Finite Group Databasebeta

← #10691477

Subgroups — #10691477

Order 768.

588 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 111 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#83298C211yesmaximal, characteristic
2#73617C211yesmaximal, characteristic, complemented
3C3×#1502C211yesmaximal, characteristic, complemented
4#71815C211yesmaximal, characteristic, complemented
5#70762C211yesmaximal, characteristic, complemented
6#67736C211yesmaximal, characteristic, complemented
7#67620C211yesmaximal, characteristic, complemented
8#9237—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×C4×D12C2211yescharacteristic
11C2×#872—21no—
12C3×#431C2211yescharacteristic
13C3×C4×C4⋊C4C2211yescharacteristic
14C2×#837—21no—
15C2×#837—21no—
16#5087C412yescomplemented
17#5072C412yescomplemented
18#5036—21no—
19#5033—21no—
20C4×#213—22no—
21#4988—21no—
22#4972—21no—
23C4×(C2×D3)⋊C4—21no—
24C42×Dic3C2211yescharacteristic
25#4925C2211yescharacteristic
26#4924C2211yescharacteristic
27#4669—22no—
28C3×#327—22no—
29#4459—22no—
30#3100—31no—
31#2293—31no—
32#1962—31no—
33#1502D311yescharacteristic, complemented
34#1486—31no—
35#1303—31no—
36#1276—31no—
37C2×C3×#41—21no—
38C22×D12D411yescharacteristic
39C22×C4×D3—21no—
40C3×C4⋊Q8C2×C411yescharacteristic
41C3×#127C2×C411yescharacteristic
42C3×#124—21no—
43C3×#121—21no—
44C3×#121—21no—
45C3×C4×D4—21no—
46C3×C4×D4—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C42—21no—
49C2×C3×C42C2311yescharacteristic
50C2×#213—22no—
51C2×(C2×D3)⋊C4—21no—
52C2×(C2×D3)⋊C4—22no—
53C2×(C2×D3)⋊C4—22no—
54C2×C4⋊Dic3D411yescharacteristic
55C2×Dic3⋊C4—22no—
56C2×C4×Dic3—21no—
57C2×C4×Dic3—21no—
58C2×C4×Dic3D411yescharacteristic
59C2×C4×Dic3D411yescharacteristic
60C2×#208—21no—
61C4×D12—21no—
62C4×D12—21no—
63C4×D12C2×C412yes—
64C4×D12D412yes—
65#872—42no—
66#872—42no—
67C3×C4⋊C8—21no—
68C3×C22⋊C8—21no—
69C3×C4×C8—21no—
70C3×#95D412yes—
71#837—42no—
72#837—42no—
73#832—22no—
74C3⋊#95—22no—
75#830—21no—
76#809—22no—
77#807—22no—
78C4⋊(C3⋊C8)—21no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)C2×C412yes—
81C4×C3⋊C8—21no—
82#532C2×D311yescharacteristic
83C2×C4×D4—31no—
84C2×#117—61no—
85#480—61no—
86#474—32no—
87#459—32no—
88#447—61no—
89C4×D8—62no—
90#431—31no—
91#431C2×D311yescharacteristic
92#423—31no—
93C2×#104—61no—
94C2×#104—61no—
95#390—61no—
96#389—61no—
97#388—62no—
98C4×C4⋊C4C2×D311yescharacteristic
99C4×C22⋊C4—61no—
100C43—31no—
101#327—22no—
102D4⋊C8—62no—
103C23×D3—21no—
104C3×#41—41no—
105C3×#41—41no—
106C3×#41—41no—
107C3×#41—41no—
108C2×C3×Q8—21no—
109C2×C3×D4—21no—
110C2×C3×D4—21no—
111C2×C3×D4—21no—
112C2×C3×D4—41no—
113C22×C3×C4—22no—
114C22×C3×C4—41no—
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C22×Dic3—21no—
119C22×Dic3—21no—
120C22×Dic3—21no—
121C2×D12—21no—
122C2×D12—42no—
123C2×D12C22⋊C412yes—
124C2×C4×D3—22no—
125C2×C4×D3—22no—
126C2×C4×D3—41no—
127C2×C4×D3—41no—
128C2×C4×D3—41no—
129C2×C4×D3—41no—
130C2×C3×C8—41no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—21no—
133C3×C4⋊C4—42no—
134C3×C22⋊C4—41no—
135C3×C22⋊C4—41no—
136C3×C42—41no—
137C3×C42—41no—
138C3×C42C22×C411yescharacteristic
139C3×C42C2×D411yescharacteristic
140C3×C42#4111yescharacteristic
141C3×C42#4111yescharacteristic
142#213—44no—
143(C2×D3)⋊C4—22no—
144(C2×D3)⋊C4—22no—
145(C2×D3)⋊C4—44no—
146(C2×D3)⋊C4—44no—
147C4⋊Dic3—21no—
148C4⋊Dic3C22⋊C412yes—
149Dic3⋊C4—44no—
150C4×Dic3—21no—
151C4×Dic3—21no—
152C4×Dic3—21no—
153C4×Dic3—42no—
154C4×Dic3—42no—
155C4×Dic3D812yescomplemented
156#208—41no—
157#208—41no—
158C2×C3⋊C8—22no—
159C2×C3⋊C8—41no—
160C2×#41—21no—
161C22×D4—31no—
162C23×C4—61no—
163C2×D8—62no—
164C2×M16—61no—
165C4⋊Q8C4×D311yescharacteristic
166#127C4×D311yescharacteristic
167#124—21no—
168#121—21no—
169#121—21no—
170C4×D4—21no—
171C4×D4—21no—
172C4×D4—32no—
173C4×D4—32no—
174C4×D4—61no—
175C4×D4—61no—
176#117—122no—
177#117—122no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—31no—
180C2×C4⋊C4—62no—
181C2×C22⋊C4—61no—
182C2×C22⋊C4—62no—
183C2×C22⋊C4—62no—
184C2×C42—21no—
185C2×C42—31no—
186C2×C42—31no—
187C2×C42—61no—
188C2×C42—61no—
189C2×C42C22×D311yescharacteristic
190#107—62no—
191C4⋊C8—21no—
192C4⋊C8—32no—
193C4⋊C8—61no—
194C4⋊C8—61no—
195#104—122no—
196#104—122no—
197#102—62no—
198#102—62no—
199C22⋊C8—21no—
200C22⋊C8—61no—
201C4×C8—21no—
202C4×C8—61no—
203#95—62no—
204#95#6712yes—
205C23×C3—41no—
206C23×C3#12011yescharacteristic
207C22×D3—22no—
208C22×D3—42no—
209C22×D3—42no—
210C22×D3—41no—
211C22×D3—41no—
212C3×Q8—41no—
213C3×Q8—41no—
214C3×D4—42no—
215C3×D4—41no—
216C3×D4—41no—
217C3×D4—81no—
218C3×D4—81no—
219C3×D4—81no—
220C2×C3×C4—22no—
221C2×C3×C4—41no—
222C2×C3×C4—42no—
223C2×C3×C4—42no—
224C2×C3×C4—41no—
225C2×C3×C4—42no—
226C2×C3×C4—81no—
227C2×C3×C4—81no—
228C2×C3×C4C2×C22⋊C411yescharacteristic
229C2×C3×C4C4×D411yescharacteristic
230C2×C3×C4C4×D411yescharacteristic
231C2×C3×C4#12111yescharacteristic
232C2×C3×C4#12111yescharacteristic
233C2×C3×C4#12311yescharacteristic
234C2×C3×C4C2×D811yescharacteristic
235C2×C3×C4#13511yescharacteristic
236C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
237C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
238C2×Dic3—22no—
239C2×Dic3—22no—
240C2×Dic3—22no—
241C2×Dic3—41no—
242C2×Dic3—41no—
243C2×Dic3—41no—
244C2×Dic3—41no—
245C2×Dic3—41no—
246C2×Dic3—41no—
247D12—22no—
248D12—22no—
249D12—42no—
250C4×D3—42no—
251C4×D3—42no—
252C4×D3—42no—
253C4×D3—42no—
254C3×C8—81no—
255C3⋊C8—42no—
256C3⋊C8—81no—
257C24—61no—
258#41—41no—
259#41—41no—
260#41—41no—
261#41—41no—
262C2×Q8—21no—
263C2×D4—21no—
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—32no—
267C2×D4—41no—
268C2×D4—61no—
269C2×D4—122no—
270C22×C4—22no—
271C22×C4—41no—
272C22×C4—61no—
273C22×C4—61no—
274C22×C4—61no—
275C22×C4—62no—
276C22×C4—62no—
277C22×C4—121no—
278C22×C4—121no—
279C22×C4—121no—
280C22×C4—121no—
281C22×C4D3×D411yescharacteristic
282C22×C4D3×D411yescharacteristic
283C22×C4C2×#6711yescharacteristic
284D8—124no—
285M16—121no—
286M16—121no—
287C2×C8—41no—
288C2×C8—62no—
289C2×C8—121no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—32no—
293C4⋊C4—42no—
294C4⋊C4—61no—
295C4⋊C4—124no—
296C22⋊C4—41no—
297C22⋊C4—41no—
298C22⋊C4—62no—
299C22⋊C4—62no—
300C22⋊C4—124no—
301C22⋊C4—124no—
302C42—32no—
303C42—41no—
304C42—41no—
305C42—61no—
306C42—61no—
307C42—61no—
308C42—122no—
309C42—122no—
310C42C2×C4×D311yescharacteristic
311C42#23511yescharacteristic
312C42D3×D411yescharacteristic
313C42#23711yescharacteristic
314C22×C3—21no—
315C22×C3—21no—
316C22×C3—81no—
317C22×C3—81no—
318C22×C3—81no—
319C22×C3#38611yescharacteristic
320C22×C3#44711yescharacteristic
321C22×C3#44911yescharacteristic
322C2×D3—42no—
323C2×D3—42no—
324C2×D3—42no—
325C2×D3—82no—
326C2×D3—82no—
327C3×C4—21no—
328C3×C4—21no—
329C3×C4—21no—
330C3×C4—82no—
331C3×C4—81no—
332C3×C4#42112yes—
333C3×C4C4×D811yescharacteristic
334C3×C4#44211yescharacteristic
335C3×C4#45911yescharacteristic
336C3×C4#46311yescharacteristic
337Dic3—42no—
338Dic3—42no—
339Dic3—42no—
340C23—41no—
341C23—62no—
342C23—122no—
343C23—122no—
344C23—121no—
345C23—121no—
346C23#93711yescharacteristic
347Q8—41no—
348Q8—41no—
349D4—41no—
350D4—41no—
351D4—42no—
352D4—62no—
353D4—62no—
354D4—81no—
355D4—81no—
356D4—81no—
357D4—122no—
358C2×C4—22no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—41no—
364C2×C4—62no—
365C2×C4—62no—
366C2×C4—62no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—121no—
370C2×C4—121no—
371C2×C4—121no—
372C2×C4—121no—
373C2×C4—121no—
374C2×C4—121no—
375C2×C4—122no—
376C2×C4—122no—
377C2×C4—122no—
378C2×C4—122no—
379C2×C4D3×C22⋊C411yescharacteristic
380C2×C4#67⋊C411yescharacteristic
381C2×C4#88311yescharacteristic
382C2×C4#88411yescharacteristic
383C2×C4D3×D811yescharacteristic
384C2×C4#91111yescharacteristic
385C2×C4#91411yescharacteristic
386C2×C4#91611yescharacteristic
387C2×C4C4×#6711yescharacteristic
388C2×C4#93811yescharacteristic
389C8—81no—
390C8—122no—
391C8—241no—
392C2×C3—21no—
393C2×C3—21no—
394C2×C3—161no—
395C2×C3#187911yescharacteristic
396C2×C3#189711yescharacteristic
397C2×C3#190611yescharacteristic
398D3—82no—
399C22—21no—
400C22—21no—
401C22—81no—
402C22—81no—
403C22—81no—
404C22—122no—
405C22—122no—
406C22—122no—
407C22—242no—
408C22—242no—
409C22#495611yescharacteristic
410C22#515611yescharacteristic
411C22#517211yescharacteristic
412C4—21no—
413C4—21no—
414C4—21no—
415C4—82no—
416C4—81no—
417C4—122no—
418C4—122no—
419C4—122no—
420C4#475611yescharacteristic
421C4#478511yescharacteristic
422C4#478611yescharacteristic
423C4#478711yescharacteristic
424C4#482112yes—
425C3#923711yescharacteristic, complemented, Sylow
426C2—21no—
427C2—21no—
428C2—161no—
429C2—242no—
430C2#7005511yescharacteristic
431C2#7020811yescharacteristic
432C2#7038311yescharacteristic
subgroupG/H
centerC2×C4#884
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1502C2
socleC22×C3#386
prime radical#83298C2
cosocleC2×C3×C42C23
hypercenter#1502D3
hypocenterC3#9237
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1502

Complements

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

NH ≅ G/Ntype
2. #73617C2semidirect
3. C3×#1502C2semidirect
4. #71815C2semidirect
5. #70762C2semidirect
6. #67736C2semidirect
7. #67620C2semidirect
16. #5087C4semidirect
17. #5072C4semidirect
33. #1502D3semidirect
155. C4×Dic3D8semidirect
425. C3#9237semidirect

Maximal subgroups

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

subgroupindex
#832982
#736172
C3×#15022
#718152
#707622
#677362
#676202
#92373