Finite Group Databasebeta

← #10695955

Subgroups — #10695955

Order 768.

568 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 99 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#79298C211yesmaximal, characteristic
2#79068C211yesmaximal, characteristic, complemented
3#74240C211yesmaximal, characteristic
4#70767C211yesmaximal, characteristic, complemented
5#70496C211yesmaximal, characteristic
6#69546C211yesmaximal, characteristic, complemented
7C3×#1286C211yesmaximal, characteristic, complemented
8#10001—31nomaximal, Sylow
9C2×#897—21no—
10C2×C4×D12C2211yescharacteristic
11#5470C2211yescharacteristic
12C2×C4×Dic6C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C3×#423C2211yescharacteristic
15C3×#382C2211yescharacteristic
16#5121—21no—
17#5120—21no—
18#4987—21no—
19#4925C2211yescharacteristic
20#4851—21no—
21#4850—21no—
22#4842—21no—
23#4841—21no—
24#4829—22no—
25#4731—21no—
26#4721—21no—
27C4×#204—22no—
28#4672—21no—
29#4670—22no—
30#4661—21no—
31#4483—22no—
32#4456—22no—
33#4452—22no—
34#3101—31no—
35#3078—31no—
36#2536—31no—
37#1518—31no—
38#1486—31no—
39#1479—31no—
40#1286D311yescharacteristic, complemented
41C22×D12D411yescharacteristic
42C22×C4×D3—21no—
43C22×Dic6D411yescharacteristic
44C2×C3×M16—21no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C42C2311yescharacteristic
47C2×(C2×D3)⋊C4—22no—
48C2×(C2×D3)⋊C4—22no—
49C2×(C2×D3)⋊C4—21no—
50#926—21no—
51C2×C4⋊Dic3—22no—
52C2×C4⋊Dic3D411yescharacteristic
53C2×C4⋊Dic3D411yescharacteristic
54#924—21no—
55#924—21no—
56C2×Dic3⋊C4—21no—
57C2×C4×Dic3—21no—
58C2×#204—22no—
59#897—42no—
60#897—42no—
61C4×D12—21no—
62C4×D12—21no—
63C4×D12—21no—
64C4×D12—21no—
65#870—21no—
66#869D411yescharacteristic
67#869D411yescharacteristic
68C4×Dic6—21no—
69C4×Dic6—21no—
70C4×Dic6—21no—
71C4×Dic6—21no—
72C4×Dic6—21no—
73C4×Dic6—21no—
74C3×C4⋊C8—21no—
75C3×C4⋊C8—21no—
76C3×C4⋊C8D412yes—
77C3×C22⋊C8—21no—
78C3×C4×C8D412yes—
79C3×#95—21no—
80C3×#95D412yes—
81C3⋊#95—22no—
82#830—21no—
83#821—22no—
84#818—22no—
85#817—22no—
86#817—22no—
87#816—22no—
88#816—22no—
89#816—22no—
90C4⋊(C3⋊C8)—21no—
91C4×C3⋊C8—21no—
92#533—31no—
93C2×C22⋊Q8—61no—
94C2×C4×Q8—31no—
95C2×C4×D4—31no—
96#483—61no—
97#479—62no—
98#477—61no—
99#476—61no—
100#475—61no—
101#461—61no—
102#450—61no—
103C4×SD16—62no—
104#431—31no—
105#431C2×D311yescharacteristic
106#428—61no—
107#423C2×D311yescharacteristic
108#419—61no—
109#398—61no—
110#394—61no—
111#388—62no—
112#382C2×D311yescharacteristic
113#380—61no—
114#328—62no—
115Q8⋊C8—62no—
116D4⋊C8—62no—
117C23×D3—21no—
118C22×C3×C4—22no—
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×Dic3—21no—
123C22×Dic3—21no—
124C22×Dic3—21no—
125C2×D12—21no—
126C2×D12—21no—
127C2×D12—42no—
128C2×C4×D3—41no—
129C2×C4×D3—41no—
130C2×C4×D3—41no—
131C2×C4×D3—41no—
132C2×C4×D3—41no—
133C2×C4×D3—41no—
134C2×Dic6—21no—
135C2×Dic6—21no—
136C2×Dic6—21no—
137C2×Dic6—21no—
138C2×Dic6—21no—
139C2×Dic6—21no—
140C2×Dic6—21no—
141C2×Dic6SD1612yescomplemented
142C3×M16—41no—
143C3×M16—41no—
144C2×C3×C8—22no—
145C2×C3×C8—22no—
146C3×C4⋊C4—41no—
147C3×C4⋊C4—41no—
148C3×C42C2×D411yescharacteristic
149C3×C42C2×D411yescharacteristic
150C3×C42C2×D411yescharacteristic
151C3×C42#4111yescharacteristic
152#217—41no—
153(C2×D3)⋊C4—44no—
154(C2×D3)⋊C4—44no—
155(C2×D3)⋊C4—41no—
156(C2×D3)⋊C4—41no—
157C4⋊Dic3—21no—
158C4⋊Dic3—21no—
159C4⋊Dic3—21no—
160C4⋊Dic3—21no—
161C4⋊Dic3—21no—
162C4⋊Dic3—21no—
163C4⋊Dic3—21no—
164C4⋊Dic3—21no—
165C4⋊Dic3—44no—
166Dic3⋊C4—41no—
167Dic3⋊C4—41no—
168Dic3⋊C4—41no—
169Dic3⋊C4—41no—
170C4×Dic3—41no—
171C4×Dic3—41no—
172C4×Dic3—41no—
173C2×C3⋊C8—41no—
174#204—44no—
175C22×Q8—31no—
176C22×D4—31no—
177C23×C4—61no—
178C2×SD16—62no—
179C2×M16—21no—
180C4⋊Q8—31no—
181C4⋊Q8—31no—
182#125—61no—
183C22⋊Q8—61no—
184C22⋊Q8—61no—
185C22⋊Q8—122no—
186C22⋊Q8—122no—
187C4×Q8—61no—
188C4×Q8—61no—
189C4×Q8—61no—
190C4×Q8—61no—
191C4×Q8—61no—
192C4×Q8—61no—
193C4×D4—61no—
194C4×D4—61no—
195C4×D4—61no—
196C4×D4—61no—
197#117—61no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—31no—
200C2×C4⋊C4—31no—
201C2×C4⋊C4—62no—
202C2×C4⋊C4—61no—
203C2×C22⋊C4—62no—
204C2×C22⋊C4—62no—
205C2×C22⋊C4—61no—
206C2×C42—61no—
207C2×C42C22×D311yescharacteristic
208#107—62no—
209#106—62no—
210#106—62no—
211C4⋊C8—21no—
212C4⋊C8—21no—
213C4⋊C8—61no—
214C4⋊C8D1212yes—
215#103—62no—
216#103—62no—
217#103—62no—
218#102—62no—
219C22⋊C8—21no—
220C22⋊C8—61no—
221C4×C8—61no—
222C4×C8D1212yes—
223#95—21no—
224#95—62no—
225#95D1212yes—
226C23×C3#12011yescharacteristic
227C22×D3—22no—
228C22×D3—41no—
229C22×D3—41no—
230C22×D3—41no—
231C22×D3—41no—
232C22×D3—41no—
233C22×D3—41no—
234C2×C3×C4—41no—
235C2×C3×C4—41no—
236C2×C3×C4—42no—
237C2×C3×C4—42no—
238C2×C3×C4—42no—
239C2×C3×C4#12011yescharacteristic
240C2×C3×C4#12011yescharacteristic
241C2×C3×C4#12111yescharacteristic
242C2×C3×C4#12111yescharacteristic
243C2×C3×C4#12111yescharacteristic
244C2×C3×C4#12711yescharacteristic
245C2×C3×C4C2×SD1611yescharacteristic
246C2×C3×C4#13511yescharacteristic
247C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
248C2×C3×C4#13711yescharacteristic
249C2×Dic3—22no—
250C2×Dic3—41no—
251C2×Dic3—41no—
252C2×Dic3—42no—
253C2×Dic3—41no—
254C2×Dic3—41no—
255C2×Dic3—41no—
256C2×Dic3—41no—
257C2×Dic3—41no—
258C2×Dic3—41no—
259D12—22no—
260D12—22no—
261D12—41no—
262D12—41no—
263C4×D3—81no—
264C4×D3—81no—
265C4×D3—81no—
266C4×D3—81no—
267Dic6—22no—
268Dic6—22no—
269Dic6—41no—
270Dic6—41no—
271Dic6—42no—
272Dic6—81no—
273C3×C8—42no—
274C3×C8—42no—
275C3⋊C8—81no—
276C24—61no—
277C2×Q8—32no—
278C2×Q8—61no—
279C2×Q8—61no—
280C2×Q8—61no—
281C2×Q8—61no—
282C2×Q8—61no—
283C2×Q8—61no—
284C2×Q8—61no—
285C2×D4—61no—
286C2×D4—61no—
287C2×D4—122no—
288C22×C4—22no—
289C22×C4—61no—
290C22×C4—61no—
291C22×C4—61no—
292C22×C4—121no—
293C22×C4—121no—
294C22×C4—121no—
295C22×C4—121no—
296C22×C4—121no—
297C22×C4—121no—
298C22×C4C2×D1211yescharacteristic
299C22×C4D3×D411yescharacteristic
300C22×C4D3×D411yescharacteristic
301SD16—124no—
302M16—41no—
303M16—41no—
304C2×C8—22no—
305C2×C8—22no—
306C2×C8—121no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—61no—
310C4⋊C4—61no—
311C4⋊C4—61no—
312C4⋊C4—61no—
313C4⋊C4—61no—
314C4⋊C4—61no—
315C4⋊C4—61no—
316C4⋊C4—61no—
317C4⋊C4—124no—
318C4⋊C4—121no—
319C4⋊C4—121no—
320C4⋊C4—121no—
321C4⋊C4—121no—
322C22⋊C4—124no—
323C22⋊C4—124no—
324C22⋊C4—121no—
325C22⋊C4—121no—
326C22⋊C4—121no—
327C42—121no—
328C42—121no—
329C42—121no—
330C42C2×D1211yescharacteristic
331C42C2×D1211yescharacteristic
332C42D3×D411yescharacteristic
333C42#23911yescharacteristic
334C22×C3—21no—
335C22×C3—21no—
336C22×C3#39211yescharacteristic
337C22×C3#45011yescharacteristic
338C22×C3#45211yescharacteristic
339C2×D3—42no—
340C2×D3—42no—
341C2×D3—42no—
342C2×D3—81no—
343C2×D3—81no—
344C2×D3—81no—
345C2×D3—81no—
346C3×C4—21no—
347C3×C4—21no—
348C3×C4—21no—
349C3×C4—21no—
350C3×C4—82no—
351C3×C4#44811yescharacteristic
352C3×C4#44911yescharacteristic
353C3×C4#46511yescharacteristic
354C3×C4#46811yescharacteristic
355Dic3—42no—
356Dic3—81no—
357Dic3—81no—
358Dic3—81no—
359Dic3—81no—
360C23—62no—
361C23—121no—
362C23—121no—
363C23—121no—
364C23—121no—
365C23—121no—
366C23—121no—
367C23#88211yescharacteristic
368Q8—62no—
369Q8—62no—
370Q8—122no—
371Q8—121no—
372Q8—121no—
373Q8—241no—
374D4—62no—
375D4—62no—
376D4—121no—
377D4—121no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—62no—
384C2×C4—121no—
385C2×C4—121no—
386C2×C4—122no—
387C2×C4—121no—
388C2×C4—121no—
389C2×C4—121no—
390C2×C4—121no—
391C2×C4—121no—
392C2×C4—121no—
393C2×C4—241no—
394C2×C4—241no—
395C2×C4—241no—
396C2×C4—241no—
397C2×C4#87411yescharacteristic
398C2×C4#88211yescharacteristic
399C2×C4#88211yescharacteristic
400C2×C4#89511yescharacteristic
401C2×C4#89511yescharacteristic
402C2×C4#89511yescharacteristic
403C2×C4#91111yescharacteristic
404C2×C4D3×SD1611yescharacteristic
405C2×C4#91611yescharacteristic
406C2×C4#91811yescharacteristic
407C8—42no—
408C8—42no—
409C8—241no—
410C2×C3—21no—
411C2×C3—21no—
412C2×C3#200011yescharacteristic
413C2×C3#200911yescharacteristic
414C2×C3#201011yescharacteristic
415D3—82no—
416C22—21no—
417C22—21no—
418C22—122no—
419C22—122no—
420C22—122no—
421C22—241no—
422C22—241no—
423C22—241no—
424C22—241no—
425C22#467111yescharacteristic
426C22#477611yescharacteristic
427C22#480811yescharacteristic
428C4—21no—
429C4—21no—
430C4—21no—
431C4—21no—
432C4—82no—
433C4—122no—
434C4—241no—
435C4—241no—
436C4—241no—
437C4—241no—
438C4#478111yescharacteristic
439C4#480611yescharacteristic
440C4#486411yescharacteristic
441C4#488611yescharacteristic
442C3#1000111yescharacteristic, complemented, Sylow
443C2—21no—
444C2—21no—
445C2—242no—
446C2#6870311yescharacteristic
447C2#6888711yescharacteristic
448C2#6903111yescharacteristic
subgroupG/H
centerC22#4671
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1286C2
socleC22×C3#392
prime radicalC22×D12D4
hypercenter#1286D3
hypocenterC3#10001
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1286

Complements

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

NH ≅ G/Ntype
2. #79068C2semidirect
4. #70767C2semidirect
6. #69546C2semidirect
7. C3×#1286C2semidirect
40. #1286D3semidirect
141. C2×Dic6SD16semidirect
442. C3#10001semidirect

Maximal subgroups

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

subgroupindex
#792982
#790682
#742402
#707672
#704962
#695462
C3×#12862
#100013