Finite Group Databasebeta

← #11780213

Subgroups — #11780213

Order 1152.

500 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 94 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
1Dic6×D12C211yesmaximal, characteristic
2C3×#4846C211yesmaximal, characteristic
3#10590384C211yesmaximal, characteristic, complemented
4#10590311C211yesmaximal, characteristic, complemented
5#10589352C211yesmaximal, characteristic, complemented
6C3×#4484C211yesmaximal, characteristic, complemented
7#10589081C211yesmaximal, characteristic
8#71114—31nomaximal
9#68273—31nomaximal
10C2×D3×Dic6—21no—
11#64078C2211yescharacteristic
12C3×C4×D12C2211yescharacteristic
13C3×#869C2211yescharacteristic
14C3×C4×Dic6C2211yescharacteristic
15#63918—21no—
16#63893—22no—
17Dic3×D12—21no—
18#63857—21no—
19C2×#3718—21no—
20C32×C4⋊C8C2211yescharacteristic
21C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
22C3×#821—21no—
23C3×#818—21no—
24C3×C4×C3⋊C8C2211yescharacteristic
25#63575—22no—
26#63565—21no—
27Q8×D12—31no—
28D4×Dic6—31no—
29C3×#474—31no—
30#5090—31no—
31#5078—31no—
32#5031—31no—
33#4846D311yescharacteristic
34#4682—31no—
35C3×Q8⋊C8—31no—
36D4⋊(C3⋊C8)—31no—
37#4486—31no—
38#4486—31no—
39#4486—61no—
40#4484D311yescharacteristic, complemented
41#4459—31no—
42C2×#634—21no—
43C2×C3×D12D411yescharacteristic
44C2×C3×C4×D3—21no—
45C2×C3×Dic6—21no—
46C2×C3×Dic6D411yescharacteristic
47C2×#627—22no—
48C2×D3×Dic3—22no—
49D3×Dic6—42no—
50D3×Dic6—42no—
51C2×C32×C8—21no—
52C32×C42C2311yescharacteristic
53C4⋊(C3⋊Dic3)—21no—
54C4⋊(C3⋊Dic3)D412yes—
55C2×C3⋊(C3⋊C8)—21no—
56C3×(C2×D3)⋊C4—21no—
57C3×C4⋊Dic3—21no—
58C3×C4⋊Dic3—21no—
59C3×C4⋊Dic3D411yescharacteristic
60C3×C4⋊Dic3D411yescharacteristic
61C3×Dic3⋊C4—21no—
62C3×C4×Dic3—21no—
63C2×C3×C3⋊C8—21no—
64Dic3⋊Dic3—22no—
65(C2×D3)⋊Dic3—22no—
66#3718—42no—
67#1649—91noSylow
68C2×D3×Q8—61no—
69C22×Dic6—61no—
70C3×C4⋊Q8—31no—
71C3×C4×Q8—31no—
72C3×C4×D4—31no—
73C2×#216—61no—
74C2×#216—61no—
75D4×Dic3—61no—
76#924—61no—
77#897—62no—
78#897—61no—
79#888—61no—
80#878—62no—
81C4×D12—61no—
82C4×D12C2×D311yescharacteristic
83#869—31no—
84#869—31no—
85#869—61no—
86#869—61no—
87#869C2×D311yescharacteristic
88C4×Dic6C2×D311yescharacteristic
89C3×#107—61no—
90C3×C4⋊C8—21no—
91C3×C4⋊C8C2×D311yescharacteristic
92C3×C4⋊C8C2×D311yescharacteristic
93C3×#102—61no—
94C3×C4×C8—31no—
95#835—61no—
96#821—21no—
97#818—21no—
98#818—62no—
99#810—61no—
100#807—62no—
101C4⋊(C3⋊C8)—31no—
102C4⋊(C3⋊C8)—31no—
103C4⋊(C3⋊C8)—61no—
104C4×C3⋊C8C2×D311yescharacteristic
105C22×C3×D3—21no—
106C2×C32×C4C2×D411yescharacteristic
107C2×C32×C4C2×D411yescharacteristic
108C2×C32×C4C2×D411yescharacteristic
109C2×C3⋊Dic3—22no—
110#634—41no—
111#634—41no—
112C2×C3×Dic3—21no—
113C2×C3×Dic3—21no—
114C2×C3×Dic3—21no—
115C2×C3×Dic3—41no—
116C3×D12—21no—
117C3×D12Q1612yescomplemented
118C3×C4×D3—41no—
119C3×C4×D3—41no—
120C3×Dic6—21no—
121C3×Dic6—41no—
122C3×Dic6—41no—
123C3×Dic6D812yes—
124#627—44no—
125D3×Dic3—44no—
126C32×C8—41no—
127C3⋊(C3⋊C8)—41no—
128C3×C3⋊C8—22no—
129D4×Q8—91no—
130#500—91no—
131#474—31no—
132#462—91no—
133#332—91no—
134Q8⋊C8—31no—
135D4⋊C8—91no—
136C2×C3×Q8—31no—
137C2×C3×Q8—61no—
138C2×C3×D4—31no—
139C22×C3×C4—61no—
140C22×Dic3—62no—
141D3×Q8—122no—
142D3×Q8—122no—
143C2×D12#6711yescharacteristic
144C2×C4×D3—21no—
145C2×C4×D3—62no—
146C2×Dic6—21no—
147C2×Dic6—62no—
148C2×Dic6—62no—
149C2×Dic6—61no—
150C2×Dic6—61no—
151C2×Dic6—121no—
152C2×Dic6—122no—
153C2×Dic6—122no—
154C2×Dic6D1211yescharacteristic
155C2×C3×C8—21no—
156C2×C3×C8—21no—
157C2×C3×C8—41no—
158C2×C3×C8—61no—
159C3×C4⋊C4—31no—
160C3×C4⋊C4—31no—
161C3×C4⋊C4—61no—
162C3×C4⋊C4—61no—
163C3×C4⋊C4—61no—
164C3×C22⋊C4—61no—
165C3×C42—21no—
166C3×C42—61no—
167C3×C42C22×D311yescharacteristic
168C3×C42C22×D311yescharacteristic
169#217—62no—
170#216—122no—
171#216—122no—
172(C2×D3)⋊C4—21no—
173(C2×D3)⋊C4—62no—
174C4⋊Dic3—21no—
175C4⋊Dic3—21no—
176C4⋊Dic3—32no—
177C4⋊Dic3—32no—
178C4⋊Dic3—62no—
179C4⋊Dic3—62no—
180C4⋊Dic3—61no—
181C4⋊Dic3—61no—
182C4⋊Dic3—121no—
183C4⋊Dic3D1211yescharacteristic
184C4⋊Dic3#6711yescharacteristic
185Dic3⋊C4—21no—
186Dic3⋊C4—62no—
187C4×Dic3—21no—
188C2×C3⋊C8—21no—
189C2×C3⋊C8—61no—
190C2×C3⋊C8—61no—
191C2×C3⋊C8—121no—
192C22×C32#12011yescharacteristic
193C2×C3×D3—22no—
194C2×C3×D3—41no—
195C2×C3×D3—41no—
196C32×C4—21no—
197C32×C4C2×D811yescharacteristic
198C32×C4C2×Q1611yescharacteristic
199C32×C4AGL(1,ℤ/8)11yescharacteristic
200C32×C4#13711yescharacteristic
201C3⋊Dic3—42no—
202C3×Dic3—22no—
203C3×Dic3—41no—
204C3×Dic3—41no—
205C3×Dic3—81no—
206C22×Q8—181no—
207C2×Q16—181no—
208C4⋊Q8—31no—
209C4⋊Q8—91no—
210C4⋊Q8—181no—
211C22⋊Q8—182no—
212C22⋊Q8—181no—
213C4×Q8—31no—
214C4×D4—31no—
215C4×D4—181no—
216#107—61no—
217#107—182no—
218C4⋊C8—91no—
219C4⋊C8D3211yescharacteristic
220#103—181no—
221#102—61no—
222C4×C8—31no—
223C23×C3—61no—
224C22×D3—21no—
225C3×Q8—32no—
226C3×Q8—61no—
227C3×Q8—121no—
228C3×Q8—121no—
229C3×D4—32no—
230C3×D4—61no—
231C2×C3×C4—21no—
232C2×C3×C4—21no—
233C2×C3×C4—21no—
234C2×C3×C4—61no—
235C2×C3×C4—61no—
236C2×C3×C4—61no—
237C2×C3×C4—121no—
238C2×C3×C4—121no—
239C2×C3×C4—121no—
240C2×C3×C4C2×D1211yescharacteristic
241C2×C3×C4D3×D411yescharacteristic
242C2×C3×C4D3×D411yescharacteristic
243C2×C3×C4D3×D411yescharacteristic
244C2×C3×C4D3×D411yescharacteristic
245C2×C3×C4C2×#6711yescharacteristic
246C2×Dic3—21no—
247C2×Dic3—21no—
248C2×Dic3—21no—
249C2×Dic3—41no—
250C2×Dic3—62no—
251C2×Dic3—62no—
252C2×Dic3—122no—
253C2×Dic3—124no—
254D12—21no—
255D12#21612yescomplemented
256C4×D3—41no—
257C4×D3—41no—
258C4×D3—124no—
259Dic6—21no—
260Dic6—41no—
261Dic6—41no—
262Dic6—124no—
263Dic6—124no—
264Dic6—121no—
265Dic6—121no—
266Dic6—121no—
267Dic6—121no—
268Dic6—241no—
269Dic6—241no—
270Dic6D2412yes—
271C3×C8—41no—
272C3×C8—41no—
273C3×C8—62no—
274C3×C8—81no—
275C3⋊C8—22no—
276C3⋊C8—121no—
277C3⋊C8—121no—
278C3⋊C8—241no—
279C2×C32#44711yescharacteristic
280C2×C32#45111yescharacteristic
281C2×C32#45611yescharacteristic
282C3×D3—42no—
283C2×Q8—31no—
284C2×Q8—61no—
285C2×Q8—182no—
286C2×Q8—181no—
287C2×Q8—362no—
288C2×Q8—362no—
289C2×D4—31no—
290C22×C4—61no—
291C22×C4—182no—
292Q16—362no—
293C2×C8—21no—
294C2×C8—61no—
295C2×C8—181no—
296C4⋊C4—31no—
297C4⋊C4—31no—
298C4⋊C4—61no—
299C4⋊C4—61no—
300C4⋊C4—61no—
301C4⋊C4—92no—
302C4⋊C4—182no—
303C4⋊C4—181no—
304C22⋊C4—61no—
305C22⋊C4—182no—
306C42—61no—
307C42C2×D3211yescharacteristic
308C22×C3—21no—
309C22×C3—62no—
310C22×C3—121no—
311C22×C3—121no—
312C22×C3#88211yescharacteristic
313C22×C3#93711yescharacteristic
314C2×D3—22no—
315C2×D3—41no—
316C2×D3—41no—
317C3×C4—21no—
318C3×C4—21no—
319C3×C4—21no—
320C3×C4—21no—
321C3×C4—21no—
322C3×C4—21no—
323C3×C4—41no—
324C3×C4—62no—
325C3×C4—121no—
326C3×C4—121no—
327C3×C4—241no—
328C3×C4C2×D2411yescharacteristic
329C3×C4#90811yescharacteristic
330C3×C4D3×D811yescharacteristic
331C3×C4#91111yescharacteristic
332C3×C4#91511yescharacteristic
333C3×C4D3×Q1611yescharacteristic
334C3×C4C2×#21611yescharacteristic
335C3×C4#95111yescharacteristic
336Dic3—22no—
337Dic3—41no—
338Dic3—41no—
339Dic3—81no—
340Dic3—122no—
341Dic3—122no—
342Dic3—242no—
343C32#164911yescharacteristic, complemented, Sylow
344C23—61no—
345Q8—32no—
346Q8—61no—
347Q8—121no—
348Q8—121no—
349Q8—364no—
350Q8—361no—
351Q8—361no—
352D4—32no—
353D4—61no—
354C2×C4—61no—
355C2×C4—61no—
356C2×C4—61no—
357C2×C4—121no—
358C2×C4—121no—
359C2×C4—121no—
360C2×C4—182no—
361C2×C4—364no—
362C2×C4D3×D1211yescharacteristic
363C2×C4#380211yescharacteristic
364C2×C4D3×#6711yescharacteristic
365C8—41no—
366C8—62no—
367C8—361no—
368C2×C3—21no—
369C2×C3—21no—
370C2×C3—21no—
371C2×C3—122no—
372C2×C3#473211yescharacteristic
373C2×C3#480911yescharacteristic
374C2×C3#482611yescharacteristic
375C2×C3#505011yescharacteristic
376C2×C3#515611yescharacteristic
377C2×C3#520511yescharacteristic
378D3—42no—
379C22—62no—
380C22—121no—
381C22—121no—
382C22#6392211yescharacteristic
383C4—21no—
384C4—62no—
385C4—121no—
386C4—121no—
387C4—241no—
388C4—362no—
389C4D3×D2411yescharacteristic
390C4#6379511yescharacteristic
391C4#6392911yescharacteristic
392C4D3×#21611yescharacteristic
393C3—21no—
394C3#6827311yescharacteristic, complemented
395C3#7111411yescharacteristic, complemented
396C2—122no—
397C2#1059060411yescharacteristic
398C2#1059094411yescharacteristic
399C2#1059101711yescharacteristic
subgroupG/H
centerC22#63922
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4⋊C8C22
socleC22×C32#120
prime radicalC2×C3×D12D4
hypercenterC4⋊C8D32
hypocenterC32#1649
LCSG ▷ C32×C42 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊C8

Complements

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

NH ≅ G/Ntype
3. #10590384C2semidirect
4. #10590311C2semidirect
5. #10589352C2semidirect
6. C3×#4484C2semidirect
40. #4484D3semidirect
117. C3×D12Q16semidirect
255. D12#216semidirect
343. C32#1649semidirect
394. C3#68273semidirect
395. C3#71114semidirect

Maximal subgroups

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

subgroupindex
Dic6×D122
C3×#48462
#105903842
#105903112
#105893522
C3×#44842
#105890812
#711143
#682733