Finite Group Databasebeta

← #10690123

Subgroups — #10690123

Order 768.

578 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 105 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
1C2×#5086C211yesmaximal, characteristic
2#74230C211yesmaximal, characteristic
3C3×#1993C211yesmaximal, characteristic, complemented
4#71788C211yesmaximal, characteristic, complemented
5C2×#4464C211yesmaximal, characteristic
6#67659C211yesmaximal, characteristic, complemented
7#67630C211yesmaximal, characteristic, complemented
8#9003—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C22×#216—21no—
11C22×#215—21no—
12C2×#875C2211yescharacteristic
13C2×C3×#97C2211yescharacteristic
14C3×#382C2211yescharacteristic
15#5086—22no—
16#5086C414yescomplemented
17#4990—21no—
18#4989—21no—
19#4936C2211yescharacteristic
20C2×#803C2211yescharacteristic
21C2×#803C2211yescharacteristic
22#4670—21no—
23#4668—21no—
24#4665—21no—
25#4646—21no—
26C3×#340—21no—
27C3×#340—21no—
28#4529—21no—
29#4529—21no—
30#4464—22no—
31C2×#497—31no—
32#2379—31no—
33#1993—31no—
34#1993D311yescharacteristic, complemented
35C2×#329—31no—
36#1295—31no—
37#1295—31no—
38C22×C3×Q8—21no—
39C22×D12D411yescharacteristic
40C22×C4×D3—21no—
41C22×Dic6D411yescharacteristic
42C22×C3×C8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8C2×C412yescomplemented
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4D411yescharacteristic
50C2×C3×C4⋊C4D411yescharacteristic
51C2×C3×C42C2311yescharacteristic
52C2×#216—24no—
53C2×#216—42no—
54C2×#216—42no—
55C2×#215—24no—
56C2×#215—42no—
57C2×#215—42no—
58C2×(C2×D3)⋊C4—21no—
59C2×(C2×D3)⋊C4—21no—
60C2×(C2×D3)⋊C4—21no—
61C2×(C2×D3)⋊C4—21no—
62C2×(C2×D3)⋊C4—21no—
63C2×C4⋊Dic3D411yescharacteristic
64C2×C4⋊Dic3D411yescharacteristic
65C2×Dic3⋊C4—21no—
66C2×C4×Dic3—21no—
67C22×C3⋊C8—21no—
68C22×C3⋊C8—21no—
69#875—21no—
70#875—21no—
71#875C2×C412yes—
72#875D412yes—
73C3×#97—21no—
74C3×#97—21no—
75C3×#95—21no—
76C3×#95—21no—
77C3⋊#95—21no—
78C3⋊#95—21no—
79C3⋊#95—21no—
80C3⋊#95—21no—
81#803—21no—
82#803—21no—
83#803—21no—
84#803C2×C412yescomplemented
85C22×Q16—61no—
86C22×SD16—61no—
87C2×C4⋊Q8C2×D311yescharacteristic
88C2×#124—31no—
89#497—34no—
90#497—62no—
91C2×#97—31no—
92C2×#97—31no—
93C2×#97C2×D311yescharacteristic
94#391—61no—
95#388—61no—
96#388—61no—
97#386—61no—
98#385—61no—
99#385—61no—
100#382—31no—
101#382C2×D311yescharacteristic
102#340—21no—
103#340—21no—
104#340—61no—
105#340—61no—
106#329—62no—
107C23×D3—21no—
108C2×C3×Q8—22no—
109C2×C3×Q8—41no—
110C2×C3×Q8—41no—
111C2×C3×Q8—41no—
112C2×C3×Q8—41no—
113C2×C3×Q8—41no—
114C22×C3×C4—21no—
115C22×C3×C4—21no—
116C22×C3×C4—41no—
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×Dic3—21no—
121C22×Dic3—21no—
122C22×Dic3—21no—
123C2×D12—21no—
124C2×D12—41no—
125C2×D12—41no—
126C2×D12C22⋊C412yes—
127C2×C4×D3—81no—
128C2×C4×D3—81no—
129C2×C4×D3—81no—
130C2×Dic6—21no—
131C2×Dic6—41no—
132C2×Dic6—41no—
133C2×Dic6C22⋊C412yes—
134C2×C3×C8—22no—
135C2×C3×C8—41no—
136C2×C3×C8—41no—
137C3×C4⋊C4—22no—
138C3×C4⋊C4—22no—
139C3×C4⋊C4—41no—
140C3×C4⋊C4—41no—
141C3×C4⋊C4—41no—
142C3×C4⋊C4—41no—
143C3×C42C22×C411yescharacteristic
144C3×C42C2×D411yescharacteristic
145C3×C42#4111yescharacteristic
146C3×C42#4111yescharacteristic
147#216—48no—
148#215—48no—
149(C2×D3)⋊C4—22no—
150(C2×D3)⋊C4—22no—
151(C2×D3)⋊C4—22no—
152(C2×D3)⋊C4—22no—
153(C2×D3)⋊C4—81no—
154(C2×D3)⋊C4—81no—
155(C2×D3)⋊C4—81no—
156C4⋊Dic3—41no—
157C4⋊Dic3—41no—
158Dic3⋊C4—81no—
159C4×Dic3—81no—
160C2×C3⋊C8—22no—
161C2×C3⋊C8—22no—
162C2×C3⋊C8—22no—
163C2×C3⋊C8—22no—
164C2×C3⋊C8—41no—
165C2×C3⋊C8—41no—
166C22×Q8—21no—
167C22×Q8—31no—
168C22×D4—31no—
169C23×C4—61no—
170C2×Q16—64no—
171C2×Q16—122no—
172C2×Q16—122no—
173C2×SD16—64no—
174C2×SD16—122no—
175C2×SD16—122no—
176C22×C8—21no—
177C22×C8—61no—
178C22×C8—61no—
179C4⋊Q8—21no—
180C4⋊Q8—21no—
181C4⋊Q8—21no—
182C4⋊Q8C4×D312yescomplemented
183#124—32no—
184#124—32no—
185#124—61no—
186#124—61no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—31no—
190C2×C4⋊C4—31no—
191C2×C4⋊C4—61no—
192C2×C4⋊C4#6711yescharacteristic
193C2×C4⋊C4#6711yescharacteristic
194C2×C22⋊C4—61no—
195C2×C22⋊C4—61no—
196C2×C22⋊C4—61no—
197C2×C22⋊C4—61no—
198C2×C22⋊C4—61no—
199C2×C42—61no—
200C2×C42C22×D311yescharacteristic
201#97—21no—
202#97—21no—
203#97—32no—
204#97—61no—
205#97—61no—
206#97—61no—
207#95—21no—
208#95—21no—
209#95—61no—
210#95—61no—
211#95—61no—
212#95—61no—
213C23×C3#12011yescharacteristic
214C22×D3—22no—
215C22×D3—41no—
216C22×D3—41no—
217C22×D3—41no—
218C22×D3—41no—
219C22×D3—41no—
220C22×D3—41no—
221C3×Q8—42no—
222C3×Q8—42no—
223C3×Q8—81no—
224C3×Q8—81no—
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4—21no—
228C2×C3×C4—21no—
229C2×C3×C4—21no—
230C2×C3×C4—21no—
231C2×C3×C4—42no—
232C2×C3×C4—41no—
233C2×C3×C4—41no—
234C2×C3×C4—41no—
235C2×C3×C4—41no—
236C2×C3×C4—41no—
237C2×C3×C4—41no—
238C2×C3×C4—81no—
239C2×C3×C4—81no—
240C2×C3×C4C2×C22⋊C411yescharacteristic
241C2×C3×C4C4×D411yescharacteristic
242C2×C3×C4C4×D411yescharacteristic
243C2×C3×C4#12111yescharacteristic
244C2×C3×C4#12111yescharacteristic
245C2×C3×C4#12311yescharacteristic
246C2×Dic3—22no—
247C2×Dic3—41no—
248C2×Dic3—41no—
249C2×Dic3—41no—
250C2×Dic3—41no—
251C2×Dic3—41no—
252C2×Dic3—41no—
253C2×Dic3—41no—
254C2×Dic3—41no—
255D12—42no—
256D12—81no—
257C4×D3—84no—
258Dic6—42no—
259Dic6—81no—
260C3×C8—42no—
261C3⋊C8—24no—
262C3⋊C8—42no—
263C24—61no—
264C2×Q8—22no—
265C2×Q8—32no—
266C2×Q8—41no—
267C2×Q8—41no—
268C2×Q8—41no—
269C2×Q8—41no—
270C2×Q8—41no—
271C2×Q8—61no—
272C2×Q8—121no—
273C2×Q8—121no—
274C2×D4—32no—
275C2×D4—61no—
276C2×D4—121no—
277C2×D4—121no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—41no—
281C22×C4—61no—
282C22×C4—61no—
283C22×C4—61no—
284C22×C4—241no—
285C22×C4—241no—
286C22×C4—241no—
287C22×C4D3×D411yescharacteristic
288C22×C4D3×D411yescharacteristic
289C22×C4C2×#6711yescharacteristic
290Q16—128no—
291SD16—128no—
292C2×C8—22no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—62no—
296C2×C8—62no—
297C2×C8—62no—
298C2×C8—62no—
299C2×C8—121no—
300C2×C8—121no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—121no—
308C4⋊C4—121no—
309C4⋊C4—241no—
310C22⋊C4—62no—
311C22⋊C4—62no—
312C22⋊C4—62no—
313C22⋊C4—62no—
314C22⋊C4—241no—
315C22⋊C4—241no—
316C22⋊C4—241no—
317C42—241no—
318C42C2×C4×D311yescharacteristic
319C42#23511yescharacteristic
320C42D3×D411yescharacteristic
321C42#23711yescharacteristic
322C22×C3#38611yescharacteristic
323C22×C3#45311yescharacteristic
324C22×C3#45311yescharacteristic
325C22×C3#45511yescharacteristic
326C22×C3#45511yescharacteristic
327C22×C3#45611yescharacteristic
328C22×C3#45611yescharacteristic
329C2×D3—42no—
330C2×D3—42no—
331C2×D3—42no—
332C2×D3—81no—
333C2×D3—81no—
334C2×D3—81no—
335C2×D3—81no—
336C3×C4—21no—
337C3×C4—21no—
338C3×C4—21no—
339C3×C4—21no—
340C3×C4—21no—
341C3×C4—21no—
342C3×C4—82no—
343C3×C4—81no—
344C3×C4—81no—
345Dic3—42no—
346Dic3—81no—
347Dic3—81no—
348C23—62no—
349C23—121no—
350C23—121no—
351C23—121no—
352C23—121no—
353C23—121no—
354C23—121no—
355C23#93711yescharacteristic
356Q8—42no—
357Q8—42no—
358Q8—81no—
359Q8—81no—
360Q8—122no—
361Q8—241no—
362D4—122no—
363D4—241no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—62no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—121no—
381C2×C4—121no—
382C2×C4—121no—
383C2×C4—121no—
384C2×C4—121no—
385C2×C4—121no—
386C2×C4—121no—
387C2×C4—121no—
388C2×C4—244no—
389C2×C4D3×C22⋊C411yescharacteristic
390C2×C4#67⋊C411yescharacteristic
391C2×C4#88311yescharacteristic
392C2×C4#88411yescharacteristic
393C2×C4C4×#6711yescharacteristic
394C2×C4#93811yescharacteristic
395C8—42no—
396C8—64no—
397C8—122no—
398C2×C3#165911yescharacteristic
399C2×C3#166011yescharacteristic
400C2×C3#166111yescharacteristic
401C2×C3#166111yescharacteristic
402C2×C3#188211yescharacteristic
403C2×C3#188211yescharacteristic
404C2×C3#188711yescharacteristic
405D3—82no—
406C22—122no—
407C22—122no—
408C22—122no—
409C22—241no—
410C22—241no—
411C22—241no—
412C22—241no—
413C22#495611yescharacteristic
414C22#506111yescharacteristic
415C22#509511yescharacteristic
416C22#519811yescharacteristic
417C22#519811yescharacteristic
418C22#520211yescharacteristic
419C22#520511yescharacteristic
420C4—21no—
421C4—21no—
422C4—21no—
423C4—21no—
424C4—21no—
425C4—21no—
426C4—81no—
427C4—81no—
428C4—82no—
429C4—122no—
430C4—241no—
431C4—241no—
432C3#900311yescharacteristic, complemented, Sylow
433C2—242no—
434C2#6969311yescharacteristic
435C2#7032311yescharacteristic
436C2#7034811yescharacteristic
437C2#7124411yescharacteristic
438C2#7124511yescharacteristic
439C2#7139111yescharacteristic
440C2#7139211yescharacteristic
subgroupG/H
centerC23#937
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1993C2
socleC23×C3#120
prime radicalC22×D12D4
cosocleC2×C3×C42C23
hypercenter#1993D3
hypocenterC3#9003
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1993

Complements

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

NH ≅ G/Ntype
3. C3×#1993C2semidirect
4. #71788C2semidirect
6. #67659C2semidirect
7. #67630C2semidirect
16. #5086C4semidirect
34. #1993D3semidirect
46. C3×C4⋊Q8C2×C4semidirect
84. #803C2×C4semidirect
182. C4⋊Q8C4×D3semidirect
432. C3#9003semidirect

Maximal subgroups

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

subgroupindex
C2×#50862
#742302
C3×#19932
#717882
C2×#44642
#676592
#676302
#90033