Finite Group Databasebeta

← #10695959

Subgroups — #10695959

Order 768.

548 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 101 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
1C3×#2580C211yesmaximal, characteristic
2#82447C211yesmaximal, characteristic, complemented
3#82296C211yesmaximal, characteristic
4#71622C211yesmaximal, characteristic, complemented
5#71597C211yesmaximal, characteristic
6#69546C211yesmaximal, characteristic, complemented
7#67631C211yesmaximal, characteristic, complemented
8#10002—31nomaximal, Sylow
9C2×C3×C4×Q8C2211yescharacteristic
10C2×C3×C4×D4C2211yescharacteristic
11#5470C2211yescharacteristic
12C3×#399—21no—
13C3×#393—21no—
14C3×#390—21no—
15C3×#389—21no—
16C3×#385—22no—
17C3×#382C2211yescharacteristic
18#5228—21no—
19#5215—21no—
20#5031—21no—
21C4×#216—22no—
22#5021—21no—
23#5009—22no—
24#5008—21no—
25#5007—21no—
26#4970—21no—
27#4964—21no—
28#4925C2211yescharacteristic
29#4925C2211yescharacteristic
30#4924C2211yescharacteristic
31#4535—22no—
32D4⋊(C3⋊C8)—22no—
33#4452—22no—
34#3104—31no—
35#3077—31no—
36#2580D311yescharacteristic
37#1518—31no—
38#1487—31no—
39#1478—31no—
40#1286—31no—
41C22×C3×Q8D411yescharacteristic
42C22×C3×D4D411yescharacteristic
43C23×C3×C4—21no—
44C3×C4×Q8—21no—
45C3×C4×Q8—21no—
46C3×C4×Q8—21no—
47C3×C4×Q8—21no—
48C3×C4×D4—21no—
49C3×C4×D4—21no—
50C3×C4×D4—21no—
51C3×C4×D4—21no—
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4—22no—
55C2×C3×C4⋊C4D411yescharacteristic
56C2×C3×C4⋊C4D411yescharacteristic
57C2×C3×C22⋊C4—22no—
58C2×C3×C22⋊C4—21no—
59C2×C3×C22⋊C4—22no—
60C2×C3×C42—21no—
61C2×C3×C42C2311yescharacteristic
62C2×#216—22no—
63#926—21no—
64#924—21no—
65#924—21no—
66C2×#208—21no—
67#870—21no—
68#869D411yescharacteristic
69#869D411yescharacteristic
70C4×Dic6—21no—
71C4×Dic6—21no—
72C3×#95—22no—
73C3×#95—21no—
74C3×#95D412yes—
75#835—22no—
76#832—22no—
77#832—22no—
78#830—21no—
79#830—21no—
80#810—22no—
81#808—22no—
82#807—22no—
83#807—22no—
84C4⋊(C3⋊C8)—21no—
85C4⋊(C3⋊C8)—21no—
86C4⋊(C3⋊C8)—21no—
87C4⋊(C3⋊C8)D412yes—
88C4×C3⋊C8—21no—
89C4×C3⋊C8D412yes—
90#533—31no—
91C2×C4×Q8C2×D311yescharacteristic
92C2×C4×D4C2×D311yescharacteristic
93#484—61no—
94#478—62no—
95#477—61no—
96#476—61no—
97#474—61no—
98#462—61no—
99#451—61no—
100C4×Q16—62no—
101#431—31no—
102#431—31no—
103#428—61no—
104#423—31no—
105#418—61no—
106#399—21no—
107#393—21no—
108#390—21no—
109#389—21no—
110#385—22no—
111#382C2×D311yescharacteristic
112#328—62no—
113Q8⋊C8—62no—
114D4⋊C8—62no—
115C24×C3—21no—
116C2×C3×Q8—21no—
117C2×C3×Q8—21no—
118C2×C3×Q8—42no—
119C2×C3×D4—21no—
120C2×C3×D4—21no—
121C2×C3×D4—21no—
122C2×C3×D4—21no—
123C2×C3×D4—21no—
124C2×C3×D4Q1612yescomplemented
125C22×C3×C4—21no—
126C22×C3×C4—22no—
127C22×C3×C4—21no—
128C22×C3×C4—21no—
129C22×C3×C4—41no—
130C22×C3×C4—41no—
131C22×C3×C4—82no—
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4C2×D411yescharacteristic
135C2×Dic6—21no—
136C2×Dic6—21no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—41no—
142C3×C4⋊C4—41no—
143C3×C4⋊C4—41no—
144C3×C4⋊C4—41no—
145C3×C4⋊C4—82no—
146C3×C22⋊C4—44no—
147C3×C22⋊C4—41no—
148C3×C22⋊C4—41no—
149C3×C22⋊C4—82no—
150C3×C42—41no—
151C3×C42—41no—
152C3×C42C2×D411yescharacteristic
153C3×C42C2×D411yescharacteristic
154C3×C42C2×D411yescharacteristic
155C3×C42#4111yescharacteristic
156#217—41no—
157#216—44no—
158C4⋊Dic3—21no—
159C4⋊Dic3—21no—
160C4⋊Dic3—21no—
161C4⋊Dic3—21no—
162Dic3⋊C4—41no—
163Dic3⋊C4—41no—
164C4×Dic3—41no—
165#208—41no—
166#208—41no—
167C2×C3⋊C8—22no—
168C2×C3⋊C8—22no—
169C2×C3⋊C8—41no—
170C22×Q8#6711yescharacteristic
171C22×D4#6711yescharacteristic
172C23×C4—21no—
173C2×Q16—62no—
174C2×M16—61no—
175C4⋊Q8—31no—
176C4⋊Q8—31no—
177#125—61no—
178C22⋊Q8—61no—
179C22⋊Q8—61no—
180C4×Q8—21no—
181C4×Q8—21no—
182C4×Q8—21no—
183C4×Q8—21no—
184C4×Q8—61no—
185C4×Q8—61no—
186C4×D4—21no—
187C4×D4—21no—
188C4×D4—21no—
189C4×D4—21no—
190#117—61no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4#6711yescharacteristic
195C2×C4⋊C4#6711yescharacteristic
196C2×C22⋊C4—22no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—22no—
199C2×C42—21no—
200C2×C42C22×D311yescharacteristic
201#107—62no—
202#107—62no—
203#106—62no—
204C4⋊C8—32no—
205C4⋊C8—61no—
206C4⋊C8—61no—
207C4⋊C8—61no—
208#103—62no—
209#103—62no—
210#102—62no—
211#102—62no—
212C22⋊C8—61no—
213C22⋊C8—61no—
214C4×C8—32no—
215C4×C8—61no—
216#95—22no—
217#95—21no—
218#95#6712yes—
219C23×C3—22no—
220C23×C3—42no—
221C23×C3—42no—
222C23×C3—41no—
223C23×C3—41no—
224C23×C3#12011yescharacteristic
225C3×Q8—22no—
226C3×Q8—22no—
227C3×Q8—41no—
228C3×Q8—41no—
229C3×D4—22no—
230C3×D4—22no—
231C3×D4—41no—
232C3×D4—41no—
233C2×C3×C4—22no—
234C2×C3×C4—41no—
235C2×C3×C4—41no—
236C2×C3×C4—41no—
237C2×C3×C4—41no—
238C2×C3×C4—42no—
239C2×C3×C4—41no—
240C2×C3×C4—42no—
241C2×C3×C4—41no—
242C2×C3×C4—42no—
243C2×C3×C4—41no—
244C2×C3×C4—41no—
245C2×C3×C4—42no—
246C2×C3×C4—82no—
247C2×C3×C4—82no—
248C2×C3×C4#12011yescharacteristic
249C2×C3×C4#12011yescharacteristic
250C2×C3×C4#12111yescharacteristic
251C2×C3×C4#12111yescharacteristic
252C2×C3×C4#12111yescharacteristic
253C2×C3×C4#12711yescharacteristic
254C2×C3×C4C2×Q1611yescharacteristic
255C2×C3×C4#13511yescharacteristic
256C2×C3×C4#13711yescharacteristic
257C2×C3×C4#13711yescharacteristic
258C2×Dic3—41no—
259C2×Dic3—41no—
260Dic6—42no—
261Dic6—81no—
262C3⋊C8—42no—
263C3⋊C8—42no—
264C3⋊C8—81no—
265C24—21no—
266C2×Q8—21no—
267C2×Q8—21no—
268C2×Q8—42no—
269C2×Q8—61no—
270C2×Q8—61no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4#21612yescomplemented
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—82no—
284C22×C4C2×#6711yescharacteristic
285C22×C4C2×#6711yescharacteristic
286C22×C4C2×#6711yescharacteristic
287Q16—124no—
288M16—121no—
289M16—121no—
290C2×C8—62no—
291C2×C8—62no—
292C2×C8—121no—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—61no—
302C4⋊C4—61no—
303C4⋊C4—61no—
304C4⋊C4—61no—
305C4⋊C4—82no—
306C4⋊C4—121no—
307C4⋊C4—121no—
308C22⋊C4—44no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—82no—
312C22⋊C4—121no—
313C42—41no—
314C42—41no—
315C42—121no—
316C42D3×D411yescharacteristic
317C42D3×D411yescharacteristic
318C42D3×D411yescharacteristic
319C42#23711yescharacteristic
320C22×C3—21no—
321C22×C3—21no—
322C22×C3—44no—
323C22×C3—42no—
324C22×C3—82no—
325C22×C3—81no—
326C22×C3—81no—
327C22×C3#39211yescharacteristic
328C22×C3#45111yescharacteristic
329C22×C3#45211yescharacteristic
330C3×C4—21no—
331C3×C4—21no—
332C3×C4—21no—
333C3×C4—21no—
334C3×C4—42no—
335C3×C4—82no—
336C3×C4—81no—
337C3×C4—81no—
338C3×C4#45111yescharacteristic
339C3×C4#45211yescharacteristic
340C3×C4#46711yescharacteristic
341C3×C4#47011yescharacteristic
342Dic3—81no—
343Dic3—81no—
344C23—22no—
345C23—42no—
346C23—42no—
347C23—41no—
348C23—41no—
349C23#95311yescharacteristic
350Q8—22no—
351Q8—22no—
352Q8—41no—
353Q8—41no—
354Q8—122no—
355Q8—241no—
356D4—22no—
357D4—22no—
358D4—41no—
359D4—41no—
360C2×C4—22no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—121no—
376C2×C4—121no—
377C2×C4#93711yescharacteristic
378C2×C4#93711yescharacteristic
379C2×C4#93811yescharacteristic
380C2×C4#93911yescharacteristic
381C2×C4#93911yescharacteristic
382C2×C4#94011yescharacteristic
383C2×C4#94211yescharacteristic
384C2×C4C2×#21611yescharacteristic
385C2×C4#95011yescharacteristic
386C2×C4#95111yescharacteristic
387C8—122no—
388C8—122no—
389C8—241no—
390C2×C3—21no—
391C2×C3—21no—
392C2×C3—82no—
393C2×C3#200111yescharacteristic
394C2×C3#200811yescharacteristic
395C2×C3#201011yescharacteristic
396C22—21no—
397C22—21no—
398C22—44no—
399C22—42no—
400C22—82no—
401C22—81no—
402C22—81no—
403C22#522311yescharacteristic
404C22#522811yescharacteristic
405C22#523911yescharacteristic
406C4—21no—
407C4—21no—
408C4—21no—
409C4—21no—
410C4—42no—
411C4—82no—
412C4—81no—
413C4—81no—
414C4—241no—
415C4—241no—
416C4#504711yescharacteristic
417C4#505011yescharacteristic
418C4#505111yescharacteristic
419C4#505211yescharacteristic
420C3#1000211yescharacteristic, complemented, Sylow
421C2—21no—
422C2—21no—
423C2—82no—
424C2#7170011yescharacteristic
425C2#7171511yescharacteristic
426C2#7185811yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#2580C2
socleC22×C3#392
prime radicalC22×C3×D4D4
hypercenter#2580D3
hypocenterC3#10002
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2580

Complements

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

NH ≅ G/Ntype
2. #82447C2semidirect
4. #71622C2semidirect
6. #69546C2semidirect
7. #67631C2semidirect
124. C2×C3×D4Q16semidirect
276. C2×D4#216semidirect
420. C3#10002semidirect

Maximal subgroups

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

subgroupindex
C3×#25802
#824472
#822962
#716222
#715972
#695462
#676312
#100023