Finite Group Databasebeta

← #13061727

Subgroups — #13061727

Order 1344.

456 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C7×C22⋊Q8C211yesmaximal, characteristic, complemented
2#10621429C211yesmaximal, characteristic, complemented
3#10621427C211yesmaximal, characteristic, complemented
4C4×#4268C211yesmaximal, characteristic
5C2×#66629C211yesmaximal, characteristic, complemented
6#10621379C211yesmaximal, characteristic, complemented
7#10621379C211yesmaximal, characteristic, complemented
8C4⋊C4×D21C211yesmaximal, characteristic, complemented
9#10621372C211yesmaximal, characteristic, complemented
10#10621372C211yesmaximal, characteristic, complemented
11#10621370C211yesmaximal, characteristic, complemented
12#10621364C211yesmaximal, characteristic
13#10621364C211yesmaximal, characteristic
14#10621361C211yesmaximal, characteristic, complemented
15#10621361C211yesmaximal, characteristic, complemented
16#90327—31nomaximal, Hall
17C2×C3×C7×Q8C2211yescharacteristic
18C22×C3×C4×C7C2211yescharacteristic
19C2×#4268C2211yescharacteristic
20C22×Dic21C2211yescharacteristic
21C22×Dic21C2211yescharacteristic
22C2×C4×D21C2211yescharacteristic
23C2×C4×D21C2211yescharacteristic
24C2×C4×D21C2211yescharacteristic
25C2×Dic42C2211yescharacteristic
26C2×Dic42C2211yescharacteristic
27C3×C7×C4⋊C4C2211yescharacteristic
28C3×C7×C4⋊C4C2211yescharacteristic
29C3×C7×C4⋊C4C2211yescharacteristic
30C3×C7×C22⋊C4C2211yescharacteristic
31C3×C7×C22⋊C4C2211yescharacteristic
32#66636C2211yescharacteristic
33#66631C2211yescharacteristic
34#66631C2211yescharacteristic
35#66631C2211yescharacteristic
36C4⋊Dic21C2211yescharacteristic
37C4⋊Dic21C2211yescharacteristic
38#66629C2211yescharacteristic
39#66629C2211yescharacteristic
40#66629C2211yescharacteristic
41#66629C2211yescharacteristic
42#66629C2211yescharacteristic
43#66629C2211yescharacteristic
44#66629C2211yescharacteristic
45#66629C2211yescharacteristic
46#66629C2211yescharacteristic
47#66629C2211yescharacteristic
48#66629C2211yescharacteristic
49C4×Dic21C2211yescharacteristic
50C4×Dic21C2211yescharacteristic
51C4×Dic21C2211yescharacteristic
52C7×C22⋊Q8D311yescharacteristic, complemented
53#6509—31no—
54#6507—31no—
55C4×C22⋊D7—31no—
56C2×Dic7⋊C4—31no—
57#6459—31no—
58#6459—31no—
59D7×C4⋊C4—31no—
60#6452—31no—
61#6452—31no—
62#6450—31no—
63(C22⋊D7)⋊C4—31no—
64(C22⋊D7)⋊C4—31no—
65#6441—31no—
66#6441—31no—
67#5636—71nomaximal, Hall
68C23×C3×C7C2311yescharacteristic
69C22×D21C2311yescharacteristic
70C3×C7×Q8—41no—
71C2×C3×C4×C7—41no—
72C2×C3×C4×C7C2311yescharacteristic
73C2×C3×C4×C7C2311yescharacteristic
74C2×C3×C4×C7C2311yescharacteristic
75C2×C3×C4×C7C2311yescharacteristic
76C2×C3×C4×C7C2311yescharacteristic
77C2×C3×C4×C7C2311yescharacteristic
78#4268Q814yescomplemented
79C2×Dic21—22no—
80C2×Dic21—41no—
81C2×Dic21C2311yescharacteristic
82C2×Dic21C2311yescharacteristic
83C2×Dic21C2311yescharacteristic
84C2×Dic21C2311yescharacteristic
85C2×Dic21C2311yescharacteristic
86C2×Dic21C2311yescharacteristic
87C2×Dic21C2311yescharacteristic
88C4×D21—22no—
89C4×D21—41no—
90C4×D21—41no—
91Dic42—22no—
92Dic42—41no—
93C2×C7×Q8C2×D311yescharacteristic
94C22×C4×C7C2×D311yescharacteristic
95C2×C22⋊D7—31no—
96C22×Dic7—31no—
97C22×Dic7—31no—
98C2×C4×D7—31no—
99C2×C4×D7—31no—
100C2×C4×D7—31no—
101C2×Dic14—31no—
102C2×Dic14—31no—
103C7×C4⋊C4C2×D311yescharacteristic
104C7×C4⋊C4C2×D311yescharacteristic
105C7×C4⋊C4C2×D311yescharacteristic
106C7×C22⋊C4C2×D311yescharacteristic
107C7×C22⋊C4C2×D311yescharacteristic
108C22⋊Dic7—31no—
109(C2×D7)⋊C4—31no—
110(C2×D7)⋊C4—31no—
111(C2×D7)⋊C4—31no—
112C4⋊Dic7—31no—
113C4⋊Dic7—31no—
114Dic7⋊C4—31no—
115Dic7⋊C4—31no—
116Dic7⋊C4—31no—
117Dic7⋊C4—31no—
118Dic7⋊C4—31no—
119Dic7⋊C4—31no—
120Dic7⋊C4—31no—
121Dic7⋊C4—31no—
122Dic7⋊C4—31no—
123Dic7⋊C4—31no—
124Dic7⋊C4—31no—
125C4×Dic7—31no—
126C4×Dic7—31no—
127C4×Dic7—31no—
128C3×C22⋊Q8D711yescharacteristic, complemented
129#946—71no—
130#944—71no—
131C4×#67—71no—
132C2×Dic3⋊C4—71no—
133#896—71no—
134#896—71no—
135D3×C4⋊C4—71no—
136#889—71no—
137#889—71no—
138#887—71no—
139#67⋊C4—71no—
140#67⋊C4—71no—
141#878—71no—
142#878—71no—
143C22×C3×C7—21no—
144C22×C3×C7—21no—
145C22×C3×C7C2×Q812yes—
146C22×C3×C7C2411yescharacteristic
147C2×D21—21no—
148C2×D21—21no—
149C2×D21C2×Q812yes—
150C3×C4×C7—21no—
151C3×C4×C7—21no—
152C3×C4×C7—21no—
153C3×C4×C7—21no—
154C3×C4×C7—21no—
155C3×C4×C7—21no—
156Dic21—21no—
157Dic21—21no—
158Dic21—21no—
159Dic21—21no—
160Dic21—21no—
161Dic21C2×Q812yes—
162Dic21#4112yes—
163#554—211noSylow
164C23×C7C22×D311yescharacteristic
165C22×D7—31no—
166C7×Q8—41no—
167C2×C4×C7—41no—
168C2×C4×C7C22×D311yescharacteristic
169C2×C4×C7C22×D311yescharacteristic
170C2×C4×C7C22×D311yescharacteristic
171C2×C4×C7C22×D311yescharacteristic
172C2×C4×C7C22×D311yescharacteristic
173C2×C4×C7C22×D311yescharacteristic
174C22⋊D7—34no—
175C2×Dic7—31no—
176C2×Dic7—31no—
177C2×Dic7—31no—
178C2×Dic7—31no—
179C2×Dic7—31no—
180C2×Dic7—31no—
181C2×Dic7—31no—
182C2×Dic7—62no—
183C2×Dic7—121no—
184C4×D7—62no—
185C4×D7—121no—
186C4×D7—121no—
187Dic14—62no—
188Dic14—121no—
189C2×C3×Q8C2×D711yescharacteristic
190C22×C3×C4C2×D711yescharacteristic
191C2×#67—71no—
192C22×Dic3—71no—
193C22×Dic3—71no—
194C2×C4×D3—71no—
195C2×C4×D3—71no—
196C2×C4×D3—71no—
197C2×Dic6—71no—
198C2×Dic6—71no—
199C3×C4⋊C4C2×D711yescharacteristic
200C3×C4⋊C4C2×D711yescharacteristic
201C3×C4⋊C4C2×D711yescharacteristic
202C3×C22⋊C4C2×D711yescharacteristic
203C3×C22⋊C4C2×D711yescharacteristic
204#217—71no—
205(C2×D3)⋊C4—71no—
206(C2×D3)⋊C4—71no—
207(C2×D3)⋊C4—71no—
208C4⋊Dic3—71no—
209C4⋊Dic3—71no—
210Dic3⋊C4—71no—
211Dic3⋊C4—71no—
212Dic3⋊C4—71no—
213Dic3⋊C4—71no—
214Dic3⋊C4—71no—
215Dic3⋊C4—71no—
216Dic3⋊C4—71no—
217Dic3⋊C4—71no—
218Dic3⋊C4—71no—
219Dic3⋊C4—71no—
220Dic3⋊C4—71no—
221C4×Dic3—71no—
222C4×Dic3—71no—
223C4×Dic3—71no—
224C2×C3×C7—22no—
225C2×C3×C7C22×Q811yescharacteristic
226C2×C3×C7C2×#4111yescharacteristic
227C2×C3×C7H(5,𝔽2)11yescharacteristic
228D21—22no—
229C4⋊Q8—211no—
230#125—211no—
231#125—211no—
232C22⋊Q8—211no—
233C22⋊Q8—211no—
234C22⋊Q8—211no—
235C22⋊Q8—211no—
236C22⋊Q8—211no—
237C22⋊Q8D2111yescharacteristic, complemented
238C4×Q8—211no—
239C4×D4—211no—
240C4×D4—211no—
241C4×D4—211no—
242C2×C4⋊C4—211no—
243C2×C4⋊C4—211no—
244C22×C7—21no—
245C22×C7—21no—
246C22×C7D3×Q812yes—
247C22×C7C23×D311yescharacteristic
248C2×D7—32no—
249C2×D7—61no—
250C2×D7—61no—
251C4×C7—21no—
252C4×C7—21no—
253C4×C7—21no—
254C4×C7—21no—
255C4×C7—21no—
256C4×C7—21no—
257Dic7—32no—
258Dic7—32no—
259Dic7—61no—
260Dic7—61no—
261Dic7—61no—
262Dic7—61no—
263Dic7—61no—
264C23×C3C22×D711yescharacteristic
265C22×D3—71no—
266C3×Q8—41no—
267C2×C3×C4—41no—
268C2×C3×C4C22×D711yescharacteristic
269C2×C3×C4C22×D711yescharacteristic
270C2×C3×C4C22×D711yescharacteristic
271C2×C3×C4C22×D711yescharacteristic
272C2×C3×C4C22×D711yescharacteristic
273C2×C3×C4C22×D711yescharacteristic
274#67—74no—
275C2×Dic3—71no—
276C2×Dic3—71no—
277C2×Dic3—71no—
278C2×Dic3—71no—
279C2×Dic3—71no—
280C2×Dic3—71no—
281C2×Dic3—71no—
282C2×Dic3—142no—
283C2×Dic3—281no—
284C4×D3—142no—
285C4×D3—281no—
286C4×D3—281no—
287Dic6—142no—
288Dic6—281no—
289C3×C7#55411yescharacteristic, complemented, Hall
290C2×Q8—211no—
291C2×Q8—211no—
292C2×Q8C2×D2111yescharacteristic
293C2×D4—211no—
294C22×C4—211no—
295C22×C4—211no—
296C22×C4—211no—
297C22×C4—211no—
298C22×C4—211no—
299C22×C4C2×D2111yescharacteristic
300C4⋊C4—211no—
301C4⋊C4—211no—
302C4⋊C4—211no—
303C4⋊C4—211no—
304C4⋊C4—211no—
305C4⋊C4—211no—
306C4⋊C4—211no—
307C4⋊C4—211no—
308C4⋊C4—211no—
309C4⋊C4—211no—
310C4⋊C4—211no—
311C4⋊C4—211no—
312C4⋊C4—211no—
313C4⋊C4C2×D2111yescharacteristic
314C4⋊C4C2×D2111yescharacteristic
315C4⋊C4C2×D2111yescharacteristic
316C22⋊C4—211no—
317C22⋊C4—211no—
318C22⋊C4—211no—
319C22⋊C4—211no—
320C22⋊C4C2×D2111yescharacteristic
321C22⋊C4C2×D2111yescharacteristic
322C42—211no—
323C42—211no—
324C42—211no—
325C2×C7—22no—
326C2×C7#100411yescharacteristic
327C2×C7C2×D3×Q811yescharacteristic
328C2×C7D3×#4111yescharacteristic
329D7—62no—
330C22×C3—21no—
331C22×C3—21no—
332C22×C3Q8×D712yes—
333C22×C3C23×D711yescharacteristic
334C2×D3—72no—
335C2×D3—141no—
336C2×D3—141no—
337C3×C4—21no—
338C3×C4—21no—
339C3×C4—21no—
340C3×C4—21no—
341C3×C4—21no—
342C3×C4—21no—
343Dic3—72no—
344Dic3—72no—
345Dic3—141no—
346Dic3—141no—
347Dic3—141no—
348Dic3—141no—
349Dic3—141no—
350C23—211no—
351C23C22×D2111yescharacteristic
352Q8—41no—
353Q8—422no—
354Q8—841no—
355D4—214no—
356C2×C4—41no—
357C2×C4—211no—
358C2×C4—211no—
359C2×C4—211no—
360C2×C4—211no—
361C2×C4—211no—
362C2×C4—211no—
363C2×C4—211no—
364C2×C4—422no—
365C2×C4—422no—
366C2×C4—841no—
367C2×C4—841no—
368C2×C4—841no—
369C2×C4C22×D2111yescharacteristic
370C2×C4C22×D2111yescharacteristic
371C2×C4C22×D2111yescharacteristic
372C2×C4C22×D2111yescharacteristic
373C2×C4C22×D2111yescharacteristic
374C2×C4C22×D2111yescharacteristic
375C7#563611yescharacteristic, complemented, Sylow
376C2×C3—22no—
377C2×C3#654811yescharacteristic
378C2×C3C2×Q8×D711yescharacteristic
379C2×C3D7×#4111yescharacteristic
380D3—142no—
381C22—21no—
382C22—21no—
383C22—212no—
384C22—421no—
385C22—421no—
386C22Q8×D2112yes—
387C22C23×D2111yescharacteristic
388C4—21no—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—212no—
395C4—212no—
396C4—421no—
397C4—421no—
398C4—421no—
399C4—421no—
400C4—421no—
401C3#9032711yescharacteristic, complemented, Sylow
402C2—22no—
403C2—422no—
404C2#1062165711yescharacteristic
405C2C2×Q8×D2111yescharacteristic
406C2#41×D2111yescharacteristic
subgroupG/H
center, FrattiniC22C23×D21
commutator, socle, cosocleC22×C3×C7C24
FittingC3×C7×C22⋊Q8C2
prime radicalC2×#4268C22
hypercenterC22⋊Q8D21
hypocenterC3×C7#554
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C3×C7×C22⋊Q8C2semidirect
2. #10621429C2semidirect
3. #10621427C2semidirect
5. C2×#66629C2semidirect
6. #10621379C2semidirect
7. #10621379C2semidirect
8. C4⋊C4×D21C2semidirect
9. #10621372C2semidirect
10. #10621372C2semidirect
11. #10621370C2semidirect
14. #10621361C2semidirect
15. #10621361C2semidirect
52. C7×C22⋊Q8D3semidirect
78. #4268Q8semidirect
128. C3×C22⋊Q8D7semidirect
237. C22⋊Q8D21semidirect
289. C3×C7#554semidirect
375. C7#5636semidirect
401. C3#90327semidirect

Maximal subgroups

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

subgroupindex
C3×C7×C22⋊Q82
#106214292
#106214272
C4×#42682
C2×#666292
#106213792
C4⋊C4×D212
#106213722
#106213702
#106213642
#106213612
#903273
#56367