Finite Group Databasebeta

← #11776222

Subgroups — #11776222

Order 1152.

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

normal subgroups only

This group has 551 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
1#10593907C211yesmaximal, characteristic
2#10592532C211yesmaximal, characteristic, complemented
3#10590256C211yesmaximal, characteristic, complemented
4C3×#4485C211yesmaximal, characteristic, complemented
5C3×#4481C211yesmaximal, characteristic, complemented
6#10589095C211yesmaximal, characteristic, complemented
7#10589090C211yesmaximal, characteristic, complemented
8#68482—31nomaximal
9#68382—31nomaximal
10C2×#3796C412yes—
11C42×C3⋊D3C2211yescharacteristic
12C3×#874C2211yescharacteristic
13C3×#869C2211yescharacteristic
14#63916—21no—
15#63916—21no—
16#63898—21no—
17#63898—21no—
18#63863C412yes—
19C32×C4⋊C8C2211yescharacteristic
20#63647—21no—
21C8×C3⋊Dic3—21no—
22C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
23C3×C4⋊(C3⋊C8)C2211yescharacteristic
24C3×C4⋊(C3⋊C8)C2211yescharacteristic
25#63557—21no—
26#63553—21no—
27#5725—31no—
28#5716—31no—
29#4833—31no—
30#4833—31no—
31#4833—31no—
32#4833—31no—
33#4833—61no—
34C3×#333—31no—
35C3×#331—31no—
36#4546—31no—
37#4544—31no—
38#4485—31no—
39#4485D311yescharacteristic, complemented
40#4481—31no—
41#4481D311yescharacteristic, complemented
42C2×C4×C3⋊D3C2×C411yescharacteristic
43C2×C4×C3⋊D3D411yescharacteristic
44C2×C4×C3⋊D3D411yescharacteristic
45C2×C3×D12—21no—
46C2×C3×D12—21no—
47C2×C3×D12C2×C412yes—
48C2×C3×Dic6C2×C412yes—
49C2×#626—22no—
50C2×D3×Dic3—22no—
51#3796—44no—
52C2×C32×C8—21no—
53C32×C42C2311yescharacteristic
54C4×C3⋊Dic3C2×C411yescharacteristic
55C4×C3⋊Dic3D411yescharacteristic
56C4×C3⋊Dic3D411yescharacteristic
57C2×C3⋊(C3⋊C8)—21no—
58C3×C4⋊Dic3—21no—
59C3×C4⋊Dic3—21no—
60C2×C3×C3⋊C8—21no—
61C2×C3×C3⋊C8—21no—
62(C2×D3)⋊Dic3—22no—
63(C2×D3)⋊Dic3—22no—
64#1543—91noSylow
65C2×#239—32no—
66C2×#237—32no—
67C3×C4⋊Q8—31no—
68C3×#127—31no—
69#947—32no—
70#938—61no—
71#938—61no—
72#936—32no—
73D4×Dic3—61no—
74D4×Dic3—61no—
75#895—61no—
76#895—61no—
77#893—61no—
78#893—61no—
79#874C2×D311yescharacteristic
80C42×D3—31no—
81C42×D3—31no—
82C42×D3—61no—
83#869C2×D311yescharacteristic
84C3×C4⋊C8—21no—
85C3×C4⋊C8—31no—
86C3×C4⋊C8—31no—
87C3×C4⋊C8C2×D311yescharacteristic
88C3×C4⋊C8C2×D311yescharacteristic
89(C2×D3)⋊C8—61no—
90(C2×D3)⋊C8—61no—
91(C2×D3)⋊C8—61no—
92(C2×D3)⋊C8—61no—
93(C2×D3)⋊C8—62no—
94C8×Dic3—61no—
95C8×Dic3—61no—
96C8×Dic3—61no—
97C8×Dic3—61no—
98C8×Dic3—62no—
99C4⋊(C3⋊C8)—31no—
100C4⋊(C3⋊C8)—31no—
101C4⋊(C3⋊C8)—61no—
102C4⋊(C3⋊C8)C2×D311yescharacteristic
103C4⋊(C3⋊C8)C2×D311yescharacteristic
104C22×C3⋊D3C22⋊C411yescharacteristic
105C22×C3×D3—22no—
106C2×C32×C4C22×C411yescharacteristic
107C2×C32×C4C2×D411yescharacteristic
108C2×C32×C4C2×D411yescharacteristic
109C2×C3⋊Dic3C22⋊C411yescharacteristic
110C2×C3⋊Dic3C22⋊C411yescharacteristic
111C2×C3⋊Dic3C22⋊C411yescharacteristic
112C4×C3⋊D3—21no—
113C4×C3⋊D3—21no—
114C4×C3⋊D3—21no—
115C4×C3⋊D3—21no—
116C4×C3⋊D3—22no—
117C2×C3×Dic3—22no—
118C3×D12—42no—
119C3×D12—41no—
120C3×D12—41no—
121C3×D12—41no—
122C3×D12—41no—
123C3×Dic6—42no—
124#626—44no—
125D3×Dic3—44no—
126C32×C8—41no—
127C3⋊(C3⋊C8)—41no—
128C3×C3⋊C8—41no—
129C3×C3⋊C8—41no—
130#532—91no—
131#431—91no—
132#431—91no—
133#333—31no—
134#333—91no—
135#331—31no—
136#331—91no—
137C2×C3×Q8—32no—
138C2×C3×D4—32no—
139C2×C3×D4—61no—
140C2×C3×D4—61no—
141C2×#67—62no—
142C22×Dic3—62no—
143#239—124no—
144#237—124no—
145C2×D12—21no—
146C2×D12—21no—
147C2×D12—62no—
148C2×D12C4×D312yes—
149C2×C4×D3—31no—
150C2×C4×D3—31no—
151C2×C4×D3—31no—
152C2×C4×D3—31no—
153C2×C4×D3—31no—
154C2×C4×D3—31no—
155C2×C4×D3—61no—
156C2×C4×D3—61no—
157C2×C4×D3—61no—
158C2×C4×D3—62no—
159C2×Dic6C4×D312yes—
160C2×C3×C8—22no—
161C2×C3×C8—21no—
162C2×C3×C8—21no—
163C2×C3×C8—61no—
164C2×C3×C8—61no—
165C3×C4⋊C4—61no—
166C3×C4⋊C4—61no—
167C3×C42—21no—
168C3×C42C22×D311yescharacteristic
169C3×C42C22×D311yescharacteristic
170#217—62no—
171#217—62no—
172(C2×D3)⋊C4—62no—
173(C2×D3)⋊C4—62no—
174C4⋊Dic3—21no—
175C4⋊Dic3—21no—
176C4×Dic3—31no—
177C4×Dic3—31no—
178C4×Dic3—31no—
179C4×Dic3—31no—
180C4×Dic3—31no—
181C4×Dic3—31no—
182C4×Dic3—61no—
183C4×Dic3—61no—
184C4×Dic3—61no—
185C2×C3⋊C8—21no—
186C2×C3⋊C8—21no—
187C2×C3⋊C8—62no—
188C2×C3⋊C8—61no—
189C2×C3⋊C8—61no—
190C22×C32C2×C22⋊C411yescharacteristic
191C2×C3⋊D3—21no—
192C2×C3⋊D3—21no—
193C2×C3⋊D3—21no—
194C2×C3×D3—42no—
195C2×C3×D3—42no—
196C2×C3×D3—42no—
197C32×C4—21no—
198C32×C4#13511yescharacteristic
199C32×C4#13511yescharacteristic
200C32×C4#13511yescharacteristic
201C32×C4#13511yescharacteristic
202C3⋊Dic3—21no—
203C3⋊Dic3—21no—
204C3⋊Dic3#10012yes—
205C3×Dic3—42no—
206C2×#41—92no—
207C4⋊Q8—31no—
208#127—31no—
209#124—92no—
210#121—181no—
211#121—181no—
212C4×D4—181no—
213C4×D4—181no—
214C2×C42—91no—
215C4⋊C8—31no—
216C4⋊C8—31no—
217C4⋊C8—91no—
218C4⋊C8D3211yescharacteristic
219C22⋊C8—181no—
220C22⋊C8—181no—
221C4×C8—181no—
222C4×C8—181no—
223C23×C3—62no—
224C22×D3—22no—
225C22×D3—31no—
226C22×D3—31no—
227C22×D3—61no—
228C3×Q8—122no—
229C3×D4—122no—
230C3×D4—121no—
231C3×D4—121no—
232C3×D4—121no—
233C3×D4—121no—
234C2×C3×C4—21no—
235C2×C3×C4—21no—
236C2×C3×C4—21no—
237C2×C3×C4—62no—
238C2×C3×C4C2×C4×D311yescharacteristic
239C2×C3×C4C2×C4×D311yescharacteristic
240C2×C3×C4D3×D411yescharacteristic
241C2×C3×C4D3×D411yescharacteristic
242C2×C3×C4D3×D411yescharacteristic
243C2×C3×C4D3×D411yescharacteristic
244#67—124no—
245C2×Dic3—22no—
246C2×Dic3—31no—
247C2×Dic3—31no—
248C2×Dic3—31no—
249C2×Dic3—31no—
250C2×Dic3—31no—
251C2×Dic3—31no—
252C2×Dic3—61no—
253C2×Dic3—61no—
254C2×Dic3—61no—
255C2×Dic3—124no—
256D12—42no—
257D12—41no—
258D12—41no—
259D12—41no—
260D12—41no—
261D12—124no—
262C4×D3—62no—
263C4×D3—62no—
264C4×D3—62no—
265C4×D3—62no—
266C4×D3—61no—
267C4×D3—61no—
268C4×D3—61no—
269C4×D3—61no—
270C4×D3—61no—
271C4×D3—61no—
272C4×D3—61no—
273C4×D3—61no—
274C4×D3—122no—
275C4×D3—121no—
276C4×D3—121no—
277C4×D3—124no—
278Dic6—42no—
279C3×C8—42no—
280C3×C8—41no—
281C3×C8—41no—
282C3×C8—121no—
283C3×C8—121no—
284C3⋊C8—41no—
285C3⋊C8—41no—
286C3⋊C8—122no—
287C3⋊C8—121no—
288C3⋊C8—121no—
289C2×C32C2×#10011yescharacteristic
290C2×C32#41611yescharacteristic
291C2×C32#41611yescharacteristic
292C3⋊D3—41no—
293C3×D3—82no—
294#41—364no—
295C2×Q8—32no—
296C2×D4—32no—
297C2×D4—61no—
298C2×D4—61no—
299C2×D4—182no—
300C22×C4—91no—
301C22×C4—91no—
302C22×C4—91no—
303C22×C4—182no—
304C2×C8—21no—
305C2×C8—61no—
306C2×C8—61no—
307C2×C8—181no—
308C4⋊C4—61no—
309C4⋊C4—61no—
310C22⋊C4—182no—
311C22⋊C4—182no—
312C42—91no—
313C42—91no—
314C42—91no—
315C42C2×D3211yescharacteristic
316C22×C3—21no—
317C22×C3—122no—
318C22×C3—122no—
319C22×C3—122no—
320C22×C3D3×C22⋊C411yescharacteristic
321C22×C3D3×C22⋊C411yescharacteristic
322C2×D3—42no—
323C2×D3—42no—
324C2×D3—42no—
325C2×D3—62no—
326C2×D3—61no—
327C2×D3—61no—
328C2×D3—61no—
329C2×D3—61no—
330C2×D3—61no—
331C2×D3—61no—
332C2×D3—121no—
333C2×D3—121no—
334C3×C4—21no—
335C3×C4—21no—
336C3×C4—21no—
337C3×C4—21no—
338C3×C4—21no—
339C3×C4—21no—
340C3×C4—41no—
341C3×C4—122no—
342C3×C4#91211yescharacteristic
343C3×C4#91211yescharacteristic
344C3×C4#91611yescharacteristic
345C3×C4#91611yescharacteristic
346C3×C4#91611yescharacteristic
347C3×C4#91611yescharacteristic
348C3×C4#91911yescharacteristic
349C3×C4#91911yescharacteristic
350Dic3—32no—
351Dic3—32no—
352Dic3—42no—
353Dic3—62no—
354Dic3—62no—
355Dic3—61no—
356Dic3—61no—
357Dic3—61no—
358Dic3—61no—
359Dic3—121no—
360C32#154311yescharacteristic, complemented, Sylow
361C23—62no—
362C23—91no—
363Q8—122no—
364D4—122no—
365D4—121no—
366D4—121no—
367D4—121no—
368D4—121no—
369D4—364no—
370C2×C4—62no—
371C2×C4—91no—
372C2×C4—91no—
373C2×C4—91no—
374C2×C4—182no—
375C2×C4—181no—
376C2×C4—181no—
377C2×C4—181no—
378C2×C4—181no—
379C2×C4—364no—
380C2×C4C4×D3211yescharacteristic
381C2×C4#380211yescharacteristic
382C2×C4#381111yescharacteristic
383C8—41no—
384C8—121no—
385C8—121no—
386C8—361no—
387C2×C3—21no—
388C2×C3—21no—
389C2×C3—21no—
390C2×C3—242no—
391C2×C3D3×#10011yescharacteristic
392C2×C3D3×#10011yescharacteristic
393C2×C3#477211yescharacteristic
394C2×C3#477211yescharacteristic
395C2×C3#480411yescharacteristic
396C2×C3#480411yescharacteristic
397D3—82no—
398D3—122no—
399D3—121no—
400D3—121no—
401C22—122no—
402C22—122no—
403C22—122no—
404C22—181no—
405C22—181no—
406C22—181no—
407C22#6392111yescharacteristic
408C4—21no—
409C4—92no—
410C4—122no—
411C4—181no—
412C4—181no—
413C4#6380911yescharacteristic
414C4#6381011yescharacteristic
415C4#6393611yescharacteristic
416C4#6395011yescharacteristic
417C3—21no—
418C3#6838211yescharacteristic, complemented
419C3#6848211yescharacteristic, complemented
420C2—242no—
421C2—361no—
422C2#1059061111yescharacteristic
423C2#1059095611yescharacteristic
424C2#1059100111yescharacteristic
subgroupG/H
centerC22#63921
commutatorC2×C32×C4C22×C4
FrattiniC42C2×D32
FittingC32×C4⋊C8C22
socleC22×C32C2×C22⋊C4
prime radical#10593907C2
cosocleC32×C42C23
hypercenterC4⋊C8D32
hypocenterC32#1543
LCSG ▷ C2×C32×C4 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊C8

Complements

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

NH ≅ G/Ntype
2. #10592532C2semidirect
3. #10590256C2semidirect
4. C3×#4485C2semidirect
5. C3×#4481C2semidirect
6. #10589095C2semidirect
7. #10589090C2semidirect
39. #4485D3semidirect
41. #4481D3semidirect
360. C32#1543semidirect
418. C3#68382semidirect
419. C3#68482semidirect

Maximal subgroups

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

subgroupindex
#105939072
#105925322
#105902562
C3×#44852
C3×#44812
#105890952
#105890902
#684823
#683823