Finite Group Databasebeta

← #11713822

Subgroups — #11713822

Order 768.

503 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 503 conjugacy classes of subgroups — too many to draw. Only its 53 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#85308C211yesmaximal, characteristic
2#85303C211yesmaximal, characteristic, complemented
3A4×#107C211yesmaximal, characteristic, complemented
4#73113C211yesmaximal, characteristic
5SL(2,ℤ/4)⋊C8C211yesmaximal, characteristic, complemented
6#73053C211yesmaximal, characteristic
7#73050C211yesmaximal, characteristic, complemented
8#26336—31nomaximal, Sylow
9C2×C4⋊S4C2211yescharacteristic
10C2×C4×S4—21no—
11C2×A4⋊Q8—21no—
12C2×C8×A4C2211yescharacteristic
13A4×C4⋊C4C2211yescharacteristic
14A4×C4⋊C4C2211yescharacteristic
15(C2×S4)⋊C4—21no—
16C4⋊SL(2,ℤ/4)C2211yescharacteristic
17SL(2,ℤ/4)⋊C4—21no—
18C4×SL(2,ℤ/4)C2211yescharacteristic
19C2×A4⋊C8C2211yescharacteristic
20#4890—41nomaximal
21C2×#474—31no—
22C22×#107D311yescharacteristic, complemented
23#2440—31no—
24#2361—31no—
25#2093—31no—
26#2086—31no—
27#2067—31no—
28#2057—31no—
29#1947—31no—
30#1918—31no—
31#1893—31no—
32#1874—31no—
33#1862—31no—
34#1852—31no—
35#1850—31no—
36C22×S4—21no—
37C2×C4×A4—21no—
38C2×C4×A4—21no—
39C2×C4×A4C2311yescharacteristic
40C2×SL(2,ℤ/4)—21no—
41C2×SL(2,ℤ/4)D411yescharacteristic
42C2×SL(2,ℤ/4)D411yescharacteristic
43C4⋊S4—21no—
44C4⋊S4Q812yescomplemented
45C4×S4—42no—
46A4⋊Q8—42no—
47#893—41no—
48#888—41no—
49C8×A4—21no—
50A4⋊C8—21no—
51C3×#107—41no—
52#821—41no—
53Dic3⋊C8—41no—
54#809—41no—
55#807—41no—
56C23×C8C2×D311yescharacteristic
57C2×C4⋊Q8—31no—
58C2×C22⋊Q8—61no—
59C2×#121—31no—
60C2×C4×D4—31no—
61C2×C4×D4—61no—
62C22×C4⋊C4C2×D311yescharacteristic
63C22×C4⋊C4C2×D311yescharacteristic
64#474—61no—
65#474—61no—
66#474—61no—
67#474—61no—
68C2×#107—31no—
69C2×#107—31no—
70C2×#107—31no—
71C2×#107—31no—
72C2×#107—31no—
73C2×#107—31no—
74C2×C4⋊C8—31no—
75C2×C4⋊C8—31no—
76C2×#102—31no—
77C2×#102—31no—
78C2×#102—31no—
79C2×#102—31no—
80C2×C22⋊C8—31no—
81#398—61no—
82#395—61no—
83#390—31no—
84#389—31no—
85#389—31no—
86#389—31no—
87#388—61no—
88#386—61no—
89#385—61no—
90#385—61no—
91#384—31no—
92#380—31no—
93C4×C4⋊C4—31no—
94C4×C22⋊C4—31no—
95#340—31no—
96#340—31no—
97#340—31no—
98#340—31no—
99#340—31no—
100#340—31no—
101#336—31no—
102#336—31no—
103C22×A4C2×D411yescharacteristic
104C2×S4—22no—
105C2×S4—41no—
106C2×S4—41no—
107C2×D12—41no—
108C2×C4×D3—81no—
109C2×Dic6—81no—
110C4×A4—41no—
111C4×A4—41no—
112C4×A4C2×Q811yescharacteristic
113C4×A4#4111yescharacteristic
114SL(2,ℤ/4)—21no—
115SL(2,ℤ/4)—41no—
116SL(2,ℤ/4)D812yes—
117C2×C3×C8—41no—
118C3×C4⋊C4—41no—
119C3×C4⋊C4—41no—
120(C2×D3)⋊C4—81no—
121C4⋊Dic3—41no—
122Dic3⋊C4—81no—
123C4×Dic3—41no—
124C2×C3⋊C8—41no—
125C22×Q8—61no—
126C22×D4—31no—
127C22×D4—61no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—61no—
131C23×C4C22×D311yescharacteristic
132C22×C8—21no—
133C22×C8—31no—
134C22×C8—31no—
135C22×C8—31no—
136C22×C8—31no—
137C22×C8—61no—
138C4⋊Q8—61no—
139C4⋊Q8—61no—
140C4⋊Q8—121no—
141C22⋊Q8—122no—
142C22⋊Q8—122no—
143#121—32no—
144#121—32no—
145#121—61no—
146#121—61no—
147C4×D4—61no—
148C4×D4—61no—
149C4×D4—121no—
150C4×D4—122no—
151C4×D4—122no—
152C2×C4⋊C4—31no—
153C2×C4⋊C4—31no—
154C2×C4⋊C4—31no—
155C2×C4⋊C4—31no—
156C2×C4⋊C4—31no—
157C2×C4⋊C4—31no—
158C2×C4⋊C4—31no—
159C2×C4⋊C4—61no—
160C2×C4⋊C4—61no—
161C2×C4⋊C4—61no—
162C2×C4⋊C4—61no—
163C2×C4⋊C4—61no—
164C2×C4⋊C4—61no—
165C2×C4⋊C4—61no—
166C2×C4⋊C4—61no—
167C2×C22⋊C4—31no—
168C2×C22⋊C4—31no—
169C2×C22⋊C4—61no—
170C2×C22⋊C4—61no—
171C2×C22⋊C4—61no—
172C2×C22⋊C4—61no—
173C2×C22⋊C4—61no—
174C2×C42—31no—
175C2×C42—31no—
176C2×C42—61no—
177#107—31no—
178#107—31no—
179#107—31no—
180#107—61no—
181#107—61no—
182#107—61no—
183#107—121no—
184#107S411yescharacteristic, complemented
185C4⋊C8—62no—
186C4⋊C8—61no—
187C4⋊C8—61no—
188#102—61no—
189#102—62no—
190#102—61no—
191#102—61no—
192#102—61no—
193#102—121no—
194C22⋊C8—61no—
195C22⋊C8—61no—
196#95—31no—
197#95—31no—
198#95—61no—
199#95—61no—
200#95—61no—
201#95—61no—
202#95—61no—
203C22×D3—81no—
204C2×A4C22⋊Q811yescharacteristic
205C2×A4C2×D811yescharacteristic
206C2×A4#13711yescharacteristic
207S4—42no—
208C2×C3×C4—41no—
209C2×C3×C4—81no—
210C2×C3×C4—81no—
211C2×Dic3—41no—
212C2×Dic3—41no—
213C2×Dic3—81no—
214D12—42no—
215D12—81no—
216C4×D3—162no—
217Dic6—162no—
218C3×C8—81no—
219C3⋊C8—81no—
220C24—61no—
221C24D3×D411yescharacteristic
222C2×Q8—121no—
223C2×Q8—122no—
224C2×Q8—121no—
225C2×Q8—241no—
226C2×D4—32no—
227C2×D4—61no—
228C2×D4—61no—
229C2×D4—61no—
230C2×D4—61no—
231C2×D4—62no—
232C2×D4—62no—
233C2×D4—61no—
234C2×D4—121no—
235C2×D4—121no—
236C2×D4—121no—
237C2×D4—121no—
238C22×C4—31no—
239C22×C4—31no—
240C22×C4—31no—
241C22×C4—31no—
242C22×C4—31no—
243C22×C4—31no—
244C22×C4—31no—
245C22×C4—31no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—61no—
249C22×C4—61no—
250C22×C4—61no—
251C22×C4—61no—
252C22×C4—61no—
253C22×C4—61no—
254C22×C4—61no—
255C22×C4—121no—
256C22×C4—122no—
257C22×C4—121no—
258C22×C4—121no—
259C22×C4—121no—
260C22×C4—241no—
261C22×C4#23511yescharacteristic
262C22×C4D3×Q811yescharacteristic
263C2×C8—32no—
264C2×C8—31no—
265C2×C8—61no—
266C2×C8—61no—
267C2×C8—61no—
268C2×C8—61no—
269C2×C8—61no—
270C2×C8—61no—
271C2×C8—61no—
272C2×C8—61no—
273C2×C8—61no—
274C2×C8C2×S411yescharacteristic
275C4⋊C4—32no—
276C4⋊C4—31no—
277C4⋊C4—31no—
278C4⋊C4—61no—
279C4⋊C4—61no—
280C4⋊C4—61no—
281C4⋊C4—61no—
282C4⋊C4—61no—
283C4⋊C4—61no—
284C4⋊C4—61no—
285C4⋊C4—61no—
286C4⋊C4—61no—
287C4⋊C4—121no—
288C4⋊C4—122no—
289C4⋊C4—122no—
290C4⋊C4—122no—
291C4⋊C4—122no—
292C4⋊C4—121no—
293C4⋊C4—241no—
294C4⋊C4C2×S411yescharacteristic
295C4⋊C4C2×S411yescharacteristic
296C22⋊C4—32no—
297C22⋊C4—32no—
298C22⋊C4—62no—
299C22⋊C4—62no—
300C22⋊C4—61no—
301C22⋊C4—61no—
302C22⋊C4—121no—
303C22⋊C4—122no—
304C22⋊C4—121no—
305C22⋊C4—121no—
306C22⋊C4—121no—
307C22⋊C4—241no—
308C42—61no—
309C42—61no—
310C42—61no—
311C42—61no—
312C42—122no—
313C22×C3—41no—
314C2×D3—82no—
315C2×D3—161no—
316C2×D3—161no—
317A4#47411yescharacteristic, complemented
318C3×C4—41no—
319C3×C4—41no—
320C3×C4—161no—
321C3×C4—161no—
322Dic3—42no—
323Dic3—81no—
324Dic3—161no—
325C23—31no—
326C23—31no—
327C23—31no—
328C23—31no—
329C23—62no—
330C23—121no—
331C23—121no—
332C23—121no—
333C23—121no—
334C23—121no—
335C23—121no—
336C23#89611yescharacteristic
337C23D3×D811yescharacteristic
338C23#91811yescharacteristic
339Q8—242no—
340Q8—242no—
341D4—62no—
342D4—62no—
343D4—121no—
344D4—122no—
345D4—122no—
346D4—122no—
347D4—122no—
348D4—121no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—31no—
352C2×C4—31no—
353C2×C4—31no—
354C2×C4—31no—
355C2×C4—31no—
356C2×C4—31no—
357C2×C4—31no—
358C2×C4—32no—
359C2×C4—32no—
360C2×C4—61no—
361C2×C4—61no—
362C2×C4—61no—
363C2×C4—61no—
364C2×C4—61no—
365C2×C4—61no—
366C2×C4—62no—
367C2×C4—61no—
368C2×C4—61no—
369C2×C4—61no—
370C2×C4—61no—
371C2×C4—61no—
372C2×C4—61no—
373C2×C4—61no—
374C2×C4—121no—
375C2×C4—121no—
376C2×C4—121no—
377C2×C4—121no—
378C2×C4—121no—
379C2×C4—121no—
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—121no—
389C2×C4—121no—
390C2×C4—242no—
391C2×C4—242no—
392C2×C4C22×S411yescharacteristic
393C8—21no—
394C8—61no—
395C8—121no—
396C8—121no—
397C2×C3—41no—
398C2×C3—41no—
399C2×C3—41no—
400D3—162no—
401C22—31no—
402C22—31no—
403C22—31no—
404C22—31no—
405C22—31no—
406C22—31no—
407C22—31no—
408C22—31no—
409C22—31no—
410C22—61no—
411C22—122no—
412C22—122no—
413C22—122no—
414C22—241no—
415C22—241no—
416C22—241no—
417C22—241no—
418C22#489011yescharacteristic, complemented
419C22D4×S411yescharacteristic
420C4—31no—
421C4—31no—
422C4—41no—
423C4—41no—
424C4—62no—
425C4—62no—
426C4—121no—
427C4—122no—
428C4—121no—
429C4—121no—
430C4—121no—
431C4—241no—
432C4—241no—
433C4#591211yescharacteristic
434C4Q8×S411yescharacteristic
435C3—41noSylow
436C2—31no—
437C2—31no—
438C2—31no—
439C2—31no—
440C2—242no—
441C2#8531111yescharacteristic
442C2D8×S411yescharacteristic
443C2#8546411yescharacteristic
subgroupG/H
centerC22D4×S4
commutator, cosocleC2×C4×A4C23
FrattiniC2×C4C22×S4
FittingC22×#107D3
socleC24D3×D4
prime radicalC2×C4⋊S4C22
hypercenter#107S4
hypocenterA4#474
LCSG ▷ C2×C4×A4 ▷ C2×A4 ▷ A4
UCSC1 ◁ C22 ◁ C2×C4 ◁ #107

Complements

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

NH ≅ G/Ntype
2. #85303C2semidirect
3. A4×#107C2semidirect
5. SL(2,ℤ/4)⋊C8C2semidirect
7. #73050C2semidirect
22. C22×#107D3semidirect
44. C4⋊S4Q8semidirect
184. #107S4semidirect
317. A4#474semidirect
418. C22#4890semidirect

Maximal subgroups

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

subgroupindex
#853082
#853032
A4×#1072
#731132
SL(2,ℤ/4)⋊C82
#730532
#730502
#263363
#48904