no—
114C3×#103—91no—
115#835—31no—
116#835—91no—
117#832—91no—
118#821—91no—
119#818—91no—
120#809—91no—
121#807—91no—
122#807—91no—
123C22×C3⋊D3—21no—
124C22×C3×D3—31no—
125C22×C3×D3—31no—
126C22×C3×D3—61no—
127C22×C3×D3—61no—
128C32×Q8—21no—
129C32×Q8—21no—
130C32×Q8—21no—
131C32×Q8—21no—
132C32×Q8—41no—
133C32×Q8—41no—
134C2×C32×C4—22no—
135C2×C32×C4—21no—
136C2×C32×C4—21no—
137C2×C32×C4—21no—
138C2×C32×C4C22×D311yescharacteristic
139C2×C32×C4C22×D311yescharacteristic
140C2×C3⋊Dic3—21no—
141C4⋊(C3⋊D3)—21no—
142C4⋊(C3⋊D3)—21no—
143C2×C3×Dic3—31no—
144C2×C3×Dic3—31no—
145C2×C3×Dic3—61no—
146C2×C3×Dic3—62no—
147C2×C3×Dic3—61no—
148C2×C3×Dic3—61no—
149C2×C3×Dic3—61no—
150C3×D12—61no—
151C3×D12—61no—
152C3×C4×D3—61no—
153C3×C4×D3—61no—
154C3×C4×D3—61no—
155C3×C4×D3—61no—
156C3×C4×D3—121no—
157C3×C4×D3—121no—
158#626—122no—
159#625—122no—
160C3⋊(C3⋊C8)—21no—
161C3⋊(C3⋊C8)#6712yescomplemented
162C3×C3⋊C8—32no—
163C3×C3⋊C8—61no—
164#468—271noSylow
165C2×H(3,𝔽3)#12111yescharacteristic
166C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
167C2×H(3,𝔽3)#13711yescharacteristic
168#272—41no—
169(C3⋊D3)⋊C3—81no—
170C2×C3×Q8—31no—
171C2×C3×Q8—31no—
172C2×C3×Q8—61no—
173C2×C3×Q8D3211yescharacteristic
174C2×C3×D4—91no—
175C22×C3×C4—91no—
176C2×#67—181no—
177C2×#67—181no—
178C2×D12—31no—
179C2×D12—91no—
180C2×D12—181no—
181C2×C4×D3—31no—
182C2×C4×D3—31no—
183C2×C4×D3—61no—
184C3×SD16—182no—
185C2×C3×C8—91no—
186C2×C3×C8—91no—
187C3×C4⋊C4—91no—
188C3×C4⋊C4—91no—
189C3×C4⋊C4—181no—
190C3×C22⋊C4—181no—
191#217—181no—
192#215—62no—
193#215—182no—
194(C2×D3)⋊C4—61no—
195(C2×D3)⋊C4—61no—
196(C2×D3)⋊C4—62no—
197(C2×D3)⋊C4—181no—
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—
207#208—182no—
208C2×C3⋊C8—31no—
209C2×C3⋊C8—31no—
210C2×C3⋊C8—91no—
211C2×C3⋊C8—91no—
212ΓL(1,𝔽25)—182no—
213ΓL(1,𝔽25)—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—62no—
221C2×C3×D3—61no—
222C2×C3×D3—61no—
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×C4#23511yescharacteristic
238C32×C4D3×D411yescharacteristic
239C32×C4#23711yescharacteristic
240C32×C4C2×#6711yescharacteristic
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—91no—
250C2×M16—271no—
251C22⋊Q8—91no—
252#121—271no—
253#107—271no—
254#103—91no—
255#102—271no—
256H(3,𝔽3)#46811yescharacteristic, 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—21no—
265C3×Q8—21no—
266C3×Q8—61no—
267C3×Q8—61no—
268C3×Q8—61no—
269C3×Q8—61no—
270C3×Q8—121no—
271C3×Q8—121no—
272C3×D4—181no—
273C3×D4—181no—
274C2×C3×C4—21no—
275C2×C3×C4—31no—
276C2×C3×C4—31no—
277C2×C3×C4—61no—
278C2×C3×C4—61no—
279C2×C3×C4—62no—
280C2×C3×C4—61no—
281C2×C3×C4—91no—
282C2×C3×C4—181no—
283C2×C3×C4—181no—
284C2×C3×C4—181no—
285C2×C3×C4—181no—
286C2×C3×C4C2×D3211yescharacteristic
287#67—362no—
288#67—362no—
289C2×Dic3—31no—
290C2×Dic3—31no—
291C2×Dic3—61no—
292C2×Dic3—61no—
293C2×Dic3—61no—
294C2×Dic3—62no—
295C2×Dic3—61no—
296C2×Dic3—181no—
297D12—61no—
298D12—61no—
299D12—181no—
300D12—181no—
301D12—362no—
302C4×D3—61no—
303C4×D3—61no—
304C4×D3—61no—
305C4×D3—61no—
306C4×D3—121no—
307C4×D3—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#91411yescharacteristic
319C2×C32#91411yescharacteristic
320C2×C32#91511yescharacteristic
321C2×C32#91811yescharacteristic
322C2×C32#93811yescharacteristic
323C3⋊D3—81no—
324C3×D3—122no—
325C3×D3—121no—
326C3×D3—121no—
327C3×D3—241no—
328C2×Q8((C3⋊D3)⋊C3)⋊C211yescharacteristic
329C2×D4—91no—
330C2×D4—541no—
331C22×C4—91no—
332SD16—182no—
333M16—542no—
334C2×C8—91no—
335C2×C8—91no—
336C4⋊C4—91no—
337C4⋊C4—91no—
338C4⋊C4—181no—
339C22⋊C4—181no—
340C22⋊C4—541no—
341C22×C3—31no—
342C22×C3—31no—
343C22×C3—61no—
344C22×C3—181no—
345C22×C3—181no—
346C22×C3—181no—
347C22×C3—361no—
348C22×C3—361no—
349C22×C3—361no—
350C22×C3#381111yescharacteristic
351C2×D3—61no—
352C2×D3—61no—
353C2×D3—61no—
354C2×D3—61no—
355C2×D3—61no—
356C2×D3—61no—
357C2×D3—62no—
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#380411yescharacteristic
380C3×C4D3×#6711yescharacteristic
381Dic3—61no—
382Dic3—61no—
383Dic3—121no—
384Dic3—121no—
385Dic3—121no—
386Dic3—122no—
387Dic3—121no—
388Dic3—361no—
389C32—21no—
390C32#481211yescharacteristic, complemented
391C32#517411yescharacteristic, complemented
392C23—91no—
393C23—181no—
394Q8—21no—
395Q8—21no—
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—361no—
418C2×C3—721no—
419C2×C3#6394111yescharacteristic
420C2×C3#6395111yescharacteristic
421C2×C3#6395811yescharacteristic
422D3—121no—
423D3—121no—
424D3—122no—
425D3—241no—
426D3—721no—
427C22—181no—
428C22—181no—
429C22—181no—
430C22—361no—
431C22—361no—
432C22—361no—
433C22#8867711yescharacteristic
434C4—41no—
435C4—181no—
436C4—361no—
437C4—361no—
438C4#8867111yescharacteristic
439C4#8867611yescharacteristic
440C3—31no—
441C3—31no—
442C3—61no—
443C3#1059159011yescharacteristic
444C2—361no—
445C2—721no—
446C2#1172122911yescharacteristic
447C2#1172123111yescharacteristic
448C2#1172130311yescharacteristic
subgroupG/H
centerC22#88677
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×Q8×H(3,𝔽3)C22
socleC22×C3#3811
prime radical#11721159C2
hypercenterC2×Q8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#468
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. #11721863C2semidirect
2. C2×#3755⋊C3C2semidirect
4. C2×#88435C2semidirect
5. #63662⋊C3C2semidirect
6. #11720308C2semidirect
7. #11720306C2semidirect
23. C2×#3755D3semidirect
32. #63662D3semidirect
53. (C3⋊(C3⋊C8))⋊C3D4semidirect
161. C3⋊(C3⋊C8)#67semidirect
256. H(3,𝔽3)#468semidirect
390. C32#4812semidirect
391. C32#5174semidirect

Maximal subgroups

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

subgroupindex
#117218632
C2×#3755⋊C32
#117211592
C2×#884352
#63662⋊C32
#117203082
#117203062
#105915893
#105915873