Finite Group Databasebeta

← #13138006

Subgroups — #13138006

Order 1728.

470 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 73 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#64106⋊C3C211yesmaximal, characteristic, complemented
2#11721156C211yesmaximal, characteristic
3C2×#88437C211yesmaximal, characteristic, complemented
4C2×#88430C211yesmaximal, characteristic, complemented
5#11720531C211yesmaximal, characteristic, complemented
6#63634⋊C3C211yesmaximal, characteristic, complemented
7#11720313C211yesmaximal, characteristic, complemented
8#10590506—31nomaximal
9#10590503—31nomaximal
10C2×C4⋊#272C2211yescharacteristic
11C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
12C2×#634⋊C3C2211yescharacteristic
13C2×#6200—21no—
14C4⋊C4×H(3,𝔽3)C2211yescharacteristic
15C2×#6180C2211yescharacteristic
16#3752⋊C3—21no—
17C4⋊((C3⋊Dic3)⋊C3)—21no—
18C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20#88451—21no—
21#88437—22no—
22#88430—22no—
23#64106D311yescharacteristic, complemented
24C3×#897—31no—
25#63916—31no—
26#63908—31no—
27C2×#3716—31no—
28C2×#3715—31no—
29C2×#3712—31no—
30C2×#3711—31no—
31#63634D311yescharacteristic, complemented
32C3×#809—31no—
33C3×#809—31no—
34C3×#809—61no—
35C3×#807—31no—
36#63567—31no—
37#63566—31no—
38C22×#272—21no—
39C22×(C3⋊D3)⋊C3D411yescharacteristic
40C2×C4×H(3,𝔽3)—21no—
41C2×C4×H(3,𝔽3)C2311yescharacteristic
42C4⋊#272—21no—
43C4⋊#272—21no—
44C2×(C3⋊Dic3)⋊C3—21no—
45C2×(C3⋊Dic3)⋊C3—21no—
46C2×(C3⋊Dic3)⋊C3D411yescharacteristic
47C4×(C3⋊D3)⋊C3—21no—
48C4×(C3⋊D3)⋊C3—21no—
49#634⋊C3—21no—
50#634⋊C3—21no—
51#6200—42no—
52#6180—21no—
53(C3⋊(C3⋊C8))⋊C3D412yescomplemented
54#5049—91no—
55#4886—91no—
56#4780—91no—
57C2×C4×C3⋊D3C2×D311yescharacteristic
58C2×#634C2×D311yescharacteristic
59C2×C3×D12—31no—
60C2×C3×D12—31no—
61C2×C3×D12—61no—
62C2×C3×C4×D3—31no—
63C2×C3×Dic6—31no—
64C2×#626—61no—
65C2×#625—61no—
66C32×C4⋊C4—21no—
67C32×C4⋊C4C2×D311yescharacteristic
68C32×C4⋊C4C2×D311yescharacteristic
69#3752—21no—
70C4⋊(C3⋊Dic3)—21no—
71C4⋊(C3⋊Dic3)C2×D311yescharacteristic
72C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
73C3×(C2×D3)⋊C4—61no—
74C3×C4⋊Dic3—31no—
75C3×C4⋊Dic3—61no—
76C2×C3×C3⋊C8—31no—
77C2×C3×C3⋊C8—31no—
78C2×C3×C3⋊C8—31no—
79C2×C3×C3⋊C8—61no—
80(C2×D3)⋊Dic3—61no—
81(C2×D3)⋊Dic3—61no—
82#3716—62no—
83#3715—62no—
84#3712—62no—
85#3711—62no—
86C22×H(3,𝔽3)C2×D411yescharacteristic
87C2×#272—41no—
88C2×#272—41no—
89C2×#272—41no—
90C2×(C3⋊D3)⋊C3—21no—
91C2×(C3⋊D3)⋊C3—21no—
92C2×(C3⋊D3)⋊C3—21no—
93C4×H(3,𝔽3)—41no—
94C4×H(3,𝔽3)C2×D411yescharacteristic
95C4×H(3,𝔽3)#4111yescharacteristic
96(C3⋊Dic3)⋊C3—21no—
97(C3⋊Dic3)⋊C3—41no—
98(C3⋊Dic3)⋊C3—41no—
99C3×C22⋊Q8—91no—
100C2×#215—91no—
101#938—91no—
102C2×#214—91no—
103#930—91no—
104C2×#208—91no—
105C2×#204—91no—
106C2×ΓL(1,𝔽25)—91no—
107C2×ΓL(1,𝔽25)—91no—
108#897—31no—
109#897—91no—
110#895—91no—
111C3×#107—91no—
112C3×#102—91no—
113#835—91no—
114#816—91no—
115#810—91no—
116#809—31no—
117#809—31no—
118#809—61no—
119#807—31no—
120#807—91no—
121C22×C3⋊D3#6711yescharacteristic
122C22×C3×D3—31no—
123C22×C3×D3—61no—
124C22×C3×D3—61no—
125C22×C3×D3—121no—
126C2×C32×C4—21no—
127C2×C32×C4—21no—
128C2×C32×C4—21no—
129C2×C32×C4—41no—
130C2×C32×C4C22×D311yescharacteristic
131C2×C32×C4C22×D311yescharacteristic
132C2×C3⋊Dic3—21no—
133C2×C3⋊Dic3—21no—
134C2×C3⋊Dic3#6711yescharacteristic
135C4×C3⋊D3—21no—
136C4×C3⋊D3—21no—
137#634—21no—
138#634—21no—
139C2×C3×Dic3—31no—
140C2×C3×Dic3—61no—
141C2×C3×Dic3—61no—
142C3×D12—62no—
143C3×D12—61no—
144C3×D12—61no—
145C3×D12—61no—
146C3×D12—61no—
147C3×D12—121no—
148C3×C4×D3—61no—
149C3×C4×D3—61no—
150C3×Dic6—61no—
151C3×Dic6—61no—
152#626—122no—
153#625—122no—
154C3⋊(C3⋊C8)D1212yescomplemented
155C3×C3⋊C8—32no—
156C3×C3⋊C8—61no—
157C3×C3⋊C8—61no—
158C3×C3⋊C8—121no—
159#468—271noSylow
160C2×H(3,𝔽3)#12111yescharacteristic
161C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
162C2×H(3,𝔽3)#13711yescharacteristic
163#272—81no—
164(C3⋊D3)⋊C3—41no—
165C2×C3×Q8—91no—
166C2×C3×D4—91no—
167C22×C3×C4—91no—
168C2×#67—181no—
169C2×#67—181no—
170C2×D12—31no—
171C2×D12—31no—
172C2×D12—61no—
173C2×D12—181no—
174C2×C4×D3—31no—
175C2×C4×D3—91no—
176C2×Dic6—31no—
177C2×Dic6—91no—
178C2×C3×C8—91no—
179C2×C3×C8—91no—
180C3×C4⋊C4—31no—
181C3×C4⋊C4—31no—
182C3×C4⋊C4—61no—
183C3×C4⋊C4—91no—
184C3×C4⋊C4—181no—
185C3×C4⋊C4D3211yescharacteristic
186C3×C22⋊C4—181no—
187#217—181no—
188#215—182no—
189#214—182no—
190(C2×D3)⋊C4—61no—
191(C2×D3)⋊C4—181no—
192(C2×D3)⋊C4—181no—
193(C2×D3)⋊C4—181no—
194C4⋊Dic3—31no—
195C4⋊Dic3—61no—
196C4⋊Dic3—91no—
197C4⋊Dic3—181no—
198#208—182no—
199C2×C3⋊C8—31no—
200C2×C3⋊C8—31no—
201C2×C3⋊C8—31no—
202C2×C3⋊C8—61no—
203C2×C3⋊C8—91no—
204#204—182no—
205ΓL(1,𝔽25)—182no—
206ΓL(1,𝔽25)—182no—
207C22×C32—21no—
208C22×C32D3×D411yescharacteristic
209C22×C32C2×#6711yescharacteristic
210C2×C3⋊D3—21no—
211C2×C3⋊D3—21no—
212C2×C3⋊D3—21no—
213C2×C3×D3—61no—
214C2×C3×D3—61no—
215C2×C3×D3—61no—
216C2×C3×D3—122no—
217C2×C3×D3—121no—
218C2×C3×D3—121no—
219C2×C3×D3—121no—
220C2×C3×D3—121no—
221C2×C3×D3—121no—
222C2×C3×D3—121no—
223C2×C3×D3—241no—
224C2×C3×D3—241no—
225C32×C4—21no—
226C32×C4—21no—
227C32×C4—41no—
228C32×C4—41no—
229C32×C4—81no—
230C32×C4C2×D1211yescharacteristic
231C32×C4D3×D411yescharacteristic
232C32×C4#23711yescharacteristic
233C32×C4#23911yescharacteristic
234C3⋊Dic3—21no—
235C3⋊Dic3—41no—
236C3⋊Dic3—41no—
237C3×Dic3—61no—
238C3×Dic3—121no—
239C3×Dic3—121no—
240C2×SD16—271no—
241C2×M16—271no—
242C22⋊Q8—91no—
243#121—271no—
244#107—91no—
245#103—271no—
246#102—91no—
247H(3,𝔽3)#46811yescharacteristic, complemented, Sylow
248C23×C3—91no—
249C23×C3—181no—
250C22×D3—31no—
251C22×D3—61no—
252C22×D3—61no—
253C22×D3—91no—
254C22×D3—121no—
255C3×Q8—181no—
256C3×Q8—181no—
257C3×D4—181no—
258C3×D4—181no—
259C2×C3×C4—21no—
260C2×C3×C4—31no—
261C2×C3×C4—31no—
262C2×C3×C4—61no—
263C2×C3×C4—61no—
264C2×C3×C4—61no—
265C2×C3×C4—91no—
266C2×C3×C4—121no—
267C2×C3×C4—181no—
268C2×C3×C4—181no—
269C2×C3×C4—181no—
270C2×C3×C4—181no—
271C2×C3×C4C2×D3211yescharacteristic
272#67—362no—
273#67—362no—
274C2×Dic3—31no—
275C2×Dic3—61no—
276C2×Dic3—61no—
277C2×Dic3—91no—
278C2×Dic3—181no—
279C2×Dic3—181no—
280D12—61no—
281D12—61no—
282D12—61no—
283D12—61no—
284D12—62no—
285D12—121no—
286D12—362no—
287C4×D3—61no—
288C4×D3—61no—
289C4×D3—181no—
290C4×D3—181no—
291Dic6—61no—
292Dic6—61no—
293Dic6—181no—
294Dic6—181no—
295C3×C8—92no—
296C3×C8—181no—
297C3⋊C8—32no—
298C3⋊C8—61no—
299C3⋊C8—61no—
300C3⋊C8—92no—
301C3⋊C8—121no—
302C2×C32—21no—
303C2×C32—21no—
304C2×C32—21no—
305C2×C32#89511yescharacteristic
306C2×C32#91111yescharacteristic
307C2×C32#91811yescharacteristic
308C2×C32#93911yescharacteristic
309C2×C32#94211yescharacteristic
310C2×C32#94911yescharacteristic
311C3⋊D3—41no—
312C3×D3—121no—
313C3×D3—242no—
314C3×D3—241no—
315C3×D3—241no—
316C2×Q8—91no—
317C2×D4—91no—
318C2×D4—541no—
319C22×C4—91no—
320SD16—542no—
321M16—542no—
322C2×C8—91no—
323C2×C8—91no—
324C4⋊C4—91no—
325C4⋊C4—181no—
326C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
327C22⋊C4—181no—
328C22⋊C4—541no—
329C22×C3—31no—
330C22×C3—31no—
331C22×C3—61no—
332C22×C3—181no—
333C22×C3—181no—
334C22×C3—181no—
335C22×C3—361no—
336C22×C3—361no—
337C22×C3—361no—
338C22×C3D3×#6711yescharacteristic
339C2×D3—61no—
340C2×D3—61no—
341C2×D3—61no—
342C2×D3—121no—
343C2×D3—121no—
344C2×D3—121no—
345C2×D3—121no—
346C2×D3—121no—
347C2×D3—121no—
348C2×D3—122no—
349C2×D3—181no—
350C2×D3—181no—
351C2×D3—181no—
352C2×D3—241no—
353C2×D3—241no—
354C3×C4—31no—
355C3×C4—31no—
356C3×C4—31no—
357C3×C4—31no—
358C3×C4—41no—
359C3×C4—61no—
360C3×C4—61no—
361C3×C4—121no—
362C3×C4—121no—
363C3×C4—181no—
364C3×C4—241no—
365C3×C4—361no—
366C3×C4—361no—
367C3×C4#379611yescharacteristic
368C3×C4D3×D1211yescharacteristic
369Dic3—61no—
370Dic3—121no—
371Dic3—121no—
372Dic3—181no—
373Dic3—361no—
374Dic3—361no—
375C32—21no—
376C32#488611yescharacteristic, complemented
377C32#504911yescharacteristic, complemented
378C23—91no—
379C23—181no—
380Q8—181no—
381Q8—181no—
382D4—181no—
383D4—181no—
384D4—1082no—
385C2×C4—21no—
386C2×C4—91no—
387C2×C4—181no—
388C2×C4—181no—
389C2×C4—181no—
390C2×C4—181no—
391C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
392C8—92no—
393C8—181no—
394C2×C3—31no—
395C2×C3—31no—
396C2×C3—31no—
397C2×C3—31no—
398C2×C3—31no—
399C2×C3—31no—
400C2×C3—61no—
401C2×C3—61no—
402C2×C3—61no—
403C2×C3—361no—
404C2×C3—721no—
405C2×C3#6388611yescharacteristic
406C2×C3#6392511yescharacteristic
407C2×C3#6394811yescharacteristic
408D3—121no—
409D3—241no—
410D3—241no—
411D3—242no—
412D3—361no—
413C22—181no—
414C22—181no—
415C22—181no—
416C22—361no—
417C22—361no—
418C22—361no—
419C22#8867611yescharacteristic
420C4—41no—
421C4—181no—
422C4—361no—
423C4—361no—
424C4#8864911yescharacteristic
425C4#8865411yescharacteristic
426C3—31no—
427C3—31no—
428C3—61no—
429C3#1059058511yescharacteristic
430C2—361no—
431C2—721no—
432C2#1172112711yescharacteristic
433C2#1172119511yescharacteristic
434C2#1172123411yescharacteristic
subgroupG/H
centerC22#88676
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC4⋊C4×H(3,𝔽3)C22
socleC22×C3D3×#67
prime radical#11721156C2
hypercenterC4⋊C4((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 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #64106⋊C3C2semidirect
3. C2×#88437C2semidirect
4. C2×#88430C2semidirect
5. #11720531C2semidirect
6. #63634⋊C3C2semidirect
7. #11720313C2semidirect
23. #64106D3semidirect
31. #63634D3semidirect
53. (C3⋊(C3⋊C8))⋊C3D4semidirect
154. C3⋊(C3⋊C8)D12semidirect
247. H(3,𝔽3)#468semidirect
376. C32#4886semidirect
377. C32#5049semidirect

Maximal subgroups

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

subgroupindex
#64106⋊C32
#117211562
C2×#884372
C2×#884302
#117205312
#63634⋊C32
#117203132
#105905063
#105905033