Finite Group Databasebeta

← #11780200

Subgroups — #11780200

Order 1152.

500 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 100 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
1C3×#5727C211yesmaximal, characteristic
2C3×#5078C211yesmaximal, characteristic, complemented
3#10590384C211yesmaximal, characteristic, complemented
4#10590269C211yesmaximal, characteristic
5#10589413C211yesmaximal, characteristic, complemented
6#10589084C211yesmaximal, characteristic, complemented
7#10589081C211yesmaximal, characteristic
8#71916—31nomaximal
9#71376—31nomaximal
10C2×C3×D3×Q8—21no—
11C32×C4⋊Q8C2211yescharacteristic
12C3×#946—21no—
13C3×#896—22no—
14C3×#893—21no—
15C3×#888—21no—
16C3×C4×D12C2211yescharacteristic
17C3×#869C2211yescharacteristic
18C3×C4×Dic6C2211yescharacteristic
19C2×#3718—21no—
20C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
21C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
22C3×#807—22no—
23C3×C4×C3⋊C8C2211yescharacteristic
24#63576—21no—
25#63565—21no—
26#63562—21no—
27C3×D4×Q8—31no—
28#5727D311yescharacteristic
29C3×#500—31no—
30#5090—31no—
31#5087—31no—
32#5078D311yescharacteristic, complemented
33#5031—31no—
34#5007—31no—
35#4547—31no—
36#4547—31no—
37#4547—61no—
38#4535—31no—
39D4⋊(C3⋊C8)—31no—
40#4459—31no—
41#4456—31no—
42C2×C32×Q8—21no—
43C3×D3×Q8—42no—
44C3×D3×Q8—42no—
45C2×C3×D12D411yescharacteristic
46C2×C3×C4×D3—22no—
47C2×C3×C4×D3—21no—
48C2×C3×Dic6—22no—
49C2×C3×Dic6—21no—
50C2×C3×Dic6D411yescharacteristic
51C32×C4⋊C4—21no—
52C32×C4⋊C4D412yes—
53C32×C42C2311yescharacteristic
54C2×C3⋊(C3⋊C8)—21no—
55C2×C3⋊(C3⋊C8)—21no—
56C3×(C2×D3)⋊C4—22no—
57C3×(C2×D3)⋊C4—21no—
58C3×C4⋊Dic3—21no—
59C3×C4⋊Dic3—21no—
60C3×C4⋊Dic3D411yescharacteristic
61C3×C4⋊Dic3D411yescharacteristic
62C3×Dic3⋊C4—22no—
63C3×Dic3⋊C4—21no—
64C3×C4×Dic3—21no—
65C2×C3×C3⋊C8—21no—
66#3718—42no—
67#1649—91noSylow
68C22×C3×Q8—61no—
69C2×D3×Q8—21no—
70C3×C4⋊Q8—21no—
71C3×C4⋊Q8—31no—
72C3×C4⋊Q8—61no—
73C3×C4⋊Q8C2×D311yescharacteristic
74C3×C4⋊Q8C2×D311yescharacteristic
75C3×C22⋊Q8—61no—
76C3×C22⋊Q8—62no—
77C3×C4×Q8—31no—
78C3×C4×D4—31no—
79C3×C4×D4—61no—
80#946—21no—
81C2×#216—61no—
82C2×#216—61no—
83#896—22no—
84#893—21no—
85#888—21no—
86C4×D12C2×D311yescharacteristic
87#869C2×D311yescharacteristic
88C4×Dic6C2×D311yescharacteristic
89C3×#107—62no—
90C3×C4×C8—31no—
91#835—61no—
92#832—61no—
93#810—61no—
94#809—61no—
95#807—22no—
96#807—61no—
97#807—61no—
98C4⋊(C3⋊C8)—31no—
99C4⋊(C3⋊C8)—31no—
100C4⋊(C3⋊C8)—31no—
101C4⋊(C3⋊C8)—31no—
102C4⋊(C3⋊C8)—61no—
103C4⋊(C3⋊C8)—61no—
104C4×C3⋊C8C2×D311yescharacteristic
105C22×C3×D3—21no—
106C32×Q8—41no—
107C32×Q8—41no—
108C2×C32×C4—22no—
109C2×C32×C4C2×D411yescharacteristic
110C2×C32×C4C2×D411yescharacteristic
111C2×C32×C4C2×D411yescharacteristic
112C2×C3×Dic3—21no—
113C2×C3×Dic3—21no—
114C2×C3×Dic3—21no—
115C2×C3×Dic3—41no—
116C3×D12—21no—
117C3×D12Q1612yescomplemented
118C3×C4×D3—44no—
119C3×C4×D3—41no—
120C3×C4×D3—41no—
121C3×Dic6—21no—
122C3×Dic6—44no—
123C3×Dic6—41no—
124C3×Dic6—41no—
125C3×Dic6D812yes—
126C3⋊(C3⋊C8)—41no—
127C3⋊(C3⋊C8)—41no—
128C3×C3⋊C8—22no—
129D4×Q8—31no—
130#500—31no—
131#474—91no—
132#462—91no—
133#332—91no—
134Q8⋊C8—91no—
135D4⋊C8—91no—
136C2×C3×Q8—21no—
137C2×C3×Q8—21no—
138C2×C3×Q8—31no—
139C2×C3×Q8—41no—
140C2×C3×Q8—62no—
141C2×C3×Q8—61no—
142C2×C3×Q8—122no—
143C2×C3×Q8—122no—
144C2×C3×D4—31no—
145C22×C3×C4—62no—
146C22×C3×C4—61no—
147D3×Q8—42no—
148D3×Q8—42no—
149C2×D12#6711yescharacteristic
150C2×C4×D3—22no—
151C2×C4×D3—21no—
152C2×Dic6—22no—
153C2×Dic6—21no—
154C2×Dic6#6711yescharacteristic
155C2×C3×C8—61no—
156C3×C4⋊C4—21no—
157C3×C4⋊C4—21no—
158C3×C4⋊C4—22no—
159C3×C4⋊C4—31no—
160C3×C4⋊C4—31no—
161C3×C4⋊C4—41no—
162C3×C4⋊C4—62no—
163C3×C4⋊C4—61no—
164C3×C4⋊C4—61no—
165C3×C4⋊C4—61no—
166C3×C4⋊C4#6712yes—
167C3×C4⋊C4#6712yes—
168C3×C22⋊C4—62no—
169C3×C22⋊C4—61no—
170C3×C42—21no—
171C3×C42—61no—
172C3×C42C22×D311yescharacteristic
173C3×C42C22×D311yescharacteristic
174#216—122no—
175#216—122no—
176(C2×D3)⋊C4—22no—
177(C2×D3)⋊C4—21no—
178C4⋊Dic3—21no—
179C4⋊Dic3—21no—
180C4⋊Dic3#6711yescharacteristic
181C4⋊Dic3#6711yescharacteristic
182Dic3⋊C4—22no—
183Dic3⋊C4—21no—
184C4×Dic3—21no—
185C2×C3⋊C8—21no—
186C2×C3⋊C8—61no—
187C2×C3⋊C8—61no—
188C2×C3⋊C8—61no—
189C2×C3⋊C8—61no—
190C2×C3⋊C8—121no—
191C2×C3⋊C8—121no—
192C22×C32#12011yescharacteristic
193C2×C3×D3—22no—
194C2×C3×D3—41no—
195C2×C3×D3—41no—
196C32×C4—21no—
197C32×C4—42no—
198C32×C4C2×D811yescharacteristic
199C32×C4C2×Q1611yescharacteristic
200C32×C4AGL(1,ℤ/8)11yescharacteristic
201C32×C4#13711yescharacteristic
202C3×Dic3—22no—
203C3×Dic3—41no—
204C3×Dic3—41no—
205C3×Dic3—81no—
206C22×Q8—61no—
207C2×Q16—181no—
208C4⋊Q8—31no—
209C4⋊Q8—61no—
210C4⋊Q8D3211yescharacteristic
211C22⋊Q8—61no—
212C22⋊Q8—62no—
213C4×Q8—31no—
214C4×D4—31no—
215C4×D4—61no—
216#107—62no—
217#107—181no—
218C4⋊C8—91no—
219C4⋊C8—91no—
220#103—181no—
221#102—181no—
222C4×C8—31no—
223C23×C3—61no—
224C22×D3—21no—
225C3×Q8—32no—
226C3×Q8—41no—
227C3×Q8—41no—
228C3×Q8—41no—
229C3×Q8—41no—
230C3×Q8—61no—
231C3×Q8—81no—
232C3×Q8—81no—
233C3×Q8—124no—
234C3×Q8—121no—
235C3×Q8—121no—
236C3×D4—32no—
237C3×D4—61no—
238C2×C3×C4—22no—
239C2×C3×C4—22no—
240C2×C3×C4—21no—
241C2×C3×C4—21no—
242C2×C3×C4—21no—
243C2×C3×C4—42no—
244C2×C3×C4—61no—
245C2×C3×C4—61no—
246C2×C3×C4—61no—
247C2×C3×C4—124no—
248C2×C3×C4—121no—
249C2×C3×C4—121no—
250C2×C3×C4—121no—
251C2×C3×C4D3×D411yescharacteristic
252C2×C3×C4D3×D411yescharacteristic
253C2×C3×C4C2×#6711yescharacteristic
254C2×C3×C4C2×#6711yescharacteristic
255C2×C3×C4C2×#6711yescharacteristic
256C2×C3×C4C2×#6711yescharacteristic
257C2×Dic3—21no—
258C2×Dic3—21no—
259C2×Dic3—21no—
260C2×Dic3—41no—
261D12—21no—
262D12#21612yescomplemented
263C4×D3—44no—
264C4×D3—41no—
265C4×D3—41no—
266Dic6—21no—
267Dic6—44no—
268Dic6—41no—
269Dic6—41no—
270Dic6#21312yes—
271C3×C8—62no—
272C3⋊C8—22no—
273C3⋊C8—121no—
274C3⋊C8—121no—
275C3⋊C8—121no—
276C3⋊C8—121no—
277C3⋊C8—241no—
278C3⋊C8—241no—
279C2×C32#44711yescharacteristic
280C2×C32#45111yescharacteristic
281C2×C32#45611yescharacteristic
282C3×D3—42no—
283C2×Q8—21no—
284C2×Q8—31no—
285C2×Q8—62no—
286C2×Q8—61no—
287C2×Q8—122no—
288C2×Q8—122no—
289C2×D4—31no—
290C22×C4—62no—
291C22×C4—61no—
292Q16—362no—
293C2×C8—61no—
294C2×C8—181no—
295C2×C8—181no—
296C4⋊C4—21no—
297C4⋊C4—31no—
298C4⋊C4—31no—
299C4⋊C4—62no—
300C4⋊C4—61no—
301C4⋊C4—61no—
302C4⋊C4—61no—
303C4⋊C4#62512yes—
304C22⋊C4—62no—
305C22⋊C4—61no—
306C42—61no—
307C42C2×D3211yescharacteristic
308C22×C3—21no—
309C22×C3—62no—
310C22×C3—121no—
311C22×C3—121no—
312C22×C3#93711yescharacteristic
313C22×C3#95311yescharacteristic
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—42no—
325C3×C4—42no—
326C3×C4—62no—
327C3×C4—82no—
328C3×C4—121no—
329C3×C4—121no—
330C3×C4—241no—
331C3×C4D3×D811yescharacteristic
332C3×C4#91111yescharacteristic
333C3×C4D3×Q1611yescharacteristic
334C3×C4#91811yescharacteristic
335C3×C4C2×#21311yescharacteristic
336C3×C4#93211yescharacteristic
337C3×C4#94211yescharacteristic
338C3×C4C2×#21611yescharacteristic
339Dic3—22no—
340Dic3—41no—
341Dic3—41no—
342Dic3—81no—
343C32#164911yescharacteristic, complemented, Sylow
344C23—61no—
345Q8—32no—
346Q8—41no—
347Q8—41no—
348Q8—61no—
349Q8—124no—
350Q8—121no—
351Q8—121no—
352D4—32no—
353D4—61no—
354C2×C4—22no—
355C2×C4—61no—
356C2×C4—61no—
357C2×C4—61no—
358C2×C4—124no—
359C2×C4—121no—
360C2×C4—121no—
361C2×C4—121no—
362C2×C4C2×#62511yescharacteristic
363C2×C4D3×#6711yescharacteristic
364C2×C4D3×#6711yescharacteristic
365C8—62no—
366C8—361no—
367C8—361no—
368C2×C3—21no—
369C2×C3—21no—
370C2×C3—21no—
371C2×C3—122no—
372C2×C3#509511yescharacteristic
373C2×C3#515611yescharacteristic
374C2×C3#518611yescharacteristic
375C2×C3#521711yescharacteristic
376C2×C3#522811yescharacteristic
377C2×C3#524611yescharacteristic
378D3—42no—
379C22—62no—
380C22—121no—
381C22—121no—
382C22#6397611yescharacteristic
383C4—21no—
384C4—42no—
385C4—62no—
386C4—121no—
387C4—121no—
388C4—241no—
389C4D3×#21311yescharacteristic
390C4#6393211yescharacteristic
391C4D3×#21611yescharacteristic
392C4#6394811yescharacteristic
393C3—21no—
394C3#7137611yescharacteristic, complemented
395C3#7191611yescharacteristic, complemented
396C2—122no—
397C2#1059144711yescharacteristic
398C2#1059159411yescharacteristic
399C2#1059164811yescharacteristic
subgroupG/H
centerC22#63976
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4⋊Q8C22
socleC22×C32#120
prime radicalC2×C3×D12D4
hypercenterC4⋊Q8D32
hypocenterC32#1649
LCSG ▷ C32×C42 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. C3×#5078C2semidirect
3. #10590384C2semidirect
5. #10589413C2semidirect
6. #10589084C2semidirect
32. #5078D3semidirect
117. C3×D12Q16semidirect
262. D12#216semidirect
343. C32#1649semidirect
394. C3#71376semidirect
395. C3#71916semidirect

Maximal subgroups

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

subgroupindex
C3×#57272
C3×#50782
#105903842
#105902692
#105894132
#105890842
#105890812
#719163
#713763