Finite Group Databasebeta

← #10697680

Subgroups — #10697680

Order 768.

526 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 105 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×#3077C211yesmaximal, characteristic
2#83373C211yesmaximal, characteristic, complemented
3#82296C211yesmaximal, characteristic
4#71826C211yesmaximal, characteristic, complemented
5#71614C211yesmaximal, characteristic
6#70501C211yesmaximal, characteristic, complemented
7#67711C211yesmaximal, characteristic, complemented
8#10258—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×C4×D4C2211yescharacteristic
11#5470C2211yescharacteristic
12C3×#478—22no—
13C3×#476—21no—
14C3×#474—21no—
15C3×#428—21no—
16C3×#423C2211yescharacteristic
17C3×#418—21no—
18#5215—21no—
19C2×#830—22no—
20#5091—21no—
21#5090—21no—
22#5051—22no—
23#5031—21no—
24#5021—21no—
25#5009—22no—
26#5008—21no—
27#5007—21no—
28#4964—21no—
29#4924C2211yescharacteristic
30C2×C4⋊(C3⋊C8)C2211yescharacteristic
31C2×C4×C3⋊C8C2211yescharacteristic
32#4535—22no—
33#4484—22no—
34#3249—31no—
35#3077—31no—
36#3077D311yescharacteristic
37#1954—31no—
38#1493—31no—
39#1493—31no—
40#1292—31no—
41C22×C3×D4D411yescharacteristic
42C23×C3×C4—21no—
43C2×C3×M16—21no—
44C3×C4⋊Q8D411yescharacteristic
45C3×C4⋊Q8D411yescharacteristic
46C3×#125—21no—
47C3×C22⋊Q8—21no—
48C3×C22⋊Q8—21no—
49C3×C4×Q8—21no—
50C3×C4×Q8—21no—
51C3×C4×D4—21no—
52C3×C4×D4—21no—
53C3×C4×D4—21no—
54C3×C4×D4—21no—
55C3×#117—21no—
56C2×C3×C4⋊C4D411yescharacteristic
57C2×C3×C22⋊C4—21no—
58C2×C3×C42C2311yescharacteristic
59C2×#216—22no—
60#926—21no—
61#924—21no—
62#924—21no—
63C2×#208—21no—
64C22×C3⋊C8—21no—
65C22×C3⋊C8D412yes—
66#870—21no—
67#869D411yescharacteristic
68#869D411yescharacteristic
69C4×Dic6—21no—
70C4×Dic6—21no—
71C3×#107—22no—
72C3×#106—22no—
73C3×C4⋊C8—21no—
74C3×C4⋊C8D412yes—
75C3×#102—22no—
76C3×#102—22no—
77#835—22no—
78#832—22no—
79#832—22no—
80#830—44no—
81#810—22no—
82#808—22no—
83#807—22no—
84#807—22no—
85C4⋊(C3⋊C8)—21no—
86C4⋊(C3⋊C8)—21no—
87C4⋊(C3⋊C8)—21no—
88C4⋊(C3⋊C8)D412yes—
89C4×C3⋊C8—21no—
90C4×C3⋊C8—21no—
91#533—31no—
92#533C2×D311yescharacteristic
93C2×C4×D4C2×D311yescharacteristic
94#478—22no—
95#478—62no—
96#477—61no—
97#477—61no—
98#476—21no—
99#476—61no—
100#474—21no—
101#474—61no—
102#467—62no—
103#462—61no—
104#462—61no—
105#428—21no—
106#428—61no—
107#423—31no—
108#423C2×D311yescharacteristic
109C2×C4⋊C8—31no—
110#418—21no—
111#418—61no—
112C2×C22⋊C8—62no—
113C2×C4×C8—31no—
114Q8⋊C8—62no—
115Q8⋊C8—62no—
116C24×C3—21no—
117C2×C3×Q8—21no—
118C2×C3×Q8—21no—
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—42no—
127C22×C3×C4—42no—
128C22×C3×C4—41no—
129C22×C3×C4—41no—
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4#4111yescharacteristic
133C2×Dic6—21no—
134C2×Dic6—21no—
135C3×M16—41no—
136C3×M16—41no—
137C2×C3×C8—22no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—21no—
142C3×C4⋊C4—21no—
143C3×C4⋊C4—21no—
144C3×C4⋊C4—41no—
145C3×C4⋊C4—41no—
146C3×C22⋊C4—41no—
147C3×C22⋊C4—41no—
148C3×C22⋊C4—41no—
149C3×C42—41no—
150C3×C42C2×D411yescharacteristic
151C3×C42C2×D411yescharacteristic
152C3×C42C2×D411yescharacteristic
153C3×C42C2×D411yescharacteristic
154#217—41no—
155#216—82no—
156C4⋊Dic3—21no—
157C4⋊Dic3—21no—
158C4⋊Dic3—21no—
159C4⋊Dic3—21no—
160Dic3⋊C4—41no—
161Dic3⋊C4—41no—
162C4×Dic3—41no—
163#208—41no—
164#208—41no—
165C2×C3⋊C8—22no—
166C2×C3⋊C8—22no—
167C2×C3⋊C8—22no—
168C2×C3⋊C8—41no—
169C2×C3⋊C8—41no—
170C2×C3⋊C8—42no—
171C22×D4#6711yescharacteristic
172C23×C4—21no—
173C2×Q16—62no—
174C2×M16—21no—
175C2×M16—61no—
176C22×C8—32no—
177C22×C8—61no—
178C4⋊Q8—31no—
179C4⋊Q8—31no—
180C4⋊Q8#6711yescharacteristic
181C4⋊Q8#6711yescharacteristic
182#125—21no—
183#125—61no—
184C22⋊Q8—21no—
185C22⋊Q8—21no—
186C22⋊Q8—61no—
187C22⋊Q8—61no—
188C4×Q8—21no—
189C4×Q8—21no—
190C4×Q8—61no—
191C4×Q8—61no—
192C4×D4—21no—
193C4×D4—21no—
194C4×D4—21no—
195C4×D4—21no—
196#117—21no—
197#117—61no—
198C2×C4⋊C4#6711yescharacteristic
199C2×C22⋊C4—21no—
200C2×C42C22×D311yescharacteristic
201#107—22no—
202#107—62no—
203#107—62no—
204#106—22no—
205#106—62no—
206C4⋊C8—21no—
207C4⋊C8—32no—
208C4⋊C8—61no—
209C4⋊C8—61no—
210C4⋊C8—61no—
211C4⋊C8#6712yes—
212#103—62no—
213#103—62no—
214#102—22no—
215#102—22no—
216#102—62no—
217#102—62no—
218C22⋊C8—124no—
219C4×C8—61no—
220C4×C8—61no—
221C23×C3—22no—
222C23×C3—42no—
223C23×C3—42no—
224C23×C3—41no—
225C23×C3—41no—
226C23×C3#12111yescharacteristic
227C3×Q8—42no—
228C3×Q8—41no—
229C3×Q8—41no—
230C3×D4—22no—
231C3×D4—22no—
232C3×D4—41no—
233C3×D4—41no—
234C2×C3×C4—21no—
235C2×C3×C4—21no—
236C2×C3×C4—41no—
237C2×C3×C4—42no—
238C2×C3×C4—41no—
239C2×C3×C4—41no—
240C2×C3×C4—41no—
241C2×C3×C4—82no—
242C2×C3×C4—81no—
243C2×C3×C4—81no—
244C2×C3×C4#12011yescharacteristic
245C2×C3×C4#12011yescharacteristic
246C2×C3×C4#12011yescharacteristic
247C2×C3×C4#12111yescharacteristic
248C2×C3×C4#12111yescharacteristic
249C2×C3×C4#12711yescharacteristic
250C2×C3×C4C2×Q1611yescharacteristic
251C2×C3×C4#13511yescharacteristic
252C2×C3×C4#13711yescharacteristic
253C2×C3×C4#13711yescharacteristic
254C2×Dic3—41no—
255C2×Dic3—41no—
256Dic6—42no—
257Dic6—41no—
258Dic6—41no—
259C3×C8—42no—
260C3⋊C8—42no—
261C3⋊C8—42no—
262C3⋊C8—42no—
263C24—21no—
264C2×Q8—21no—
265C2×Q8—21no—
266C2×Q8—61no—
267C2×Q8—61no—
268C2×D4—21no—
269C2×D4—21no—
270C2×D4—21no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4#21612yescomplemented
274C22×C4—21no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4D3×D411yescharacteristic
280C22×C4#23711yescharacteristic
281C22×C4C2×#6711yescharacteristic
282Q16—242no—
283M16—41no—
284M16—41no—
285M16—121no—
286M16—121no—
287C2×C8—22no—
288C2×C8—62no—
289C2×C8—62no—
290C2×C8—62no—
291C2×C8—121no—
292C2×C8—121no—
293C2×C8—122no—
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—61no—
303C4⋊C4—61no—
304C4⋊C4—61no—
305C4⋊C4—61no—
306C4⋊C4—121no—
307C4⋊C4—121no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—121no—
312C42—41no—
313C42—121no—
314C42D3×D411yescharacteristic
315C42D3×D411yescharacteristic
316C42C2×#6711yescharacteristic
317C42C2×#6711yescharacteristic
318C22×C3—21no—
319C22×C3—21no—
320C22×C3—44no—
321C22×C3—42no—
322C22×C3—82no—
323C22×C3—81no—
324C22×C3—81no—
325C22×C3#39211yescharacteristic
326C22×C3#46711yescharacteristic
327C22×C3#47011yescharacteristic
328C3×C4—21no—
329C3×C4—21no—
330C3×C4—21no—
331C3×C4—81no—
332C3×C4—81no—
333C3×C4—81no—
334C3×C4#45111yescharacteristic
335C3×C4#45111yescharacteristic
336C3×C4#45211yescharacteristic
337C3×C4#45211yescharacteristic
338C3×C4#45311yescharacteristic
339C3×C4#45411yescharacteristic
340Dic3—81no—
341Dic3—81no—
342C23—22no—
343C23—42no—
344C23—42no—
345C23—41no—
346C23—41no—
347C23#93911yescharacteristic
348Q8—42no—
349Q8—41no—
350Q8—41no—
351Q8—122no—
352Q8—121no—
353Q8—121no—
354D4—22no—
355D4—22no—
356D4—41no—
357D4—41no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—41no—
361C2×C4—42no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—82no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—121no—
369C2×C4—121no—
370C2×C4#93711yescharacteristic
371C2×C4#93711yescharacteristic
372C2×C4#93811yescharacteristic
373C2×C4#93911yescharacteristic
374C2×C4#94011yescharacteristic
375C2×C4#94211yescharacteristic
376C2×C4C2×#21611yescharacteristic
377C2×C4#95011yescharacteristic
378C2×C4#95111yescharacteristic
379C2×C4#95311yescharacteristic
380C8—42no—
381C8—122no—
382C8—122no—
383C8—122no—
384C2×C3—21no—
385C2×C3—21no—
386C2×C3—82no—
387C2×C3#200111yescharacteristic
388C2×C3#200611yescharacteristic
389C2×C3#201711yescharacteristic
390C22—21no—
391C22—21no—
392C22—44no—
393C22—42no—
394C22—82no—
395C22—81no—
396C22—81no—
397C22#505111yescharacteristic
398C22#505211yescharacteristic
399C22#522311yescharacteristic
400C4—21no—
401C4—21no—
402C4—21no—
403C4—81no—
404C4—81no—
405C4—81no—
406C4—241no—
407C4—241no—
408C4#504711yescharacteristic
409C4#505011yescharacteristic
410C4#519811yescharacteristic
411C4#520111yescharacteristic
412C4#522811yescharacteristic
413C4#523911yescharacteristic
414C3#1025811yescharacteristic, complemented, Sylow
415C2—21no—
416C2—21no—
417C2—82no—
418C2#7170011yescharacteristic
419C2#7209811yescharacteristic
420C2#7212111yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3077C2
socleC22×C3#392
prime radicalC22×C3×D4D4
hypercenter#3077D3
hypocenterC3#10258
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3077

Complements

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

NH ≅ G/Ntype
2. #83373C2semidirect
4. #71826C2semidirect
6. #70501C2semidirect
7. #67711C2semidirect
124. C2×C3×D4Q16semidirect
273. C2×D4#216semidirect
414. C3#10258semidirect

Maximal subgroups

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

subgroupindex
C3×#30772
#833732
#822962
#718262
#716142
#705012
#677112
#102583