Finite Group Databasebeta

← #13140596

Subgroups — #13140596

Order 1728.

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

normal subgroups only

This group has 612 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
1#11721850C211yesmaximal, characteristic, complemented
2C2×#3753⋊C3C211yesmaximal, characteristic, complemented
3C2×#3754⋊C3C211yesmaximal, characteristic, complemented
4#11721113C211yesmaximal, characteristic, complemented
5C2×#88437C211yesmaximal, characteristic, complemented
6C2×#88436C211yesmaximal, characteristic, complemented
7#11720303C211yesmaximal, characteristic, complemented
8#10591472—31nomaximal
9#10591469—31nomaximal
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C2×#6235—22no—
12C2×C4⋊#272C2211yescharacteristic
13C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
14C2×#634⋊C3C2211yescharacteristic
15C4×#1084C2211yescharacteristic
16#3753⋊C3—22no—
17#3754⋊C3—22no—
18C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20#88450—22no—
21#88437—22no—
22#88436—22no—
23C2×#3754D311yescharacteristic, complemented
24C2×#3753D311yescharacteristic, complemented
25C3×#940—31no—
26C3×#940—31no—
27C3×#940—61no—
28C2×C3×#214—31no—
29C2×C3×#213—31no—
30#63864—31no—
31#63860—31no—
32C2×#3716—31no—
33C2×#3715—31no—
34C2×#3714—31no—
35C2×#3713—31no—
36#63554—31no—
37#63554—31no—
38C23×H(3,𝔽3)—21no—
39C22×#272—21no—
40C22×(C3⋊D3)⋊C3—21no—
41D4×H(3,𝔽3)—21no—
42D4×H(3,𝔽3)—21no—
43C2×C4×H(3,𝔽3)C2311yescharacteristic
44#6235—44no—
45C2×#1084D412yes—
46C4⋊#272—21no—
47C4⋊#272—21no—
48C2×(C3⋊Dic3)⋊C3—21no—
49C4⋊((C3⋊D3)⋊C3)—21no—
50C4⋊((C3⋊D3)⋊C3)—21no—
51#634⋊C3—21no—
52#634⋊C3—21no—
53(C3⋊(C3⋊C8))⋊C3D412yescomplemented
54(C3⋊(C3⋊C8))⋊C3D412yescomplemented
55#5176—91no—
56#5154—91no—
57#5074—91no—
58C2×C32×D4—21no—
59C2×C32×D4C2×D311yescharacteristic
60C2×C32×D4C2×D311yescharacteristic
61C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
62C2×#634C2×D311yescharacteristic
63C2×C3×#67—62no—
64C2×C3×#67—62no—
65C2×C3×#67—64no—
66C2×C3×D12—31no—
67C2×C3×D12—31no—
68C2×C3×D12—31no—
69C2×C3×D12—61no—
70C2×C3×Dic6—31no—
71#3754—22no—
72#3753—22no—
73C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
74C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
75C3×#214—62no—
76C3×#213—62no—
77C3×C4×Dic3—31no—
78C3×C4×Dic3—31no—
79C3×C4×Dic3—61no—
80C2×C3×C3⋊C8—31no—
81C2×C3×C3⋊C8—31no—
82#3722—62no—
83(C2×D3)⋊Dic3—62no—
84#3716—62no—
85#3715—62no—
86#3714—62no—
87#3713—62no—
88C22×H(3,𝔽3)—41no—
89C22×H(3,𝔽3)—42no—
90C22×H(3,𝔽3)C2×D411yescharacteristic
91C2×#272—41no—
92C2×#272—42no—
93C2×(C3⋊D3)⋊C3—41no—
94C2×(C3⋊D3)⋊C3—42no—
95C4×H(3,𝔽3)C2×D411yescharacteristic
96C4×H(3,𝔽3)C2×D411yescharacteristic
97#1084—22no—
98(C3⋊Dic3)⋊C3—41no—
99C2×C3×SD16—91no—
100C2×C3×D8—91no—
101C3×#127—91no—
102#947—91no—
103C2×#215—91no—
104#940—31no—
105#940—31no—
106#940—61no—
107#936—91no—
108C2×#214—31no—
109C2×#214—91no—
110C2×#214—91no—
111C2×#213—31no—
112C2×#213—91no—
113C2×#213—91no—
114C2×#213—91no—
115C2×D24—91no—
116C2×#204—91no—
117#875—91no—
118#815—91no—
119#815—91no—
120#803—91no—
121C23×C32—22no—
122C23×C32—21no—
123C23×C32—21no—
124C22×C3⋊D3—21no—
125C22×C3×D3—62no—
126C22×C3×D3—61no—
127C22×C3×D3—61no—
128C22×C3×D3—61no—
129C32×D4—21no—
130C32×D4—21no—
131C32×D4—21no—
132C32×D4—21no—
133C32×D4—41no—
134C32×D4—41no—
135C2×C32×C4—21no—
136C2×C32×C4C22×D311yescharacteristic
137C2×C32×C4C22×D311yescharacteristic
138C2×C3⋊Dic3—21no—
139C4⋊(C3⋊D3)—21no—
140C4⋊(C3⋊D3)—21no—
141#634—21no—
142#634—21no—
143C3×#67—128no—
144C3×#67—124no—
145C3×#67—124no—
146C2×C3×Dic3—32no—
147C2×C3×Dic3—32no—
148C2×C3×Dic3—62no—
149C2×C3×Dic3—61no—
150C3×D12—61no—
151C3×D12—61no—
152C3×D12—61no—
153C3×D12—61no—
154C3×D12—61no—
155C3×D12—61no—
156C3×D12—121no—
157C3×D12—121no—
158C3×Dic6—61no—
159C3×Dic6—61no—
160C3⋊(C3⋊C8)#6712yescomplemented
161C3⋊(C3⋊C8)#6712yescomplemented
162C3×C3⋊C8—32no—
163C3×C3⋊C8—32no—
164#496—271noSylow
165C2×H(3,𝔽3)—81no—
166C2×H(3,𝔽3)#12711yescharacteristic
167C2×H(3,𝔽3)AGL(1,ℤ/8)12yes—
168#272—81no—
169(C3⋊D3)⋊C3—81no—
170C2×C3×Q8—91no—
171C2×C3×D4—31no—
172C2×C3×D4—31no—
173C2×C3×D4—61no—
174C2×C3×D4—91no—
175C2×C3×D4—91no—
176C2×C3×D4—182no—
177C2×C3×D4D3211yescharacteristic
178C2×#67—62no—
179C2×#67—62no—
180C2×#67—64no—
181C2×D12—31no—
182C2×D12—31no—
183C2×D12—31no—
184C2×D12—61no—
185C2×D12—91no—
186C2×Dic6—31no—
187C2×Dic6—91no—
188C3×SD16—182no—
189C3×D8—182no—
190C2×C3×C8—91no—
191C2×C3×C8—91no—
192C3×C42—91no—
193#217—182no—
194#215—182no—
195#214—62no—
196#214—182no—
197#214—182no—
198#213—62no—
199#213—182no—
200#213—182no—
201#213—182no—
202(C2×D3)⋊C4—182no—
203(C2×D3)⋊C4—182no—
204C4×Dic3—31no—
205C4×Dic3—31no—
206C4×Dic3—61no—
207C2×C3⋊C8—31no—
208C2×C3⋊C8—31no—
209C2×C3⋊C8—91no—
210C2×C3⋊C8—91no—
211D24—182no—
212#204—182no—
213C22×C32—21no—
214C22×C32—44no—
215C22×C32—42no—
216C22×C32—42no—
217C22×C32—42no—
218C22×C32—41no—
219C22×C32—41no—
220C22×C32D3×D411yescharacteristic
221C22×C32D3×D411yescharacteristic
222C2×C3⋊D3—42no—
223C2×C3⋊D3—41no—
224C2×C3×D3—122no—
225C2×C3×D3—124no—
226C2×C3×D3—122no—
227C2×C3×D3—122no—
228C2×C3×D3—122no—
229C2×C3×D3—121no—
230C2×C3×D3—121no—
231C2×C3×D3—121no—
232C32×C4—21no—
233C32×C4—21no—
234C32×C4D3×D411yescharacteristic
235C32×C4D3×D411yescharacteristic
236C32×C4C2×#6711yescharacteristic
237C32×C4C2×#6711yescharacteristic
238C3⋊Dic3—41no—
239C3×Dic3—64no—
240C3×Dic3—62no—
241C3×Dic3—62no—
242C3×Dic3—121no—
243C2×SD16—91no—
244C2×SD16—271no—
245C2×D8—91no—
246C2×D8—271no—
247#127—91no—
248#124—271no—
249#97—271no—
250H(3,𝔽3)#49611yescharacteristic, complemented, Sylow
251C23×C3—21no—
252C23×C3—61no—
253C23×C3—61no—
254C23×C3—62no—
255C23×C3—181no—
256C23×C3—181no—
257C22×D3—61no—
258C22×D3—61no—
259C22×D3—62no—
260C22×D3—61no—
261C22×D3—181no—
262C3×Q8—181no—
263C3×Q8—181no—
264C3×D4—21no—
265C3×D4—21no—
266C3×D4—61no—
267C3×D4—61no—
268C3×D4—61no—
269C3×D4—61no—
270C3×D4—121no—
271C3×D4—121no—
272C3×D4—181no—
273C3×D4—181no—
274C3×D4—181no—
275C3×D4—181no—
276C3×D4—364no—
277C2×C3×C4—31no—
278C2×C3×C4—31no—
279C2×C3×C4—61no—
280C2×C3×C4—92no—
281C2×C3×C4—181no—
282C2×C3×C4C2×D3211yescharacteristic
283#67—124no—
284#67—124no—
285#67—128no—
286C2×Dic3—32no—
287C2×Dic3—32no—
288C2×Dic3—62no—
289C2×Dic3—61no—
290C2×Dic3—181no—
291D12—61no—
292D12—61no—
293D12—61no—
294D12—61no—
295D12—61no—
296D12—61no—
297D12—121no—
298D12—121no—
299D12—181no—
300D12—181no—
301Dic6—61no—
302Dic6—61no—
303Dic6—181no—
304Dic6—181no—
305C3×C8—92no—
306C3×C8—92no—
307C3⋊C8—32no—
308C3⋊C8—32no—
309C3⋊C8—92no—
310C3⋊C8—92no—
311C2×C32—22no—
312C2×C32—21no—
313C2×C32—82no—
314C2×C32—81no—
315C2×C32—81no—
316C2×C32#91112yes—
317C2×C32#91412yes—
318C2×C32#94011yescharacteristic
319C2×C32#94011yescharacteristic
320C3⋊D3—81no—
321C3×D3—242no—
322C3×D3—241no—
323C3×D3—241no—
324C3×D3—241no—
325C2×Q8—91no—
326C2×D4—91no—
327C2×D4—91no—
328C2×D4—182no—
329C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
330SD16—182no—
331SD16—542no—
332D8—182no—
333D8—542no—
334C2×C8—91no—
335C2×C8—91no—
336C22⋊C4—542no—
337C42—91no—
338C22×C3—31no—
339C22×C3—31no—
340C22×C3—42no—
341C22×C3—41no—
342C22×C3—61no—
343C22×C3—122no—
344C22×C3—122no—
345C22×C3—121no—
346C22×C3—121no—
347C22×C3—124no—
348C22×C3—122no—
349C22×C3—361no—
350C22×C3—362no—
351C22×C3—362no—
352C22×C3—361no—
353C22×C3#381111yescharacteristic
354C2×D3—121no—
355C2×D3—121no—
356C2×D3—122no—
357C2×D3—122no—
358C2×D3—122no—
359C2×D3—124no—
360C2×D3—122no—
361C2×D3—121no—
362C2×D3—361no—
363C2×D3—362no—
364C3×C4—31no—
365C3×C4—31no—
366C3×C4—31no—
367C3×C4—31no—
368C3×C4—61no—
369C3×C4—61no—
370C3×C4—182no—
371C3×C4—361no—
372C3×C4D3×#6711yescharacteristic
373C3×C4D3×#6711yescharacteristic
374Dic3—62no—
375Dic3—62no—
376Dic3—64no—
377Dic3—121no—
378Dic3—361no—
379C32—21no—
380C32#515411yescharacteristic, complemented
381C32#517611yescharacteristic, complemented
382C23—21no—
383C23—181no—
384C23—181no—
385Q8—181no—
386Q8—181no—
387D4—21no—
388D4—21no—
389D4—181no—
390D4—181no—
391D4—181no—
392D4—181no—
393D4—364no—
394C2×C4—92no—
395C2×C4—181no—
396C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
397C8—92no—
398C8—92no—
399C2×C3—31no—
400C2×C3—31no—
401C2×C3—32no—
402C2×C3—32no—
403C2×C3—61no—
404C2×C3—62no—
405C2×C3—81no—
406C2×C3—241no—
407C2×C3—241no—
408C2×C3—242no—
409C2×C3—721no—
410C2×C3—721no—
411C2×C3#6393712yes—
412C2×C3#6397911yescharacteristic
413D3—241no—
414D3—241no—
415D3—242no—
416D3—241no—
417D3—721no—
418C22—41no—
419C22—42no—
420C22—361no—
421C22—361no—
422C22—362no—
423C22—362no—
424C22#8867711yescharacteristic
425C4—182no—
426C4—361no—
427C4#8867611yescharacteristic
428C4#8867611yescharacteristic
429C3—31no—
430C3—31no—
431C3—61no—
432C3#1059147511yescharacteristic
433C2—81no—
434C2—721no—
435C2—721no—
436C2#1172119312yes—
437C2#1172131711yescharacteristic
subgroupG/H
centerC22#88677
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×D4×H(3,𝔽3)C22
socleC22×C3#3811
hypercenterC2×D4((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×D4

Complements

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

NH ≅ G/Ntype
1. #11721850C2semidirect
2. C2×#3753⋊C3C2semidirect
3. C2×#3754⋊C3C2semidirect
4. #11721113C2semidirect
5. C2×#88437C2semidirect
6. C2×#88436C2semidirect
7. #11720303C2semidirect
23. C2×#3754D3semidirect
24. C2×#3753D3semidirect
53. (C3⋊(C3⋊C8))⋊C3D4semidirect
54. (C3⋊(C3⋊C8))⋊C3D4semidirect
160. C3⋊(C3⋊C8)#67semidirect
161. C3⋊(C3⋊C8)#67semidirect
250. H(3,𝔽3)#496semidirect
380. C32#5154semidirect
381. C32#5176semidirect

Maximal subgroups

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

subgroupindex
#117218502
C2×#3753⋊C32
C2×#3754⋊C32
#117211132
C2×#884372
C2×#884362
#117203032
#105914723
#105914693