Finite Group Databasebeta

← #11780202

Subgroups — #11780202

Order 1152.

608 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 608 conjugacy classes of subgroups — too many to draw. Only its 98 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
1Q8×C4⋊(C3⋊D3)C211yesmaximal, characteristic
2C3×#5031C211yesmaximal, characteristic, complemented
3#10590313C211yesmaximal, characteristic
4#10590311C211yesmaximal, characteristic, complemented
5C3×#4535C211yesmaximal, characteristic, complemented
6#10589094C211yesmaximal, characteristic, complemented
7#10589082C211yesmaximal, characteristic
8#71065—31nomaximal
9#68273—31nomaximal
10C2×Q8×C3⋊D3—21no—
11C32×C4×Q8C2211yescharacteristic
12#64106—21no—
13#64106—22no—
14C4×C4⋊(C3⋊D3)—21no—
15C4×C4⋊(C3⋊D3)C2211yescharacteristic
16#64078—21no—
17#64078C2211yescharacteristic
18C2×C3×#216—21no—
19C3×#869C2211yescharacteristic
20C3×#810—21no—
21C3×C4⋊(C3⋊C8)C2211yescharacteristic
22C3×C4⋊(C3⋊C8)C2211yescharacteristic
23C3×C4×C3⋊C8C2211yescharacteristic
24#63575—22no—
25#63575—21no—
26#63564—21no—
27Q8×D12—31no—
28Q8×D12—31no—
29Q8×D12—61no—
30C3×#462—31no—
31#5087—31no—
32#5078—31no—
33#5031D311yescharacteristic, complemented
34#4846—31no—
35#4682—31no—
36C3×Q8⋊C8—31no—
37#4535D311yescharacteristic, complemented
38#4486—31no—
39#4483—31no—
40#4482—31no—
41#4459—31no—
42C2×C32×Q8D411yescharacteristic
43Q8×C3⋊D3—42no—
44Q8×C3⋊D3—42no—
45C2×C4⋊(C3⋊D3)D411yescharacteristic
46C2×C4×C3⋊D3—22no—
47C2×C4×C3⋊D3—21no—
48C2×#634—22no—
49C2×#634—21no—
50C2×C3×Dic6—21no—
51C32×C4⋊C4—21no—
52C32×C4⋊C4D411yescharacteristic
53C32×C42—21no—
54C32×C42C2311yescharacteristic
55#3752—22no—
56#3752—21no—
57C4⋊(C3⋊Dic3)—22no—
58C4⋊(C3⋊Dic3)—21no—
59C4⋊(C3⋊Dic3)D411yescharacteristic
60C4⋊(C3⋊Dic3)D412yes—
61C3×#216—42no—
62C3×C4⋊Dic3—21no—
63C3×C4⋊Dic3—21no—
64C2×C3×C3⋊C8—21no—
65C2×C3×C3⋊C8—21no—
66C2×C3×C3⋊C8—21no—
67#1649—91noSylow
68C2×D3×Q8—62no—
69C2×D3×Q8—61no—
70C2×D3×Q8—61no—
71C2×C3×Q16—61no—
72C3×C4⋊Q8—31no—
73C3×C4×Q8—21no—
74C3×C4×Q8C2×D311yescharacteristic
75C3×C4×Q8C2×D311yescharacteristic
76C2×#216—21no—
77#897—62no—
78#897—62no—
79#897—64no—
80#897—62no—
81#897—61no—
82#897—61no—
83C4×D12—31no—
84C4×D12—31no—
85C4×D12—61no—
86C4×D12—61no—
87C4×D12—61no—
88C4×D12—62no—
89#869—31no—
90#869—31no—
91#869—61no—
92#869—61no—
93#869—61no—
94#869—62no—
95#869C2×D311yescharacteristic
96C3×C4⋊C8—31no—
97C3×C4⋊C8—31no—
98C3×#103—61no—
99C3×C4×C8—31no—
100#821—61no—
101#818—62no—
102#818—61no—
103#810—21no—
104#809—61no—
105#807—62no—
106#807—61no—
107C4⋊(C3⋊C8)C2×D311yescharacteristic
108C4⋊(C3⋊C8)C2×D311yescharacteristic
109C4×C3⋊C8C2×D311yescharacteristic
110C22×C3⋊D3—21no—
111C32×Q8—21no—
112C32×Q8D812yes—
113C2×C32×C4—21no—
114C2×C32×C4—21no—
115C2×C32×C4C2×D411yescharacteristic
116C2×C32×C4C2×D411yescharacteristic
117C2×C32×C4C2×D411yescharacteristic
118C2×C3⋊Dic3—21no—
119C2×C3⋊Dic3—22no—
120C4⋊(C3⋊D3)—21no—
121C4⋊(C3⋊D3)Q1612yescomplemented
122C4×C3⋊D3—44no—
123C4×C3⋊D3—41no—
124C4×C3⋊D3—41no—
125#634—44no—
126#634—41no—
127#634—41no—
128C2×C3×Dic3—41no—
129C3×Dic6—41no—
130C3×Dic6—41no—
131C3×C3⋊C8—22no—
132C3×C3⋊C8—41no—
133C3×C3⋊C8—41no—
134D4×Q8—91no—
135#500—91no—
136#474—91no—
137#462—31no—
138#332—91no—
139Q8⋊C8—31no—
140D4⋊C8—91no—
141C2×C3×Q8—21no—
142C2×C3×Q8—61no—
143C2×C3×Q8D1211yescharacteristic
144C2×C3×Q8#6711yescharacteristic
145D3×Q8—124no—
146D3×Q8—122no—
147D3×Q8—122no—
148D3×Q8—124no—
149D3×Q8—122no—
150D3×Q8—122no—
151C2×D12—31no—
152C2×D12—31no—
153C2×D12—61no—
154C2×C4×D3—64no—
155C2×C4×D3—62no—
156C2×C4×D3—62no—
157C2×C4×D3—62no—
158C2×C4×D3—61no—
159C2×C4×D3—61no—
160C2×Dic6—21no—
161C2×Dic6—64no—
162C2×Dic6—62no—
163C2×Dic6—62no—
164C2×Dic6—62no—
165C2×Dic6—61no—
166C2×Dic6—61no—
167C3×Q16—122no—
168C2×C3×C8—61no—
169C2×C3×C8—61no—
170C2×C3×C8—61no—
171C3×C4⋊C4—22no—
172C3×C4⋊C4—21no—
173C3×C4⋊C4—21no—
174C3×C4⋊C4—21no—
175C3×C4⋊C4—61no—
176C3×C4⋊C4—61no—
177C3×C4⋊C4D1211yescharacteristic
178C3×C4⋊C4#6711yescharacteristic
179C3×C42—22no—
180C3×C42—21no—
181C3×C42—21no—
182C3×C42—21no—
183C3×C42C22×D311yescharacteristic
184C3×C42C22×D311yescharacteristic
185#216—42no—
186(C2×D3)⋊C4—64no—
187(C2×D3)⋊C4—62no—
188(C2×D3)⋊C4—62no—
189(C2×D3)⋊C4—62no—
190(C2×D3)⋊C4—61no—
191(C2×D3)⋊C4—61no—
192C4⋊Dic3—21no—
193C4⋊Dic3—21no—
194C4⋊Dic3—32no—
195C4⋊Dic3—31no—
196C4⋊Dic3—32no—
197C4⋊Dic3—31no—
198C4⋊Dic3—61no—
199C4⋊Dic3—62no—
200C4⋊Dic3—62no—
201C4⋊Dic3—64no—
202C4⋊Dic3—62no—
203C4⋊Dic3—62no—
204C4⋊Dic3—61no—
205C4⋊Dic3—61no—
206C2×C3⋊C8—21no—
207C2×C3⋊C8—21no—
208C2×C3⋊C8—21no—
209C22×C32#12011yescharacteristic
210C2×C3⋊D3—22no—
211C2×C3⋊D3—41no—
212C2×C3⋊D3—41no—
213C32×C4—22no—
214C32×C4—21no—
215C32×C4—41no—
216C32×C4C2×D811yescharacteristic
217C32×C4C2×Q1611yescharacteristic
218C32×C4AGL(1,ℤ/8)11yescharacteristic
219C32×C4#13711yescharacteristic
220C3⋊Dic3—41no—
221C3⋊Dic3—42no—
222C3×Dic3—81no—
223C22×Q8—181no—
224C2×Q16—61no—
225C4⋊Q8—31no—
226C4⋊Q8—91no—
227C4⋊Q8—181no—
228C22⋊Q8—182no—
229C22⋊Q8—181no—
230C4×Q8D3211yescharacteristic
231C4×D4—91no—
232C4×D4—181no—
233#107—181no—
234#107—182no—
235C4⋊C8—31no—
236C4⋊C8—31no—
237#103—61no—
238#102—181no—
239C4×C8—31no—
240C22×D3—62no—
241C22×D3—61no—
242C22×D3—61no—
243C3×Q8—22no—
244C3×Q8—21no—
245C3×Q8—21no—
246C3×Q8—22no—
247C3×Q8—121no—
248C3×Q8—121no—
249C3×Q8D2412yes—
250C3×Q8#21312yes—
251C2×C3×C4—22no—
252C2×C3×C4—21no—
253C2×C3×C4—21no—
254C2×C3×C4—22no—
255C2×C3×C4—21no—
256C2×C3×C4—21no—
257C2×C3×C4—21no—
258C2×C3×C4—21no—
259C2×C3×C4—21no—
260C2×C3×C4—121no—
261C2×C3×C4C2×D1211yescharacteristic
262C2×C3×C4D3×D411yescharacteristic
263C2×C3×C4D3×D411yescharacteristic
264C2×C3×C4D3×D411yescharacteristic
265C2×C3×C4D3×D411yescharacteristic
266C2×C3×C4C2×#6711yescharacteristic
267C2×Dic3—41no—
268C2×Dic3—62no—
269C2×Dic3—64no—
270C2×Dic3—62no—
271C2×Dic3—61no—
272C2×Dic3—62no—
273C2×Dic3—61no—
274D12—32no—
275D12—32no—
276D12—62no—
277D12—62no—
278D12—61no—
279D12—61no—
280C4×D3—128no—
281C4×D3—124no—
282C4×D3—124no—
283C4×D3—122no—
284C4×D3—122no—
285C4×D3—121no—
286C4×D3—121no—
287C4×D3—121no—
288C4×D3—121no—
289Dic6—41no—
290Dic6—41no—
291Dic6—128no—
292Dic6—124no—
293Dic6—124no—
294Dic6—122no—
295Dic6—122no—
296Dic6—121no—
297Dic6—121no—
298Dic6—121no—
299Dic6—121no—
300C3×C8—62no—
301C3×C8—121no—
302C3×C8—121no—
303C3⋊C8—22no—
304C3⋊C8—41no—
305C3⋊C8—41no—
306C2×C32#44711yescharacteristic
307C2×C32#45111yescharacteristic
308C2×C32#45611yescharacteristic
309C3⋊D3—42no—
310C2×Q8—61no—
311C2×Q8—182no—
312C2×Q8—181no—
313C2×Q8—362no—
314C2×Q8—362no—
315C2×Q8#62611yescharacteristic
316C2×D4—91no—
317C22×C4—182no—
318C22×C4—181no—
319Q16—122no—
320C2×C8—61no—
321C2×C8—61no—
322C2×C8—61no—
323C4⋊C4—21no—
324C4⋊C4—61no—
325C4⋊C4—61no—
326C4⋊C4—91no—
327C4⋊C4—92no—
328C4⋊C4—182no—
329C4⋊C4—181no—
330C4⋊C4#62611yescharacteristic
331C22⋊C4—182no—
332C22⋊C4—181no—
333C42—21no—
334C42C2×D3211yescharacteristic
335C22×C3—21no—
336C22×C3#88211yescharacteristic
337C22×C3#93711yescharacteristic
338C2×D3—64no—
339C2×D3—62no—
340C2×D3—62no—
341C2×D3—122no—
342C2×D3—122no—
343C2×D3—121no—
344C2×D3—121no—
345C2×D3—121no—
346C2×D3—121no—
347C3×C4—24no—
348C3×C4—21no—
349C3×C4—22no—
350C3×C4—22no—
351C3×C4—21no—
352C3×C4—21no—
353C3×C4—21no—
354C3×C4—21no—
355C3×C4—21no—
356C3×C4—41no—
357C3×C4—42no—
358C3×C4—41no—
359C3×C4—41no—
360C3×C4—241no—
361C3×C4C2×D2411yescharacteristic
362C3×C4#90811yescharacteristic
363C3×C4#91511yescharacteristic
364C3×C4D3×Q1611yescharacteristic
365C3×C4D3×Q1611yescharacteristic
366C3×C4#91811yescharacteristic
367C3×C4C2×#21311yescharacteristic
368C3×C4#94911yescharacteristic
369Dic3—81no—
370Dic3—124no—
371Dic3—122no—
372Dic3—121no—
373Dic3—122no—
374Dic3—121no—
375Dic3—122no—
376C32#164911yescharacteristic, complemented, Sylow
377C23—181no—
378Q8—21no—
379Q8—121no—
380Q8—121no—
381Q8—364no—
382Q8—361no—
383Q8—361no—
384Q8#371412yes—
385D4—92no—
386D4—181no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—121no—
390C2×C4—182no—
391C2×C4—181no—
392C2×C4—364no—
393C2×C4—361no—
394C2×C4—361no—
395C2×C4C2×#62611yescharacteristic
396C2×C4#381111yescharacteristic
397C2×C4#381111yescharacteristic
398C8—62no—
399C8—121no—
400C8—121no—
401C2×C3—21no—
402C2×C3—21no—
403C2×C3—21no—
404C2×C3#473211yescharacteristic
405C2×C3#480911yescharacteristic
406C2×C3#482611yescharacteristic
407C2×C3#503611yescharacteristic
408C2×C3#518611yescharacteristic
409C2×C3#520511yescharacteristic
410D3—124no—
411D3—122no—
412D3—122no—
413C22—182no—
414C22—361no—
415C22—361no—
416C22#6398111yescharacteristic
417C4—21no—
418C4—22no—
419C4—41no—
420C4—241no—
421C4—362no—
422C4—361no—
423C4C2×#371411yescharacteristic
424C4#6382511yescharacteristic
425C4#6394411yescharacteristic
426C4#6395111yescharacteristic
427C3—21no—
428C3#6827311yescharacteristic, complemented
429C3#7106511yescharacteristic, complemented
430C2—362no—
431C2#1059116811yescharacteristic
432C2#1059160311yescharacteristic
433C2#1059166811yescharacteristic
subgroupG/H
centerC22#63981
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4×Q8C22
socleC22×C32#120
prime radicalC2×C4⋊(C3⋊D3)D4
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×#5031C2semidirect
4. #10590311C2semidirect
5. C3×#4535C2semidirect
6. #10589094C2semidirect
33. #5031D3semidirect
37. #4535D3semidirect
121. C4⋊(C3⋊D3)Q16semidirect
376. C32#1649semidirect
428. C3#68273semidirect
429. C3#71065semidirect

Maximal subgroups

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

subgroupindex
Q8×C4⋊(C3⋊D3)2
C3×#50312
#105903132
#105903112
C3×#45352
#105890942
#105890822
#710653
#682733