Finite Group Databasebeta

← #10691911

Subgroups — #10691911

Order 768.

632 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 632 conjugacy classes of subgroups — too many to draw. Only its 107 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#83302C211yesmaximal, characteristic
2#74153C211yesmaximal, characteristic, complemented
3C3×#1517C211yesmaximal, characteristic, complemented
4#71816C211yesmaximal, characteristic, complemented
5#70774C211yesmaximal, characteristic, complemented
6#67737C211yesmaximal, characteristic, complemented
7#67631C211yesmaximal, characteristic, complemented
8#9299—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×#893—21no—
11C2×#874C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#382C2211yescharacteristic
14C2×#837—21no—
15C2×#837—21no—
16#5088C412yescomplemented
17#5073C412yescomplemented
18#5036—22no—
19#5015—22no—
20#4988—21no—
21#4932C2211yescharacteristic
22#4925C2211yescharacteristic
23#4924C2211yescharacteristic
24#4665—22no—
25#4665—22no—
26C3×#327—22no—
27#4485—22no—
28#3205—31no—
29#2328—31no—
30#1963—31no—
31#1545—31no—
32#1517D311yescharacteristic, complemented
33#1286—31no—
34#1274—31no—
35C2×C3×#41—21no—
36C22×D12—21no—
37C22×D12—21no—
38C22×D12D412yes—
39C22×C4×D3—22no—
40C3×C4⋊Q8C2×C411yescharacteristic, complemented
41C3×#127C2×C411yescharacteristic, complemented
42C3×#124—21no—
43C3×#121—21no—
44C3×#121—21no—
45C3×C4×D4—21no—
46C3×C4×D4—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C42C2311yescharacteristic
49C2×#213—24no—
50C2×(C2×D3)⋊C4—22no—
51C2×(C2×D3)⋊C4—22no—
52C2×(C2×D3)⋊C4—22no—
53C2×C4×Dic3D411yescharacteristic
54C2×C4×Dic3D411yescharacteristic
55C2×#208—21no—
56#893—42no—
57#893—42no—
58#874—21no—
59#874—21no—
60#874C2×C412yes—
61#874D412yes—
62C3×C4⋊C8—21no—
63C3×C22⋊C8—21no—
64C3×C4×C8—21no—
65C3×#95—21no—
66C3×#95D412yes—
67#837—42no—
68#837—42no—
69C3⋊#95—21no—
70C3⋊#95—21no—
71#830—21no—
72#809—24no—
73C4⋊(C3⋊C8)—21no—
74C4⋊(C3⋊C8)—21no—
75C4⋊(C3⋊C8)—21no—
76C4×C3⋊C8C2×C412yescomplemented
77#532C2×D311yescharacteristic
78C2×#127—31no—
79C2×C4×D4—61no—
80#493—32no—
81#486—32no—
82#459—62no—
83#447—62no—
84#431—31no—
85#431C2×D311yescharacteristic
86#423—31no—
87C2×#104—61no—
88C2×#104—61no—
89#390—61no—
90#386—62no—
91#386—62no—
92#382C2×D311yescharacteristic
93#376—31no—
94#331—62no—
95#327—22no—
96C23×D3—22no—
97C3×#41—41no—
98C3×#41—41no—
99C3×#41—41no—
100C3×#41—41no—
101C2×C3×Q8—21no—
102C2×C3×D4—21no—
103C2×C3×D4—21no—
104C2×C3×D4—21no—
105C2×C3×D4—41no—
106C22×C3×C4—22no—
107C22×C3×C4—41no—
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4C2×D411yescharacteristic
111C22×Dic3—21no—
112C22×Dic3—21no—
113C2×D12—22no—
114C2×D12—22no—
115C2×D12—22no—
116C2×D12—22no—
117C2×D12—22no—
118C2×D12—41no—
119C2×D12—41no—
120C2×D12—41no—
121C2×D12—41no—
122C2×D12—82no—
123C2×D12—82no—
124C2×D12C22⋊C414yes—
125C2×C4×D3—44no—
126C2×C4×D3—82no—
127C2×C4×D3—82no—
128C2×C3×C8—41no—
129C3×C4⋊C4—21no—
130C3×C4⋊C4—21no—
131C3×C4⋊C4—41no—
132C3×C4⋊C4—41no—
133C3×C22⋊C4—41no—
134C3×C22⋊C4—41no—
135C3×C42C22×C411yescharacteristic
136C3×C42C2×D411yescharacteristic
137C3×C42#4111yescharacteristic
138C3×C42#4111yescharacteristic
139#213—84no—
140(C2×D3)⋊C4—44no—
141(C2×D3)⋊C4—82no—
142(C2×D3)⋊C4—82no—
143C4×Dic3—21no—
144C4×Dic3—21no—
145C4×Dic3—21no—
146C4×Dic3D812yescomplemented
147#208—41no—
148#208—41no—
149C2×C3⋊C8—22no—
150C2×C3⋊C8—41no—
151C2×#41—21no—
152C22×D4—32no—
153C22×D4—61no—
154C22×D4—61no—
155C23×C4—62no—
156C2×D8—64no—
157C2×M16—61no—
158C4⋊Q8C4×D311yescharacteristic, complemented
159#127—32no—
160#127—32no—
161#127—61no—
162#127—61no—
163#127C4×D311yescharacteristic, complemented
164#124—21no—
165#121—21no—
166#121—21no—
167C4×D4—21no—
168C4×D4—21no—
169C4×D4—122no—
170C4×D4—122no—
171C2×C4⋊C4—21no—
172C2×C22⋊C4—62no—
173C2×C22⋊C4—62no—
174C2×C22⋊C4—62no—
175C2×C42—31no—
176C2×C42—31no—
177C2×C42C22×D311yescharacteristic
178C4⋊C8—21no—
179C4⋊C8—61no—
180C4⋊C8—61no—
181C4⋊C8—61no—
182#104—122no—
183#104—122no—
184#102—64no—
185C22⋊C8—21no—
186C22⋊C8—61no—
187C4×C8—21no—
188C4×C8—32no—
189#95—21no—
190#95—61no—
191#95—61no—
192#95#6712yes—
193C23×C3—41no—
194C23×C3#12011yescharacteristic
195C22×D3—24no—
196C22×D3—42no—
197C22×D3—42no—
198C22×D3—42no—
199C22×D3—42no—
200C22×D3—42no—
201C22×D3—42no—
202C3×Q8—41no—
203C3×Q8—41no—
204C3×D4—42no—
205C3×D4—41no—
206C3×D4—41no—
207C3×D4—81no—
208C3×D4—81no—
209C3×D4—81no—
210C2×C3×C4—41no—
211C2×C3×C4—41no—
212C2×C3×C4—42no—
213C2×C3×C4—42no—
214C2×C3×C4—41no—
215C2×C3×C4—42no—
216C2×C3×C4—81no—
217C2×C3×C4—81no—
218C2×C3×C4C2×C22⋊C411yescharacteristic
219C2×C3×C4C4×D411yescharacteristic
220C2×C3×C4C4×D411yescharacteristic
221C2×C3×C4#12111yescharacteristic
222C2×C3×C4#12111yescharacteristic
223C2×C3×C4#12311yescharacteristic
224C2×C3×C4C2×D811yescharacteristic
225C2×C3×C4#13511yescharacteristic
226C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
227C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
228C2×Dic3—22no—
229C2×Dic3—41no—
230C2×Dic3—41no—
231C2×Dic3—41no—
232C2×Dic3—41no—
233C2×Dic3—41no—
234D12—44no—
235D12—42no—
236D12—42no—
237D12—42no—
238D12—42no—
239D12—82no—
240D12—82no—
241D12—82no—
242C4×D3—88no—
243C3×C8—81no—
244C3⋊C8—42no—
245C3⋊C8—81no—
246C24—62no—
247#41—41no—
248#41—41no—
249#41—41no—
250#41—41no—
251C2×Q8—21no—
252C2×D4—21no—
253C2×D4—21no—
254C2×D4—21no—
255C2×D4—34no—
256C2×D4—41no—
257C2×D4—62no—
258C2×D4—62no—
259C2×D4—62no—
260C2×D4—62no—
261C2×D4—62no—
262C2×D4—121no—
263C2×D4—121no—
264C2×D4—121no—
265C2×D4—121no—
266C2×D4—242no—
267C2×D4—242no—
268C22×C4—22no—
269C22×C4—41no—
270C22×C4—61no—
271C22×C4—61no—
272C22×C4—124no—
273C22×C4—242no—
274C22×C4—242no—
275C22×C4D3×D411yescharacteristic
276C22×C4D3×D411yescharacteristic
277C22×C4C2×#6711yescharacteristic
278D8—244no—
279M16—121no—
280M16—121no—
281C2×C8—41no—
282C2×C8—62no—
283C2×C8—121no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C22⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—124no—
291C22⋊C4—242no—
292C22⋊C4—242no—
293C42—32no—
294C42—61no—
295C42—61no—
296C42—61no—
297C42C2×C4×D311yescharacteristic
298C42#23511yescharacteristic
299C42D3×D411yescharacteristic
300C42#23711yescharacteristic
301C22×C3—21no—
302C22×C3—21no—
303C22×C3—81no—
304C22×C3—81no—
305C22×C3—81no—
306C22×C3#38611yescharacteristic
307C22×C3#44711yescharacteristic
308C22×C3#44911yescharacteristic
309C2×D3—44no—
310C2×D3—44no—
311C2×D3—44no—
312C2×D3—82no—
313C2×D3—82no—
314C2×D3—82no—
315C2×D3—82no—
316C3×C4—21no—
317C3×C4—21no—
318C3×C4—21no—
319C3×C4—21no—
320C3×C4—82no—
321C3×C4—81no—
322C3×C4C4×D811yescharacteristic
323C3×C4#44211yescharacteristic
324C3×C4#45911yescharacteristic
325C3×C4#46311yescharacteristic
326Dic3—42no—
327Dic3—81no—
328C23—41no—
329C23—64no—
330C23—122no—
331C23—122no—
332C23—122no—
333C23—122no—
334C23—122no—
335C23—122no—
336C23#93711yescharacteristic
337Q8—41no—
338Q8—41no—
339D4—42no—
340D4—41no—
341D4—41no—
342D4—81no—
343D4—81no—
344D4—81no—
345D4—124no—
346D4—122no—
347D4—122no—
348D4—122no—
349D4—122no—
350D4—242no—
351D4—242no—
352D4—242no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—62no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—121no—
363C2×C4—121no—
364C2×C4—121no—
365C2×C4—121no—
366C2×C4—121no—
367C2×C4—248no—
368C2×C4D3×C22⋊C411yescharacteristic
369C2×C4#67⋊C411yescharacteristic
370C2×C4#88311yescharacteristic
371C2×C4#88411yescharacteristic
372C2×C4D3×D811yescharacteristic
373C2×C4#91111yescharacteristic
374C2×C4#91411yescharacteristic
375C2×C4#91611yescharacteristic
376C2×C4C4×#6711yescharacteristic
377C2×C4#93811yescharacteristic
378C8—81no—
379C8—122no—
380C8—241no—
381C2×C3—21no—
382C2×C3—21no—
383C2×C3—161no—
384C2×C3#187911yescharacteristic
385C2×C3#190411yescharacteristic
386C2×C3#191111yescharacteristic
387D3—84no—
388C22—21no—
389C22—21no—
390C22—81no—
391C22—81no—
392C22—81no—
393C22—124no—
394C22—124no—
395C22—124no—
396C22—242no—
397C22—242no—
398C22—242no—
399C22—242no—
400C22#495611yescharacteristic
401C22#515611yescharacteristic
402C22#517211yescharacteristic
403C4—21no—
404C4—21no—
405C4—21no—
406C4—21no—
407C4—82no—
408C4—81no—
409C4—122no—
410C4—241no—
411C4#475611yescharacteristic
412C4#478511yescharacteristic
413C4#478611yescharacteristic
414C4#478711yescharacteristic
415C3#929911yescharacteristic, complemented, Sylow
416C2—21no—
417C2—21no—
418C2—161no—
419C2—244no—
420C2#7004911yescharacteristic
421C2#7020811yescharacteristic
422C2#7040811yescharacteristic
subgroupG/H
centerC22#4956
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1517C2
socleC22×C3#386
prime radical#83302C2
cosocleC2×C3×C42C23
hypercenter#1517D3
hypocenterC3#9299
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1517

Complements

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

NH ≅ G/Ntype
2. #74153C2semidirect
3. C3×#1517C2semidirect
4. #71816C2semidirect
5. #70774C2semidirect
6. #67737C2semidirect
7. #67631C2semidirect
16. #5088C4semidirect
17. #5073C4semidirect
32. #1517D3semidirect
40. C3×C4⋊Q8C2×C4semidirect
41. C3×#127C2×C4semidirect
76. C4×C3⋊C8C2×C4semidirect
146. C4×Dic3D8semidirect
158. C4⋊Q8C4×D3semidirect
163. #127C4×D3semidirect
415. C3#9299semidirect

Maximal subgroups

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

subgroupindex
#833022
#741532
C3×#15172
#718162
#707742
#677372
#676312
#92993